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Examining the issues surrounding violating the assumption of independent observations in reliability generalization studies :
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ABSTRACT: Because both validity and reliability indices are a function of the scores on a given administration of a measure, their values can often vary across samples. It is a common mistake to say that a test is reliable when in fact it is not the test that is reliable but the scores on the test that are reliable. In 1998, vachahaase proposed a fixedeffects metaanalytic method for evaluating reliability that is similar to validity generalization studies called reliability generalization (rg). This study was conducted to evaluate alternative analysis strategies for the metaanalysis method of reliability generalization when the reliability estimates are not statistically independent. Five approaches for handling the violation of independence were implemented: ignoring the violation and treating each observation as independent, calculating one mean or median from each study, randomly selecting only one observation per study, or using a mixed effects model.^ This Monte Carlo study included five factors in the method. These factors were (a) the coefficient alpha, (b) sample size in the primary studies, (c) number of primary studies in the rg study, (d) number of reliability estimates from each, and (e) the degree of violation of independence where the strength of the dependence is related to the number of reliability indices (i.e. coefficient alpha) derived from a simulated set of examines and the magnitude of the correlation between the journal studies (with intraclass correlation icc = 0, .0l .30, and .90). These factors were used to simulate samples under known and controlled population conditions. In general, the results suggested that the type of treatment does not have a noticeable impact on the accuracy of the reliability results but that researchers should be cautious when the intraclass correlation is relatively large. In addition, the simulations in this study resulted in very poor confidence band coverage.^ This research suggested that RG metaanalysis methods are appropriate for describing the overall average reliability of a measure or construct but the RG researcher should be careful in regards to the construction of confidence intervals.
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Dissertation (Ph.D.)University of South Florida, 2007.
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Examining the Issues Surrounding Viola ting the Assumption of Independent Observations in Reliability Generalization Studies: A Simulation Study by Jeanine L. Romano A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Measurement and Evaluation College of Education University of South Florida Major Professor: Jeffrey D. Kromrey, Ph.D. Michael T. Brannick, Ph.D. Robert F. Dedrick, Ph.D. Anthony J. Onwuegbuzie, Ph.D. Date of Approval February 26, 2007 Keywords: VachaHaase, Research Synthesis, Metaanalysis, Intraclass Correlation, Statistical Assumptions Copyright 2007, Jeanine L. Romano
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Dedication This dissertation is dedicated to the memo ry of my mother June Alice Romano, the smartest person I have ever known. She was a woman who raised six children and loved us all unconditionally. She lived her live with passion. I aspire every day to be just like her. My mother taught me that the key to success is perseverance. There were plenty times during this journey that I had my doubts about my ability to complete my PhD. During the times where I felt great self doubt and somewhat overwhelmed, I would find comfort in remembering what my mother alwa ys said to me when I was so very young: Â“Anything is possible if you put your mind to itÂ” For the graduate student that might be perusing this document looking for encourag ement for their dissertation, I hope her words will inspire you as well!
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Acknowledgements This dissertation would not have been possible if it hadnÂ’t been for the many individuals that have contributed to my learning along the way. First, I want to thank my brothers and sisters: Geno, Joanne, George, Vincent and Patty, they ha ve been there to encourage me along the way. Second, I would also like to thank my boss and dear friend, Dr. Linda Devine, who has supported me both professionally and personall y. In addition, she has been the ultimate role model for women in the field of Higher Education; I want to be just like her when I grow up! Third, I would like to thank my friends Lisa, David, and Janene th at allowed me to invade their computers with my strange SAS software in order to run some of the conditions for this study. Fourth, IÂ’d like to thank my USF Â“familyÂ” my fellow graduate student s: Moya, Melinda, Gianna, Kris, Peggy, Tom, Susan, Bruce, Ha, Dorian, and Jesse; our departme nt office manager: Lisa Adkins; and the other department faculty members: Dr Bruce Hall, Dr. Lou Carey, Dr. Connie Hines, and Dr. John Ferron. They have contributed to my learning in ways that I can not even begin to measure! Finally, IÂ’d like to thank the members of my committee. Michael Brannick for showing me that research should be exciting. Robert De drick for showing me th at research should be comforting. Anthony J. Onwuegbuzie for showing me that research should be precise. Finally, IÂ’d like to thank Jeffrey D. Kromrey, my mentor and major professor. Even when the early drafts of this document where barely comprehendible, he would always say, Â“ItÂ’s a good start.Â” I will always be grateful for his encouraging words a nd his unlimited patience! He taught me so much about research and even more about being a kind and patient human being. It has truly been an honor having him as my major professor and colleague.
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i TABLE OF CONTENTS List of Tables................................................................................................................. ....iii List of Figures................................................................................................................ ....vi Abstract....................................................................................................................... .....viii Chapter One: Introduction...................................................................................................1 Methodological Issues in RG Studies......................................................................2 Purpose of the Study................................................................................................4 Research Questions..................................................................................................5 Hypotheses...............................................................................................................6 Limitations of the Study...........................................................................................8 Definitions of Terms................................................................................................9 Importance of the Study.........................................................................................12 Chapter Two: Literature Review.......................................................................................13 MetaAnalysis........................................................................................................13 The Reliability of Measures...................................................................................18 RG Studies.............................................................................................................21 Issues in the Debate on Re liability Generalization................................................34 Independence.........................................................................................................39 Summary................................................................................................................45 Chapter Three: Method......................................................................................................48 Purpose...................................................................................................................48 Research Questions................................................................................................49 Sample....................................................................................................................49 Method...................................................................................................................52 Simulation of the data................................................................................53 Simulation of intraclass correlation..........................................................55 Conduct of RG analyses.............................................................................58 Evaluation of the results.............................................................................59 Chapter Four: Results........................................................................................................62 How the Results were Evaluated...........................................................................62 Box Plots................................................................................................................64 Summary of Box Plot Results................................................................................68 2 Analysis.............................................................................................................69 Bias........................................................................................................................71
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ii Root Mean Squared Error......................................................................................75 Band Coverage.......................................................................................................80 Band Width............................................................................................................89 A Deeper Look at Band Coverage.........................................................................96 Summary of Results.............................................................................................102 Chapter Five: Conclusions...............................................................................................108 Summary of the Study.........................................................................................108 Research Questions..............................................................................................111 Summary of Study Results...................................................................................112 Limitations of the Study.......................................................................................118 Implications..........................................................................................................119 Importance of the Study...........................................................................119 Importance in Regards to Future RG Studies..........................................119 Suggestions for Future Research.........................................................................121 References..................................................................................................................... ...126 Appendices:.................................................................................................................... ..143 Appendix A: SAS Code for Monte Carlo Simulation.........................................144 About the Author...................................................................................................End Page
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iii List of Tables Table 1 Reliability Generalization Studies........................................................................22 Table 2 Number of Items Needed to Generate Reliability Parameter...............................50 Table 3 Population Item Parame ters Used for Simulations...............................................51 Table 4 Results of the Simulation of Intraclass Correlation.............................................55 Table 5 Correlation Between Dependent Variables...........................................................63 Table 6 2 Analysis of the Effects of F actors in the RG Simulation..................................70 Table 7 Bias in Estimated Mean Reliability by Treatment and Intraclass Correlation.............................................................................................................71 Table 8 Bias in Estimated Mean Reliability by Treatment and Coefficient Alpha......................................................................................................................72 Table 9 Bias in Estimated Mean Reliability by Treatment and Average Sample Size.........................................................................................................................73 Table 10 Bias in Estimated Mean Reliability by In traclass Correlation and Coefficient Alpha...................................................................................................74 Table 11 RMSE of Estimated Mean Reliability by Treatment and Coefficient Alpha......................................................................................................................76 Table 12 RMSE of Estimated Mean Reliability by Treatment and Average Sample Size.........................................................................................................................76 Table 13 RMSE for Estimated Mean Reliability by Treat ment and Intraclass Correlation.............................................................................................................77
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iv Table 14 RMSE for Estimated Mean Reliability Average Sample Size by Coefficient Alpha...................................................................................................78 Table 15 RMSE for Estimated Mean Reliability Intraclass Correlation by Coefficient Alpha...................................................................................................79 Table 16 Band Coverage of Estimated Mean Reliability by Treatment and Intraclass Correlation............................................................................................81 Table 17 Band Coverage of Estimated Mean Reliability for Intraclass Correlation and Treatment by Coefficient Alpha..................................................83 Table 18 Band Coverage of Estimated Mean Reliab ility for Average Sample Size and Treatment by Coefficient Alpha......................................................................86 Table 19 Band Coverage of Estimated Mean Reliability for Number of Primary Studies by Treatment.............................................................................................88 Table 20 Band Coverage of Estimated Mean Reliability for Coefficient Alpha by Treatment...............................................................................................................88 Table 21 Band Width of Estimated Mean Reliability by Treatment and Average Sample Size............................................................................................................90 Table 22 Band Width of Estimated Mean Reliability by Treatment and Coefficient Alpha......................................................................................................................90 Table 23 Band Width of Estimated Mean Re liability by Treatment and Number of Primary Studies......................................................................................................91 Table 24 Band Width of Estimated Mean Reliability for Average Sample Size by Coefficient Alpha..................................................................................................92
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v Table 25 Band Width of Estimated Mean Reliability for Number of Primary Studies by Average Sample Size...........................................................................93 Table 26 Band Width of Estimated Mean Reliability fo r Number of Primary Studies by Coefficient Alpha.................................................................................95 Table 27 Large Band Coverage by Type of Treatment.....................................................97 Table 28 Percentage of Large Band Coverage fo r Intraclass Correlation by Treatment...............................................................................................................98 Table 29 Percentage of Large Band Coverage for Co efficient Alpha by Treatment...............................................................................................................99 Table 30 Percentage of Large Band Coverage for Average Sample Size by Treatment.............................................................................................................100 Table 31 Percentage of Large Band Coverage for the Nu mber of Primary Studies by Treatment........................................................................................................102 Table 32 Percentage of Large Band Coverage for Numb er of Reliabilities Per Study by Treatment..............................................................................................102
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vi List of Figures Figure 1. The model of the simulation for the study..........................................................54 Figure 2. A model for how the Â“testtakingÂ” was simulated..............................................57 Figure 3. Distribution of bias estimates for reliability co efficients for all five types of treatments...........................................................................................................65 Figure 4. Distribution of RMSE estimates for reliability coefficients for all five types of treatments.................................................................................................66 Figure 5. Distribution of band coverage for reliability coefficients for all five types of treatments...............................................................................................67 Figure 6. Distribution of band width for re liability coefficients for all five types of treatments.................................................................................................68 Figure 7. Bias in estimated mean reliability by treatment and coefficient alpha.......................................................................................................................72 Figure 8. Bias in estimated mean reli ability by intraclass correlation and coefficient..............................................................................................................74 Figure 9. RMSE for estimated mean reliab ility average sample size by coefficient alpha.............................................................................................................. .........78 Figure 10. RMSE for estimated mean re liability intracl ass correlation by oefficient alpha.......................................................................................................79 Figure 11. Band coverage of estimated mean reliability intraclass orrelation by coefficient alpha...............................................................................84
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vii Figure 12. Band coverage of estimated mean reliability for average sample size by coefficient alpha............................................................................................... .87 Figure 13. Band width of estimated mean re liability for average sample size by coefficient alpha.....................................................................................................92 Figure 14. Band width of estimated mean relia bility for number of primary studies by average sample size..........................................................................................94 Figure 15. Band width of estimated mean relia bility for number of primary studies by coefficient alpha................................................................................................ 95
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viii Examining the Issues Surrounding Viola ting the Assumption of Independent Observations in Reliability Generalization Studies: A Simulation Study Jeanine L. Romano ABSTRACT Because both validity and reliability indi ces are a function of the scores on a given administration of a measure, their valu es can often vary across samples. It is a common mistake to say that a test is reliable when in fact it is not the test that is reliable but the scores on the test that are reliable In 1998, VachaHaase proposed a fixedeffects metaanalytic method for evaluating reliability that is similar to validity generalization studies called reliability ge neralization (RG). This study was conducted to evaluate alternative analysis strategies for the meta analysis method of reliability generalization when the reliability estimates are not statistically independent. Five approaches for handling the violation of independence were implemented: ignoring the violation and treating each observation as independent, calculating one mean or median from each study, randomly selecting only one observation per study, or using a mixed effects model. This Monte Carlo study included five factors in the method. These factors were (a) the coefficient alpha, (b) sample si ze in the primary studies, (c) nu mber of primary studies in the RG study, (d) number of reliability estim ates from each, and (e) the degree of violation of independence wher e the strength of the depende nce is related to the number of reliability indices (i.e. coefficient alpha) derived from a simulated set of examines and the magnitude of the correlation between the journal studies (with intraclass correlation
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ix ICC = 0, .0l .30, and .90). These factors we re used to simulate samples under known and controlled population conditi ons. In general, the results suggested that the type of treatment does not have a noticeable impact on the accuracy of the reliability results but that researchers should be cautious when the intr aclass correlation is relatively large. In addition, the simulations in th is study resulted in very poor confidence band coverage. This research suggested that RG metaanalysi s methods are appropriate for describing the overall average reliability of a measure or construct but the RG researcher should be careful in regards to the constr uction of confidence intervals.
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1 Chapter One: Introduction Ideally, social science res earch is conducted using meas urement instruments that will produce valid and reliable information. When these tests are first developed to measure a certain construct (e.g., depression), they are usually evaluated in regard to the validity and reliability of their scores. A lthough these procedures are conducted for the development of the instrument, the fact that both validity and reliability can fluctuate across samples, as both indices are a function of the scores on a gi ven administration of a measure, is often overlooked. It is a common mistake to report that a test is reliable when in fact it is not the test that is reliable but the scores on a test that are reliable (VachaHaase, Kogan, & Thompson, 2000). Because reliability can fluc tuate across studies, it has been recommended that researchers should always evalua te the reliability of their sc ores and report the results. The American Psychology Association (APA) Task Force on Statistical Inference in their 1999 report stated: It is important to remember that a test is not reliable or unreliable. Reliability is a Property of the scores on a test for a pa rticular population of examineesÂ…Thus, authors should provide reliability coeffici ents of the scores for the data being analyzed even when the focus of their re search is not psychometric (Wilkinson & APA Task Force on Statistical Inference, 1999, p. 596).
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2 Validity generalization studies have been conducted to desc ribe the extent to which validity evidence for scores are generalizab le across research contexts (Hunter & Schmidt, 1990; Schmidt & Hunter, 1977). In 1998, VachaHaase proposed a fixedeffects metaanalytic method for evaluating reliability, similar to validity generalization studies, this is called reliability generalization (RG). RG studies can be used to investigate the distribution of reliabili ty estimates across stud ies and to identify st udy characteristics that may be related to variation in reliability estimat es, such as sample size type of reliability estimate (coefficient alpha vs. testretest), diffe rent forms of an instrument, or participant characteristics (Henson, 2001; VachaHaase, 1998). This method is recommended for describing estimated measurement error in a te stÂ’s scores across studi es and can also be used to analyze measurement error in different scales that measure the same construct. Methodological Issues in RG Studies Potential methodological problems are evid ent in RG studies, and the debate about their solution has only just begun (Helms, 1999; Sawilowsky, 2000; Thompson & VachaHaase, 2000). Major contr oversies include (a) approaches for treatment of large proportions of missing data in the publis hed literature, (b) the use of nonlinear transformations of sample reliability estim ates, (c) the need to weight the observed sample statistics to account for differences in sampling error across studies (d) the differences between analyses of reliability coefficients and analyses of the estimated standard errors of measurement (SEM), a nd (e) appropriate analyses of reliability estimates that are not sta tistically independent (Sawilowsky, 2000; Thompson & VachaHaase, 2000).
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3 This research primarily focused on appropriate analysis of reliability estimates that are not statistically i ndependent. Several RG studies have been conducted that included samples that did not represent inde pendent observations. For example, in their study on the StateTrait Anxiet y Inventory (STAI; Spielberger, Gorsuch, & Lushene, 1970), Barnes, Harp, and Jung (2002) obtaine d 117 reliability coefficients from 45 articles where each subgroup of participants was treated as an observation. When Capraro and Capraro (2002) conducted an RG study on the MyersBriggs Type Indicator scale (Myers & McCaulley, 1985), they included 70 reliability coefficients from only 14 published studies. Yin and FanÂ’s (2000) RG study on the Beck Depression Inventory (BDI; Beck, Ward, Mendelson, Mock, & Erbaugh, 1961) included 164 reliability coefficients from 90 studies. Similarly, V achaHaaseÂ’s (1998) RG study on the Bem Sex Role Inventory (BSRI; Bern, 1974) used 87 reliability coeffi cients from 57 studies, and CarusoÂ’s (2000) RG study on the NEO persona lity scale (Costa & McCrae, 1985) used 51 reliability estimates from 37 studies. Clearl y, these are violations of independence of observations. The assumption of independence of observa tions is commonly vi olated in metaanalytic research (Becker, 2000; Hedges & Olkin, 1985; Hunter & Schmidt, 1990). Violations can lead to biased results in which Type I error (rejecting a true null hypotheses) and Type II error (failing to reject a false null hy potheses) rates are inaccurate (Barcikowski, 1981; Scariano & Da venport, 1987). The problem of violation of independence has been investigated in re gard to various statistical techniques (e.g., Barcikowski, 1981; Bock, 1975; HewittGe rvais & Kromrey, 1999; Kenny & Judd, 1986; Kromrey & Dickinson, 1996; Raudenbush & Bryk, 1987; Scariano & Davenport, 1987).
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4 Further, several studies have been conducte d concerning the consequences of dependent observations in metaanalysis (e.g., Becker & Kim, 2002; Beretvas & Pastor, 2003; Cooper, 1979; Greenhouse & Iyengar, 1994; Hedges & Olkin, 1985; Landman & Dawes, 1982; Raudenbush, Becker, & Kalaian, 1988; Ro senthal & Rubin, 1986; Tracz, Elmore, & Pohlmann, 1992). In general, this body of research has indicated that ignoring the assumption of independence can impact the ma gnitude of statistica l significance. There are several approaches to dealing with the viol ation of independence that have been recommended by researchers (B ecker, 2000). These appr oaches include, ignoring it and treating each observation as independent (e.g., Smith, Glass, & Miller, 1980), calculating one mean or median from each study (e.g., Tracz et al., 1992), selecting only one observation per study (e.g., Rosenthal & Rubin, 1986), and using a mixed effects model (e.g., Beretvas & Pastor, 2003). As the available literature suggests, viol ating the assumption of independence is a serious issue. Because the RG study method is a relatively new technique, it is imperative that the consequences of vi olating independence be investigated. Even more important, the research techniques that have been used in previous treatments to control for violation of independence need to be investigated in the RG study environmen t to investigate the extent to which Type I error is impacted. Purpose of the Study This studyÂ’s purpose was to examine the potential impact of selected methodological factors on the validity of RG study conclusions. Although all of the controversies described previously are im portant, this study focused on the issues surrounding violating the assumption that th e observations are independent and the
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5 methods that researchers have devised to handle dependent data in a metaanalysis. Factors such as (a) the magn itude of coefficient alpha, (b ) sample size (i.e., number of examinees), (c) number of studies, (d) the numbe r of reliabilities incl uded in each journal study, and (e) the magnitude of the intracla ss correlation among journa l studies (i.e. the degree of dependence among journal studies) we re also considered. The impact of these factors on the accuracy of estimating reliability was investigated when four approaches to violation of independence were used: (a) trea ting dependent observati ons as independent, (b) randomly selecting a reliability index from each study, (c) calculating a mean or a median, and (d) using a twolevel mixed eff ects model. In other words, for certain method factors, does violation of independe nce significantly impact the accuracy of estimating the true reliability parameter? Research Questions In RG studies, the dependent variable in the analyses is the reliability estimate (Henson & Thompson, 2001). This research fo cused on how certain study methods, in regards to violation of independence, affect the estimated mean reliability of scores calculated across studies. The key questions that were addressed in this study were: 1. What is the effect on point and interval estimates of mean re liability of ignoring violation of independence of observations in RG studies (i.e., treating all reliability coefficients as indepe ndent observations)? 2. What is the effect on point a nd interval estimates of mean reliability of using a mean or median reliability from each study as part of a sample in a RG study? 3. What is the effect on point and interval estimates of mean re liability of randomly selecting a reliability estimate from each st udy as a part of a sample in a RG study?
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6 4. What is the effect on point and interval es timates of mean reliab ility of using a two level mixedeffects model for RG studies (i.e ., reliabilities are nested within studies)? 5. In regard to violations of independence, wh at impact do factors such as the magnitude of coefficient alpha, sample size, number of journal studies, number of reliability coefficients from each study, and the magnitude of the intraclass correlation (ICC) of the studies (i.e., the magnitude of the viol ation of independence) have when any of the methods discussed in the four rese arch questions above are investigated? Hypotheses 1. Of the five approaches to dealing with vi olation of independence examined in this research, ignoring the dependence among studies provides the worst point and interval estimates of the reliability in RG metaanalysis compared to the other treatments used; confidence interval cove rage will be grossly underestimated when dependence is ignored. 2. Randomly selecting one reliability estimat e from each study as a means to control for dependence provides better point and interval estimates of the reliability in the RG metaanalysis than ignoring the dependence; confidence interval coverage will be less problematic when randomly selecting one re liability coefficient from each study than when the dependence is ignored. 3. Calculating a mean or a median reliability from each study as a means to control for dependence provides better point and interval estimates of the reliability in the RG metaanalysis than randomly selecting one reliability estimate from each study and even better point and interv al estimates than ignoring th e violation of independence;
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7 confidence interval coverage will be less problematic using this method than when using the other previous methods (ignoring, randomly selecting). 4. The use of a twolevel mixed model provides better point and interval estimates of the reliability than the other four approaches examined in this research; the twolevel mixed model is the best approach for conf idence interval coverage in regards to violation of independence in RG metaanalysis. 5. While ignoring the dependence is the worst approach and the use of the twolevel mixed model is the best approach for estim ating point and interval estimates of the reliability, the extent to which the above methods are tenable will be moderated by the following characteristics in the RG metaanalysis. a. Point and interval estimates generated from population with larger reliability coefficients are less biased than are t hose estimates generated from populations with smaller reliability coefficients; as th e reliability estimate increases the bias of the point and interval estimates decreases. b. Point and interval estimates genera ted from populations where the mean sample size of groups is small are mo re biased than are those estimates generated from populations where the mean sample size is large; as the mean sample size of groups increases the bias of the point and interval estimates decreases. c. Point and interval estimates generate d from populations where the number of journal studies is large are less biased than those generated from populations where the number of journal studies is small; as the number of journal studies increases the bias of the point a nd interval estimates decreases.
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8 d. Point and interval estimates generate d from populations where the number of reliabilities is large are more biased than those generated from populations where the number of reliabilities is sm all; as the number of reliabilities from each study increases the biased of the poi nt and interval estimates increases. e. Point and interval estimates generated from populations where the intraclass correlation is large are more biased th an are those estimates from populations where the intraclass correla tion is small or zero; as the intraclass correlation increase the bias of the point and interval estimates increases. Limitations of the Study The limitations of this study are relate d to the Monte Carlo method for the study. While the Monte Carlo method was used to simula te RG studies, the va lues of the factors used in the simulation were fixed for each study. Because the data for this study were simulated, the number of reliability indices from each simulated study was a fixed value in each of the simulations as each study contributed the same number of reliability indices per study. While it is obvious that seve ral of the RG studies conducted previously treated reliability coefficients from the same study as independent, not all of the studies contributed equal numbers of reliability coefficients. In several of the RG studies conducted prev iously, testretest re liability estimates given are very rarely and seldom evaluate d. Because coefficient alpha is the most common reliability coefficient reported, this was the only in dex used in the study. It is important to remember, however, that coeffi cient alpha has a tendency to under estimate the actual reliabili ty index (Crocker & Algina, 1986).
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9 Definitions of Terms The following terms are used throughout this study: Classical Test Theory A model used in testing where an individualÂ’s observed score (X) on a measure is composed of the sum of his or her true score (T) and error score (E), i.e. ; Crocker & Algina,1986 XTE Effect Size The magnitude of the effect of a treatment. According to Cohen (1988) it is Â“the degree to which a phenom enon is present in a populationÂ” (p. 78). Types of Effect Size There are many ways to calculate an effect size. However, Rosenthal (1994) states they basi cally fall into two Â“familiesÂ”: the d family and the r family. The d family is based on CohenÂ’s d where d is the sample effect size that estimate the population effect size such that EC C and EC C X X d s The r family refers to the Pearson pro duct moment correlation. IntraClass Correlation (ICC) The statis tical index that measures the magnitude of the dependence among observations such that: () ((1))bw bwMSMS ICC M SiMS where MSb is the mean squares between studies, MSw is the mean squared within studies, and i is the number of reliabilities for each study (S tevens, 1999). This value can range from 1 1 n to 1. The larger the ICC th e higher the degree of the de pendence (Kenny & Judd, 1986). In mixed models, where there is a twolevel hierarchy, it is define d as the proportion of variance in the dependent variab le that is between the secondlevel units (Kreft & de Leeuw, 1998). Specifically, in an RG study le vel one would model the variance within
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10 studies and level two would model the vari ance between studies. Therefore, the ICC would represent the proportion of variance in reliability that is between studies. Independence of Observations The assu mption that observations in a study are independent means there is no correlati on or relationship between them (Glass & Hopkins, 1996). Kenny and Judd (1986) define i ndependence of observa tions in term of conditional probabilities: Â“If tw o observations are independent of each other, then the conditional probability of one of them, given the other, is not different from the unconditional probabi lityÂ” (p. 422). If i X and j X are samples from an infinite population with a mean of and a variance of 2 then the observations i X and j X are said to be independent if the expected value of the product of the distance of i X to the mean and the distance of j X to the mean is equal to zero i.e. 0ijEXX (Kenny & Judd, 1986). In the case of metaana lytic research, observa tions are considered to be independent when the value of any sta tistic (when it is included in a metaanalysis) is in no way predictable from the value of a ny other included statistic in the same metaanalysis study (Tracz et al., 1992). MetaAnalysis This is the method developed by Glass (1976 ) that uses statistical procedures to combine results of multiple studies to make inferences in regards to an overall measure of an index (e.g ., reliability) across studies. Mixed Effects Models This is a model that is used for multiple levels of measurements that analyzes data in a cluste red or nested structur e (Kreft & de Leeuw, 1998). It is often referred to as Hi erarchical Linear Modeling (HLM).
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11 Reliability This is the degree to which the scores of a measure (i.e. test) are consistent over repeated administrations of th e same test or parallel forms of the test (Crocker & Algina, 1986). Reliability Index This is the correlation that represents the strength of the relationship between true and obs erved scores. It is the ratio of the standard deviation of true scores to the standard deviation of observed scores XT T x (Crocker & Algina, 1986). Reliability Coefficient This is the ratio of true score variance to observed score variance and is the square of the reliability index. It is also de fined as the correlation between two scores on parallel tests 122 X 2 T X X (Crocker & Algina, 1986). There are two main types of reliability coefficients: TestRetest This is the correlation between scores on two separate administrations of the same measure given to same group of individuals. InternalConsistency This is the co rrelation that is based on a single administration of a test. The types of reliability coefficients will be discussed in detail in chapter 2. Reliability Generalization (RG) Studies This is a metaanalysis study method that was developed by VachaHaase to make ge neralizations about the average reliability of a measure or construct (VachaHaase, 1998). Reliability Induction When authors report reliability from previous samples or test manuals when defending the reliability of the data in their studies (VachaHaase, 1998).
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12 Importance of the Study Whereas Thompson and VachaHaase (2000) have argued that a series of RG studies could reveal that, across samples, the reliability of scores for a given scale are relatively stable, they also reported that it is possible that such analys es could reveal that the variation in reliability is not related to tr eatment factors. It is important to recognize that to comprehend what RG studies may re veal, the consequences of the method flaws of RG studies must first be considered. This research will address the consequences of violating the assumption of independence and offer some suggestions for handling these issues. Not only will the results of this re search contribute to the future RG study methods, it will also serve as a reminder of the consequences of ignoring important assumptions in all research.
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13 Chapter Two: Literature Review This literature review is divided into four parts. First, metaanalysis and reliability are briefly discussed. Second, literature on specific points of inte rest on the 32 published RG studies that have been conduc ted to date is presented. Thir d, the issues that have been addressed by scholars in response to RG st udies method are presented. Finally, the literature about violation of i ndependence is presented. MetaAnalysis Metaanalysis, sometimes referred to as research synthesis, is a quantitative research approach that converts individual study outcomes to a common metric, such as effect sizes, and compares them across studi es. Each study is considered one observation from a hypothetical universe of studies. In 1976, Glass originated th e term Â‘metaanalysisÂ’ and defined it as Â“the statistical analysis of a large collection of analysis results from individual studies for the purpo se of integrating findingsÂ” (p. 3). Metaanalysis is a secondary analysis that can be used to summarize quantitatively large bodies of literature. When a large number of studies are aggregated, metaanalysis can investigate factors that were not investigated in the primary studies and detect the effect of possible moderating vari ables. Since it was first introduced, several approaches have been developed. There are fi ve basic approaches: classical or Glassian metaanalysis, study effects metaanalysis, homogeneity testbased metaanalysis, validity generalization metaanalysis, and ps ychometric metaanalysis (BangertDrowns, 1986; Hunter & Schmidt, 1990).
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14 The classical or Glassian metaanalysis pr ocedure calculates the mean effect size, ( d), as an estimate of th e population effect size, ( ), across studies for the entire universe of studies. This method has been criticized for being too liberal when determining which studies to include in a meta analysis. Glass argued that all the studies related to a given topic should be in the sample regardless of quality of a study because study quality is related to the variance of treatment effects in each study. In GlassÂ’s method, the unit of analysis is the study findi ng such that effect si zes can be calculated for each comparison between groups or sub groups for the different criteria from each individual study. In GlassÂ’s method, effect sizes also can be aver aged from different dependent variables that ma y measure different construc ts (BangertDrowns, 1986; Hunter & Schmidt, 1990). The Glassian method has been strongly cr iticized for several reasons: (a) it clearly violates the assumption of independence of observations by including several effect sizes from a single research study, which, in turn, leads to rather large inflated total sample size; (b) all studies are include d regardless of the quality of the method (i.e., Â“garbage in garbage outÂ”); and (c) the method has a te ndency to mix different independent and dependent variables (BangertDrowns, 1986; Hedges, 1982; Hunter & Schmidt, 1990). Glass defended his methodology by stating the pur pose of a metaanalysis is to present a very broad overview of a specific research in terest. For example, Glass investigated the impact of all types of psychotherapy (Smith, Glass, & Miller, 1977) and the effects of class size on all types of achievement (Glass & Smith, 1979). The study effect metaanaly sis is very similar to th e Glassian method except the criteria for the inclusion of studies are mu ch more selective. If a studyÂ’s methods are
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15 flawed, the study is not included. Another impor tant difference is the study is the unit of analysis; thus, only one eff ect size is calculated for each study. This method was suggested by Mansfield and Busse (1977) and has been applied in several metaanalyses since 1979 (see Bangert & Drowns, 1986). Some researchers advocate inclusion ru les that are more selective (Henson, Kogan, & VachaHaase, 2001; Landman & Dawes, 1982; Mansfield & Busse, 1977; Thompson & VachaHaase, 2000; VachaHaase, 1998; Wortman & Bryant, 1984). Studies with serious methodologi cal flaws are excluded. The di fficulty with this approach is that reviewer bias can influence decisions about which studies should be included. This, in turn, may distort th e findings of the metaanalysis in regard to the true population. Glass argued that a ll the studies related to a gi ven topic should be in the sample, regardless of quality. The distribution of effect sizes should then be corrected for sampling error, measurement error, and restriction of range. Study reports or publications, however, do not always contai n information necessary for making these corrections. Metaanalysts using this appr oach may average effects from different dependent variables, even when these effect s measure different constructs. The problem is that when study findings ar e used as the units of anal ysis, nonindependent data are produced and greater weight is given to studi es with more comparisons. This may cause a bias towards statistically significant results (i .e., inflation of Type I error rate; BangertDrowns, 1986). The test of homogeneity metaanalysis me thod is used to evaluate how much of the variance among effect sizes is due to sampli ng error. In this met hod, statistical tests are used to determine if the variability in st udy outcomes is statistically significant. If the
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16 tests are statistically signifi cant, then this would be the basis for detecting moderating variables (BangertDrowns, 1986; Hedges, 1982; Hunter & Schmidt, 1990; Rosenthal & Rubin, 1982). A major criticism of this approa ch is that it based on only the estimated error and lacks the power to detect differ ences. Hunter and Schmidt (1990) contended that there could be other arti facts that might be sources of variance. Hedges and Olkin (1985) contended that even if the variance across studies is statistically significant and not due to artifactual sources, it is often sm all in magnitude and usually not practically significant. They cautioned researchers to i nvestigate the actual size of the variance. Schmidt and Hunter (1977) developed a pr ocedure usually referred to as validity generalization to address the problem of artifacts that can affect variance in observed effect sizes. In this particular method, co rrelations are used to measure effect sizes. Schmidt and Hunter argued that the mean ef fect size should be corrected because it was attenuated by unreliability a nd possible range restriction. In this method, they test for statistical artifacts. There are 11 statistical artifacts wh ich Hunter and Schmidt (1990) have identified that could di stort the size of th e study correlation. These are (a) sampling error, (b) error of measurement in the depende nt variable, (c) error of measurement in the independent variable, (d) dichotomization of a continuous dependent variable, (e) dichotomization of a continuous independent variable, (f) rang e variation in the independent variable, (g) range variation in the dependent variable, (h) deviation from perfect construct validity in the independent variable, (i) de viation from perfect construct validity in the dependent vari able, (j) reporting or transcription error, and (k) variance due to extraneous factors. Hunter and Schmidt (1990) argue that if th e first three of these artifacts account for 75% or more of the obser ved variance of the effect sizes, the residual
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17 of the observed variance is due to the other eight. This led to conclusion that the true observed variance was actually zero (i .e., the effect sizes from each study are homogeneous). If this is not the case, the ne xt step should involve testing for moderating variables. RG studies involve looking at measurement error acr oss studies in an attempt to characterize the psychometric pr operties of the hypothetical un iverse of studies that may employ a particular measure. Such properties may include the mean reliability coefficient obtained in such a population, the variance of the reliability coefficient across studies, and treatment factors that may influence the magnitude of the coefficient (i.e., moderating variables). In the aggregation of research resu lts through metaanalysis, fundamental questions typically focus on (a) point and interval estimation of the mean effect size and (b) the relationship between the mean effect si ze and treatment factors. Estimates of mean effect sizes and relationships between effect sizes and other variable s usually are obtained using weighted least squares, in which effect sizes of individual st udies are weighted by the inverse of their sampling varian ce (Hedges & Olkin, 1985). That is, 1 Âˆ vari i wherei = weight for the i effect size, and Âˆ vari = estimated sampling variance of the i effect size. The argument for using such weights in statisti cal estimates is that the weights will give greater credibility to the e ffect sizes obtained from studi es with less sampling error. These studies typically have larger sample sizes.
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18 Glass argued that literature reviews should be as system atic as primary research and should interpret the results of individual studies in the context of distributions of findings, partially determined by study charac teristics and partially random. Since that time, metaanalysis has become a widely acc epted research tool encompassing a family of procedures used in a variety of discipline s. In a metaanalysis, research studies are collected, coded, and interpre ted using statistical methods similar to those used in primary data analysis. The result is an in tegrated review of fi ndings that is more systematic and exact than a narrative review. The Reliability of Measures Reliability refers to dependability or consistency. In educational and psychological research, tests are used to quantif y the relative standing of an individual on a psychological trait or ability. In educationa l and psychological research when attempts are made to measure a trait or ability more than once for an individual, it is very unusual for that individual to score ex actly the same for every admini stration, unlike the physical sciences. What can be measured is the degr ee to which a test score is consistent. The challenge is that when individuals take a test there are systematic and random errors that can occur when a test is repeated. In addition, repeated administrations of a test are not always feasible. In classical test theory, the reliability coefficient, x x is defined as the correlation between scores on parallel tests (Crocker & Aligina, 1986). According to classical test theory, an examineeÂ’s observed score, X, can be expressed as the sum of his/her true score and random error: X = T + E
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19 The reliability coefficient is the proportion of the observed variance in scores that represents true score varian ce rather than random error: 2 2 true xx total where x x is the ratio of the true score variance to total score variance. The most common approaches for estimati ng the reliability of scores include administering the same test twice to the sa me examinees (testretest reliability) or administering the test once and estimating score reliability from the intercorrelation of test items (internal consistency reliability ). Testretest reliability is estimated by calculating the correlation coefficient between the scores obtained on the two administrations of the test. Internal consiste ncy reliability is estimated by calculating the correlations between subsets of items on the test (Crocker & Algina, 1986). There are several indices that can be used to measure internal consistency: Coefficient Alpha Also known as Cr onbachÂ’s alpha, it can be calculated as follows: 2 21 1i xs k ks where k is the number of items on a test and 2 is is the variance of item i, and 2 x s is the total test variance (Crocker & Algina, 1986). Kuder Richardson Formulas (KR21 and KR20) These are indices of homogeneity that Kuder and Richards on (1937) developed that are based on the proportion of correct and incorre ct answers to each of the items on the test. Kuder Richardson Formulas are used when a test is scored dichotomously. KR20 can be calculated as follows:
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20 KR20 = 21 1x p q k ks where k is the number of items on the test 2 x s is the variance of scores on the total test, p is the proportion of correct answers, and q is the proportion of incorrect answers. KR 21 is si milar to KR20 except with KR21 it is assumed that all items on a measur e are equally difficult. KR21 can be calculated as follows: KR21 = 2() 1 1xkXkX kks where k is the number of items on the test, 2 x s is the variance of scores on the total test, and X is the mean of the scores. Splithalf Method Reliability is estimated by artificially splitting a measurement in half and calculati ng the correlation between the two halves. It has been argued that this produces a reliability coefficient that underestimates the true reliability (C rocker & Algina, 1986), therefore the Spearman Brown prophecy formula can be employed to calculate a corrected estimate. It can be calculated as follows: '2 1AB xx AB where x x is the predicted reliability coefficient for the fulllength of the test and AB is the correlation between the two halves.
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21 RG Studies Since 1998, 32 RG studies have been publishe d; these have been labeled with an asterisk in the reference section and are listed in Table 1. In addition the scales in which these RG studies have examined are labele d with a double asterisk in the reference section. Of these, only th ree (Henson et al., 2001; Rees e, Kieffer, & Briggs, 2002; Viswesvaran & Ones, 2000) have examined reli ability generalization in terms of multiple measures of the same construct. The follo wing paragraphs highlight some of the key characteristics of these 32 studies. Inclusion criteria are key characteristics of RG studies. In most cases, before a study was examined in terms of its reliability reporting, studies had to be in English and published. There were only three RG studies that allowed nonEng lish studies into the sample (Barnes, Harp, & Jung 2002; Beretv as, Meyers, & Leite, 2002; De Ayala, VonderharrCarlson, & Kim, 2005) and only five articles that included dissertations in the sample (Barnes et al., 2002; Beretvas et al., 2002; Capraro & Capraro, 2002; Nilsson, Schmidt, & Meek, 2002; OÂ’Rourke, 2004). Very little explanation was given in any of the RG studies to support the inclusion cr iteria. When Caruso and Edwards (2001); Caruso, Witkeiwitz, BelcourtDittloff and Gottlieb (2001) and Leach, Henson, Odom, and Cagle (2006) conducted their RG studies no t only did they use only published studies in English, they also eliminated any testre test coefficients. It was not clear in these articles if the testretest coefficients came fr om the same articles as the alphas in the RG study or if they were from other published st udies. The authors simp ly argued that there were not enough testretest coeffici ents to conduct a valid study.
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22Table 1 Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Barnes, Harp, & Jung (2002) Spielberger StateTrait Anxiety Inventory Published articles only Allowed nonEnglish articles 46 articles 6 % of the articles found Total number of samples 117. 59 were state (52 alpha, 7 testretest) 58 were trait (51 alpha, 7 testretest) State Alpha M= .91; Md= .92; SD = .05 Testretest M= .70; Md = .68; SD = 0.20 Trait Alpha M=.89; Md = .90; SD = .05 Testretest M= .88; Md = .88; SD = 0.05 Descriptive statistics for both alpha and testretest presented separately Correlation for alpha only Beretvas, Meyers, & Leite (2002) MarloweCrowne Social Desirability Scale Published articles and dissertations Allowed nonEnglish articles 72 articles 8.7% of the studies found Total number of sample 182 149 CronbachÂ’s alpha 3 Spearman Brown 9 KR20 21 testretest Mixed effects model M= .726; SE = .0248 Fixed effects mode l M= .68; SE = .0059 Median not reported Fisherz transformation applied Mixed effect models Internal consistencies grouped together Only 123 internal consistency reliabilities were used in the mixed effect model Capraro & Capraro (2002) MyersBriggs Type Indicator Published articles and dissertations No mention of nonEnglish versions 14 articles 7% of the articles found Total number of samples 70 50 alpha 20 testretest Alpha M= .816; SD = .082 Testretest = .813; SD = .098 EI scale M= .838; SD = .052 SN scale M= .843; SD = .052 TF scale M= .764; SD =.122 JP scale M= .822; SD =.073 Median not reported Descriptive statisti cs and box plots for alpha and testretest presented. Capraro, Capraro, & Henson (2001) Mathematics Anxiety Rating Scale Published articles no mention of dissertations or nonEnglish versions 17 articles 25% of the articles found Total number of samples 35 28 alpha 7 testretest Alpha M= .915; SD = .083 Testretest M= .841; SD =.073 Median not reported 4 regression models Descriptive statistics presented for these regression models Caruso (2000) NEO personal ity scales Published articles only 37 articles 15% of the articles found Total number of samples 51 47 alpha 4 testretest NEO scales N scale M= .88; Md = .88; SD = .07 E scale M= .83; Md = .83; SD = .09 O scale M= .79; Md = .79; SD =.13 A scale M= .75; Md = .77; SD = .10 C scale M= .83; Md = .84; SD = .47 Fisherz transformation applied Used correction for restriction of range formula Descriptive statistics reported Correlations Analysis of Variance (ANOVA)
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23Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Caruso & Edwards (2001) Junior Eysenck Personality Questionnaire Published articles only Non English versions omitted Testretest omitted 23 articles 7% of the articles found Total number of samples 44 All were alpha P scale M= .68; Md = .68; SD = .09 E scale M= .73; Md = .73; SD = .07 N scale M= .78; Md = .80; SD =.08 L scale M= .77; Md = .79; SD = .10 Fisherz transformations applied Descriptive statistics for scales and predictor variables Regression Analysis Caruso, Witkiewitz, BelcourtDittloff, & Gottlieb (2001) Eysenck Personality Questionnaire De Ayala, VonderharrCarlson, & Kim (2005) Beck Anxiety Inventory scores Published articles only Non English versions omitted Testretest omitted Published articles only Some nonEnglish versions omitted 44 articles 2.9% of the article found 47 articles 32.4% of the articles found Total number of samples 69 for three of the scales and 65 for one. All were alpha 43 alpha 12 testretest P scale M= .66; Md = .68; SD = .13 E scale M= .82; Md = .82; SD = .05 N scale M= .83; Md = .83; SD =.04 L scale M= .77; Md = .78; SD = .05 Alpha M= .91; SD = .03 Testretest =.66; SD = .22 Medians not reported Fisherz transformations applied Descriptive statistics for scales and predictor variables Multiple regression analysis Descriptive statistics Box plot Bivariate correlation analysis Alpha and testretest analyzed separately DeditiusIsland & Caruso. (2002) ZuckermanÂ’s Sensation Seeking Scale, form V Published articles only 21 articles 8.6% of the articles found Total number of samples 113 All were alpha TAS scale M= .75; Md = .75 ; SD = .07 ES scale M= .69; Md = .66; SD = .10 DIS scale M= .69; Md = .71; SD = .08 BS scale M= .62; Md= .61; SD = .16 Total M= .76; Md = .78; SD = .10 Fisherz transformation applied. Descriptive statistics presented Correlation analysis Hanson, Curry, & Bandalos (2002) Working Alliance Inventory Published articles only 25 articles 38% of articles found Total number of samples 73 67 alpha 6 interrater reliability (observer version) Client M= .93; SD = .04 Client Short M= .95; SD = .03 Therapist M= .91; SD = .05 TherapistShort M= .92; SD = .04 Observer M =.79; SD = .12 Medians not reported Descriptive statistics Stem and leaf display, Box Plots Bivariate correlation analysis Helms (1999) White Racial Identity Attitude Scale Studies from a previous metaanalysis study 38 articles 28 alphas for all five scales 3 alphas for four scales Contact M= .51 Disintegration M= .75 Reintegration M= .76 Pseudo M= .66 Autonomy M= .59 Median and standard deviation not reported Onetail chi squared analysis UX test Pearson correlation analysis
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24Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Hellman, Fuqua & Worley ( 2006) Survey of Perceived Organization Support Published articles from a previous metaanalysis and additional published studies found in a search 56 articles 90.3 % of articles found Total number in the sample 77 All were alpha Mean = .88; Md = .90; SD = .10 Descriptive statistics Box plot Bivariate correlation analysis Henson & Hwang (2002) KolbÂ’s Learning Style Inventory Published articles only 34 articles 30.9 % of the articles found Total number of samples 388 206 alpha 182 testretest Alpha CE Med .= 75; error = .25 RO Med =.79 ; error = .21 AE Med =.81; error = .19 AC Med = .80; error = .20 Test Â–retest CE Med .= 40; error = .60 RO Med =.52 ; error = .48 AE Med =.55; error = .45 AC Med = .56; error = .44 Means and standard deviations not reported Descriptive stat istics and Box Plots alpha and testretest were displayed separately Multiple regression Henson, Kogan, & VachaHaase (2001) Teacher Efficacy Scale, Science Teaching Efficacy Be lief Instrument, Teacher Locus of Control, and Responsibility for Student Achievement Published articles only 52 articles 5.3% of the article found Total number in the sample 86 All alpha RSA+ M= .76; SD = .03 RSAM= .84; SD = .04 TLCI+ M= .74; SD = .02 TLCIM= .70; SD = .13 PSTE M= .88 ; SD = .05 STOE M= .761; SD =.025 PTE M = .778; SD = .057 GTE M= .696; SD = .072 Medians not reported Descriptive statistics Box plots Bivariate correlations Kieffer & Reese (2003) Geriatric Depression Scale Published articles only 98 articles 28.99% of the article found 117 articles reported means and standard deviations that were used to calculate KR 21 Total number in sample 267 100 alpha 33 test retest 134 calculated KR21 Overall without KR21 estimates M= .85; SD = .09 Over all with KR21 estimates M= .8027; SD =.14 Alpha M= .8522; SD = .09 Testretest M= .83; SD = .08 KR21 estimates M= .76; SD = .14 Medians not reported Descriptive statistics were presented for each of the reliability types separate and together. Box Plots for the 133 coefficients Compared to the 267 (added KR21) Multiple regression separately for the 133 reliabilities and the 267 coefficients
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25Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Lane, White, & Henson (2002) Coopersmith SelfEsteem Inventory Published articles in English only 33 articles 11.97 % of the articles found 107 articles reported means and standard deviations that were used to calculate KR 21 Total number in the sample 683 66 KR20/ alpha 69 testretest 548 calculated KR21 KR20/alpha M= .729; SD =.14 Testretest M= .55; SD = .172 KR21 M= .67; SD =.31 Medians not reported Descriptive statistics Scatter graph depicting the relationship between KR21 and KR20/alpha Box plots Regression analysis ANOVA Leach, Henson, Odom, & Cagle (2006) SelfDescription Questionnaire. Published articles only Testretest omitted 56 articles 50% of the articles found 813 alphas Three subscales evaluated separately SDQ I (n =29), SDQ II (n =13), SDQ III ( n = 24) SDQ I M= .92; SD = .04 SDQ II math M= .93; SD = .01 Verbal M= .85; SD = .04 GS M= .85; SD = .01 GSC M= .86; SD = .03 SDQ III Not reported Medians not reported Descriptive statistics Regression analysis ANOVA Nilsson, Schmidt, & Meek (2002) Career DecisionMaking SelfEfficacy Scale Published articles and dissertations 20 articles/dissertations 41% of the articles found Total number in the sample 20 19 alpha 1 testretest CDMSE Mean = .95; SD = .04 CMESEshort form Mean = .94; SD = .01 Medians not reported Descriptive statistics Bivariate correlations ANOVA OÂ’Rourke, (2004) Center for Epidemiologics StudiesDepression (CESD) Scale Published articles and dissertation 106 articles/dissertations 68% of the articles found Total number in the sample 141 11 testretest 130 alpha Mean = .88, Md = .89; SD = .05 Descriptive statistics presented for alpha and testretest separately Testretest sample (n = 11) was only evaluated using a Correlation coefficient. Descriptive statistics, box plot and regression analysis for alpha only ( n = 130) Reese, Kieffer, & Briggs (2002) Adult Attachment Scale Bell Object Relations Inventory Inventory of Parent and Peer Attachment Parental Attachment Questionnaire Parental Bonding Instrument Published articles only 53 articles 34.4% of the articles found Total number in the sample 53 44 alpha 9 testretest Combined AAS M= .75; SD = .07 BORI M= .77; SD = .08 IPPA M= 87; SD = .08 .PAQ M= .89; SD = .05 PBI M= .82; SD = .11 Medians not reported Descriptive statistics presented for alpha and testretest separately and combined Box Plots of combined reliabilities for each subscale Bivariate correlations
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26Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Ross, Blackburn, & Forbes (2005) Patterns of Adaptive Learning Survey Published articles only 30 articles 47 % if the articles found Total number in the sample 103 alphas Overall M= .77; SD = .07 EX scale M= .68; SD = .07 TG scale M= .79; SD = .05 PAP scale M= .79; SD = .07 PAV scale M= .81; SD = .04 Medians not reported Descriptive statistics were displayed using Box Plots separating the four scales. Scales were averaged separately and together. Regression analysis Ryngala,Shields, & Caruso (2005). Reliability Generalization of the Revised ChildrenÂ’s Manifest Scale Partitioned normative sample from previous study NA 48 alphas for each of the 4 subscales from 48 sub samples Overall M= .79; Md = .81; SD = .06 Phy scale M= .59; Md = .61; SD = .13 W&S scale M= .7 6; Md = .77; SD = .06 Consent scale M=.62; Md = 63; SD = .11 Lie scale M= .70; Md = .72; SD = .10 Fisher z transformations applied Descriptive statistics Hierarchical multiple regression analysis Shields & Caruso (2003) Alcohol Use Disorders Identification Test Published articles in English only 17 articles 16.3% of the articles found Total number in the sample 24 All alpha Mean = .79; Md = .81; SD = .10 Fisher z transformations applied Descriptive statistics Multiple regression analysis Hierarchical regression analysis Shields, & Caruso (2004) Cage Questionnaire Published English only articles 13 articles 15 of the articles found 22 alphas Mean = .73; Md = .74; SD = .09 Descriptive statistics Bivariate correlation and point Â– biseriral correlation. Thompson & Cook (2002) LibQUAL+TM scores The survey was administered to 20,416 persons from 43 universities in the US and Canada NA 43 alphas from all 43 universities All alpha Overall M= .94; SD = .02 S_Affect scale M= .94; SD = .01 Li_Place scale M= .90; SD = .03 Pers_Com scale M= .86; SD = .04 Info_Acc scale M= .72; SD = .07 Medians not reported Alpha for each of the 43 university is displayed Descriptive statistics Regression analysis VachaHaase(1998) Bem Sex Role Inventory Published articles only 57 articles 9 % of the articles found Total number in sample 87 pairs for male and female. The article reports that alpha, KR20 and testretest were found but no Â“nÂ” was reported. Box Plots were displayed but no specific values for mean, median or standard deviations were reported Descriptive statistics were displayed using Box Plots separating Male and female reliabilities. Alpha and test Â–retest are not analyzed separately. Regression analysis Canonical correlation
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27Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis VachaHaase, Kogan, Tani, & Woodall (2001) Minnesota Multiphasic Personality Inventory (clinical) Published articles only 153 articles 7.8 % of the articles found 10 scales had an average 49 reliability coefficients. The article reports that alpha and testretest were found but no Â“nÂ” was reported. Hs scale M= .72; Md = .76 ; SD = .13 D scale M= .70; Md = .73; SD = .17 Hy scale M= .65; Md = .70 ; SD = .16 Pd scale M= .66; Md = .68 ; SD = .16 Mf scale M= .67; Md = .72 ; SD = .20 Pa scale M= .64; Md = .68 ; SD = .15 Pt scale M= .72; Md = .78 ; SD = .18 Sc scale M= .73; Md = .79 ; SD = .18 Ma scale M= .69; Md = .72 ; SD = .14 Si scale M= .81; Md = .85 ; SD = 14 Descriptive statistics were displayed using Box Plots separating the 10 scales. Alpha and test Â–retest are not analyzed separately. Multiple regression analysis VachaHaase, Tani, Kogan, Woodall, & Thompson (2001) Minnesota Multiphasic Personality Inventory (validity) Published articles only For three of the scales L, F and K 153 articles 7.8 % of the articles found 37 articles specifically for the L, F, and K scales 47 coefficients for the L scale 48 coefficients each for the F and K scales The arti cle reports that alpha, and testretest were found but no Â“nÂ” was is reported. L scale M= .68; Md = .71; SD = .16 F scale M= .68; Md = .72; SD = .18 K scale M= .73; Md = .76; SD = .13 Descriptive statistics were displayed using Box Plots separating the three scales. Alpha and testÂ–retest are not analyzed separately. Regression analysis Viswesvaran & Ones (2000) Â“Big Five FactorsÂ” Published technical manuals 28 technical manuals Total number in the sample 2207 1359 alpha 848 testretest Alpha Emotional Stability M= .78; SD = .11 Extraversion M= .78; SD = .09 Open to Experience M= .73; SD = .12 Agreeableness M= .75 ; SD = .11 Conscientious M= .78; SD = .10 Testretest Emotional Stability M= .75 ; SD = .10 Extraversion M= .76; SD = .12 Open to Experience M= .71; SD = .13 Agreeableness M= .69; SD = .14 Conscientious M= .72; SD = .13 Descriptive stat istics and Box plots for alpha and testretest reported separately Wallace & Wheeler (2002) Life Satisfaction Index Published articles only 30 articles 19.11% of the articles found Total number in the sample 34 All alpha Mean = .79; Md = .79; SD = .10 Fisher z transformations applied Descriptive statistics Bivariate Correlations
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28Table 1 (continued) Reliability Generalization Studies Study and Instrument ( or construct) investigated Additional inclusion criteria Number of articles found that reported reliability for the data Â“in handÂ” Number of samples and type of reliability Reliability Mean, Median and Standard Deviation. Type of Analysis Yin & Fan (2000) Beck Depression Inventory Published English only articles 90 articles 7.5% of the articles found Total number in the study 165 142 alpha 23 testretest 121 SEM were also calculated Overall M= 82; SD = .008 Alpha M= .84; SD = .007 Testretest M= .69; SD = .009 Medians not reported. Descriptive statistics for different types of reliability are reported (separately and combined) and for SEM. Eta squared calculated for an effect size Correlation analysis Youngstrom & Green (2003) Differential Emotions ScalesÂ—IV Secondary analysis of published studies only None of the studies identified reported reliability. Raw data was retrieved from 30 studies Total number in the study 30 All alpha Fear M= .77; SD = .09 Selfhostility M= .74; SD = .15 Shyness M= .73; SD = .11 Sadness M= .73; SD = .10 Enjoyment M= .71; SD = .13 Anger M= .71; SD = .12 Guilt M= .63; SD = .15 Shame M= 63; SD = .13 Contempt M= .58; SD = .15 Disgust M= .61; SD = .11 Surprise M= .56; SD = .19 Interest M= .56; SD = .19 Negative affect M= .92; SD = .02 Hostility M= .77; SD = .08 Positive affect = .71; SD = .12 Medians not reported Descriptive Statistics Biviariate correlations
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29 Not all RG studies have been conducte d on articles found through searches of published articles. For example, Helms (1999) used 38 studies from a previous metaanalysis of the White Raci al Identity Scale (Helms & Carter, 1990). In their study evaluating scores for the LibQual measure, Thompson and Cook (2002) administered the scale to 20,416 persons from 43 different universities. Coefficient alpha then was calculated for all 43 universities, and these es timates were used in the RG evaluation of the scale. In their study on the Revised Ch ildrenÂ’s Manifest Anxi ety Scale (RCMAS; Reynolds & Paget, 1983), Ryngala, Shields, and Caruso (2005) used a normative sample derived from 13 states and 80 sc hool districts across the United States such that they had a sample size of 4, 972 children ranging in ages between 6 and 19 years old. This information came from a study that wa s conducted by Reynolds and Paget (1983). Ryngala et al. (2005) include d 48 subsamples (2 gender x 2 ethnic x 12 age groups = 48) from Reynolds and PagetÂ’s data for their RG study. Coefficient alpha was calculated using each of these 48 subsamples for each of the four subscales of the instrument. When Youngstrom and Green (2003) attempted to conduct a RG study on the Differential Emotions Scales IV (Izard, Libero, Putnam & Haynes, 1993) they found no studies that reported reliability estimates. They actually contacted several author s from their search and calculated coefficient alpha from the raw data of 30 different st udies. For their RG study on the Perceived Organization Support ( Eisenberger, Huntington, Hutchison, & Sowa, 1986) Hellman, Fuqua, and Worley (2006) used published articles from a previous metaanalysis and additional articles found in their own search. When Viswesvaran and Ones (2002) wanted to examine the reliability of score measuring the Â“Big Five FactorsÂ” (Barrick & Mount, 1991) they us ed 28 technical manuals as a data source. Other than
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30 Viswesvaran and OnesÂ’s study, all of the RG studies used reliability coefficients that were either calculated or reported for the actu al data from the studies. Thus, the criteria for inclusion can contribute to possible bias in the statistical analysis. Seventeen of the RG studies included bot h testretest reliability and internal consistency estimates, and 15 examined only inte rnal consistency estimates. Within most of these studies, even when both testretest and internal consistenc y estimates were used, the number of studies that used internal cons istency to estimate the reliability was always higher (Table 1). For example, in their study on the MarloweCrowne Social Desirability Scale (Crowne & Marlowe, 1960), Beretvas et al. (2002) found 93 articles that reported reliability but had a total of 182 observations, 21 of which were testretest coefficients. As mentioned earlier several of the studies (Caruso & Edwards, 2001; Caruso et al., 2001; Leach et al., 2006) decided to omit testret est reliabilities all together. Several of the RG studies also used a derived KR21 for dichotomously scored measures using means and construct standard deviations re ported in the given studies to estimate the reliability (Henson et al., 2001; Kieffer & R eese, 2003; Lane et al., 2002).These authors argued that the use of KR21 was a possible solu tion to estimating reliability indices that were not given in the original study. Many types of analysis have been used in RG studies. For almost all of the studies, descriptive statistics were available, such as the mean, sample size, and standard deviation of the scale(s). Se veral studies also displayed box plots (Capraro & Capraro, 2002; De Ayala et al., 2005; Hanson et al., 2002; Henson & Hwang, 2002; Hellman et al., 2006; Henson et al., 2001; Kieffer & Reese, 2003; Lane et al., 2002; Resse et al., 2002; Ross et al., 2005;VachaHaase, 1998; Vach aHaase, Kogan, Tani, & Woodall, 2001;
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31 VachaHaase, Tani, Kogan, Woodall, & Thompson, 2001; Viswesvaran & Ones, 2000). In their RG study on the Working Alliance Inventory (Horvath & Greenberg, 1989), Hanson et al. (2002) provided a stemandleaf pl ot of the score reliabilities. If a measure had several scales within it, the descriptive statistics were reported separately for each scale of the measure. In several studies, te stretest and coefficient alpha were analyzed together. For example, in all three of he r studies (VachaHaase, 1998; VachaHaase, Kogan, et al., 2001; VachaHaase, Tani, et al., 2001), VachaHaase coded testretest and coefficient alpha separately but did not dist inguish the two when calculating descriptive statistics. In all three of thes e articles, box plots were used to display the distributions of reliability coefficients for each scale of the measure she was investigating. The articles did not indicate how many of the reliability coefficients were testretest and how many were coefficient alpha. Some of the articles displayed descriptive statistics for the two types of reliabilities separately and together (Reese et al., 2002; Yin & Fan, 2000). When Kieffer and Reese (2003) and Lane et al. ( 2002) used data from st udies to calculate KR21, they reported the descriptiv e statistics using all of the reliabilities together and all of the reliabilities that were not calculated from the data. In other words, the testretest coefficients were not separated from the intern al consistency reliabi lities. In their study on the Coopersmith SelfEsteem Inventory (C oopersmith, 1967), Lane et al. (2002) had a total of 683 reliability coefficients, 66 were KR20/coefficient alpha, 69 were testretest, and 548 were calculated KR21. In their article, two box plots were displayed next to each other for comparison, one without the 548 calculated KR21 and one including them. The 69 testretest coefficients were not analyzed separately. The failure to analyze testretest and internal consistency reliability estimates separately represents a major limitation of
PAGE 44
32 available RG studies. Examining reliability over time (testÂ–retest) and examining reliability in terms of internal consistency (coefficient alpha) represent different aspects of reliability (Cro cker & Algina, 1986; Henson, 2001). For several of the studies, bivariate correlations were calculated between characteristics such as sample size and reli ability, mean age and reliability, gender and reliability, and scale variance and reliability (e.g., Barnes et al., 2002; Caruso, 2000; DeditiusIsland & Caruso, 2002; Hanson et al., 2002; Henson et al., 2001; Nilsson et al., 2002; Reese et al., 2002; Wallace & Wheeler 2002; Youngstrom & Green, 2003). Some of the studies involved the use of regression analysis with reliability as the dependent variable (e.g., Capraro et al., 2001; Caruso & Edwards, 2001; Lane et al., 2002; Thomson & Cook, 2002; VachaHaase, 1998; VachaHaas e, Tani, et al., 2001). When possible, many studies employed multiple regression (Caruso et al., 2001; Henson & Hwang, 2002; Kieffer & Reese, 2003; Shields & Caru so, 2003; VachaHaase, Kogan, et al., 2001). Only a few of the RG studies employe d analysis of variance (ANOVA; Caruso, 2000; Lane et al., 2002; Nilss on et al., 2002). Only one of the studies applied mixed models (Beretvas et al., 2002). The number of journal article s for each of the RG studies conducted so far have ranged from 14 to 153 ( see Table 1). As noted earlier, there have been five RG studies that did not use articles found through searches of pub lished articles (Helms, 1999; Ryngala et al., 2005; Thompson & Cook, 2002; Viswesvaran & Ones, 2000; Youngstrom & Green, 2003). While all of the studies indicated the number of reliabilities included and the number of articles used, only one of the RG studies actually indicated the frequency of reliabilities for each arti cle, the RG study conducted by Be retvas et al. (2002), on the
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33 MarloweCrowne Social Desi rability Scale (Cro wne & Marlowe, 1960). In this study they displayed a table that indicated the fr equency of reliability estimates per study. These values ranged from 1 to 11. Fiftytw o of the studies reported one estimate of internal consistency, 10 of the studies reporte d two, three of the st udies reported three, four of the studies reported four, one study re ported seven, one st udy reported eight, and one study reported 11 internal c onsistency estimates. Eleven of the studies reported one testretest reliability estimate, one of the studies reported two, a nd one of the studies reported 11 testretest reliability estimates. A similar issue was seen with sample sizes. Five of the studies reported a sample size per study range (Capraro & Capraro, 2002; Caruso, 2000; Caruso & Edwards, 2001; De Ayla et al., 2005; Wallace & Wheeler, 2002). The ranges of sample sizes for the five studies were: 343 to 1078, 21 to 3856, 70 to 20,968, 40 to 1172, and 20 to 1574, respectively. Only four RG studies included information on a mean sample size (Capraro et al., 2001; Hanson et al., 2002; Lane et al., 2002; VachaHa ase, Kogan, et al., 2001). The mean sample sizes and standard devi ations from the other four studies were: M = 366.23, SD =393.04; M = 56, SD = 35; M= 79.33, SD =106.33; M = 81.74; SD =84.16, respectively. For the other RG studies, sample size information was not given. In most studies, the magnitude of the reli abilities tended to be high, usually with means in the .80Â’s or higher. However, st udies reported reliabilities as low as .02 (Youngstrom & Green, 2003) and as high as .98 (Wallace & Wheeler, 2002). These extreme estimates were rare, and in most cas es the values ranged from approximately .40 to .90. It is important to note that only reported reliabilities we re part of the sample. It is
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34 possible that the reason why the means were so high was because, in most cases, only the studies that had high reli ability were published. Issues in the Debate on Re liability Generalization Regardless of the outcome of these studies almost all of the studies had some discussion of the importance of reporting reli ability and the problems with studies using what VachaHaase (1998) refers to as Â“reliability induction. Â” (p. 7). Reliability induction refers to the reporting of reliability estimates from a previous study or a test manual, not from actual study data. This type of repor ting is only marginally acceptable if two conditions are met. First, rese archers must explicitly compar e the characteristics of their samples with the characteristics of the sample from which they obtained the reliability reported (e.g., the sample used to calculate the reliability reported in the test manual). Second, the standard deviation of the scores for their sample must be similar to those from the study from which they are inducting re liability. If both of these criteria are met such that there are similariti es in the sample, it would be marginally reasonable to induce reliability (VachaHaase, 1998). Thompson and VachaHaase (2000) emphasized the need to recal l reliability is based on the scores from a test and not th e test itself. Dawis (1987) argues Â“Because reliability is a function of sample as well as an instrument, it should be evaluated on a sample from the intended populationÂ—an obvious but sometimes overlooked pointÂ” (p. 486). It is also important to note that reliability coefficients ar e used to correct effect sizes estimates for attenuation (Baugh, 2002) and to make inferences about the scores on the test. The APA Task Force (Wilkinson & APA Ta sk Force on Statistical Inference, 1999) argued that Â“Interpreting the size of an observed effect re quires an assessment of the
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35 reliability scoresÂ” (p. 596). Several meas urement textbooks concur (e.g., Crocker & Algina, 1986; Gronlund & Linn, 199 0; Pedhazur & Schmelkin, 1991). Sawilowsky (2000) argued that when Th ompson and VachaHaase (2000) refer to the reliability of the data in hand or score reliability, they are implementing Â“datametrics.Â” In his article he argued, Â“If reliability only relates to the set of scores that a test publisher obtained in a pilot, field test or norming procedure then what purpose do the Mental Measurement Yearbook and Test in Print serve?Â” (p. 117). He agreed that reliability should be reported from the research erÂ’s sample but the reliability from the test manual also should be reported as well. Not only are RG studies similar to validity generalization in term s of method, they also are similar in terms of publication bi as. In metaanalysis, publication bias is sometimes referred to as th e Â“filedrawer problemÂ” (Rosenthal, 1979, p. 260). In most cases, metaanalyses are conducted using only published studies that may be biased towards statistically significant results. The missing data problem is exacerbated in RG studies because information on reliability ofte n is not reported or th e reported reliability estimates are based on instrumentsÂ’ technical manuals rather than based on the sample used in the research. The tendency for publis hed research not to include estimates of score reliability yields data sources with very large proportions of missing information. For example, in their RG study of the Beck Depression Inventory (BDI; Beck et al., 1961) scores, Yin and Fan (2000) found that ou t of 1,200 studies that used the BDI, 80.1% (961) did not mention reliability at all, 5.6% (67) mentioned it with no citation of the estimateÂ’s source, and 6.8% (82) cited re liability from the published test manuals or other sources, leaving only 7.5% (90) of the studies that reported reliability coefficients
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36 for the data used in the actual studies. Thus, the lack of re porting of reliability coefficients for the data in hand is a common occurrence (Thompson & Snyder, 1998; VachaHaase, Ness, Nilsson, & Reetz, 1999). To compensate for the small amount of reported reliability estimates, some RG re searchers have used KR21 derived from reported studies to estimate reliability and in crease the sample size for the RG analysis. For example, in the RG study on the Coope rsmith Selfesteem Inventory (CSEI; Coopersmith, 1967), Lane et al. (2002) found 33 studies that reported some form of reliability for the data in hand; however, 107 reported sufficient descriptive information to compute a KR21 reliability estimate. Becau se the CSEI is a dichotomous instrument, these authors derived 548 KR21 coefficients to add to the pool of reliability estimates. Some concern has been voiced re garding the use of FisherÂ’s z transformation to normalize reliability esti mates when conducting an RG study (Sawilowsky, 2000). Thompson and VachaHaase (2000) have argued that reliability coefficients are a squared metric (i.e., the squared correlation between observed scores and Â“trueÂ” scores) and consequently the FisherÂ’s z transformation is unnecessary. This issue has been recently explored using testrest reliability (Rom ano & Kromrey, 2002) and coefficient alpha (Romano & Kromrey, 2004). Results of these studies suggested the use of FisherÂ’s z transformation of the reliability estimates pr ovided a modest increase in the accuracy of the estimation of the population mean score reliabi lity coefficient. This has also been the case in RG studies that implemented the Fish erÂ’s z transformation (B eretvas et al., 2002; Caruso et al., 2001; Wallace & Wheeler, 2002; Shields & Caruso, 2003). With regards to the issue of sample weighting, Hunter and Schmidt (1990) developed a method in which the weighted mean correlation is computed with the
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37 individual correlations weighted in terms of their sample sizes. More weight is given to the results of studies with larger samples because these estimates have smaller sampling errors. While this method was used in Yin and FanÂ’s (2000) RG st udy, it is not common practice in RG studies. In his RG study on the NEO personality scales, Caruso (2000) addressed this issue but argued that because the sample sizes ranged from n= 21 to n = 3,856, the large samples would have much more influence than would small samples. He also stated that because he found no statis tically significant correlation between sample size and reliability, sample weighting was unn ecessary. Finally, he indicated that he conducted an analysis using sample size weight s, and the results were no different than those obtained from the unweighted analysis In their investiga tion of this issue simulating testretest reliability estimates (2002) and internal coefficient alpha (2004) Romano and Kromrey found the use of weighted estimates provided better confidence band coverage than the use of unweighted estimates. Interest has developed in the similariti es and differences between RG analyses based on reliability coefficients and those based on the standard error of measurement or SEM. For example, in their RG study on th e BDI, Yin and Fan (2000) argue that the standard error should be reported because SE M is a function of bot h group variability and the reliability estimate. They argued that ther e is not an inverse re lationship between the SEM and the reliability estimate, that is, Â“Â… a lower reliability estimate does not necessarily mean the corresponding SEM will be largerÂ” (p. 206). While Thompson and VachaHaase (2000) agreed th at an RG study can be accomplished using the SEM, they indicated that the SEM is Â“rather crudeÂ” b ecause it estimates an individualÂ’s observed score variation in the population (i.e., holdi ng constant the true score). When examining
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38 the distribution of the SEM, examinees that sc ore above the mean are more likely to have a positive error of measurement and examinees that score below the mean are more likely to have a negative erro r of measurement. Another consider ation is the further away from the mean that an individual scores on a given measure, the larger the error of measurement (Hopkins, 1998). Dimi trov (2002) also points out that relationship between reliability and SEM is based on the assumption that the error variance is the same for all scores. Finally, Thompson and VachaHaase ( 2000) pointed out that even if one chooses to use the SEM in an RG study, it can only be useful when the same scale and form is used across studies because SEM is a function of the scale. In other words, it would not make sense to look at the SEM if one was comp aring studies that used different forms of a particular scale (forms with different variances) or if one was comparing multiple measures of the same construct. Another concern with RG studies is that many of the reliabilities are not only based on different sample sizes, they are al so based on different scale lengths. For example, Caruso (2000) in his study of th e NEO personality scale (Costa & McCrae, 1985) used a Spearman Brown formula to adjust alpha for the different number of items. Dimitrov (2002) cautions that th e splithalf appro ach requires that the estimates have equal variances. Researchers conducting RG st udies do not have access to raw data and therefore cannot test for equal variances. An analysis of all of the previ ously mentioned studi es suggested the generalization of the reliability of the study being analyzed was secondary. In some ways, it seemed that the purpose of these RG studies was to encour age researchers to evaluate the reliability of the measures that they em ploy. Most of the studies discussed in detail
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39 the percentage of studies from the articles identified that reported reliability for the data Â“inhandÂ” (see Table 1). Most of the authors also noted that even after the publication of Wilkinson and APA Task Force on Statistical Inference (1999), numer ous authors still do not report the reliability of the sc ores in the indi vidual studies. Independence Another important issue to consider is the fact that in several of the RG studies the samples used in the study did not repr esent independent ob servations. For all statistical procedures there are basic assumptions that unde rlie them (Glass & Hopkins, 1996; Pedhazur, 1982; Stevens, 1999). When examining mean differences (e.g., ANOVA, ttest) the main assumpti ons about the populations are: 1. The observations in each group are normally distributed. 2. The population variances are homogeneous (i.e., for n groups, 222 12....n ) 3. The observations are independent. These assumptions are important because the violation of any of them can lead to an increase in the probability of making a Type I or Type II error (Stevens. 1999). When statistical techniques are used to conduct research, a sample is collected to make inferences about a population. For these inferenc es to be tenable, the treatment of a study should comply with these assumptions. The irony is that in most research, violating these assumptions is somewhat unavoidable. Clearly, it is not possible for every set of observations in a given study to be indepe ndent and normally distributed with equal variance. As Stevens (1999) points out, the question is not Â“Are the assumptions being violatedÂ” but, rather, Â“How radically must a given assumption be violated before it has a serious effect on Type I or Type II error rates?Â” (p. 75). Since most metaanalyses
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40 typically involve the use of t and F tests, these assumptions are inherent in metaanalysis research. As Hedges (1982) points out, Â“If th e assumptions for the validity of the ttest are met, it is possible to derive the properties of estimators of the effect sizes exactlyÂ” (p. 13). Research has been conducted to investig ate these assumptions (e.g., Barcikowski, 1981; Bock, 1975; Glass, Peckman, & Sande rs, 1972; Kenny & Judd, 1986; Landman & Dawes, 1982; Raudenbush & Bryk, 1987; S cariano & Davenport, 1987; Tracz et al., 1992). In their literature revi ew of the first two assu mptions (i.e., normality and homogeneity of variances) Glass et al. (1972) conc luded that nonnormality only slighted impacts the alpha le vel of a study, even in cases where the distribution is skewed; given a large enough sample, the violat ion of the assumption of normality is not problematic (i.e., the statistical analysis is r obust with larger sample sizes). The research also indicated that violating the assump tion of homogeneity of variances was only problematic when group sizes are unequal such that the larger n divided by the smaller n is greater than 1.5 (Stevens, 1999). Even though they state that violating the assumption of independence is Â“Â….far more seriousÂ…Â” (p 242), Glass et al. (1972) did not investigate it in their research. Stevens also argues that in regards to the assumption of independence, Â“Â…it is by far the most important assumptionÂ” (p 77). Kenny and Judd (1986) investigated the consequences of violating independence in ANOVA. Their research demonstrated that for the F test, the mean squared within and the mean squared between are considerably biased when nonindependence is ignored. Both Scariano and Davenport (1987) and Barcikowski (1981) investigated the impact that dependence has on the inflation of the
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41 alpha level (i.e., Type I error) of a study. The intraclass correlation (ICC) was used to measure the extent that dependence is present among observati ons in a study (see Definitions in Chapter 1). Both studies i ndicated that even when the intraclass correlation was as low as .01, the larger th e number of observations in a group, the higher the Type I error rate. For example, both st udies indicated that when the intraclass correlation was .01 and the number of observa tions within a group was 10, the actual alpha level was .06 not the assumed value of .05. When the number of observations was 100, the actual level was inflat ed to approximately .17. The larger intraclass correlation turned out to be even more problematic. For example, Scariano and Davenport (1987) simulated two groups of sample size, n = 100. When the ICC was .30, the actual alpha level was approximately .77. In other words, given a study with thes e characteristics, the researcher has 77% chance of making a Type I error. This happens because when observation are correlated (i.e., dependent), then the standard error is actually smaller then if they are not correlated (i.e., independe nt). This is an issue that should not be ignored. Tracz et al. (1992) investigated the eff ect of violation of the assumption of independence when combining correlation coeffici ents in a metaanalysis. In their study they investigated the effect of the violat ion of the assumption of independence on the distribution of r and the distribution of co rrelation after a FisherÂ’s z transformation. They conducted a Monte Carlo study us ing the following parameters: (a) sample size within a study (n = 20, 50, 100), (b) the number of predictors (p = 1, 2, 3, 5), the population intercorrelation am ong predictors (rho(i) = 0,. 30,.70), and the population correlation between predictors and criterion (rho(p)= 0, .03, .07). All possi ble combinations of these
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42 parameters were used to produce the pred ictors and criteri on variables and the correlations between the predic tor and criterion variables we re calculated. The population intercorrelation was used as an index of dependence (i.e., when rho(i) =0 or when p = 1, the assumption of independence was not violated ). For all the combinations of parameters and for the r and z, means, medians, and standard deviations were calculated. The FisherÂ’s z transformed values of population co rrelation were evaluated for all combinations of parameters using 90%, 95%, and 99% confidence intervals. Their research suggested that nonindependence was not a major source of error in regards to means, medians, standard deviat ions, and confidence intervals. Landman and Dawes (1982) identified five different types of violation of assumption of independence: 1. Multiple measures of outcomes obtained from the same participant within single studies; 2. Measures taken at multiple point s from the same participant; 3. Nonindependence of scores within a single outcome measure. Both the complete score on the entire measure and the scores of separate scales of the measure are treated as independent; 4. Nonindependence of studies with in a single article; and 5. Nonindependent samples across articles. Considering the RG studies, two of these violations have occurred thus far in published studies: nonindependan ce of scores within a sing le measure (Caruso, 2000; Nilsson et al., 2002) and nonindepe ndence of studies within a si ngle article. For example, Yin and FanÂ’s (2000) RG study on the BDI incl uded 164 reliability coefficients from 90
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43 studies. Similarly, VachaHaaseÂ’s (1998) RG study on the Bem Sex Role Inventory (BSRI) used 87 reliability coefficients from 57 studies; and Caruso Â’s (2000) RG study on the NEO personality scale used 51 reliability estimates from 37 studies. These represent clear violations of indepe ndence of observations. There are many approaches to handling dependence of observations in a study. One approach is to ignore it. This seems to be what has been pr acticed in most RG studies, with multiple observations creat ed from one study. In a RG study, the observations have characteristic s in common such as the scale used but are different in the way that the observations are grouped (e.g ., gender or type of reliability index). The main difficulty with this appr oach is that if some studies have more outcomes than others, they can influence the combined results acro ss studies. One way that researchers have approached this problem is to weight each outcome by the inverse of the number of outcomes in a study (Becker, 2000). Although this may help in controlling for the influence that one study may have over a nother, it does not address dependence. Another approach that has been recomme nded is sensitivity analysis (Gleser & Olkin, 1994; Greenhouse & Iyengar, 1994). This involves first analyzing the studies independently with only one outcome per study and then repeating the analysis by adding in other outcomes from each study. The idea is that if the results to the metaanalysis are similar then the dependence can be ignored. It has also been suggested that when a study has multiple outcomes the researcher should average across the outcomes or use the median when the outcomes are parallel measures of a single construct (Raudenbush et al., 19 88; Tracz et al., 1992). Rosenthal and Rubin (1986) have suggested that if a study had a rather large sample size
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44 and small differences in the intercorrela tions between the outcomes, then a common composite outcome measure based on a commo n level of intercor relation should be used. Similarly, Gleser and Olkin (1994) s uggested deriving a composite outcome within studies by using individual intercorrelations among the outcome variables. Tracz et al. (1992) suggested that combini ng the statistics from nonindepe ndent data in a correlated metaanalysis does not have a negative eff ect in terms of estimating means, median, standard deviations, and confidence interv als. However, they did acknowledge that violation of independence could inflate Type I error in regards to testing of mean effect sizes. Finally, Beretevas and Pa stor (2003) argued that a mi xed effects model should be used to model dependence of multiple re liability estimates within a study while estimating how reliability estimates vary ac ross or between studies. They used a threelevel model where variability at the first or der represented the sa mpling variability among estimates using a known variance. The s econdlevel modeled the variability among samples within the same study. The thirdle vel modeled the variab ility in reliability estimates among studies. Beretevas and PastorÂ’s study investigat ed the same studies that Yin and Fan (2000) used for their RG study on the Beck Depression Inventory (BDI; Beck et al., 1961). The fixed effect models determined that there were three pred ictors (form, student proportion, and age) that were significant. In contrast, the mixed effects model found that only two predictors were significan t (student proportion and age). MixedÂ–effects model also were used in the Beretevas et al. (2002) study on the MarloweCrowne Social Desi rability Scale (MCSDS; Crowne & Marlowe, 1960). They
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45 also used Level 2 and Level 3 models and comp ared their results to a fixed effects model. The standard error estimates in the fixed eff ects model were found to be lower. They did caution researchers that smaller sample sizes (usually the case with RG studies) can have a negative impact on the estimation of the random effects at the within and betweenstudies levels. Keeping this in mind, they argued that the mixed effect does provide a better model to investigate the variability of score reliabilities. Summary Through the aggregation of a large number of studies, metaanalysis is a useful technique to generalize acro ss studies. While there are ma ny approaches to conducting a metaanalysis, these approaches also have lim itations that should be considered. Over the years, many researchers have evaluated th e issues surrounding several of the metaanalysis methods (e.g., Barcikowski, 1981; Bock, 1975; Glass et al., 1972; Kenny & Judd, 1986; Landman & Dawes, 1982; Raudenbush & Bryk, 1987; Scariano & Davenport, 1987; Tracz et al., 1992). The Re liability Generalization metaanalysis method has been used in 32 studies to eval uate the distri bution of relia bility across studies; yet, very little research has been conducted to address the possible methodological issues involving this technique. It is important to note that for the results of these RG studies to be cr edible, the method used to comb ine the results across studies must be statistically valid. For all statistical procedures, there ar e basic underlying a ssumptions (Glass & Hopkins, 1996; Pedhazur, 1982; Stevens, 1999). When examining mean differences there are three main assumptions a bout the populations: observation s are normally distributed, variances are homogeneous, and observations are independent. The research conducted
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46 on these assumptions has suggested that with larger sample sizes the violation of the assumption of normality does not have much of an impact on inflating Type I error rates. Similar conclusions have been made in re gard to the assumption of homogeneity of variances. This assumption is only a problem wh en group sizes differ such that the larger n is more than 1.5 time larger than the smaller n (Stevens, 1999). The assumption of independence, however, is the most problematic Even when the intraclass correlation is as low as .01, Type I error rates are drasti cally inflated (Barcikowski, 1981; Scariano & Davenport, 1987). In general, most RG studies have viol ated independence by i gnoring the fact that many of the score reliabilities in the sample are from the same study. Although there are many approaches that have been used to handle dependence of observations in metaanalysis, most RG researchers have chos en to ignore the dependency in their observations. In most of the RG studies each reliability coefficient is treated as independent even though it is quite common that more than one coefficient from each study was used. None of the RG studies calculated a mean or median reliability as a means to control for violation of independen ce. None of the RG studies investigated reliability using sensitivity analysis or chos e at random a reliability estimate to represent each study. So far the only original RG study th at has applied a mixed effect model is the Beretevas et al. (2002) study on the MarloweCrowne Social Desirabilty Scale (Crowne & Marlowe, 1960). In addition, Beretevas and Pastor (2003) used a mixed effect model method for their study that investigated the sa me studies that Yin a nd Fan (2000) used for their RG study on the Beck Depression Inve ntory (BDI; Beck et al., 1961). In both
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47 studies it was argued that the mi xed effect model provide a be tter model to investigate the variability of score reliabilities. The impact of ignoring the possible depe ndence in the reliability coefficients used in RG studies should be examined along with the other approach es to dealing with dependence. Thus, this research investigated the impact of violati ng the assumption that the observations are independent In addition, the methods th at researchers have devised to deal with dependent data in a metaanaly sis also were investig ated. It was expected that investigating the impact of these appr oaches would provide important guidelines for future RG studies such that the treatmen t of these studies is not compromised.
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48 Chapter Three: Method This chapter outlines the experimental method and how the data were simulated to represent a typical RG study. The methodology used in the study was intended to address the stated purpose of the study. Purpose The purpose of this research was to exam ine the potential impact of selected methodological factors on the validity of c onclusions from RG studies. Although all of the controversies described in Chapter 2 are important, this study focused on the issues surrounding violating the assu mption that the observations are independent and the methods that researchers have devised to deal with dependent data in a metaanalysis. Factors such as (a) the magnitude of coeffi cient alpha, (b) sample size (i.e., number of examinees), (c) number of studies, (d) the numbe r of reliabilities included in each journal study and (e) the intraclass correlation between journal studies (i.e., the degree of dependence between journal studies) were also considered. These factors were used in the method to investigate whether the magnit ude of these factors had an impact on the accuracy of estimating reliability when four approaches to addressing the violation of independence were used: (a) treating depende nt observations as independent, (b) randomly selecting a reliability index from each study, (c) calculating a mean or a median, and (d) using a twolevel Mixed Eff ects model. In other words, for certain method factors, does violation of independe nce significantly impact the accuracy of estimating the true reliability parameter?
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49 Research Questions In RG studies the dependent variable in the analyses is the reliability estimate (Henson & Thompson, 2001). This research focused on how certain study methods, in regards to violation of independence, affect the estimated mean reliability of scores calculated across studies. The key questions that were addressed in this study were: 1. What is the effect on point and interval estimates of mean reliability of ignoring violation of independence of observations in RG studies (i.e. tr eating all reliability coefficients as independent observations)? 2. What is the effect on point and interval estim ates of mean reliabil ity of using a mean or median reliability from each study as part of a sample in a RG study? 3. What is the effect of randomly selecting a reliability estimate from each study as a part of a sample in a RG study? 4. What is the effect on point and interval es timates of mean reliability of using a two level mixedeffects model for RG studies (i.e reliabilities are nested within studies)? 5. In regard to violations of independence, wh at impact do factors such as the magnitude of coefficient alpha, sample size, number of journal studies, number of reliability coefficients from each study, and the magnitude of the intraclass correlation (ICC) of the studies (i.e., the magnitude of the viol ation of independence) have when any of the methods discussed in the four rese arch questions above are investigated? Sample Samples of primary studies were gene rated using population parameters from a threeparameter Item Response Theory (IRT) model (Table 3) that were developed by Hanson and Beguin (1999). The data in their study came from two forms, A and Z, of a
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50 60 item American College Testing (ACT) Mathematics Assessment. The two forms did not have any items in common. Randomly equivalent groups of examinees took the assessment such that 2696 took Form A and 2670 took Form Z. These values were used to simulate scores of examinees and in turn generate coefficient alpha for various sample sizes of examinees and various test lengths. From these simulated examinee responses, subsets of items were selected that yielded the target values of coefficient al pha. These target values, computed from the simulated examinees were used as the populati on values to which the subsequent sample estimates were compared. The coefficient alpha values were generated using the information from the threeparameter model. The following table displays the number of items that were selected to simulate the population parameters: Table 2 Number of Items Needed to Ge nerate Reliability Parameter x x Number of Items .30 3 .50 6 .70 11 .90 50
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51 Table 3 Population Item Paramete rs Used for Simulations Parameters Parameters Item A BCItemAB C 1 0.642 2.5220.187 510.9570.192 0.194 2 0.806 1.9020.149 521.2690.683 0.15 3 0.956 1.3510.108 531.6641.017 0.162 4 0.972 1.0920.142 541.5111.393 0.123 5 1.045 0.2340.373 550.5611.865 0.24 6 0.834 0.3170.135 560.7280.678 0.244 7 0.614 0.0370.172 571.6650.036 8 0.796 0.2680.101 581.4010.117 0.057 9 1.171 0.5710.192 591.3910.031 0.181 10 1.514 0.3170.312 601.2590.259 0.229 11 0.842 0.2950.211 610.8042.283 0.192 12 1.754 0.7780.123 620.7341.475 0.233 13 0.839 1.5140.17 631.5230.995 0.175 14 0.998 1.7440.057 640.721.068 0.128 15 0.727 1.9510.194 650.8920.334 0.211 16 0.892 1.1520.238 661.2170.29 0.138 17 0.789 0.5260.115 670.8910.157 0.162 18 1.604 1.1040.475 680.9720.256 0.126 19 0.722 0.9610.151 691.2060.463 0.269 20 1.549 1.3140.197 701.3540.122 0.211 21 0.7 2.1980.184 710.9350.061 0.086 22 0.799 1.6210.141 721.4380.692 0.209 23 1.022 0.7610.439 731.6130.686 0.096 24 0.86 1.1790.131 741.1991.097 0.032 25 1.248 0.610.145 750.7861.132 0.226 26 0.896 0.2910.082 761.0410.131 0.15 27 0.679 0.0670.161 771.2850.17 0.077 28 0.996 0.7060.21 781.2190.605 0.128 29 0.42 2.7130.171 791.4731.668 0.187 30 0.977 0.2130.28 801.3340.53 0.075 31 1.257 0.1160.209 810.9651.862 0.152 32 0.984 0.2730.121 820.711.589 0.138 33 1.174 0.840.091 830.5231.754 0.149 34 1.601 0.7450.043 841.1340.604 0.181 35 1.876 1.4850.177 850.7090.68 0.064 36 0.62 1.2080.191 860.4960.443 0.142 37 0.994 0.1890.242 870.9790.181 0.124 38 1.246 0.3450.187 880.970.351 0.151 39 1.175 0.9620.1 890.5242.265 0.22 40 1.715 1.5920.096 900.9440.084 0.432 41 0.769 1.9440.161 910.8330.137 0.202 42 0.934 1.3480.174 921.1270.478 0.199 43 0.496 1.3480.328 930.8930.496 0.1 44 0.888 0.8590.199 941.2150.867 0.076 45 0.953 0.190.212 951.0790.486 0.264 46 1.022 0.1160.158 960.9320.45 0.259 47 1.012 0.4210.288 971.1410.344 0.071 48 1.605 1.3770.12 981.0680.893 0.153 49 1.009 1.1260.133 991.2171.487 0.069 50 1.31 0.0670.141 1001.311.186 0.153 Found in: Hanson and Beguin (1999, April). Obtaining a common scale for IRT item paramet ers using separate versus concurrent estimation in the common item nonequivalent groups equating design.
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52 Once the samples of examinees scores were generated, dependence was simulated by taking random samples from each simulated test administration and generating samples from populations with these reliability parameters. Method The research was conducted using a M onte Carlo simulation study method in which random samples were simulated under known and controlled population conditions. In the Monte Carlo study, RG st udies were simulated by generating samples in primary studies, estimating reliability of sc ores in these samples, and then aggregating the sample reliability estimates in the RG studies. Figure 1, below, is a model for the simulation of the study. The Monte Carlo study included five factors in the method. These factors were (a) the coefficient alpha (with xx = 0.30, 0.50, 0.70, and 0.90), (b) sample size in the primary studies (with average sample sizes, n, of 10, 50, 100, and 500), (c) number of primary studies (NPS) in the RG study (with k = 15, 50, 100, and 150) (d) number of reliability estimates from each study (with i = 1, 2, 3, 10, and 50) and (e) the degree of violation of independence wher e the strength of the depende nce is related to the number of reliability indices (i.e., coefficient alpha) derived from a simulated set of examinees and the magnitude of the correlation between the journal studies (with intraclass correlation ICC = 0, .0l .30, and .90). The values chosen for each of these factors are based in part on observed factors of actual RG studies, in part on factors of the Tracz, Elmore, and Pohlmann (1992) simulation study, and mostly on values that represent a range that is reasonable a nd typical in RG studies.
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53 Simulation of the data. The research was conducted using SAS/IM L version 9.1. Conditions for the study were run under Windows XP. Normally dist ributed random variables were generated using the RANNOR random number generator in SAS. A different seed value for the random number generator was used in each execution of the program, and the program code was verified by handchecking results from benchmark datasets. The target values, computed from the simulated examinees, were used as the population values to which the subsequent sa mple estimates were compared. For each condition investigated, several RG analyses ranging from 1,000 to 10,000 replications, were simulated. The number of replications that was chosen for each condition varied because of the amount of time the simulations took to run on the computer. Larger values of alpha took much longer to simulate such that when alpha was 90, 10,000 replications would take over three months to simulate. In this study, 48.44% of the conditions had 1,000 replications, less than 1% had 2,000, 6.25% had 5,000, and 44.53% had 10,000. The use of 1,000 to 10,000 replications provide s adequate precision for the investigation of the bias in the reliability parameter estimates. For example, 10,000 samples provide a maximum 95% confidence interval width around an observed proportion that is .0098 (Robey & Barcikowski, 1992).
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54 The impact of the four factors and the five ways of dealing with violation of independence were evaluated in terms of 1) The bias of the mean estimate 2) The RMSE of the mean estimate 3) The confidence band coverage 4) The average confidence band width. Average reliability across k studies was estimated each of the following 5 ways: 95% confidence bands were constructed around ea ch of the 5 estimates of average reliability. This was repeated 1,000, 5,000 or 10 000 times 5. Using a 2level mixed model where: At Level 1 the estimate, ikY, is considered a function of the true parameter k and sampling error ikr is modeled by: (1) 0 kikikrY Where ikY represents the ith observed value of reliability for study k and 0 k represents the expected value of the parameter for study k and ikr represents the withinstudy error term for the ith reliability in the kth study. At Level 2, the variability of the studiesÂ’ expected reliablities, around the mean reliability is model by: (2) 0000 kku 1 Averaging all i reliabilities (i.e. ignoring the violation) 2 Randomly selecting a reliability from each of the k studies in the RG study and averaging k reliabilities 4 Calculating a median of the i reliabilities for each of the k studies and averaging the k reliabilities. A value from each of these 5 factors was selected: *these values fluctuated slightly depending on the reliability index ** z transformation of these were used. Reliability (i.e. coefficient alpha)** xx Number of items .30 3 .50 6 .70 11 .90 50 Number of reliabilities from each study: i = 1, 2, 3, 10, 50 Sample size, n (i.e. # of examinees) n=10, 50, 100,500 Number of Journal studies included in RG study. k = 15, 50, 100, 150 Intraclass correlation between reliabilities from each journal ICC = 0, .01, .30, .90* Violation of Independence occured (when ICC 0 ) 3 Calculating a mean of the i reliabilities for each of the k studies and averaging the k reliabilities. F igure 3. The model of the si mulation for the study
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55 Simulation of intraclass correlation. Intraclass correlation for coefficient alpha in the simulations was generated using th e data from Hanson and Beguin (1999) Threeparameter IRT model. Recall that there were fo ur different test lengths used to generate the four population reliability pa rameters (see Table 3) for th is study. Basically for each simulation the number of journal studies was se t to 250 and number of reliabilities within each journal study was set to 50. Each reliabi lity coefficient that was generated for each of the journal studies was based on 2,000 admi nistrations (i.e., 2,000 examinees) of each of the tests simulated using SAS 9.1/PROC IML. The variance within of theta each journal study was held consta nt at 1. The variance for alpha among journal studies was adjusted by manipulating the variance of theta (i.e., scalar ability) so that the desired levels of ICC and for the resulting set of al pha coefficients. Table 4 shows the results of these simulations. Table 4 Results of the Simulation of Intraclass Correlation Items Var between for theta Var within of theta MSb for Alpha MSw for Alpha ICC Mean Alpha What Alpha Should Be 3 0.001 1 0.00151944 0.00117117 0.01 0.33 0.30 3 0.05 1 0.02349124 0.00117762 0.27 0.33 0.30 3 3 1 0.807093 0.0020887 0.89 0.27 0.30 6 0.001 1 0.00048951 0.00030898 0.01 0.54 0.50 6 0.05 1 0.00807874 0.0003155 0.33 0.54 0.50 6 0.99 1 0.29202275 0.00059354 0.91 0.49 0.50 11 0.02 1 0.00016364 0.0000965 0.01 0.69 0.70 11 0.11 1 0.00238788 0.0000998 0..31 0.68 0.70 11 0.6 1 0.07001637 0.00013581 0.91 0.67 0.70 50 0.0005 1 0.0000218 0.00000708 0.04 0.90 0.90 50 0.005 1 0.00018575 0.00000721 0.33 0.90 0.90 50 0.1 1 0.00314615 0.0000078 0.89 0.90 0.90
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56 Figure 2 is a model for how the Â“testta kingÂ” was simulated. Once a mean theta was simulated and an examineeÂ’s theta value was simulated then the test was administered to the examinee such that the examineeÂ’s score was a function of the three parameters for each item and the examineeÂ’s si mulated ability level. This was repeated for each of the n examinees for each test administration and i reliabilities were generated for each of the j studies. For each of the RG simulations there are j journals and for each of the j journals a mean theta (i.e., ability le vel) was simulated from standard normal distribution. The variance between each of the j journal studies was fixed at a value depending on the desired intraclass correlati on and coefficient alpha. For example, if a simulation was run such that the intraclass correlation was .01 and coefficent alpha was approximately .30, the variance among the mean thetas for each of the j journals was set to 0.001. Along with simulating a mean theta for each of the j studies, a theta value was simulated for each of the n examinees. The variance of th e theta values within each admistration of the simulated test was fixed at 1. The ICC was then generated by using the following formula () ((1))bw bwMSMS ICC M SiMS where MSb is the mean square between studies, MSw is the mean square within studies, and i is the number of reliabilitie s for each study (Stevens, 1999).
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57 1 22r 12r 32r 2 ir 2 r 11r 31r 1 ir k 2 k r 1 kr 3 k r ikr Within each of the k studies SAS randomly selects a theta* for each of the n examinees such that the variance of the examineesÂ’ thetasis1 For each of the k studies SAS randomly generates a mean theta (ability)* and the variance among the k studies mean thetas is adjusted depending on the desired ICC and xx using rannor(seed1) creates a normal distribution with a mean of 0 and a standard deviation of 1 F igure 4. A model for how the Â“testtakingÂ” was simulated
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58 Conduct of RG analyses. Each RG analysis was conducted using the obtained sample reliability estimates from k studies. Coefficient alpha estimates were investigated using the z transformation for coefficient alpha, ln1 z to normalize the sampling distributions. This transformed value of coefficient alpha is approximately normally distributed with a variance of k/{2(k1)(N2)}, where k = the number of items on the instrument and N is the average sample size for each study (Bonett, 2002). Weighted least squares analys es were conducted (Ful ler & Hester, 1999; Raudenbush, 1994; Hedges & Olkin, 1985). RG analyses were conducted on the k studies using various approaches to address the violation of independence that were disc ussed in Chapter Two. First, the dependence was ignored and an RG analysis was conduc ted. Then, a mean and a median of the reliabilities from each of the k studies were calculated and an RG analysis was conducted on these averages and medians. Next, a reliab ility index was randomly selected from each of the k studies and these were the sample for an RG analysis. Finally a mixedeffects model was executed using a twolevel mixed model where: At Level 1 the estimate, ikY, is considered a functio n of the true parameter k and sampling error ikr and is modeled by: 0 kikikrY Where ikY represents the i observed value of reliability for study k and 0k represents the estimated value of the parameter for study k and ikr represents the withinstudy error term for the i reliability in the k study.
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59 At Level 2, the variability of the expect ed reliabilities of the study, around the mean reliability is model by: 0000kku Where0k is expressed as00 the overall mean reliability in the k studies, and 0ju represents the betweenstudy error term. The SAS PROC MIXED procedure was used to estimate parameters from these multilevel models. The results of these approaches were evaluated in rega rds to their accuracy in the estimation of coefficien t alpha. This was undertaken by using procedures such as PROC MEANS and PROC CORR in SAS with the output generated from the code written in PROC IML. Evaluation of the results. Multiple combinations of the five met hod factors along with the four ways of dealing with dependence within journal studies were used to simulate an RG study. Each simulated RG study was used to obtain an estim ated mean reliability. In addition, a 95% confidence band was constructed around each p opulation estimate. For the construction of confidence bands, the sampling error of each estimate of score dependability index was calculated: 2) 1)(N 2(k k 2k where 2 k is the estimated sampling variance of ztransformed rxx The standard error used for construction of the confidence band for the mean index of score dependability was obtained as:
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60 1 2 11K k kSE where 2 kis the sampling error variance for an index (i.e., the transformed coefficient alpha) in the k study and the summation is across the studies included in the RG analysis. The impact of the treatmen t factors was evaluated based upon the bias in the mean estimates, root mean square error, the confidence band coverage, and the average confidence band width. Bias was estimated as the difference between the average sample estimate and the known population value of th e reliability coefficient. That is, Âˆ ÂˆR i iBias R where Âˆi = the sample estimate from the i RG study, = the population value, and the summation is over the R simulated RG studies. Root mean square error estimates were calcu lated to evaluate the efficiency of the estimators. This value is calculated using the formula: 2Âˆ ÂˆR i iRMSE R Confidence band coverage probabiliti es were estimated by computing the proportion of confidence bands in the R simulated RG studies that contained the parameter of interest. Similarly, confidence band width was computed as the average width of confidence bands from the R simulated RG studies. Each analysis was used to obtain an estimated mean reliability and a 95% confidence band around this population estimate. Re sults of this resear ch are presented as
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61 graphs of the bias, confidence band coverage, and confidence band width as functions of the method factors employed in the Monte Carlo study.
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62 Chapter Four: Results The results of this study are presented in detail and are organized in the order of the research questions. The following research questions were addressed by these results: 1. What is the effect on point and interval estimates of mean reliability of ignoring violation of independence of observations in RG studies (i.e., treating all reliability coefficients as independent observations)? 2. What is the effect on point and interval estim ates of mean reliabil ity of using a mean or median reliability from each study as part of a sample in a RG study? 3. What is the effect on point and interval estimates of mean re liability of randomly selecting a reliability estimate from each st udy as a part of a sample in a RG study? 4. What is the effect on point and interval es timates of mean reliability of using a two level mixedeffects model for RG studies (i.e ., reliabilities are nested within studies)? 5. In regard to violations of independence, wh at impact do factors such as the magnitude of coefficient alpha, sample size, number of journal studies, number of reliability coefficients from each study, and the magnitude of the intraclass correlation (ICC) of the studies (i.e., the magnitude of the viol ation of independence) have when any of the methods discussed in the four rese arch questions above are investigated? How the Results were Evaluated There were 6,400 conditions simulated us ing the five factor s of this Monte Carlo study generated from th e coefficient alpha (with xx = 0.30, 0.50, 0.70, and 0.90),
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63 sample size in the primary studies (with average sample sizes, n, of 10, 50, 100, and 500), number of primary studies in the RG study (with k = 15, 50, 100, and 150) number of reliability estimates from each study (with i = 1, 2, 3, 10, and 50), and the degree of violation of independence ( ICC = 0, .01, .30, .90). In addition, the choice of treatment (ignoring the dependence, Violation; random selection of a reliability from each journal study, Random; calculating a mean from each journal study, Mean; calculating a median from each journal study, Median; and using a twolevel mixed model, HLM) was also an independent variable for the study. Thus, this yielded 4 (xx) x 4(n) x 4(k) x 5(i) x 4 (ICC) x 5 (treatment) = 6,400 RG conditions. Interc orrelation analysis was conducted between the independent variables and all were equa l to 0. This was because the design is a balanced factorial arrangement of factors. The results for the intercorrelation for the dependent variables are listed in Table 5. The correlation was largest in magnitude between Bias and RMSE such that r = .89 and smallest in magnitude between RMSE and Band Coverage such that r = .17. It was surprising to see that the correlation between Band Coverage and Band Width was only .29. Table 5 Correlation Between Dependent Variables Bias RMSE Band Coverage Band Width Bias .89 .29 .46 RMSE .17 .74 Band Coverage .29 Band Width Note: All correlations were significant at the = .01 level
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64 First, the Bias, root mean square error (RMSE), confidence Band Coverage, and confidence Band Width were evaluated for each of the treatments. This was undertaken by creating box plots for all the conditions. Thes e are displayed in Figures 36. Then, the results of the simulation were evaluated using PROC GLM in SAS such that the dependent variables were Bias, RMSE, Band Coverage, and Band Width and the independent variables were the five types of factors and the choice of treatment. The effect size,2 ,was calculated to measure the degr ee of the associa tion between the independent variables main effects and the de pendent variables along with the firstorder interaction effects between the independent variables and the depe ndent variables. Etasquared is the proportion of th e total variance that in the case of this study, can be attributed to one of the factor s (or type of treatment) or an interaction between two of the factors (or an interaction between the type of research method and one of the factors). It is calculated as the ratio of the effect variance (SSeffect) to the total variance (SStotal). 2 effect totalSS SS Box Plots To address the first four research questions, box plots were created for Bias, RMSE, Band Coverage, and Band Width to examine the results of each of the treatments. Figure 3 displays the results for the Bias in all five treatments. From this figure one can see that all five of the methods behave fa irly the same way. The studies in which a median was calculated for each journal study do appear to have a few cases where the Bias was much larger in magnitude (minim um = .27 and median = .01), but the
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65 quartiles and the median values were simila r across all five methods such that the Bias had a rather small range. The median value fo r the other types of treatment was 0. In general Bias was relatively very small and mostly nega tive, that is, the reliability from the simulations only slightly underestim ated the population parameters. Figure 5. Distribution of bias estimates for reliability coefficients for all five types of treatments In Figure 4 the results for RMSE in the study are displayed. From this figure one can see that the pattern is very simila r to the results found when examining the Bias. The studies where a median was calculated for each journal study also appear to have a few conditions where the RMSE is a bit larger in magnitude (maximum = .27). The maximum value for the rest of the conditions was appr oximately .12. Once ag ain, the quartiles and the medians are relatively equal for all five treatments. For all five types of treatments the minimum value was close to 0 as was the firs t quartile. The median value was also the Results for Bias in RG Simulation HLM Mean Median Random Violation 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4
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66 same for all five treatments (median =.01). These results indicate that the estimates were very efficient regardless of the treatment. Results for RMSE in RG Simulation HLM Mean Median Random Violation0.40.200.20.4 Figure 6. Distribution of RMSE estimates for reliability coefficients for all five types of treatments In Figure 5, confidence Band Coverage is displayed. These values represent the proportion of times that the population reliabi lity (i.e., coefficient alpha) fell within a 95% confidence interval for each simulation. While the range for all the treatment conditions ranged from 0 to 1, there was a wi der range for simulati ons that ignored the dependence (Violation) and used mixed models (HLM) than the other three methods. The median Band Coverage for Violation was .54 and for HLM the median Band Coverage was .64. In contrast, the median Band Coverage for Random was .84, for Median, it was .89, and for Mean it was .94. In addition the interquartile range for Violation was .86 and for HLM it was .88. For the other three treatments the interquartile range was .44 for Random, .54 for Median and .77 for Mean. These results suggest calculating a mean of
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67 the reliabilities from each journal study provided better Band Coverage than the other four treatments. Also, thes e results also suggest that use of mixed models (HLM) provides very poor Band Coverage that was similar to violating independence. Results for Band Coverage in RG simulation HLM Mean Median Random Violation 0.10.10.30.50.70.91.1 Figure 7. Distribution of band coverage for reliability coefficients for all five types of treatments In Figure 6, the mean values of the estimated confidence Band Widths for reliability estimates are displa yed. As with the results for Bias and RMSE, the results for all five treatments are very similar. All fi ve methods had a minimum value of 0 and the median values for all five conditions were sim ilar in size (.01, .03, .03, .03, and .02 for Violation, Random, Median, Median, and HLM, respectively). The largest Band Width was .59 and was found when median values were used in the simulation; the second largest was .47 using HLM. The other three types of tr eatment produced a maximum value of .45. As apparent in Figure 6, the in terquartile ranges were relatively the same for all five treatments, ranging from .03 (for Violation) to .06 (for Median, Mean and
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68 Random). While it is obvious that the interquartile range of the Band Width for Median, Mean, and Random is twice that of Violation and almost twice as much as that of HLM, (.04), these values are still small. Regardless of the treatment that was applied, the results produced very narrow bands. Results for Band Width in RG Simulation HLM Mean Median Random Violation0.10.10.30.50.70.91.1 Figure 8. Distribution of band width fo r reliability coefficients for all five types of treatments Summary of Box Plot Results The four box plot figures sugge st that the five treatments used for dealing with the violation of independence do not have great impact on the variability in Bias, RMSE or the Band Width. The Bias across methods for the most part was relatively small and never exceeded .01. The RMSE analysis produced similar results. In general, the reliability estimates from the simulations only slightly underestimated the population parameter. While there is difference in the ranges of the treatments, the Band Width was typically rather small. There was a slightly smaller median for Violation and the interquartile
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69 range for Mean, Median and Random (.06) was twice as large as the interquartile range for Violation (.03) and almost twice as large for HLM (.04). The type of treatment did seem to have an impact on the variability of Band Coverage. These results suggest i gnoring the dependence (Violation) or the use of mixed models (HLM) provides very liberal Band Coverage and using a mean reliability for each study as the unit of analysis (Mean) seems to provide better Band Coverage. 2 Analysis In addition to box plots, 2 was calculated to measure the degree of the association between the independent variablesÂ’ main ef fects (true alpha, average sample size from each study, number of primary studies, number of reliability estimates from each study, the degree of violation, and the treat ment), and the dependent variables (Bias, RMSE, Band Coverage, and Band Width), along with the firstorder interaction effects between the independent variables and the dependent va riables. The results of this analysis are presented in Table 6. The 2 values ranged from 0 to .28. According to Cohen (1988), 2 = .05 is considered a medium effect. Using th is criterion, tables and graphs were created for factors where values of 2 were greater than or equal to .05. Even though it is clear from this analysis that the treatments for controlling for non independence only had a significant effect on Band Coverage, because they were addressed in the research questions, the treatments were also included in all of the analysis and presentation of the main effects results.
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70 Table 6 2 Analysis of the Effects of Factors in the RG Simulation BIAS RMSE Band Coverage Band Width Factor 2 Factor 2 Factor 2 Factor 2 ICC 0.21 xx 0.21 ICC 0.20 n 0.28 xx 0.17 n 0.19 ICC X xx 0.12 xx 0.20 N 0.12 ICC 0.14 xx X n 0.11 NPS 0.12 ICC X xx 0.08 xx X n 0.06 NPS 0.08 xx X n 0.10 TR X xx 0.05 ICC X xx 0.05 xx 0.07 n X NPS 0.07 TR X n 0.03 TR X xx 0.03 TR 0.06 xx X NPS 0.05 xx X n 0.02 NPS 0.02 TR X NR 0.04 TR 0.03 TR 0.02 TR X n 0.02 n 0.04 TR X n 0.02 TR X NR 0.01 TR 0.01 ICC X n 0.03 TR X NR 0.02 xx X NR 0.01 xx X NPS 0.01 NR 0.02 TR X xx 0.01 ICC X n 0.01 TR X NR 0.01 TR X xx 0.02 NR 0.01 NR X n 0.01 ICC X n 0.01 TR X n 0.01 TR X NPS 0.01 NR 0.00 n X NPS 0.01 ICC X NPS0.01 NR X n 0.00 TR X ICC 0.00 NR X NPS 0.00 xx X NPS 0.01 xx X NR 0.00 n X NPS 0.00 NR 0.00 n X NPS 0.01 NR X NPS 0.00 NPS 0.00 TR X NPS 0.00 TR X ICC 0.00 ICC 0.00 ICC X NR 0.00 xx X NR 0.00 NR X n 0.00 TR X ICC 0.00 xx X NPS 0.00 NR X n 0.00 TR X NPS 0.00 ICC X xx 0.00 TR X NPS 0.00 ICC X NR 0.00 xx X NR 0.00 ICC X NPS0.00 NR X NPS 0.00 TR X ICC 0.00 ICC X NR 0.00 ICC X n 0.00 ICC X NPS 0.00 ICC X NPS0.00 NR X NPS 0.00 ICC X NR 0.00 Note. ICC = intraclass correlation, NR = number of reliability pe r primary journal study, NPS = number of primary studies, n = average sample size, TR = Treatment, and xx = coefficient alpha
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71 Bias Table 6 indicates that factors for Bias where 2 0.05 were ICC ( 2 = .21), xx (2 = .17), n ( 2 = .12), the interaction between ICC and xx ( 2 = .08) and the interaction between the treatment and xx ( 2 = .05). The results using average Bias as an outcome and these factors as predictors are presente d in Table 7 through Ta ble 10. In addition, Figure 7 displays the interactions between ICC and xx in regards to Bias and Figure 8 displays the interaction between the treatment and xx in regards to Bias. In Table 7 information about the extent to which the magnitude of the intraclass correlation (ICC) is associated with the Bias in estimated mean reliability by treatment is presented. The Bias was as little as approximately 0 and as large as .04 in magnitude when ICC =.90 and the treatment was Median. The averages of the magnitude of Bias for ICC ranged from .01 to .03 such that for ICC = 0, .01, and .30 the average Bias was .01 and for ICC = .90 the average Bias was .03. In regards to the types of treatment there was very little difference in the average Bias. This was of course not surprising given that the 2 was only .02 for treatment. While the Bias was slightly larger for ICC =.90, it was still very small and the average Bias was never positive; that is, average reliability was never overestimated. Table 7 Bias in Estimated Mean Reliability by Treatment and Intraclass Correlation Average of BIAS ICC Treatment 0.00 0.01 0.30 0.90 Average Violation 0.01 0.01 0.01 0.03 0.01 Random 0.01 0.01 0.01 0.03 0.01 Median 0.01 0.01 0.02 0.04 0.02 Mean 0.01 0.01 0.01 0.03 0.01 HLM 0.00 0.00 0.01 0.03 0.01 Average 0.01 0.01 0.01 0.03 0.01
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72 In Table 8 and in Figure 7 information about the extent to which the magnitude of the reliability parameter, xx, contributes to the Bias in estimated mean reliability by treatment is presented. The averages of the magnitude of Bias for xx ranged from 0 to .02 such that for xx = .33 and .54 the average Bias was .02, for xx = .69 the average Bias was .01 and for xx =.90 the average Bias was 0. While the Bias was slightly larger for xx =.33 and a Median treatment, it was still very sma ll. As with the results for ICC the average Bias was never positive; that is, averag e reliability was never overestimated. Table 8 Bias in Estimated Mean Reliability by Treatment and Coefficient Alpha Average of BIAS xx Treatment 0.33 0.54 0.69 0.90 Average Violation 0.02 0.02 0.01 0.00 0.01 Random 0.02 0.02 0.01 0.00 0.01 Mean 0.02 0.02 0.01 0.00 0.01 Median 0.04 0.03 0.01 0.00 0.02 HLM 0.01 0.02 0.01 0.00 0.01 Average 0.02 0.02 0.01 0.00 0.01 0 3 3 0 5 4 0 6 9 0 9 00 3 3 0 5 4 0 6 9 0 9 00.05 0.04 0.03 0.02 0.01 0.00 0.01 0.300.500.700.90 AlphaBias Violation Random Mean Median HLM Figure 9. Bias in estimated mean reliability by treatment and coefficient alpha
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73 In Table 9 information about the extent to which the magnitude of the average sample size, n, from each primary study contributes to the Bias in estimated mean reliability by treatment is presented. The Bias was as little as .01 and as large as .05 in magnitude (.05 when n = 10, and the treatment was Median). The averages of the magnitude of Bias for n ranged from .01 to .03 such that for n = 50, 100, or 500 the average Bias was .01 and for n = 10 the average Bias was .03. Like the previous results for ICC and xx, the magnitude of the average sample size had very little impact on the Bias in the estimated mean reliability. Table 9 Bias in Estimated Mean Reliability by Treatment and Average Sample Size Average of BIAS Average Sample Size Treatment 10 50 100 500 Average Violation 0.02 0.01 0.01 0.01 0.01 Random 0.02 0.01 0.01 0.01 0.01 Median 0.05 0.01 0.01 0.01 0.02 Mean 0.02 0.01 0.01 0.01 0.01 HLM 0.02 0.01 0.01 0.01 0.01 Average 0.03 0.01 0.01 0.01 0.01 Along with the interaction between tr eatment and coefficient alpha, another notable interaction was present between intr aclass correlation and coefficient alpha. Table 10 and Figure 8 display the de tails of this interaction. The Bias in this interaction ranged from 0 to .05 in magnitude. When ICC = .90 the Bias was as much as five times as much as for the other smaller values of ICC considered in this study. The Bias for ICC = 0, .01 and .30 were relatively small and did not indicate that the magnitude of xx had an impact on Bias for these values of ICC. Howeve r, there was a notable difference when xx= .90. In this case, regardless of the ICC the Bias was zero.
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74 Table 10 Bias in Estimated Mean Reliability by In traclass Correlation and Coefficient Alpha Average of BIAS xx ICC 0.33 0.54 0.69 0.90 Average 0 0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.30 0.01 0.02 0.01 0.00 0.01 0.90 0.05 0.04 0.03 0.00 0.03 Average 0.02 0.02 0.01 0.00 0.01 0 0 0 1 0 3 0 0 9 00 0 0 1 0 3 0 0 9 00 0 0 1 0 3 0 0 9 00.06 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.90 ICCBias 0.33 0.54 0.69 0.9 Figure 10. Bias in estimated mean reliability by intraclass correlation and coefficient alpha
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75 Root Mean Squared Error Table 6 indicates that factors for RMSE where 2 0.05 were xx ( 2 = .21), n ( 2 = .19), ICC ( 2 = .14), the interaction between xx and n ( 2 = .06), and the interaction between xx and ICC ( 2 = .15). The results using average RMSE as an outcome and these factors as predictors are presented in Table 11 through Tabl e 15. In addition, Figure 9 displays information about the in teraction between average sample size and coefficient alpha and Figure 10 displays in formation about the interaction between the intraclass correlation and coefficient alpha. In Table 11, information about the ex tent to which the magnitude of the reliability parameter, xx, contributes to the RMSE of estimated mean reliability by treatment is presented. The RMSE ranged from approximately 0 to .05. Like the results for the Bias, the RMSE was largest when xx = .33 and the treatment was Median. The averages of the magnitude of RMSE for xx ranged from 0 to .04 such that for xx = .33 the average RMSE was .04, for xx = .54 it was .03, for xx = .69 it was .02, and for xx =.90 the average RMSE was 0. These results suggest that smaller values of xx will have a slightly larger RMSE compared to larger values of xx. In general, the RMSE was quite small which would suggest that the reliability estimates were rather stable regardless of the magnitude of the populati on reliability parameter. In Table 12, information about the extent to which the magnitude of the average sample size from the primary studies, n, contributes to the RMSE of estimated mean reliability by treatment is presented. The RMSE ranged from approximately .01 to .06. The RMSE was largest, .06 when n = 10 and the treatment was Median. The averages of
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76 the magnitude of RMSE for n ranged from .01 to .04 such that for n = 10 the average RMSE was .04, for n =50 or 100 it was .02, and for n = 500 the average RMSE was .01. Table 11 RMSE of Estimated Mean Reliability by Treatment and Coefficient Alpha Average of RMSE xx Treatment 0.33 0.54 0.69 0.90 Average Violation 0.03 0.03 0.02 0.00 0.02 Random 0.04 0.03 0.02 0.01 0.02 Median 0.05 0.03 0.02 0.00 0.03 Mean 0.03 0.03 0.02 0.00 0.02 HLM 0.03 0.03 0.02 0.00 0.02 Average 0.04 0.03 0.02 0.00 0.02 These results suggest that larger samples si zes have a slightly smaller and somewhat more stable RMSE than the smaller sample sizes. Overall, the RMSE was never very large which would suggest that the reliability estimates were somewhat stable regardless of the magnitude of the average sample. Table 12 RMSE of Estimated Mean Reliability by Treatment and Average Sample Size Average of RMSE Average Sample Size Treatment 10 50 100 500 Average Violation 0.03 0.02 0.01 0.01 0.02 Random 0.04 0.02 0.02 0.01 0.02 Median 0.06 0.02 0.02 0.01 0.03 Mean 0.03 0.02 0.01 0.01 0.02 HLM 0.03 0.02 0.01 0.01 0.02 Average 0.04 0.02 0.02 0.01 0.02 In Table 13, information about the extent to which the magnitude of the intraclass correlation contributes to the RMSE of estimated mean reliability by treatment is presented. The RMSE ranged from approximately .01 to .04. When the ICC was .90 the RMSE = .04 regardless of the type of treatment. For the other smaller values of ICC the variability of RMSE was negligible across treatments. The averages of the magnitude of
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77 RMSE for ICC ranged from .02 to .04 such that for ICC = 0, 0.01, and .30 the average RMSE was .02, and for ICC = .90 the average RMSE was .04. These results suggest that larger values of ICC will have a larger RMSE than smaller values of ICC regardless of the treatment. Overall, the RMSE was never very large, which would suggest that the reliability estimates were somewhat stable regardless of the magnitude of the ICC. Table 13 RMSE for Estimated Mean Reliability by Treatment and Intraclass Correlation Average of RMSE ICC Treatment 0.00 0.01 0.30 0.90 Average Violation 0.01 0.01 0.01 0.04 0.02 Random 0.02 0.02 0.02 0.04 0.02 Median 0.02 0.02 0.02 0.04 0.03 Mean 0.01 0.01 0.01 0.04 0.02 HLM 0.01 0.01 0.01 0.04 0.02 Average 0.02 0.02 0.02 0.04 0.02 The interaction between coefficient alpha and average sample size had a significant impact on the variability in the RMSE. Table 14 and Figure 9 display information about these results. The RMSE for these data ranged from 0 to .05 in magnitude. The RMSE was largest (.05), when xx = .33 or .54 and n = 10. This was five times a much as when xx = .90 and n =10. As n increased RMSE usually decreased for any given value of xx however when xx = .90 there was not much variability such that the RMSE was approximately 0 for all values of n > 10. In general, while there was a significant interaction between xx and n, the RMSE for the estimated mean reliability was small.
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78 Table 14 RMSE for Estimated Mean Reliability Aver age Sample Size by Coefficient Alpha Average of RMSE xx Average Sample Size 0.33 0.54 0.69 0.90 Average 10 0.05 0.05 0.04 0.01 0.04 50 0.03 0.03 0.01 0.00 0.02 100 0.03 0.02 0.01 0.00 0.02 500 0.02 0.01 0.01 0.00 0.01 Average 0.03 0.03 0.02 0.00 0.02 1 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 00.00 0.01 0.02 0.03 0.04 0.05 0.06 0 500 Average Sample SizeRMSE 0.33 0.54 0.69 0.90 Figure 11. RMSE for estimated mean reliability average sample size by coefficient alpha The other interaction that was significantly large for RMSE was the interaction between coefficient alpha and intraclass correlation. Inform ation about this interaction is displayed in Table 15 and Figure 10. The RMSE was largest (.06) when xx = .33 and
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79 the ICC= .90. The results for this interaction were very similar to the results for the interaction between xx and n such that when xx =.90 there was very little variability in RMSE; actually, regardless of the value of ICC when xx =.90, the RMSE was approximately 0. For the other values of xx when ICC = 0, .01, or .30 the RMSE was relatively stable regardless of the magnitude of xx. When ICC = .90 and xx =.30 the RMSE was twice as large as when ICC was smalle r. Overall, the larger value of ICC had the biggest impact on the magnitude of RMSE for smaller values of xx and the magnitude of ICC had no impact on the variability in RMSE when xx =.90. Table 15 RMSE for Estimated Mean Reliability Intr aclass Correlation by Coefficient Alpha Average of RMSE xx ICC 0.33 0.54 0.69 0.90 Average 0.00 0.03 0.02 0.01 0.00 0.02 0.01 0.03 0.02 0.01 0.00 0.02 0.30 0.03 0.02 0.01 0.00 0.02 0.90 0.06 0.05 0.03 0.00 0.04 Average 0.04 0.03 0.02 0.00 0.02 0 0 0 1 0 3 0 0 9 00 0 0 1 0 3 0 0 9 00 0.01 0.30 0.900 0 0 1 0 3 0 0 9 00.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.90 ICCRMSE 0.33 0.54 0.69 0.90 Figure 12. RMSE for estimated mean reliability intraclass correlation by coefficient alpha
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80 Band Coverage Table 6 indicated that factors for Band Coverage where 2 0.05 were ICC (2 = .20), the interaction between ICC and xx ( 2 = .12), the interaction between xx and n ( 2 = .11), the number of primary studies ( 2 = .08), xx ( 2 = .07), and the treatment ( 2 = .06). Notice that, unlike Bias, RMSE and Band Width, these results indicate that the type of treatment had a notable impact on the vari ability in the confidence band coverage of the mean reliability estimates. These results, using average Band Coverage as an outcome and these factors as predictors, are presented in Table 16 through Table 20. In addition, in Figures 11 information a bout the interaction between ICC and xx is presented and in Figure 12 informa tion about the interaction between n and xx is presented. In Table 16 information about the ex tent to which the magnitude of ICC contributes to the Band Coverage of estimated mean reliability by treatment is presented. The Band Coverage ranged from approximately .20 to .85. Band Coverage was .20 when ICC = .90 and the treatment was Violation and was .85 when the ICC = 0 and the treatment was Mean. The averages of the magnitude of Band Coverage for ICC ranged from .32 to .73 such that for ICC = 0 and .01 the average Band Coverage was .73, for ICC = .30 it was .66, and for ICC =.90 the average Band Coverage was .32. These results suggest that larger values of ICC will have a much smaller Band Coverage compared to smaller values of ICC. There was also some notable variability in Band Coverage in terms of the type of treatment. It was not surprisi ng that out of the five treatm ents explored in this study, ignoring the dependence, Violation, had the smallest average Band Coverage (.47).
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81 What was interesting was the fact that HLM had the second smallest average Band Coverage (.52) and that the largest average Band Coverage was for the treatment Mean. The type of treatment does not seem improve the size of the Band Coverage as ICC increases. In fact regardless of the treatment when ICC =.90 the Band Coverage was only as large as .38 (when the treatment was Mean) and as small as .20 (when the treatment was Violation). Table 16 Band Coverage of Estimated Mean Reliability by Treatment and In traclass Correlation Average of Band Coverage ICC Treatment 0 0.01 0.30 0.90 Average Violation 0.60 0.60 0.49 0.20 0.47 Random 0.79 0.78 0.73 0.37 0.67 Median 0.79 0.79 0.75 0.33 0.67 Mean 0.85 0.83 0.80 0.38 0.72 HLM 0.62 0.62 0.54 0.30 0.52 Average 0.73 0.73 0.66 0.32 0.61 In Table 17 information about the exte nt to which the magnitude of the interaction between the intraclass correlation and coefficien t alpha contributes to the variability Band Coverage of estimated mean reliability by treatment is presented. In Figure 11 information about the interaction between in traclass correlation and coefficient alpha also is presented. The Band Coverage ranged from approximately .07 to .97. Band Coverage was smallest, .07, when ICC = .90, xx =.69 and the treatment was Violation. It was at its largest value, .97, twice, when ICC = 0 or and when ICC= .30, xx =.33 and the treatment was Mean. The averages of the magnitude of Band Coverage for ICC by xx ranged from .14 (when ICC = .90 and xx = .69), to .89, (when ICC = 0 and xx.= .33). Surprisingly, the range of the Band Coverage for xx = .90 was only from .42 to .61. The coverage increased for xx = .90 as the ICC increased with an
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82 average Band Coverage of .47. The average Band Coverage was largest, .74, for xx = .33. As is apparent in Figure 11, for xx = .33, .54, and .69 the Band Coverage is fairly similar across values of ICC such that when ICC = 0 the Band Coverage ranges from .86 to .89 when ICC = .01, the Band Coverage ranges from .75 to .88, and when ICC = .30, the Band Coverage ranges from .67 to .87. When ICC = .90 the Band Coverage for these three values of xx drops down significantly where the Band Coverage ranges from .14 to .32. For xx =.90 a completely different pattern was seen. For this value of xx, the Band Coverage was rather small, .42, and increased only when ICC = .90. Notice that this behavior was di fferent than what was seen with the other values of xx. Clearly, the impact of the magnitude of coefficient alpha on Band Coverage depends on the magnitude of the intraclass correlation between the studies. While it was not necessarily surprising that there was an observabl e interacti on between ICC and xx, it was surprising to see that Band Coverage for xx =.90 was as small as it was and that larger ICC resulted in an increase in Band Coverage.
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83 Table 17 Band Coverage of Estimated Mean Reliab ility for Intraclass Correlation and by Coefficient Alpha Average of Band Coverage xx ICC Research Design 0.33 0.54 0.69 0.90 Average Violation 0.84 0.54 0.73 0.30 0.60 Random 0.89 0.84 0.91 0.51 0.79 Median 0.83 0.88 0.97 0.49 0.79 Mean 0.97 0.93 0.96 0.52 0.85 0 HLM 0.90 0.59 0.72 0.28 0.62 Average for ICC= 0 0.89 0.75 0.86 0.42 0.73 Violation 0.84 0.54 0.73 0.30 0.60 Random 0.89 0.83 0.91 0.51 0.78 Median 0.83 0.87 0.96 0.49 0.79 Mean 0.93 0.93 0.96 0.52 0.83 0.01 HLM 0.90 0.59 0.73 0.28 0.62 Average for ICC= .01 0.88 0.75 0.86 0.42 0.73 Violation 0.79 0.45 0.43 0.30 0.49 Random 0.88 0.77 0.77 0.51 0.73 Median 0.82 0.82 0.86 0.50 0.75 Mean 0.97 0.87 0.84 0.53 0.80 0.30 HLM 0.91 0.53 0.45 0.29 0.54 Average for ICC =.30 0.87 0.69 0.67 0.42 0.66 Violation 0.21 0.12 0.07 0.40 0.20 Random 0.38 0.27 0.19 0.68 0.37 Median 0.25 0.22 0.18 0.70 0.33 Mean 0.39 0.27 0.17 0.72 0.38 0.90 HLM 0.37 0.19 0.11 0.53 0.30 Average for ICC =.90 0.32 0.21 0.14 0.61 0.32 Average 0.74 0.60 0.63 0.47 0.61
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84 0 0 0 1 0 3 0 0 9 00 0 0 1 0 3 0 0 9 00 0 0 1 0 3 0 0 9 00.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.02 0.90 ICCBand Coverage 0.33 0.54 0.69 0.9 Figure 13. Band coverage of estimated mean reliability intraclass correlation by coefficient alpha In Table 18, information about the ex tent to which the magnitude of the interaction between average sample size, n, and the population reliability parameter, xx, contributes to the Band Coverage of estimated mean reliability by treatment is presented. In Figure 12 information about th e extent to which th e magnitude of the interaction between average sample size, n, and the population reliability parameter ( xx) contributes to the Band Coverage of estimated of mean reliab ility also is presented. The Band Coverage ranged from approximately .03 to .98. Band Coverage was smallest, .03, when n = 500, xx =.90 and the treatment was Violation. It was at its largest value, .98, twice, when xx =.33, n= 10 and the treatment was Mean, and when xx =.90, n = 10, and the treatment was Median.. The averages of the magnitude of Band Coverage for n by xx ranged from .06, where xx =.90 and n = 500, to .85, where xx =.90 and n = 10. The overall average of Band Coverage for xx ranged from .47, for xx =.90 to .74, for xx
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85 =.30. The overall average Band Coverage for n ranged from .50 (for n = 500) to .70 (for n = 10). As displayed in Figure 12, the Band Coverage had a wider range for smaller values of n such that for n = 10, when xx = .33 the average Band Coverage was .82, when xx = .54 the average Band Coverage was .59, when xx = .69 the average Band Coverage was .86, and when xx= 90 it was .62. As the average sample size increased the average Band Coverage for xx = .33, .54, and .69 did not change that drastically. This was not the case, however, for xx = .90. In this case, the average Band Coverage for n = 50, 100, and 500 went from .60 to .33 to .06, respectively. This would explain the interaction effect between average sample size and coefficient alpha. These results suggest that for smaller values of xx, the Band Coverage is less affected by an increase in sample size than for larger values of xx.
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86 Table 18 Band Coverage of Estimated Mean Reliability for Average Sample Size and Treatment by Coefficient Alpha Average of Band Coverage xx Sample Size Research Design 0.33 0.54 0.69 0.90 Average Violation 0.83 0.38 0.33 0.68 0.56 Random 0.93 0.75 0.66 0.92 0.81 Median 0.55 0.63 0.74 0.98 0.72 Mean 0.98 0.83 0.70 0.97 0.87 10 HLM 0.83 0.37 0.32 0.67 0.55 Average for Sample Size = 10 0.82 0.59 0.55 0.85 0.70 Violation 0.65 0.40 0.52 0.38 0.49 Random 0.76 0.66 0.74 0.73 0.72 Median 0.70 0.73 0.78 0.72 0.73 Mean 0.77 0.73 0.78 0.80 0.77 50 HLM 0.76 0.47 0.54 0.39 0.54 Average for Sample Size = 50 0.73 0.60 0.67 0.60 0.65 Violation 0.58 0.44 0.58 0.19 0.45 Random 0.70 0.67 0.72 0.43 0.63 Median 0.75 0.73 0.76 0.40 0.66 Mean 0.77 0.73 0.76 0.42 0.67 100 HLM 0.71 0.51 0.60 0.21 0.51 Average for Sample Size = 100 0.70 0.62 0.69 0.33 0.58 Violation 0.62 0.44 0.52 0.03 0.41 Random 0.66 0.64 0.65 0.08 0.51 Median 0.73 0.70 0.69 0.05 0.55 Mean 0.74 0.70 0.69 0.05 0.55 500 HLM 0.78 0.54 0.55 0.07 0.49 Average for Sample Size = 500 0.71 0.60 0.62 0.06 0.50 Average 0.74 0.60 0.63 0.47 0.61
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87 1 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 00.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 500 Average Sample SizeBands Coverage 0.33 0.54 0.69 0.90 Figure 14. Band coverage of estimated mean reliability for average sample size by coefficient alpha Unlike Bias and RMSE, the variability in Band Coverage was significantly influenced by the number of primary studies in the RG simulations. Table 19 displays information about Band Coverage in regards to the number of primary studies (NPS) and the treatments. For these conditions, the Band Coverage was as little as .36 (NPS= 150 and the treatment was Violation) and as large as .85 (NPS = 15 and the treatment was Median). When the NPS= 15, the Band Coverage was much larger than when NPS was 150. In fact when NPS = 15 and the treatment was Violation the Band Coverage was .63, however, when NPS = 150 and the treatment was Violation, the Band Coverage was almost half as much at .36. A simila r pattern was seen for the other research methods such that as the NPS increased the Band Coverage decreased.
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88 Table 19 Band Coverage of Estimated Mean Reliability for Number of Primary Studies by Treatment Average of Band Coverage Number of Primary Studies Treatment 15 50 100 150 Average Violation 0.63 0.49 0.41 0.36 0.47 Random 0.80 0.70 0.61 0.56 0.67 Median 0.85 0.74 0.67 0.60 0.71 Mean 0.83 0.70 0.60 0.53 0.66 HLM 0.73 0.54 0.43 0.38 0.52 Average 0.77 0.63 0.54 0.49 0.61 Finally, the magnitude of coefficient alpha also had a significant impact on the variability in Band Coverage. This information is displayed in Table 20. In this situation the Band Coverage ranged from .32, when xx = .90 and the treatment was Violation to .81, when xx = .90 and the treatment was Mean. Overall, as xx increased the Band Coverage decreased. When the treatment was Violation and xx = .90 the Band Coverage (.32) was more than half the size than when xx = .33 and the treatment was Violation. A similar result was seen when the treatment was HLM; when xx = .90 the Band Coverage was .34 but when xx = .33 it was .77, more than twice as much. Obviously, the magnitude of xx has an impact on Band Coverage especially when dependence is ignored or when mixed models are applied. Table 20 Band Coverage of Estimated Mean Reliabili ty for Coefficient Alpha by Treatment Average of Band Coverage xx Treatment 0.33 0.54 0.69 0.90 Average Violation 0.67 0.41 0.49 0.32 0.47 Random 0.76 0.68 0.69 0.54 0.67 Median 0.68 0.70 0.74 0.54 0.66 Mean 0.81 0.75 0.73 0.56 0.71 HLM 0.77 0.47 0.50 0.34 0.52 Average 0.74 0.60 0.63 0.46 0.61
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89 Band Width Table 6 indicates that factors for Band Width where 2 0.05 are n ( 2 = .28), xx ( 2 = .20), NPS ( 2 =.12), the interaction between xx and n ( 2 =.10), the interaction between n and NPS ( 2 =.07), and the interaction between xx and NPS ( 2 = .05). Notice that this dependent variable, average Band Width, had the largest 2 in this analysis. Also, note that for the other three dependent variables, Bias, RMSE, and Band Coverage, the 2 value for ICC was always larger than .05. In contrast, for Band Width the 2 for ICC was approximately 0. The results using average Band Width an outcome and these factors as predicto rs are presented in Table 21 through Table 26. In addition, the interaction between xx and n is displayed in Figure 13, the interaction between n and NPS is displayed in Figure 14, and the interaction between xx and NPS is displayed in Figure 15. In Table 21 information about the extent to which the magnitude of the average sample size, n, contributes to the Band Width of estimated mean reliability by treatment is presented. The average Band Width ranged from .01 (where n = 500 and the treatment was Violation) to .14 (where n = 10 and the treatment was Median). The overall average Band Width for sample size ranged from .02 to .11, such that when n = 500, the average Band Width was .02, for n = 100, it was .03, for n= 50 it was .05 and for n = 10 it was .11. It was not surprising that there was an inverse relationship between average sample size and Band Width given that standard error is a f unction of sample size and also has an inverse relationship; that is all things being equal, the larger the average sample size the smaller the standard error. Even though there was some variab ility across average
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90 sample size, overall the Band Width was relatively small; that is, the confidence bands were on average very narrow. Table 21 Band Width of Estimated Mean Reliability by Treatment and Average Sample Size Average of Band Width Average Sample Size Treatment 10 50 100 500 Average Violation 0.07 0.03 0.02 0.01 0.03 Random 0.13 0.05 0.04 0.02 0.06 Mean 0.13 0.05 0.04 0.02 0.06 Median 0.14 0.05 0.04 0.02 0.06 HLM 0.08 0.04 0.03 0.02 0.04 Average 0.11 0.05 0.03 0.01 0.05 In Table 22 information about the exte nt to which the magnitude of the population reliability parameter, xx, contributes to the Band Width of estimated mean reliability by treatment is presented. The average Band Width ranged from .01, where xx = .90 for all treatments, to .12, where xx = .33 and the treatment was Median. The overall average Band Width for xx ranged from .01 to .10, such that when xx =.90, the average Band Width was .01, for xx = .69, it was .04, for xx = .54 it was .06, and for xx = .33 it was .10. These results suggest that there is an inverse relationship between the magnitude of the population reliab ility parameter and the average Band Width. Table 22 Band Width of Estimated Mean Reliabili ty by Treatment and Coefficient Alpha Average of Band Width xx Treatment 0.33 0.54 0.69 0.90 Average Violation 0.06 0.04 0.02 0.01 0.03 Random 0.11 0.07 0.05 0.01 0.06 Mean 0.11 0.07 0.05 0.01 0.06 Median 0.12 0.07 0.05 0.01 0.06 HLM 0.08 0.05 0.03 0.01 0.04 Average 0.10 0.06 0.04 0.01 0.05
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91 In Table 23 information about the extent to which the magnitude of the number of primary studies contributes to the mean reliability by treatment is presented. The average Band Width ranged from .02, where NPS = 150 and the treatment was Violation to .11, where NPS = 15 and the treatments were, Random, Median, and Mean. The overall average Band Width for NPS ranged from .03 to .09, such that when NPS = 150 or 100, the average Band Width was .03, for NPS = 50, it was .05, and for NPS = 15 it was .09. As with the results for average sa mple size and population reliability parameter, these results suggest that there is an inve rse relationship between the magnitude of the number primary studies in each RG study and the average Band Width. Table 23 Band Width of Estimated Mean Reliability by Treatment and Number of Primary Studies Average of Band Width Number of Primary Studies Treatment 15 50 100 150 Average Violation 0.06 0.03 0.02 0.02 0.03 Random 0.11 0.06 0.04 0.03 0.06 Mean 0.11 0.06 0.04 0.03 0.06 Median 0.11 0.06 0.04 0.03 0.06 HLM 0.08 0.04 0.03 0.02 0.04 Average 0.09 0.05 0.03 0.03 0.05 In Table 24 and Figure 13 information about the extent to which the interaction between average sample size, n, and the population reliability parameter, xx, contributes to the Band Width of estimated mean reliability is presented. The average of the magnitude of Band Width for n by xx ranged from 0, where xx =.90 and n = 500, to .21, where xx =.33 and n = 10. As displayed in Figure 19, while the Band Width for xx = .33 was always larger than the other values of xx, it had a wider range for smaller values of n; specifically, for n = 10, the Band Width ranged from .02 to .21, but for n = 500 the
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92 Band Width only ranged from approximately 0 to .03. These results concur with the separate results for n, presented in Table 20 and for xx, presented in Table 21 such that these results suggest an inverse relationship between Band Width and these two predictors. However the additional interaction suggests that there is less variability for larger values of n and larger values of xx. Table 24 Band Width of Estimated Mean Reliability for Average Sample Si ze by Coefficient Alpha Average of Band Width xx Average Sample Size 0.33 0.54 0.69 0.90 Average 10 0.21 0.13 0.08 0.02 0.11 50 0.09 0.05 0.03 0.01 0.05 100 0.06 0.04 0.02 0.01 0.03 500 0.03 0.02 0.01 0.00 0.01 Average 0.10 0.06 0.04 0.01 0.05 1 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 01 0 5 0 1 0 0 5 0 00.05 0.10 0.15 0.20 0.25 0 500 Average Sample SizeBandwidth 0.33 0.54 0.69 0.90 Figure 15. Band width of estimated mean reliability for average sample size by coefficient alpha
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93 The interaction between average sample si ze and number of primary studies also had a significant impact on the variability in Band Width. This information is displayed in Table 25 and in Figure 14. For this interaction the Band Width ranged from .01 when n = 500 and NPS = 50, 100, or 150 to .20 when n = 10 and NPS = 15. The results for this interaction were similar to those seen for the interaction between n and xx such that smaller values of n had larger Band Width. In addition, when n =10 there was more variability in Band Width across the values of NPS than when n=500. For example, when n = 10 and NPS = 15, the Band Width was .20 and when NPS = 150 the Band Width was .06, almost onefourth of the size. In contrast, when n = 500 and NPS = 15 the Band Width was .03 and when NPS =150 the Band Width was .01, one third the size. Table 25 Band Width of Estimated Mean Re liability for Number of Primary Studies by Average Sample Size Average of Band Width Average Sample Size Number of Primary Studies 10 50 100 500 Average 15 0.20 0.08 0.06 0.03 0.09 50 0.11 0.04 0.03 0.01 0.05 100 0.08 0.03 0.02 0.01 0.03 150 0.06 0.03 0.02 0.01 0.03 Average 0.11 0.05 0.03 0.01 0.05
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94 1 5 5 0 1 0 0 1 5 01 5 5 0 1 0 0 1 5 01 5 5 0 1 0 0 1 5 01 5 5 0 1 0 0 1 5 00.00 0.05 0.10 0.15 0.20 0.25 050100150 Number of Primary studiesBandwidth 10 50 100 500 Figure 16. Band width of estimated mean reliability for number of primary studies by average sample size The interaction between coefficient alpha a nd the number of prim ary studies also was significant in regards to the Band Width. The information for these results is displayed in Table 26 and in Figure 15. For this interaction the Band Width ranged from .01 when xx = .90 and NPS = 50, 100, or 150 to .17 when xx = .33 and NPS = 15. These results were very similar to the results for the interaction between xx and n and the interaction between n and NPS; smaller values produce wider confidence bands. In addition, when xx = .33 there was much more variability across the values of NPS than when xx =.90. For example, when xx = .33 and NPS =15 the Band Width was .17 and when NPS = 150 the Band Width was .05, almost one third the size. When xx = .90 and NPS =15 the
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95 Band Width was .02 and when NPS was any other value for xx= .90 the Band Width was .01, about onehalf the size. Also note that when xx = .33 and NPS =15, the Band Width, .17, is 17 times larger than the Band Width when xx = .90 and NPS =15. Table 26 Band Width of Estimated Mean Re liability for Number of Primary Studies by Coefficient Alpha Average of Band Width xx Number of Primary Studies 0.33 0.54 0.69 0.90 Average 15 0.17 0.11 0.07 0.02 0.09 50 0.09 0.06 0.04 0.01 0.05 100 0.07 0.04 0.03 0.01 0.03 150 0.05 0.03 0.02 0.01 0.03 Average 0.10 0.06 0.04 0.01 0.05 1 5 5 0 1 0 0 1 5 01 5 5 0 1 0 0 1 5 01 5 5 0 1 0 0 1 5 00.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 050100150 Number of Primary studiesBandwidth 0.33 0.54 0.69 0.90 Figure 17. Band width of estimated mean reliability for number of primary studies by coefficient alpha
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96 A Deeper Look at Band Coverage When the results for average Band Coverage were first examined there were a noticeable number of the simulations where the average Band Coverage was quite small. Recall that for each condition investigated, se veral RG analyses were simulated such that the value of the average Band Coverage is the average proportion of times (for the 1,000 to 10,000 replications that were simula ted) that the actual population parameter was within a 95% confidence band around the mean reliability estimateÂ’s value. In other words, if the average Band Coverage for a particular estimate was .30, this means that for the RG analysis for that particular se t of factors 30% of the confidence bands contained the population coefficient alpha a nd 70% of them did not. As a means to evaluate these results the Band Coverage was divided into thr ee categories such that Band Coverage that was less than .50 was considered Â“smallÂ” and Band Coverage greater than or equal to .50 and less than .925 was considered Â“mediumÂ” and Band Coverage greater than or equal to .925 and le ss than or equal to 1 was considered Â“large.Â” The Â“cut offÂ” values chosen fo r Â“largeÂ” were based on BradleyÂ’s (1978) approach to defining robustness. Using thes e categories, approximately 34.8% of all 6,400 conditions had a small average Band Coverage, approximately 34.47% had medium average Band Coverage, and 30.73% had large Band Coverage. In addition to investigating the overall pr oportions of small, medium and large coverage, the extent to which the type of tr eatment by each factor re sulted in robust (i.e., large) Band Coverage also was analyzed. Table 27 displays the results for type of treatment. These values in the table repres ent the percentage of the total number of conditions run for each type of treatment. For example, in Table 27, the value 11.72%
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97 appears in the cell that is the intersection of Violation and large Band Coverage. This value represents the proportion of all of the 1,280 conditions that were simulated for the treatment Violation that had Band Coverage that was greater than or equal to .925 and less than 1. When the treatment was HLM about 23.28% of the Band Coverage was large. When the treatment was Random only about 21.48% was large. The treatments Mean and Median had the largest percentage of Band Coverage that was large, 51.95% and 45.23%, respectively. Table 27 Large Band Coverage by Type of Treatment Treatment Band Coverage Violation Random Median Mean HLM Total Large 11.72% 21.48% 45.23% 51.95% 23.28% 30.73% Total 1280 1280 1280 1280 1280 6400 Next, the percentage of large Band Coverage for factors by treatment is presented. Table 28 displays the results fo r intraclass correlation by treatment. The values in each cell repres ent the percent of large Band Coverage for all the conditions simulated that shared those characteristics. For example, in Table 28, the cell where ICC = 0 and the treatment is Violation contains the value 18.44%. In this study there were 320 conditions simulated for each value of ICC and treatment. The 18.44% represents the percentage of those 320 conditions that were simulated such that ICC = 0 and the treatment was Violation. For each value of ICC there were 1,600 conditions that were simulated. The percentage for each row in the last column represents the percentage of the 1,600 conditions where the ICC had large Band Coverage. For example, 43.19% of the 1,600 conditions simulated where ICC = 0 had large Band Coverage. Notice in this table that as the intraclass correla tion increased the percentage of large Band Coverage
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98 decreased such that when ICC = .90 only 7.69% of the 1,600 conditions had large Band Coverage. Out of the 320 simulations each for Violation and HLM only 1.25% and 1.56%, respectively, had large Band Coverage when ICC = .90. While the Mean treatment seemed to have the highest percentage of large Band Coverage it was still only 69.69% when ICC = 0 and was as small as 17.50% when ICC= .90. Probably the most disconcerting result was the f act that even when ICC = 0 (i.e., no proportion of variance in reliability th at is between studies), only 43.19% of the 1,600 conditions simulated had large Band Coverage. Table 28 Percentage of Large Band Coverage for Intraclass Correlation by Treatment Treatment ICC Violation Random Median Mean HLM Total 0.00 18.44% 33.44% 61.25% 69.69% 33.13% 43.19% 0.01 19.06% 31.25% 59.69% 65.63% 34.38% 42.00% 0.30 8.13% 15.94% 47.19% 55.00% 24.06% 30.06% 0.90 1.25% 5.31% 12.81% 17.50% 1.56% 7.69% Total 11.72% 21.48% 45.23% 51.95% 23.28% 30.73% The results for percentage of large Band Coverage for coefficient alpha by treatment are presented in Table 29. As with the results for the intraclass correlation and treatment, the Mean treatment had the highest percentage of large Band Coverage for each value of coefficient alpha. When the treatment was HLM and xx = .33, 55.00% of the 320 conditions that were simulated had large Band Coverage; this was more than three times as much as when the treatment was Violation. Notice that out of the 320 conditions that were simulated such that xx = .54 and the treatment was Violation, only 0.94% had large Band Coverage. In general, when xx = .33 or .69 the percentage of large Band Coverage was almost twice as much as when xx = .54 or .90. In addition, when xx = .90 and the treatment was HLM only 7.19% of the 320 conditions simulated
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99 had large band coverage. Not only was this a smaller value than when the treatment was Violation, it was the smallest percentage of large Band Coverage when xx = .90. Though xx = .90 is usually considered a desirabl e value for coefficient alpha, in this study only 22.00% of the 1,600 cond itions that were simulated had Band Coverage that would be considered robust. Table 29 Percentage of Large Band Coverage for Coefficient Alpha by Treatment Treatment xx Violation Random Median Mean HLM Total 0.33 16.88% 20.31% 45.94% 65.00% 55.00% 40.63% 0.54 0.94% 4.38% 41.56% 47.50% 9.38% 20.75% 0.69 21.56% 40.00% 58.13% 56.56% 21.56% 39.56% 0.90 7.50% 21.25% 35.31% 38.75% 7.19% 22.00% Total 11.72% 21.48% 45.23% 51.95% 23.28% 30.73% The results for percentage of large Band Coverage for average sample size by treatment are presented in Table 30. Once again, the Mean treatment had the highest percentage of Band Coverage for each of the average sa mple size values. However, when the average sample size was 500, the percentage of large Band Coverage for the Mean treatment was only slightly larger than that for the Median treatment (41.25% and 40.94%, respectively). When the average sample size was equal to 10 the percentage of large Band Coverage for the HLM treatment was slightly less than the percentage for Violation and more than half that for Random. However, for the larger values of average sample size the HLM treatment had a higher percentage of large Band Coverage than Violation and Random. When the average sample size was 500, the percentage of large Band Coverage for HLM was almost three times as much as that for Random and more than four times as much as that for Violation. In general, the smaller average sample
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100 sizes had a higher pe rcentage of large Band Coverage than did the larger average sample size. When the average sample size was 10, the percentage of large Band Coverage was still only 40.94% and was only 26.44% wh en the average sample size was 500. Table 30 Percentage of Large Band Coverage for Average Sample Size by Treatment Treatment Average Sample Size Violation Random Median Mean HLM Total 10 24.69% 43.44% 50.94% 66.56% 19.06% 40.94% 50 5.63% 16.56% 44.69% 51.88% 21.88% 28.13% 100 9.06% 14.06% 44.38% 48.13% 21.56% 27.44% 500 7.50% 11.88% 40.94% 41.25% 30.63% 26.44% Total 11.72% 21.48% 45.23% 51.95% 23.28% 30.73% The results for percentage of large Band Coverage for the number of primary studies by treatment are presented in Table 31. As with results for the other factors presented thus far, the Mean treatment had the highest percentage of large Band Coverage for each of the primary studies values. Under these conditions the Median treatment had parentages that were not much smaller than when the Mean treatment was used. Also, the treatment HLM had percentages that were always slightly higher than those for the Random and almost twice as much as when the treatment was Violation. In general, the larger the number of primary studies the lower the percentage of large Band Coverage. When the number of primary studies was 150 only 19.94% of the 1,600 conditions that were simulated had large Band Coverage and when the number of primary studies was 15 only 46.75% of the 1,600 conditions had large Band Coverage. The results for the percentage of large Band Coverage for the number of reliabilities for each primary study by treatment are presente d in Table 32. Because the number of reliabilities had five possible values the number of conditions generated for each cell
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101 was 256 not 320, (e.g., there were 256 conditi ons simulated such that the number of reliabilities was 1 and the treatment was Violation). In addition, there were 1,280 conditions simulated, not 1,600, for each numbe r of reliabilities value. When the number of reliabilities was 1 and the treatment was HLM, the percentage of large Band Coverage was 30.47%; for the other treatments it was 21.88%. As the number of reliabilities increased, th e percentage of large Band Coverage increased for the Mean treatment. When the treatment was Median and the number of reliabilities was 2, the percentage of large Band Coverage was more than double the percentage when the number of reliabilities was 1. However, the percentage of large Band Coverage was slightly smaller for the Median treatment when the number of reliabilities was 3 and then increased when the number of reliabili ties was 10 and then 50. When the treatment was HLM and the number of reliabilities was 2 the percentage of large Band Coverage was only slightly larger than when the number of reliabilities was 1. When the treatment was Random the percentage of large Band Coverage decreased as the number of reliabilities increased except when the number of reliabilities increased from 10 to 50. In this case, the percentage of large Band Coverage went from 19.92% to 21.88%, respectively. When the number of reli abilities was 3 and the treatment was HLM the percentage of large Band Coverage decreased to 28.52% and wa s even smaller when the number of reliabilities was increased. When the number of reliabi lities was 50 and the treatment was HLM only 8.20% of the 256 conditions simulated had large Band Coverage. Finally, when the treatment was Violation the percentage of large Band Coverage decreased as the number of reliabi lities increased. When the number of
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102 reliabilities was 50 and the treatment was Violation only 2.73% of the 256 conditions simulated had large Band Coverage. Overall, when the number of reliabilities was 1, only 23.59% of the 1,280 conditions simulated had large Band Coverage. For the other number of reliabili ties values the percentage of large Band Coverage was about one third of the 1,280 conditions simulated for each reliability value. Summary of Results These results were evaluated by first l ooking at the five choices of treatments (Violation, Mean, Median, Random, and HLM) addressed in the research questions for Table 31 Percentage of Large Band Coverage for the Number of Primary Studies by Treatment Treatment Number of Primary Studies Violation Random Median Mean HLM Total 15 19.06% 37.50% 66.88% 69.69% 40.63% 46.75% 50 11.56% 19.38% 47.81% 55.94% 22.50% 31.44% 100 9.06% 17.81% 35.63% 45.31% 16.25% 24.81% 150 7.19% 11.25% 30.63% 36.88% 13.75% 19.94% Total 11.72% 21.48% 45.23% 51.95% 23.28% 30.73% Table 32 Percentage of Large Band Coverage for Number of Reliabilities Per Study by Treatment Treatment Number of Reliabilities Violation Random Median Mean HLM Total 1 21.88% 21.88% 21.88% 21.88% 30.47% 23.59% 2 14.45% 22.27% 48.44% 48.44% 32.03% 33.13% 3 10.94% 21.48% 42.58% 56.64% 28.52% 32.03% 10 8.59% 19.92% 54.69% 64.84% 17.19% 33.05% 50 2.73% 21.88% 58.59% 67.97% 8.20% 31.88% Total 11.72% 21.48% 45.23% 51.95% 23.28% 30.73%
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103 this study. This was executed by creating box plots for these treatments and each of the outcomes: Bias, RMSE, Band Coverage and Band Width. The box plots indicated that regardless of the type of treatment used, the results were about the same for Bias, RMSE, and Band Width; that is, treatment does not impact th e variability in these three outcome variables. However, the type of research design did appear to have an impact on the variability in Band Coverage. This was confirmed when etasquared was calculated using PROC GLM in SAS such that the dependent variables were Bias, Root Mean Square Error, Band Coverage, and Band Width and the independent variables were the five types of factors (magnitude of coeffi cient alpha, average sample size, number of journal studies, number of reliability coefficients from each journal study, and the magnitude of the intraclass correlation), and the choice of treatment. For all four of the dependent variables, etasquared was calculate d for the main effect along with the firstorder interactions of the independent variables. Even though the choice of treatment was only a significant main effect for Band Coverage, the impact of treatment was included in the results for the evalua tion of all of the significant main effects and firstorder interactions for Bias, Root Mean Square Error, Band Coverage, and Band Width because their impact was the main focus of this research study. Even though the etasquared results indicated that ICC, xx, and n, in addition to the interaction between ICC and xx, and the interaction between treatment and xx all had an impact on the variability in Bias, overall, the magnitude of the Bias was not large, and in all cases the estimated mean reliability was never overestimated. While these results did indicate that the Median treatment resulted in slightly larger values of Bias, these values never exceeded .05 in magnitude. These results for Bias suggest that
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104 regardless of the treatment or the other factors investigated in this simulation, the estimated mean reliability was not overestim ated and was only slightly underestimated. The main effect factors that had a si gnificant impact on the variability in RMSE were xx, n, and ICC. In addition the interaction between xx and n, the interaction between xx and ICC, and the interaction between xx and the type of treatment also were shown to have a significant impact on the variability in RMSE. These results suggest that smaller values of xx will have a slightly larger RMSE compared to larger values of xx and larger samples sizes have a slight ly smaller and somewhat more stable RMSE than the smaller sample sizes. Even though when ICC = .90 the average RMSE was slightly larger than the results for the smaller values of ICC, overall the magnitude of ICC did not appear to have a large impact on the variability in RMSE. Overall, the RMSE was never very large, which would suggest that the reliability estimates were somewhat stable regardless of the magnit ude of any of these factors. When Band Coverage was examined, the main effect factors that had a significant impact on the variability in Band Coverage were ICC and the number of primary studies. In addition, the choice of tr eatment also had a signi ficant impact on the variability in Band Coverage. The firstorder interactions that were significant were the interaction between ICC and xx, and the interaction between xx and n. These results suggest that larger values of ICC will have a much smaller Band Coverage compared to smaller values of ICC. More important, the type of treatment does not seem to improve the size of the Band Coverage as ICC increases. However, th ese results also suggest that there is a noticeable interaction between ICC and xx when it comes to Band Coverage. When xx = .33, .54, or .69, the band coverage was somewhat the same and consistent in
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105 behavior (i.e., the smaller the ICC the larger the Band Coverage); this was not the case for xx =.90. The average Band Coverage for xx =.90 was very small, .47, and the larger ICC resulted in an increase in Band Coverage. A similar pattern was seen for the interaction between xx and average sample size. As sample size increased the average Band Coverage for xx = .33, .54, and .69 was relatively stable. For xx = .90, the average Band Coverage for n = 50, 100, and 500 went from .60 to .33 to .06, respectively. The main effects factors that ha d an impact on the variability in Band Width were n, xx, and the number of primary studies. Ironica lly, the firstorder interaction effects that were significant were all some pairing of these three factors: the interaction between xx and n, the interaction between n and NPS, and the interaction between xx and NPS. In contrast to the other three dependent variables, Bias, RMSE, and Band Coverage, the 2 value for ICC was approximately 0. The vari ability in ICC did not have much of an impact on the variability in Band Width. These results suggest an inverse relationship between average sample size and Band Width such that the larger the average sample size, the smaller the Band Width. This was also the case when evaluating the impact of the magnitude of the population reliability pa rameter and the number of primary studies. The larger the magnitude of xx or the larger the number of primary studies, the smaller the Band Width. There was also a noticeable interaction between xx and average sample size for Band Width such that when the average sample size is small and xx is small, there was much more variability in Band Width than when average sample size was large and xx. Similar results were seen for the interaction for these two factors, n and xx, with the number of primary studies. In ge neral, smaller values produced wider bands.
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106 Overall, the Band Width, regardless of the factors, was quite small. Band Width never exceeded .27 and on average was .05. The confidence bands were very narrow. Because there was a noticeable number of simulations in which the average Band Coverage was quite small (i.e., less than .50), the Band Coverage was evaluated by dividing the results into thr ee categories such that the Band Coverage was considered small if it was greater than zero but less than .50; Band Coverage greater than or equal to .50 and less than .925 was considered Â“medium,Â” and Band Coverage greater than or equal to .925 and less than or equal to 1 wa s considered Â“large.Â” The Â“cut offÂ” values chosen for Â“largeÂ” were based on BradleyÂ’s ( 1978) approach to de fining robustness such that the percentage of large Band Coverage would be those conditions whose results were fairly robust to Type I Error. Using these categories, approximately 34.8% of all 6400 conditions had a small average Band Coverage, approximately 34.47 % had medium average Band Coverage and 30.73% had large Band Coverage. In addition to examining the overall pe rcentage of small, medium, and large Band Coverage, the parentage of conditions that resulted in Band Coverage for each treatment and for each factor by treatmen t also was evaluated. Out of the 1,280 conditions simulated for each treatment, the Mean treatment had the highest percentage of large Band Coverage (51.95%), and the lowest percentage was for the Violation treatment (11.72%). When these treatments were paired with the factors investigated in this study (i.e., intraclass correlation, coeffi cient alpha, average sample size, number of primary studies, and number of reliabilities per study), the Mean treatment usually had the highest percentage of large Band Coverage. This was true for every factor regardless of the value with two exceptions. First, when xx = .69, the Median treatment had a
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107 slightly higher large Band Coverage than the Mean treatment (58.13% and 56.56%, respectively). Second, when the number of reliabilities per study wa s equal to 1, the HLM treatment had the highest percentage of large Band Coverage out of the five treatments and the other four treatments ha d band coverage that were all equal to 21.88%. In general, however, the HLM treatment usually had very small percentage of large Band Coverage and in many instances the results were very similar to the results when the treatment was Violation. For each of the five factors evaluated even when the value for each was at a Â“desirableÂ” level, the percentage of large Band Coverage was remarkably small. For example, even when ICC = 0 only 43.19% of the 1,600 conditions simulated had large Band Coverage. When xx = .90, only 22.00 % of the 1,600 conditions simulated had large Band Coverage. When the average sample size was equal to 500, only 26.44% of the conditions simulated had large Band Coverage. When the number of primary studies was equal to 150 only 19.94% of the conditions had large Band Coverage. Finally, when the number of reliabilities per study wa s 1 (a somewhat desirable number) the percentage of large Band Coverage was only 23.59%. One might debate what values for the factors examined would be considered Â“desirableÂ” for an RG metaanalysis. However, it is quite obvious that when onl y 30.73% of the conditions simulated had Band Coverage that was robust none of the values for the factors and none of the treatments really had very Â“desirableÂ” results.
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108 Chapter Five: Conclusions Summary of the Study Both validity and reliability indices are a function of the scores on a measure, and the magnitude of these indices can fluctu ate across administrations of a measure. It is a common mistake to say that a test is relia ble when in fact it is not the test that is reliable, but the scores on a test that are re liable. Because reliabil ity can fluctuate across studies, it has been recommended that researchers should always evaluate the reliability of their measure and report the results (Wilkinson & APA Task Force on Statistical Inference, 1999). In 1998, VachaHaase addr essed this issue when she proposed a fixedeffects metaanalytic method for evaluating reli ability, similar to va lidity generalization studies, called reliability gene ralization (RG). Validity genera lization studies have been conducted to describe the extent to which va lidity evidence for scor es are generalizable across research contexts (Hunter & Schm idt, 1990; Schmidt & Hunter, 1977). In a similar fashion, RG studies can be used to investigate the distri bution of reliability estimates across studies and to identify study characteristics that may be related to variation in reliability estimates, such as sample size, type of reliability estimate (coefficient alpha vs. testretest), different forms of an instrument, or participant characteristics (Henson, 2001; VachaHaas e, 1998). This method is recommended for describing estimated measurement error in a te st scores across studies and also can be used to analyze measurement error in differences.
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109 This research primarily focused on appropriate analysis of reliability estimates that are not statistically i ndependent. The assumption of i ndependence of observations is commonly violated in metaanalytic rese arch (Becker, 2000; Hedges & Olkin, 1985; Hunter & Schmidt, 1990). As the available l iterature suggests, violating the assumption of independence is a serious issue. There are several approaches to dealing with the viol ation of independence that have been recommended by researchers (see Be cker, 2000). Some of these are, ignoring it and treating each observation as independe nt (e.g., Smith, Glass, & Miller, 1980), calculating one mean or median from each st udy (e.g., Tracz et al., 1992), selecting only one observation per study (e.g., Rosenthal & Rubin, 1986), or using a mixed effects model (e.g., Beretevas & Pastor, 2003). For this study the samples of primary studies were generated using population parameters from a threeparameter IRT mode l (Table 2) that was developed by Hanson and Beguin (1999). From these simulated ex aminee responses, subsets of items were selected that yielded the target values of coefficient alpha. These target values, computed from the simulated examinees, were used as the population values to which the subsequent sample estimates were compared. The coefficient alpha values were generated using the information from the threeparameter model using the item information from the ACT Mathematics Assessment. The research was conducted using a M onte Carlo simulation study method in which random samples were simulated under known and controlled population conditions. In the Monte Carlo study, RG st udies were simulated by generating samples in primary studies, estimating reliability of sc ores in these samples, and then aggregating
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110 the sample reliability estimates in the RG studies. The Monte Ca rlo study included five factors. These factors were (a) the ma gnitude of the coefficient alpha (with xx = 0.30, 0.50, 0.70, and 0.90), (b) sample size in the primary studies (average sample sizes, n, of 10, 50, 100, and 500), (c) number of pr imary studies in the RG study (with k = 15, 50, 100, and 150), (d) number of reliabil ity estimates from each study (with i = 1, 2, 3, 10 and 50) and (e) the degree of violation of independence where the strength of the dependence is related to the num ber of reliability indices (i.e ., coefficient alpha) derived from a simulated set of examines and the magnitude of the correlation between the journal studies (intraclass correlation = 0, .0l .30, and .90). The values chosen for each of these factors were based in part on obs erved factors of actual RG studies, in part on factors of the Tracz et al. (1992) simu lation study, and mostly on values that represent a range that is reasonable and typical in simulation studies. In addition, there were five types of tr eatments that were applied: first, the dependence was ignored, referred to as Violation. Second, a reliability index was randomly selected from each of the simu lated journal studies, referred to as Random. Third, a mean was calculated from each simulated journal study, referred to as Mean. Fourth, a median was calculated from each simulated journal study, referred to as Median. Finally, a twolevel mixed model was used to calculate the estimated mean reliability using a null model such that the intercept value was the average reliability; this was referred to as HLM. The research was conducted using SAS/IM L version 9.1. Conditions for the study were run under Windows XP. Normally dist ributed random variables were generated using the RANNOR random number generator in SAS. A different seed value for the
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111 random number generator was used in each execution of the program and the program code was verified by handchecking results from benchmark datasets. The SAS PROC MIXED procedure was used to generate the two level null models used in this study. The impact of the treatm ent factors was evaluated ba sed upon the bias in the mean estimates, root mean square estimat es, the confidence band coverage, and the average confidence band width. Research Questions In RG studies the dependent variable in the analyses is the reliability estimate (Henson & Thompson, 2001). This research focused on how certain study methods, in regards to violation of independence, affect the estimated mean reliability of scores calculated across studies. The key questions that were addressed in this study were: 1. What is the effect on point and interval estimates of mean reliability of ignoring violation of independence of observations in RG studies (i.e., treating all reliability coefficients as independent observations)? 2. What is the effect on point and interval estim ates of mean reliabil ity of using a mean or median reliability from each study as part of a sample in a RG study? 3. What is the effect on point and interval estimates of mean re liability of randomly selecting a reliability estimate from each st udy as a part of a sample in a RG study? 4. What is the effect on point and interval es timates of mean reliability of using a two level mixedeffects model for RG studies (i.e ., reliabilities are nested within studies)? 5. In regard to violations of independence, wh at impact do factors such as the magnitude of coefficient alpha, sample size, number of journal studies, number of reliability coefficients from each study, and the magnitude of the intraclass correlation (ICC) of
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112 the studies (i.e., the magnitude of the viol ation of independence) have when any of the methods discussed in the four rese arch questions above are investigated? Summary of Study Results Because the first four research questions addressed the impact of the type of treatment, these results were evaluated by first looking at the impact of the treatments used in this study (Violation, Mean, Median, Random, and HLM). This was carried out by creating box plots for these treatm ents and each of the outcomes, Bias, RMSE, Band Coverage and Band Width. The box plots indicated that th e types of treatment does not impact the variability in Bias, RMSE, and Band Width but did seem to have an impact on Band Coverage. This was later confirmed when etasquared was calculated in regards to the type of treatment and their interaction with the other factors investigated in this study. Etasquared was calculated using PROC GLM in SAS such that the dependent variables were Bias, RMSE, Band Coverage, and Band Width and the independent variables were the five types of factors (magnitude of coefficient alpha, average sample size, number of journal studies, number of re liability coefficients from each journal study, and the magnitude of the intraclass co rrelation), and the choice of treatment. For all four of the dependent va riables etasquared was calculated for the main effects and firstorder interactions of the independent variables. A cut off value of 2 0.05 was used to determine which factors had an im portant impact on the dependent variables. While the type of treatment was only a main effect for Band Coverage, because the impact of treatment is addr essed in the research questi ons in this study, treatment
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113 included in all of the results for the evaluation of Bias, RMSE, Band Coverage, and Band Width. Etasquared results indicated that ICC, xx, and n, in addition to the interaction between ICC and xx, and the interaction treatment and xx all had an impact on the variability in Bias; overall, the magnitude of the Bias was not large and in all cases the estimated mean reliability was never overestimated. The main effect factors that ha d an impact on the variability in RMSE were xx, n, and ICC. In addition, the interaction between xx and n, the interaction between xx and ICC, and the interaction between xx and the type of treatment, also were shown to have an impact on the variability in RMSE. These results suggest that smaller values of xx had a slightly larger RMSE compared to larger values of xx and larger samples sizes have a slightly smaller and somewhat more stable RMSE than the smaller sample sizes. Even though when ICC = .90 the average RM SE was slightly larger than were the results for the smaller values of ICC, one c ould argue that ICC was not that influential on RMSE. Overall, the RMSE was never very large, wh ich would suggest that the reliability estimates were somewhat stable regardless of the magnitude of any of these factors. When Band Coverage was examined, the main eff ect factors that showed a significant impact on the variability in Band Coverage were ICC, and the number of primary studies. In addition, the type of treatment also had a significant impact on the variability in Band Coverage. The firstorder interactions that were significant were the interaction between ICC and xx, and the interaction between xx and n. These results suggest that larger values of ICC had a much smaller Band Coverage compared to
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114 smaller values of ICC. In addition, the type of treatment did not seem to improve the size of the Band Coverage as ICC increases. However, th ese results also suggest that there is a noticeable interaction between ICC and xx when it comes to Band Coverage. A similar pattern was seen for the interaction between xx and average sample size. As sample size increased the average Band Coverage for xx = .33, but when xx = .90 the average Band Coverage had much more variability and decreased substantially as the average sample size increased. The main effects factors that ha d an impact on the variability in Band Width were n, xx, and the number of primary studies. The fi rstorder interaction effects that were significant were all some pairing of thes e three factors: the interaction between xx and n, the interaction between n and NPS, and the interaction between xx and NPS. In contrast to the other th ree dependent variables, Bias, RMSE, and Band Coverage, the variability in ICC did not have much of an impact on the variability in Band Width. These results suggest that there was an inve rse relationship between average sample size and Band Width such that the larger the average sample size the smaller the Band Width. This was also the case when evaluating the impact of the magnitude of the population reliability parameter and the number of prim ary studies. The larger the magnitude of xx or the larger the number of primary studies the smaller the Band Width. There was also a noticeable interaction between xx and average sample size for Band Width such that when the average sample size is small and xx is small there was much more variability in Band Width then when average sample size was large and xx. Similar results were seen for the interaction for these two factors, n and xx, with the number of primary studies. In general, smaller values produced wider bands. Overall the Band Width,
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115 regardless of the factors, was quite small. Band Width never exceeded .27 and on average was .05. The confidence bands were very narrow. Because there was a noticeable number of simulations where the average Band Coverage was quite small (i.e., less than .50), the Band Coverage was evaluated by dividing the results into thr ee categories such that the Band Coverage was considered small if it was greater than zero but less than .50; Band Coverage greater than or equal to .50 and less than .925 was considered Â“medium,Â” and Band Coverage greater than or equal to .925 and less than or equal to 1 wa s considered Â“large.Â” The Â“cut offÂ” values chosen for Â“largeÂ” were based on BradleyÂ’s ( 1978) approach to de fining robustness such that the percentage of large Band Coverage would be those conditions whose results were fairly robust to Type I Error. Using these categories, approximately 34.8% of all 6,400 conditions had a small average Band Coverage, approximately 34.47 % had medium average Band Coverage, and 30.73% had large Band Coverage. In regards to the different types of treatments, the Mean research had the la rgest percentage of Band Coverage that was robust (51.95%). When the treatments were paired with the factors investigated in this study the Mean treatment usually still had th e highest percentage of large Band Coverage. This was true for every factor regardless of th e value with two exceptions. First, when xx = .69, the Median treatment had a slig htly higher large Band Coverage than the Mean treatment (58.13% and 56.56%, respectively). Second, when the number of reliabilities per study was equal to 1, the HLM treatment had the highest percentage of large Band Coverage out of the five treatments and the other four treatments had band coverage that were all equal to 21.88%. In general, however, the HLM treatment usually had very
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116 small percentage of large Band Coverage and in many instances the results were very similar to the results when the treatment was Violation. For each of the five factors evaluated, even when the value for each was at a Â“desirableÂ” value, the percentage of large Band Coverage was still very small. For example, even when ICC = 0 only 43.19% of the 1,600 conditions simulated had large Band Coverage. When xx = .90, only 22.00 % of the 1,600 conditions simulated had large Band Coverage. When the average sample size was equal to 500 only 26.44% of the conditions simulated had large Band Coverage. When the number of primary studies was equal to 150 only 19.94% of the conditions had large Band Coverage. Finally, when the number of reliabilities per study wa s 1 (a somewhat desirable number) the percentage of large Band Coverage was only 23.59%. In general because only 30.73% of all the conditions simulated had Band Coverage that was robust it could be argued that most of the values for the factors a nd most of the treatments did not have very Â“desirableÂ” results. Discussion It was expected, base d on previous research (Beretevas & Pastor, 2003), that HLM would provide better point estimates and be tter interval estimates than the rest of the treatments applied; however, this was not the case with this study. When the type of treatment was investigated as a part of the other factors, at times HLM behaved more like Violation than any of the other treatments. This could be because of the five types of treatments investigated these two methods were the only two that used all the observations as a part of estimating the mean re liability. In general, there was very little Bias in the results and the RMSE results were relatively small. In addition, the Band
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117 Width was overall very small which would explain the overall poor Band Coverage, i.e., narrow bands would Â“captureÂ” fewer estimates. When the Band Coverage was evaluated only 30.73%, or less than one third of all the conditions simulated, had Band Coverage that was considered robust. The fact that HLM and Violation both had results where the percentage of large Band Coverage was very small would indi cate that these two types of treatments are likely to produce reliability that are less likely to fall within a 95% confidence interval. Based on these results calculating a mean from each study seemed to produce the most robust Band Coverage. Even though it was better, the average Band Coverage for this type of treatment was still only .72. These results did suggest that the magn itude of ICC, the magnitude of the population reliability parameter, and the magn itude of the average sample size do have an impact on the point and interval estimates results. The number of primary studies had some impact in regards to Band Width but the number of reliabilities from each study was not seen to be a contributing factor. Ba sed on these results it could be argued that the point and interval estim ates are impacted the most when ICC, the population reliability parameter, and the average sample size are rather large. As was seen in these results, when this occurred the Band Coverage was quite small. However, for Bias and RMSE, even though significant differences in th e variability of the factors were found, overall, these values were rather small. Another value that was consistently small was the Band Width. These results suggest that while factors like population reliability parameter and the average sample size do have an impact on the variability of the outcomes, the overall averages for these outcomes was rather small. However, the
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118 magnitude of the ICC alone and its interacti on with these factors can have an impact on the point and interval estimates of reliability. Limitations of the Study The limitations of this study are related to the Monte Carlo method for the study. While the Monte Carlo method was used to si mulate RG studies, the values of the factors used in the simulation were fixed fo r each study. Specifically, because the data for this study were simulated the number of reliability indices from each simulated study was a fixed value in each of the simulati ons (i.e., each study contributed the same number of reliability indices per study). While it is obvious that several of the RG studies conducted so far are treating reliability coefficien ts from the same study as independent, it is also obvious that not al l of the studies cont ribute equal amounts of reliability coefficients. In addition, because the models in this study are fixedeffects models small sample sizes should not be a concern (Randenbush & Bryk, 2002). In several of the RG studies conducted so far testretest reliability estimates given are very rare and seldom evaluated. Because coefficient alpha is the most common reliability coefficient reported, this was the on ly index used in the study. It is important to note however, that coefficient alpha has a tendency to underestimate the actual reliability index (Crocker & Algina, 1986). The data for this study were simulated using information from a test of ability. All of the RG studies that b een conducted thus far have in vestigated reliability in the context of an instrument that measures some type of psychological construct. Measures of ability have a tendency to have more variability than measures of psychological constructs. It is possible that in the actual RG studies there may have been less
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119 variability in the results of the measures inve stigated. This difference in variability could have an impact on the mean estimates of the reliability indices. Another possible limitation to this study is the fact that in each RG analysis estimates were investigated using z transformation for coefficient alpha, ln1 z to normalize the sampling distributions where the transformed value of coefficient alpha is approximately normally di stributed with a variance of k/{2(k1)(N2)}(Bonett, 2002). According to Felt and Charter (2006), ther e are many ways of averaging reliability across studies; perhaps another method ma y have led to different results. Implications Importance of the Study. Researchers have suggested that the use of HLM should provide a better model to investigate the variab ility of score reliabilities (Beretevas et al., 2002; Beretevas & Pastor 2003). The result s of this study sugge st that while the magnitude of the intraclass correlation has a significant impact on the variability in the Bias and the RMSE, the impact on both of these inde pendent variables is negligible. When independence is violated, the point estim ates are still relatively stable and only slightly underestimated, regardless of the t ype of treatment that might be used to estimate the mean coefficient alpha. This rese arch does indicate that researchers should be careful in regards to construc ting confidence intervals because the Band Width was on average .06, the average Band Coverage was only .61, and only 30.73 % of the simulations had coverage that was robust. Importance in Regards to Future RG Studies. While this research seems to indicate that using HLM is not necessarily the best solution for controlling for
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120 dependence, it is possible that the use of mixed models ma y provide more power in RG analyses such that this method may provide more control of Type II error for testing differences in group means. More research ne eds to be done in this area to investigate the impact this method may have on tests of group differences. Because the magnitude of ICC does seem to impact the stability of the results, future RG studies should cons ider the magnitude of the I CC that might be present. While it is usually not possible to calculate the population parameter for this index, one could still estimate it from the sample. Regard less of the type of treatment employed, this research still supported the assumption that the larger the ICC the more problematic the results. While these results indicated that the point estimates calculated from a RG analysis have very little bias regardless of the magnitude of the factors or the type of treatment, the RG researcher should probabl y not use these point estimates to build confidence intervals for inferential statistic s. As Felt and Charter (2006) point out, the average reliability obtained from averaging acr oss studies is not the same as the average that would be obtained if all of the raw data from the groups of interest were available and the researcher calculated coefficient al pha from the combination of the groups. They argue that the average reliability obtained from averaging across studies should never be the value used to construc t confidence intervals for tests of significance among coefficients. They do suggest that there ar e methods for combining coefficient alpha across that would produce the same value as if one did have all the raw data (see Charter 2003).These results suggest that RG metaanaly sis may be useful in estimating what is
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121 typical reliability for a given measure or cons truct but should not be used when creating confidence bands. Suggestions for Future Research While this study did explore several fact ors and different types of treatments in regards to reliability generalizations studies this research only explored the average reliability across studies w ithout considering possible moderating variables. This research should be reproduced such that va lues of common moderati ng variables that are present in typical RG studies can be explored such as sample size, different forms of an instrument, or participan t characteristics (Henson, 2001; VachaHaase, 1998). Because the data for this analysis were generated from a dichotomously scored measure of mathematics ability, this research should be replicated using simulated data from a measure of a psychological construct. In addition, this study also could be duplicated using actual data from an RG study where moderating variables and a measure of a psychological construct were evalua ted. Also this research did not consider the issues in regards to reliabi lity in longitudinal studies. It is possible that longitudinal studies will produce rather large intracl ass correlation (DeS hon, Ployhart, & Sacco, 1998). Another suggestion for future research w ould be to investigate other methods for transforming alpha. Instead of using the tr ansformation recommended by Bonett (2002) another possible way to transform alpha is to apply the FisherÂ’s (1925) formula: r r z1 1 log 1513 1 or r r z1 1 ln 2 1 It is possible that a different transformati on might produce different result; however; in their Monte Carlo study using se ven different approaches to average reliability, Feldt
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122 and Charter (2006) found very little diffe rence among the averages for six of the approaches they investigated. The seventh a pproach they investigated was significantly different but this approach was applicable fo r alternativeform coefficients. They also caution the reader that these methods are for calculating the averag e coefficient, which should only be used as a descriptive stat istic. Charter (2003) recommends using the formula: 2 1 2 2 2 2 2 2 X Y N X X N X Y N r and rcombined = r2 where in N i iX n X 2 2 2i i iSD X n X 2 2 2 i i i iSD r X n Y and where r is the combined reliability index, rcombined is the combined reliability coefficient, ni iX, SDi and ri are the i group sample size, mean, standard deviation, and reliability coefficient, respectively. In these calculations it is assumed that the standard deviation was derived by dividing by n (for a sample standard deviation) and not dividing by n 1 (for a population estimate). Charter (2003) points out that if the group sample size is larger than 50 the use of e ither type of standard deviation would be acceptable. Perhaps future research using th is formula to average reliability would produce better interval estimates. It is also possible that the use of a nonparametric sampling method such as a bootstrap method to generate the confidence intervals might provide better estimates. There are several types of bootstrap methods that might be applied to construct confidence intervals for coefficient alpha (see Hess & Kromrey, 2003). Probably the
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123 most common bootstrap method is the percentile method where samples are repeatedly drawn of size n with replacement from a single sample of n observations. Each bootstrap sample provides an estimate of coefficient alpha and the set of es timates (probably at least 1,000) would result in a distribution of point estimates of mean coefficient alpha. The 2.5 percentile and the 97.5 percentile would be the end points for a 95% confidence interval. Yuan, Guarnacci, and Hapslip (2003) investigated three methods of evaluating the distribution of the sample coefficien t alpha: the existing normaltheorybased distribution, a newly propos ed distribution based on fourthorder moments, and the bootstrap empirical distribution. The results of their research suggest that using the percentile method is not a good bootstrapping approach fo r constructing confidence intervals around an estimate of coefficient alpha. Instead they recommend the bias corrected accelerated method that adjusts fo r the asymmetry in the sampling distribution and the changes in the distri bution of alpha derived using the bootstrap method. In this method, the proportion of the sampling distribution that is less than th e mean alpha is an estimate of asym metry and the estimate is includ ed in the endpoints of the 95% confidence interval. Future researchers might want to consider this method but should keep in mind that while this method may result in better interval estimates of coefficient alpha, the computation is very complex and time consuming. Finally, this study used a fi xedeffects model such that the assumption was that there is no true population varian ce in coefficient alpha in the RG metaanalysis. In this model variability of an infinite sample of e ffect size is not considered. The idea is that variance is assumed to be zero after accounting for modera tors (Shadish & Haddock, 1994). This is how most RG studi es have been conducted.
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124 For a random effects model, the studies included in a metaanalysis study are really a sample from a hypothetical collecti on of studies such ther e are two sources of variance: the variability in effect size parameters and sampling error. There is a strong argument that a random effects model might be more appropriate in terms of generalizing about reliability over studies because the resear cher is probably interested in generalizing the reliability of all possible studies that would use a particular measure or investigate a particular construct. Raudenbush (1994) does caution the researcher that if the number of studies used in a metaana lysis is small the random effects model would not be a good choice because the random eff ects variance would be a very poor estimate of the population variance. Future research should be conducted i nvestigating the use of random effects models to generate the interval estimates for reliability estimates in RG studies. The researcher would assume that the total variance of the observed study reliability estimates ivis made up of the conditional variance iv around the mean population reliability and the random variance 2xxsuch that i iv vxx 2 (Shadish & Haddock, 1994). In this study for the construction of c onfidence bands, the sampling error of each estimate of score dependability index was calculated: 2) 1)(N 2(k k 2k where 2k is the estimated sampling variances of z transformed rxx.. The standard error used for construction of the confidence band for the mean index of score dependability was obtained as:
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125 1 2 11K k kSE where 2 kis the sampling error variance for an index (i.e., transformed coefficient alpha) in the k study and the summation is across the studies included in the RG analysis. In a random effects model the additional random variance 2xx would be added to the SE and then multiplied by 1.96 to construct the interval estimates for coefficient alpha. Because of the addition of the 2xx the confidence bands would be wider and therefore result in better coverage. Thompson and VachaHaase (2000) suggest th at RG studies have the potential to describe the stability across samples of the reliability of scores for a given scale and such an analysis also could reveal that the variation in reliability is not related to the research design factors. Before RG studies can be used to investigate these issues the design of the RG studies must first be improve d to insure that the inferences made are accurate. This current study indicates that fu ture RG researchers could use this method to describe the average reliability of scores for a given measure but should not assume that this method is appropriate for interval estimates. This resear ch clearly indicates, contrary to the popular viewpoi nt, that the use of mixed models (i.e., HLM) does not necessarily alleviate the issues related to the violation of independence. More research needs to be conducted to determine the appropri ate treatment of the data. This is true not only for RG studies, but for all research in ge neral. Regardless of possible future uses and outcomes of the RG method, for these ou tcomes to have credibility, the RG study design must have credibility.
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143 Appendices:
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Appendix A: SAS Code for Monte Carlo Simulation options ls= 132 ; proc printto log = 'C:\rg\mylogyes' ; proc printto print = 'C:\rg\IC01r90n10k15.txt' ; ++ RG_Block Alpha.SAS: Simulates conditions for an entire block of the design matrix 30 September 2001: Modified the weights used in weighted means for rxx and SEM. The sample value of the statistic is no longer a part of the weight. 10 July 2003: Added subroutine for analysis of coefficient alpha 22 July 2003: Simplified the output section: matrices instead of scalars Simplified subroutines for weighted and unweighted mean calculations ++; data iosif; input item_no a_3pl b_3pl c_3pl; poolid = _n_; *if item_no < 4; *3 items for .3; *if item_no<7; *6 items for .5; *if item_no < 12; *11 items for .7; *if item_no <51; *50 items for .90; cards ; 1 0.642 2.522 0.187 2 0.806 1.902 0.149 3 0.956 1.351 0.108 4 0.972 1.092 0.142 5 1.045 0.234 0.373 6 0.834 0.317 0.135 7 0.614 0.037 0.172 8 0.796 0.268 0.101 9 1.171 0.571 0.192 10 1.514 0.317 0.312 11 0.842 0.295 0.211 12 1.754 0.778 0.123 13 0.839 1.514 0.17 14 0.998 1.744 0.057 15 0.727 1.951 0.194 16 0.892 1.152 0.238
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 145 17 0.789 0.526 0.115 18 1.604 1.104 0.475 19 0.722 0.961 0.151 20 1.549 1.314 0.197 21 0.7 2.198 0.184 22 0.799 1.621 0.141 23 1.022 0.761 0.439 24 0.86 1.179 0.131 25 1.248 0.61 0.145 26 0.896 0.291 0.082 27 0.679 0.067 0.161 28 0.996 0.706 0.21 29 0.42 2.713 0.171 30 0.977 0.213 0.28 31 1.257 0.116 0.209 32 0.984 0.273 0.121 33 1.174 0.84 0.091 34 1.601 0.745 0.043 35 1.876 1.485 0.177 36 0.62 1.208 0.191 37 0.994 0.189 0.242 38 1.246 0.345 0.187 39 1.175 0.962 0.1 40 1.715 1.592 0.096 41 0.769 1.944 0.161 42 0.934 1.348 0.174 43 0.496 1.348 0.328 44 0.888 0.859 0.199 45 0.953 0.19 0.212 46 1.022 0.116 0.158 47 1.012 0.421 0.288 48 1.605 1.377 0.12 49 1.009 1.126 0.133 50 1.31 0.067 0.141 51 0.957 0.192 0.194 52 1.269 0.683 0.15 53 1.664 1.017 0.162 54 1.511 1.393 0.123
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 146 55 0.561 1.865 0.24 56 0.728 0.678 0.244 57 1.665 0.036 0.109 58 1.401 0.117 0.057 59 1.391 0.031 0.181 60 1.259 0.259 0.229 61 0.804 2.283 0.192 62 0.734 1.475 0.233 63 1.523 0.995 0.175 64 0.72 1.068 0.128 65 0.892 0.334 0.211 66 1.217 0.29 0.138 67 0.891 0.157 0.162 68 0.972 0.256 0.126 69 1.206 0.463 0.269 70 1.354 0.122 0.211 71 0.935 0.061 0.086 72 1.438 0.692 0.209 73 1.613 0.686 0.096 74 1.199 1.097 0.032 75 0.786 1.132 0.226 76 1.041 0.131 0.15 77 1.285 0.17 0.077 78 1.219 0.605 0.128 79 1.473 1.668 0.187 80 1.334 0.53 0.075 81 0.965 1.862 0.152 82 0.71 1.589 0.138 83 0.523 1.754 0.149 84 1.134 0.604 0.181 85 0.709 0.68 0.064 86 0.496 0.443 0.142 87 0.979 0.181 0.124 88 0.97 0.351 0.151 89 0.524 2.265 0.22 90 0.944 0.084 0.432 91 0.833 0.137 0.202 92 1.127 0.478 0.199
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 147 93 0.893 0.496 0.1 94 1.215 0.867 0.076 95 1.079 0.486 0.264 96 0.932 0.45 0.259 97 1.141 0.344 0.071 98 1.068 0.893 0.153 99 1.217 1.487 0.069 100 1.310 1.186 0.153 ; proc iml symsize = 500 ; ++ Define parameters for execution of the simulation ++; replicat= 10000 ; N of metaanalyses to simulate This value will be set to 10,000; icc= .01 ; *N1 njs = 10; average sample size in study; *N2 njs = 50; *N3 njs = 100; *N4 njs = 250; *N5 njs = 500; *N6;* njs = 1500; mu1= 0 ; Pop mean; sds = 1 ; *++ 3, May 2005 Subroutine to calculate a mean rxx where ind is violated ++ Subroutine to calculate vector of variabilities for coefficient alpha, Both original alpha metric and Fishers z are used Inputs to the subroutine are ri_by_k a matrix of sample alpha estimates where ri is the number of rows(i.e. #alphas per study) k is the nunmber of columns (i.e. # of studies) items number of items on the test (scalar value) N_vec_mtx matrix of sample sizes corresponding to each reliability
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 148 Outputs are (29, April 2005 some of these variables are not needed for J9 dis) Z_w_mean weighted mean Fisher Z SE_Z = Standard error of mean Fisher Z ++; start calc_alphaVI(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); note J9 deleted values from sub routine that are not used in new study; calculate variance for each reliability estimate, Fisher Z and variance of the Z; Have to chance this so that I indexing using rows and columns of a matrix that is nr by n_studies; *print 'IV'; *print ri_by_k; *print n_vec; k=ncol(ri_by_k); ie number of studies; q = nrow(ri_by_k); ie number of reliabilties per study; Z_alpha=J(q,k, 0 ); var_Z =J(q,k, 0 ); do i = 1 to k; do v = 1 to q; ++ Be sure the Rxx values are between .01 and .99 ++; if ri_by_k[v,i] > .99 then ri_by_k [v,i] = .99 ; if ri_by_k[v,i] < .99 then ri_by_k[v,i] = .99 ; ++ Fisher Z transformation, from Bonett, 2002 ++; Z_alpha[v,i] = log( 1 abs(ri_by_k[v,i])); if ri_by_k[v,i] < 0 then Z_alpha[v,i] = Z_alpha[v,i]* 1 ; *new code added for N_vec_matrix; N_vec[1,i] = n_vec[1,i] + N_vec_mtx[v,i]; var_Z[v,i] = ( 2 #items)/((items 1 ) # (N_vec_mtx[v,i] 2 )); end ; q end;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 149 end ; k end; ++ Calculate weighted mean alpha and mean Z ++; *Rxx_w_mean = 0; Z_W_mean= 0 ; Sum_wt = 0; Sum_wtz = 0 ; do i = 1 to k; do v = 1 to q; Rxx_w_mean = Rxx_w_mean + ri_by_k[i,1]/var_alpha[i,1]; Sum_wt = sum_wt + var_alpha[i,1]##1; Z_W_mean = Z_W_mean + Z_alpha[v,i]/var_Z[v,i]; Sum_wtz = sum_wtz + var_Z[v,i]##1 ; end ; q end; end ; k end; *print z_w_mean sum_wtz; *Rxx_w_mean = Rxx_w_mean/sum_wt; Z_W_mean = Z_W_mean/sum_wtz; *print z_w_mean; ++ Calculate standard errors of the mean alpha and mean Z ++; *SE_Rxx = sqrt(sum_wt##1); SE_Z = sqrt(sum_wtz##1 ); ++ Calculate unweighted mean alpha and mean Z ++; Rxx_U_mean = (J(1,k,1)*ri_by_k)/k; Z_U_mean = (J(1,k,1)*Z_alpha)/k; finish ; *+End of Vio Ind subroutine Begining subroutine for calc mean for each study *+; *@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@; *++
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 150 3, May 2005 Subroutine to calculate a mean rxx where the mean of each study is the unit of analysis ++ Subroutine to calculate vector of variabilities for coefficient alpha, Both original alpha metric and Fishers z are used Inputs to the subroutine are ri_by_k a matrix of sample alpha estimates where ri is the number of rows(i.e. #alphas per study) k is the nunmber of columns (i.e. # of studies) items number of items on the test (scalar value) n_vec vector of sample sizes corresponding to each reliability Outputs are (29, April 2005 some of these variables are not needed for J9 dis) Z_w_mean weighted mean Fisher Z SE_Z = Standard error of mean Fisher Z ++; start calc_kalpha_mean(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); note J9 deleted values from sub routine that are not used in new study; calculate variance for each reliability estimate, Fisher Z and variance of the Z; Have to chance this so that I indexing using rows and columns of a matrix that is nr by n_studies q by k); *print 'In CALC_KALPHA_MEAN'; k=ncol(ri_by_k); q = nrow(ri_by_k); Z_alpha=J(q,k, 0 ); var_Z =J(q,k, 0 ); mean_vec= J( 1 ,k, 0 ); n_vec=J( 1 ,k, 0 ); do i = 1 to k; do v = 1 to q; ++ Be sure the Rxx values are between .01 and .99 ++;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 151 if ri_by_k[v,i] > .99 then ri_by_k [v,i] = .99 ; if ri_by_k[v,i] < .99 then ri_by_k[v,i] = .99 ; ++ Fisher Z transformation, from Bonett, 2002 ++; Z_alpha[v,i] = log( 1 abs(ri_by_k[v,i])); if ri_by_k[v,i] < 0 then Z_alpha[v,i] = Z_alpha[v,i]* 1 ; mean_vec[ 1 ,i] = mean_vec[ 1 ,i]+ Z_alpha[v,i]; N_vec[ 1 ,i] = n_vec[ 1 ,i] + N_vec_mtx[v,i]; end ; q end; end ; k end; mean_vec= mean_vec/q; N_vec= n_vec/q; do i= 1 to k; var_Z[ 1 ,i] = ( 2 #items)/((items 1 ) # (N_vec[ 1 ,i] 2 )); end ; *k end; ++ Calculate weighted mean alpha and mean Z ++; Z_W_mean= 0 ; Sum_wtz = 0 ; do i = 1 to k; Z_W_mean = Z_W_mean + mean_vec[ 1 ,i]/var_Z[ 1 ,i]; Sum_wtz = sum_wtz + var_Z[ 1 ,i]##1 ; end ; k end; Z_W_mean = Z_W_mean/sum_wtz; ++ Calculate standard errors of the mean alpha and mean Z ++; SE_Z = sqrt(sum_wtz##1 ); *print z_w_mean se_z; finish ;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 152 *++ End of mean subroutine Beginning of Median subroutine where the median of each study is the unit of analysiss *++; *@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@; 4, June 2005 Subroutine to calculate a median rxx ++ Subroutine to calculate vector of variabilities for coefficient alpha Both original alpha metric and Fishers z are used *Inputs to the subroutine are ri_by_k a matrix of sample alpha estimates where ri or q is the number of rows(i.e. #alphas per study) k is the nunmber of columns (i.e. # of studies) items number of items on the test (scalar value) N_vec_mtx matrix of sample sizes corresponding to each reliability Outputs are (29, April 2005 some of these variables are not needed for J9 dis) Z_w_mean weighted mean Fisher Z SE_Z = Standard error of mean Fisher Z ++; start calc_kalpha_Med(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); note J9 deleted values from sub routine that are not used in new study; calculate variance for each reliability estimate, Fisher Z and variance of the Z; Have to chance this so that I indexing using rows and columns of a matrix that is nr by n_studies; *print 'In calc_kalpha_Md'; k=ncol(ri_by_k); q = nrow(ri_by_k); var_alpha=J(q,k, 0 ); Z_alpha=J(q,k, 0 ); var_Z =J(q,k, 0 ); Md_vec= J( 1 ,k, 0 ); N_vec = J( 1 ,k, 0 ) ; do i = 1 to k;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 153 do v = 1 to q; ++ Be sure the Rxx values are between .01 and .99 ++; if ri_by_k[v,i] > .99 then ri_by_k [v,i] = .99 ; if ri_by_k[v,i] < .99 then ri_by_k[v,i] = .99 ; ++ Fisher Z transformation, from Bonett, 2002 ++; Z_alpha[v,i] = log( 1 abs(ri_by_k[v,i])); if ri_by_k[v,i] < 0 then Z_alpha[v,i] = Z_alpha[v,i]* 1 ; end ; q end; end ; k end; ++ Compute upper and lower endpoints of the confidence interval suggested by Feldt et al.. (1987) ++; if q = 1  q = 3 then do; w = (q+ 1 )/ 2 ; *print w; do i = 1 to k; r= rank(Z_alpha[,i]); *print r w; do v = 1 to q; if r[v]=w then Md_vec[ 1 ,i] = Z_alpha[v,i]; end ; end ; end ; if q = 2  q = 10  q = 50 then do; *FIXED THIS; m1 = q/ 2 ; m2= (q+ 2 )/ 2 ; do i = 1 to k; r= rank(Z_alpha[,i]); *print r m1 m2; BEGIN NEW PART OF CODE; do v = 1 to q;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 154 if r[v]=m1 then Md_part1 = Z_alpha[v,i]; if r[v]=m2 then Md_part2 = Z_alpha[v,i]; end ; *print Md_part1 Md_part2; Md_vec[ 1 ,i] = (Md_part1 + Md_part2)/ 2 ; END NEW PART OF CODE; end ; end ; *ADDED THIS END; *print 'vector of medians' Md_vec; do i= 1 to k; *+code for mean of sample size+; do v = 1 to q; N_vec[ 1 ,i] = n_vec[ 1 ,i] + N_vec_mtx[v,i]; end ; end ; N_vec= n_vec/q; do i= 1 to k; var_Z[ 1 ,i] = ( 2 #items)/((items 1 ) # (N_vec[ 1 ,i] 2 )); end ; k end; ++ Calculate weighted mean alpha and mean Z ++; Z_W_mean= 0 ; Sum_wtz = 0 ; do i = 1 to k; Z_W_mean = Z_W_mean + Md_vec[ 1 ,i]/var_Z[ 1 ,i]; Sum_wtz = sum_wtz + var_Z[ 1 ,i]##1 ; end ; k end; Z_W_mean = Z_W_mean/sum_wtz; SE_Z = sqrt(sum_wtz##1 ); *print z_w_mean se_z; finish ; *++ End Median Rutine begin Random routine ++; *@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 155 4, June 2005 Subroutine to calculate a random rxx from each study where the unit of analysis is a randomly selected rxx from each of the k studies. ++ Subroutine to calculate vector of variabilities for coefficient alpha Both original alpha metric and Fishers z are used *Inputs to the subroutine are ri_by_k a matrix of sample alpha estimates where ri or q is the number of rows(i.e. #alphas per study) k is the nunmber of columns (i.e. # of studies) items number of items on the test (scalar value) N_vec_mtx matrix of sample sizes corresponding to each reliability Outputs are (29, April 2005 some of these variables are not needed for J9 dis) Z_w_mean weighted mean Fisher Z SE_Z = Standard error of mean Fisher Z ++; start calc_kalpha_rand(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); calculate variance for each reliability estimate, Fisher Z and variance of the Z; Have to chance this so that I indexing using rows and columns of a matrix that is nr by n_studies; k=ncol(ri_by_k); q = nrow(ri_by_k); var_alpha=J(q,k, 0 ); Z_alpha=J(q,k, 0 ); var_Z =J(q,k, 0 ); N_vec = J( 1 ,k, 0 ) ; rand_vec= J( 1 ,k, 0 ); do i = 1 to k; do v = 1 to q; ++ Be sure the Rxx values are between .01 and .99 ++; if ri_by_k[v,i] > .99 then ri_by_k [v,i] = .99 ; if ri_by_k[v,i] < .99 then ri_by_k[v,i] = .99 ;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 156 end ; q end; end ; k end; ++ Compute upper and lower endpoints of the confidence interval suggested by Feldt et al.. (1987) ++; s=J(q, 1 0 ); do i= 1 to k; seed1 = round( 1000000 *ranuni( 0 )); do v = 1 to q; s[v, 1 ]= rannor(seed1); end ; r=rank(s); *print s r; do v = 1 to q; if r[v] = q then do; rand_vec[ 1 ,i] = ri_by_k[v,i]; N_vec[ 1 ,i]=n_vec_mtx[v,i]; end ; end ; q end; end ; k end; do i= 1 to k; *do v = 1 to q; upper = 1 (1 mean_vec[1,i])#FINV(.05,(n_vec[v,1] 1),((n_vec[v,1] 1) # (items 1))); lower = 1 (1 mean_vec[1,i])#FINV(.95,(n_vec[v,1] 1),((n_vec[v,1] 1) # (items 1))); ++ Use the width of the confidence interval to compute an equivalent variance for alpha. If alpha was normally distributed, this would be the SE. ++; *var_alpha[v,i]= (abs(upper lower)/(2#1.96))##2;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 157 ++ Fisher Z transformation, from Bonett, 2002 ++; Z_alpha[ 1 ,i] = log( 1 abs(rand_vec[ 1 ,i])); if rand_vec[ 1 ,i] < 0 then Z_alpha[ 1 ,i] = Z_alpha[ 1 ,i]* 1 ; N_vec[1,i] = n_vec[1,i] + N_vec_mtx[v,i]; *new code added for N_vec; var_Z[ 1 ,i] = ( 2 #items)/((items 1 ) # (N_vec[ 1 ,i] 2 )); *print Z_alpha; *end; q end; end ; k end; ++ Calculate weighted mean alpha and mean Z ++; *Rxx_w_mean = 0; Z_W_mean= 0 ; Sum_wt = 0; Sum_wtz = 0 ; do i = 1 to k; Rxx_w_mean = Rxx_w_mean + ri_by_k[i,1]/var_alpha[i,1]; Sum_wt = sum_wt + var_alpha[i,1]##1; Z_W_mean = Z_W_mean + Z_alpha[ 1 ,i]/var_Z[ 1 ,i]; Sum_wtz = sum_wtz + var_Z[ 1 ,i]##1 ; end ; k end; *Rxx_w_mean = Rxx_w_mean/sum_wt; Z_W_mean = Z_W_mean/sum_wtz; ++ Calculate standard errors of the mean alpha and mean Z ++; *SE_Rxx = sqrt(sum_wt##1); SE_Z = sqrt(sum_wtz##1 ); finish ; START IRT3PL (THETA, A, B, C, SEED1, SUM_P, SCORE3PL); ++
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 158 Subroutine to compute probabilities of correct responses under 3PL model. INPUTS: Theta = scalar ability A, B, C = column vectors of item parameters SEED1 = seed for random number generator OUTPUTS: SUM_P = true number correct score (sum of true pvalues) SCORE3PL = row vector of scored items (0,1) ++; n_items = NROW(A); SUM_P = 0 ; SCORE3PL = J( 1 ,n_items, 0 ); do i = 1 to n_items; AVAL = 1.702 A[i, 1 ]; DAB = AVAL (THETA B[i, 1 ]); IF DAB > 120 THEN P = C[i, 1 ]; IF DAB < 100 THEN P = .99999 ; IF DAB >= 100 & DAB<= 120 then do; DIV = 1 + EXP(DAB); P = C[i, 1 ] + ( 1.0 C[i, 1 ])/DIV; END ; end DAB; RANVAR = RANUNI(SEED1); IF RANVAR <= P THEN SCORE3PL[ 1 ,i] = 1 ; IF RANVAR > P THEN SCORE3PL[ 1 ,i] = 0 ; SUM_P = SUM_P + P; end ; *end n_items; FINISH ; ++ Main program Generates samples, calls subroutines, computes means and confidence band coverage. ++; ++ Reading in the item pool information ++;
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Appendix A: (continued) SAS C ode for Monte Carlo simulation 159 use iosif; read all var {a_3pl} into ta_3PL; read all var {b_3pl} into tb_3PL; read all var {c_3pl} into tc_3PL; read all var {poolid} into poolid; *print ta_3PL tb_3pl tc_3PL; *do pop_sds = 1 to 4; if pop_sds = 1 then sds = 1; if pop_sds = 2 then sds = 2; if pop_sds = 3 then sds = 4; if pop_sds = 4 then sds = 8; do rel_items = 1 to 1 ; if rel_items =1 then true_alpha = .33; *if rel_items = 1 then true_alpha = .54; *if rel_items = 3 then true_alpha = .69; if rel_items = 1 then true_alpha = .90 ; if true_alpha = .33 then a_3pl = ta_3pl[ 1 : 3 ]; if true_alpha = .33 then b_3pl = tb_3pl[ 1 : 3 ]; if true_alpha = .33 then c_3pl = tc_3pl[ 1 : 3 ]; if true_alpha= .54 then a_3pl = ta_3pl[ 1 : 6 ]; if true_alpha= .54 then b_3pl = tb_3pl[ 1 : 6 ]; if true_alpha= .54 then c_3pl = tc_3pl[ 1 : 6 ];
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Appendix A: SAS Code for Monte Carlo Simulation if true_alpha = .69 then a_3pl = ta_3pl[ 1 : 11 ]; if true_alpha = .69 then b_3pl = tb_3pl[ 1 : 11 ]; if true_alpha = .69 then c_3pl = tc_3pl[ 1 : 11 ]; if true_alpha = .90 then a_3pl = ta_3pl[ 1 : 50 ]; if true_alpha = .90 then b_3pl = tb_3pl[ 1 : 50 ]; if true_alpha = .90 then c_3pl = tc_3pl[ 1 : 50 ]; items = nrow(A_3PL); do njs_cond = 1 to 1 ; average sample size in study; if njs_cond = 1 then njs = 10 ; actual value 10 changed to check rxx; *if njs_cond = 1 then njs = 50; *if njs_cond = 3 then njs = 100; *if njs_cond = 4 then njs = 500; *if njs_cond = 1 then njs = 1500; do k_cond = 1 to 1 ; N of studies in each metaanalysis; if k_cond = 1 then n_studies = 15 ; actual value 15 changed to check rxx; *if k_cond = 2 then n_studies = 50; *if k_cond = 1 then n_studies = 100; *if k_cond = 4 then n_studies = 150; 30, April 2005 Jeanine added index for nr; do Num_alpha = 1 to 5 ; *i reliabilities in each of the k journals; if Num_alpha = 1 then nr = 1 ; if Num_alpha = 2 then nr = 2 ; if Num_alpha = 3 then nr = 3 ; if Num_alpha = 4 then nr = 10 ; if Num_alpha = 5 then nr = 50 ; ++ Initialize counters 5, June 2005 Note: only weighted Z alpha is needed Columns is Zalpha Rows are: ignoring dep alpha,mean alpha per study,median alpha per study, random alpha per studynot sure about HLM level 2 alpha
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Appendix A (continued) SAS Code for Monte Carlo Simulation 161 ++; Mean Values; Means = J( 4 1 0 ); Confidence Band Coverage; InBand = J( 4 1 0 ); Confidence Band Width; WideBand = J( 4 1 0 ); sumrxx= J( 4 1 0 ); rmse=J( 4 1 0 ); bias = J( 4 1 0 ); nsamples= 0 ; seed1 = round( 1000000 *ranuni( 0 )); do rep= 1 to replicat; This starts the big do loop; rep_vec = J(n_studies#NR, 1 ,rep); if rep = 1 & njs_cond = 1 & k_cond = 1 & num_alpha= 1 & rel_items = 1 then do; add reli loop; create ICCout3 from rep_vec[colname = 'meta' ]; append from rep_vec; end ; if rep > 1  njs_cond > 1  k_cond > 1  num_alpha> 1 rel_items > 1 then do; add reli loop; setout ICCout3; append from rep_vec; end ; *do study = 1 to n_studies; Inner loop for primary studies;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 162 randomly generate a sample size for each study; *do simulee = 1 to n1; Number of examinees to generate; seed1=round( 100000000 *ranuni( 0 )); idn2 = simulee; **++ 29, April 2005 added to Generate a variance between and variance within to simulate intra class corr between reliabilities. ++; do studies = 1 to n_studies; mean_theta = rannor(seed1); mean_theta = mean_theta#sqrt( .0005 ); *since this value varies I'll just type it in; do numrel = 1 to nr; n1=rannor( 0 )#( .20 #njs) + njs; n1=round(n1); if n1< 4 then n1= 4 ; *n1=njs; do simulee = 1 to n1; theta= rannor(seed1); theta = theta#sqrt( 1 )+ mean_theta; ++ 29, Aril 2005 only one administration needed for alpha Administer the test twice to each examinee: to allow both Cronbach alpha and testretest estimates ++; run IRT3PL (THETA, A_3PL, B_3PL, C_3PL, SEED1, True_P, SCORE3PL); *run IRT3PL (THETA, A_3PL, B_3PL, C_3PL, SEED1, True_P, SCORE2); ++ Build matrix of scores for examinees ++; if simulee = 1 then out3pl = score3pl;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 163 if simulee > 1 then out3pl = out3pl//score3pl; end ; ++ Computation of Cronbach Alpha ++; mu1 = J(items, 1 0 ); Changed this!; var = J( 1 ,items, 0 ); Changed this!; do k = 1 to items; do i= 1 to n1; mu1[k, 1 ] = mu1[k, 1 ] + out3pl[i,k]; end ; end n1; var[ 1 ,k]=(mu1[k, 1 ]/n1)*( 1 mu1[k, 1 ]/n1); var of items; print mu1 var; sumvar= 0 ; do k = 1 to items; sum of the item variances sumvar = sumvar + var[ 1 ,k]; end ; *end items; print sumvar; rowsum = J(n1, 1 0 ); rowsum2= J(n1,1,0); do p = 1 to n1; do k = 1 to items; rowsum[p, 1 ]=rowsum[p, 1 ] + out3pl[p,k]; *calculate the row sum for each examinee; rowsum2[p,1]=rowsum2[p,1] + out2[p,k]; end ; end n1; end ; *end items; *print rowsum; sumscore = 0 ; sumscore2 = 0 ; do p = 1 to n1; sumscore = sumscore + rowsum[p, 1 ];
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Appendix A (continued) SAS Code for Monte Carlo Simulation 164 sumscore2= sumscore2 + rowsum[p, 1 ]## 2 ; end ; end n1; vartotal= (sumscore2(sumscore## 2 /n1))/(n1); *var of all examinees total score; ++ Be sure we have some score variance before going any further ++; if vartotal > 0 then do; *print sumscore sumscore2 vartotal; rxx = (items/(items 1 ))*((vartotalsumvar)/vartotal); This is Cronbach alpha!; *print n1 out3pl rxx; if rxx < 0.00001 then rxx = .00001 ; need to confirm rxx; if rxx> .9999 then rxx = .9999 ; *Jeff change; *print 'Check Values of rxx'; *print studies numrel rxx; ***********************************************************************************; *Jeanine Add code to create matrix; if (studies = 1 & numrel = 1 ) then do; *xbartheta = mean_theta; t_alpha_vec= true_alpha; No_alpha_vec = nr; njs_vec=njs; n_studies_vec=n_studies; study = studies; est_rel = numrel; z_rxx = log( 1 abs(rxx)); rxx_vec = rxx; sd_vec = (( 2 #items)/((items 1 ) #(n1 2 )))##1 ; n_vec= n1; *sd_vec = sd; print true_alpha rxx_vec; end ; end studies = 1; if (studies > 1  numrel > 1 ) then do;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 165 *xbartheta = xbartheta//mean_theta; t_alpha_vec= t_alpha_vec//true_alpha; No_alpha_vec = No_alpha_vec //nr; njs_vec=njs_vec//njs; n_studies_vec=n_studies_vec//n_studies; study = study//studies; est_rel = est_rel//numrel; z_rxx = z_rxx//log( 1 abs(rxx)); rxx_vec = rxx_vec//rxx; n_vec= n_vec//n1; sd_vec = sd_vec//(( 2 #items)/((items 1 ) #(n1 2 )))##1 ; end ; *end studies >1; *print mean_theta studies numrel rxx; *print studies; *print rxx; end ; *end vartotal; if vartotal = 0 then Numrel= numrel1 ; end ; end big n_studies loop; end ; end big nr loop; ri_by_k= J(nr,n_studies, 0 ); N_vec_mtx=J(nr,n_studies, 0 ); *print 'first'; *print ri_by_k; do v = 1 to n_studies; do i = 1 to nr; w =nr#(v1 )+ i; ri_by_k[i,v]=rxx_vec[w, 1 ]; N_vec_mtx[i,v]=n_vec[w, 1 ]; end ; end N_studies above; end ; end nr above; *print N_vec_mtx; *print rxx_vec;* z_rxx; if rep = 1 & njs_cond = 1 & k_cond = 1 & num_alpha= 1 & rel_items = 1 then do; add reli loop;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 166 create ICCout1 from study[colname = 'study' ]; append from study; create ICCout2 from sd_vec[colname = 'weightv' ]; append from sd_vec; create ICCout4 from z_rxx[colname = 'rxx_vec' ]; append from z_rxx; create ICCout5 from t_alpha_vec[colname = 'true_alpha' ]; append from t_alpha_vec; create ICCout6 from No_alpha_vec[colname = 'num_rel' ]; append from No_alpha_vec; create ICCout7 from njs_vec[colname = 'njs' ]; append from njs_vec; create ICCout8 from N_studies_vec[colname = 'N_studies' ]; append from N_studies_vec; end ; if rep > 1  njs_cond > 1 k_cond > 1  num_alpha> 1  rel_items > 1 then do; add reli loop; setout ICCout1; append from study; setout ICCout2; append from sd_vec; setout ICCout4; append from z_rxx; setout ICCout5; append from t_alpha_vec; setout ICCout6; append from No_alpha_vec; setout ICCout7; append from njs_vec; setout ICCout8; append from N_studies_vec; end ;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 167 *+ calculate sample standard deviation and SEM; *ss1 = (J(1,n1,1)*(rowsum##2)) ((J(1,n1,1)*rowsum)##2/n1); sd = sqrt(ss1/(n11)); *________________________________________________________________; collect KR21, rxx, SEM, n and sd in vectors; if study = 1 then KR_vec= KR; if study >1 then KR_vec = KR_vec//KR; if study = 1 then retest_vec = rxx_retest; if study > 1 then retest_vec = retest_vec//rxx_retest; if study = 1 then rxx_vec = rxx; if study > 1 then rxx_vec = rxx_vec//rxx; if study = 1 then n_vec = n1; if study > 1 then n_vec = n_vec//n1; if study = 1 then sd_vec = sd; if study > 1 then sd_vec = sd_vec//sd; *end; end the 'if vartotal > 0 then do' loop; ; *end; end the studies loop; print rxx_vec retest_vec KR_vec n_vec sd_vec; *+Ignore dep partThe calc_alpha calculates a vector of alphas for all the studies +; compute mean reliability for the sample of studies; run calc_alphaVI(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); *print 'Ignore' z_w_mean; ++ Compute mean values, bandwidths and band coverage here for all studies i.e ignoring dependence
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Appendix A (continued) SAS Code for Monte Carlo Simulation 168 Only using weighted Fishers z for this study ++; means[1,1] = means[1,1] + w_alpha; means[ 1 1 ] = means[ 1 1 ] + ( 1 exp(z_w_mean)); *wideband[1,1] = wideband[1,1] + ((W_alpha + 1.96#SE_alpha) (W_alpha 1.96#SE_alpha)); wideband[ 1 1 ] = wideband[ 1 1 ] + ( 1 exp(z_w_mean 1.96 #SE_z)) ( 1 exp(z_w_mean + 1.96 #SE_z)); if (true_alpha > ( 1 exp(z_w_mean + 1.96 #SE_Z)) & true_alpha < ( 1 exp(z_w_mean 1.96 #SE_Z))) then inband[ 1 1 ] = inband[ 1 1 ] + 1 ; sumrxx[ 1 1 ]=sumrxx[ 1 1 ]+(( 1 exp(z_w_mean))true_alpha)## 2 ; bias[ 1 1 ] =bias[ 1 1 ]+ (( 1 exp(z_w_mean)) true_alpha); *+ 29, April 2005 changed for J9 dis *; free z_w_mean SE_z; end; End analysis for ingnoring dep; *+calculating mean alpha per study part The calc_alpha calculates a vector of alphas for all the studies +; compute mean reliability for the sample of studies; run calc_kalpha_mean(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); print 'mean' z_w_mean; ++ Compute mean values, bandwidths and band coverage here for all studies i.e ignoring dependence Only using weighted Fishers z for this study ++; means[1,1] = means[1,1] + w_alpha; means[ 2 1 ] = means[ 2 1 ] + ( 1 exp(z_w_mean)); *wideband[1,1] = wideband[1,1] + ((W_alpha + 1.96#SE_alpha) (W_alpha 1.96#SE_alpha)); wideband[ 2 1 ] = wideband[ 2 1 ] + ( 1 exp(z_w_mean 1.96 #SE_z)) ( 1 exp(z_w_mean + 1.96 #SE_z));
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Appendix A (continued) SAS Code for Monte Carlo Simulation 169 if (true_alpha > ( 1 exp(z_w_mean + 1.96 #SE_Z)) & true_alpha < ( 1 exp(z_w_mean 1.96 #SE_Z))) then inband[ 2 1 ] = inband[ 2 1 ] + 1 ; sumrxx[ 2 1 ]= sumrxx[ 2 1 ]+ (( 1 exp(z_w_mean))true_alpha)## 2 ; bias[ 2 1 ] =bias[ 2 1 ]+ (( 1 exp(z_w_mean)) true_alpha); *+ 29, April 2005 changed for J9 dis *; free z_w_mean SE_z; *end; End analysis for calculating one mean per study; *+calculating Md alpha per study part +; compute mean reliability for the sample of studies; run calc_kalpha_Med(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); *print 'median' z_w_mean; ++ Compute mean values, bandwidths and band coverage here for all studies i.e ignoring dependence Only using weighted Fishers z for this study ++; means[1,1] = means[1,1] + w_alpha; means[ 3 1 ] = means[ 3 1 ] + ( 1 exp(z_w_mean)); *wideband[1,1] = wideband[1,1] + ((W_alpha + 1.96#SE_alpha) (W_alpha 1.96#SE_alpha)); wideband[ 3 1 ] = wideband[ 3 1 ] + ( 1 exp(z_w_mean 1.96 #SE_z)) ( 1 exp(z_w_mean + 1.96 #SE_z)); if (true_alpha > ( 1 exp(z_w_mean + 1.96 #SE_Z)) & true_alpha < ( 1 exp(z_w_mean 1.96 #SE_Z))) then inband[ 3 1 ] = inband[ 3 1 ] + 1 ; sumrxx[ 3 1 ]=sumrxx[ 3 1 ]+ (( 1 exp(z_w_mean))true_alpha)## 2 ; bias[ 3 1 ] =bias[ 3 1 ]+ (( 1 exp(z_w_mean)) true_alpha); *+ 29, April 2005 changed for J9 dis *; free z_w_mean SE_z;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 170 *end; End analysis for calvulating one median per study; *+calculating rand alpha per study part +; compute mean reliability for the sample of studies; run calc_kalpha_rand(ri_by_k,items,N_vec_mtx,Z_W_mean,SE_Z); ++ Compute mean values, bandwidths and band coverage here for all studies i.e ignoring dependence Only using weighted Fishers z for this study ++; means[1,1] = means[1,1] + w_alpha; means[ 4 1 ] = means[ 4 1 ] + ( 1 exp(z_w_mean)); *wideband[1,1] = wideband[1,1] + ((W_alpha + 1.96#SE_alpha) (W_alpha 1.96#SE_alpha)); wideband[ 4 1 ] = wideband[ 4 1 ] + ( 1 exp(z_w_mean 1.96 #SE_z)) ( 1 exp(z_w_mean + 1.96 #SE_z)); if (true_alpha > ( 1 exp(z_w_mean + 1.96 #SE_Z)) & true_alpha < ( 1 exp(z_w_mean 1.96 #SE_Z))) then inband[ 4 1 ] = inband[ 4 1 ] + 1 ; sumrxx[ 4 1 ]=sumrxx[ 4 1 ]+ (( 1 exp(z_w_mean))true_alpha)## 2 ; bias[ 4 1 ] = bias[ 4 1 ]+(( 1 exp(z_w_mean)) true_alpha); *+ 29, April 2005 changed for J9 dis *; free z_w_mean SE_z; *end; End analysis for calculating one median per study; nsamples=nsamples+ 1 ; print means; end ; *end the big loop rep end; *print means; ++ Convert sums into means
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Appendix A (continued) SAS Code for Monte Carlo Simulation 171 ++; do row = 1 to 4 ; if means[row, 1 ] ^= then means[row, 1 ] = means[row, 1 ]/nsamples; if InBand[row, 1 ] ^= then InBand[row, 1 ] = InBand[row, 1 ]/nsamples; if WideBand[row, 1 ] ^= then WideBand[row, 1 ] = WideBand[row, 1 ]/nsamples; if Bias[row, 1 ]^= then Bias[row, 1 ] = Bias[row, 1 ]/nsamples; if sumrxx[row, 1 ]^= then sumrxx[row, 1 ] = sumrxx[row, 1 ]/nsamples; rmse[row, 1 ]= sqrt(sumrxx[row, 1 ]); end ; end row; *print 'Reliability Generalization'; label1= 'Violation' ; label2 = 'Mean' ; label3 = 'Median' ; label4 = 'Random' ; labels = label1//label2//label3//label4; print labels icc true_alpha nr njs n_studies Bias RMSE InBand WideBand means nsamples; end ; end the k_cond loop; end ; end the njs_cond loop; end ; end the rel_items loop; end ; end the num_alpha loop; data allout; merge iccout1 iccout2 iccout3 iccout4 iccout5 iccout6 iccout7 iccout8; *proc print data=allout; *proc print data =allout; *proc means data =allout; *var rxx_vec; proc datasets ; delete iccout1 iccout2 iccout3 iccout4 iccout5 iccout6 iccout7 iccout8; proc sort data = allout; by meta true_alpha num_rel njs N_studies; Proc mixed noclprint covtest noitprint noinfo ; by meta true_alpha num_rel njs N_studies;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 172 class study; weight weightv; model rxx_vec= / solution CL ; random intercept/sub =study; ods output solutionF= rgsim (keep= meta true_alpha num_rel njs N_studies estimate lower upper); ods output FitStatistics= rgCI; ods listing close ; *after that; run ; *title 'proc mixed random'; *proc print data = rgsim; ODS LISTING ; *Jeff change; proc sort data =rgsim; by true_alpha num_rel njs N_studies; data rgsim2; set rgsim; *proc transpose data = rgsim out= rgsim2; *PROC CONTENTS DATA =RGSIM2; dm 'log; clear;' continue; proc means noprint data = rgsim2; by true_alpha num_rel njs N_studies; var estimate; OUTPUT OUT = MIX mean = Zrxx_est; *proc contents data= mix; data mix_trans; MERGE mix RGSIM2; BY true_alpha num_rel njs N_studies; orig_r_mean= 1 exp(zrxx_est); *++++++transforms zmean back to alpha this is the estimate of true_alpha+++++; *wideband[2,1] = wideband[2,1] + (1 exp(z_w_mean 1.96#SE_z)) (1 exp(z_w_mean + 1.96#SE_z)); *untrans = upperlower;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 173 upper_rxx = 1 exp(upper); *Jeff change; lower_rxx = 1 exp(lower); wideband=abs(upper_rxxlower_rxx); if true_alpha < lower_rxx & true_alpha> upper_rxx then inband = 1 ; else inband = 0 ; bias = orig_r_meantrue_alpha; *DATA RMSE_CAL; *SET MIX_TRANS; sumrxx =(( 1 exp(estimate))true_alpha)** 2 ; *Jeff change again; *proc print; *var upper lower wideband wideband3 upper_rxx lower_rxx; *PROC CONTENTS DATA = MIX_TRANS; *proc print data=mix_trans; *run; proc means noprint data = mix_trans; by true_alpha num_rel njs N_studies; var wideband; output out = rgmeans1 mean = av_wideband; proc means noprint data = mix_trans; by true_alpha num_rel njs N_studies; var inband; output out = rgmeans2 mean = av_inband; proc UNIVARIATE noprint data = mix_trans; by true_alpha num_rel njs N_studies; var sumrxx; output out = rgmeans3 mean = rxx_sum; data rgmeans;
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Appendix A (continued) SAS Code for Monte Carlo Simulation 174 icc= .01 ; merge mix_trans rgmeans1 rgmeans2 rgmeans3; by true_alpha num_rel njs N_studies; *rxx_sum= (sum(sumrxx)); Rmse= sqrt(rxx_sum); *Jeff change; if first.njs or first.n_studies; Jeff change; proc print data = rgmeans; by num_rel njs; var icc true_alpha num_rel njs N_studies bias rmse av_inband av_wideband orig_r_mean; run ;
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About the Author Jeanine Romano received a bachelor degree (BS) in Mathematics education in 1994 from the University of South Florida and a Masters degree (MA) in Mathematics education in 1996. She had worked as an instructor and the Coordinator of Institutional Research and Assessment at The University of Tampa. At the University of Tampa she taught lower level mathematics courses and statistics. In addition, she has taught undergraduate measurement course both face to face and on line. Her research has been nominated for the Florida Educational Resear ch Association disti nguished paper five times and had won the award both in 2004 and in 2006. Her research was recently recognized as Best Paper for the 2007 Florid a Association of Institutional Research Conference.
