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Educational policy analysis archives.
n Vol. 9, no. 40 (October 08, 2001).
Tempe, Ariz. :
b Arizona State University ;
Tampa, Fla. :
University of South Florida.
c October 08, 2001
High school size, achievement equity, and cost : robust interaction effects and tentative results / Robert Bickel, Craig Howley, Tony Williams, [and] Catherine Glascock.
Arizona State University.
University of South Florida.
t Education Policy Analysis Archives (EPAA)
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1 of 32 Education Policy Analysis Archives Volume 9 Number 40October 8, 2001ISSN 1068-2341 A peer-reviewed scholarly journal Editor: Gene V Glass, College of Education Arizona State University Copyright 2001, the EDUCATION POLICY ANALYSIS ARCHIVES Permission is hereby granted to copy any article if EPAA is credited and copies are not sold. Articles appearing in EPAA are abstracted in the Current Index to Journals in Education by the ERIC Clearinghouse on Assessment and Evaluation and are permanently archived in Resources in Education .High School Size, Achievement Equity, and Cost: Robust Interaction Effects and Tentative Results Robert Bickel Marshall University Craig Howley ERIC Clearinghouse on Rural Education & Small Schoo ls and Ohio University Tony Williams Marshall University Catherine Glascock Ohio UniversityCitation: Bickel, R., Howley, C., Williams, T. and Glascock, C. (2001, October 8). High School Size, Achievement Equity, and Cost: Robust Interact ion Effects and Tentative Results. Education Policy Analysis Archives 9 (40). Retrieved [date] from http://epaa.asu.edu/epa a/v9n40.html.Abstract The past decade has occasioned a dramatic increase in research on relationships between school size and a variety of outcomes, including measured achievement, high school completion rates, and postsecondary
2 of 32enrollment rates. An interesting interaction effect which has been found in replications across seven very different states is that as school size increases, the "achievement test score costs" assoc iated with the proportion of economically disadvantaged students e nrolled in a school also increase. In short, as schools get larger, ave rage achievement among schools enrolling larger proportions of low socioec onomic-status students suffers. A traditional argument against sm aller schools, however, is that they are simply too expensive to operate (regardless of proven benefits). Large consolidated schools--often with narrowly specialized grade spans--are typically proposed and constructed as necessary to "save money" and to meet the "developm ental needs" of certain age groupings. This article has two objecti ves. First, to determine if the size-by-socioeconomic status interaction eff ect proves robust across alternative regression model specifications, as it did across differing states. Second, to make a tentative judgm ent as to whether the equity gains associated with smaller schools are in compatible with the need for fiscal efficiency. The analyses (based on our Texas data set) suggest that the answer to the first question is "y es" and the answer to the second question is "no." In particular, the K-12 "u nit school" configuration in Texas is shown to be both educatio nally effective and cost effective. Educational researchers and policy makers rarely m eet an issue they are willing to resolve once and for all. School size is a case in point. Interest in school size as an explanatory factor waxes and wanes, but never dies. The effect of variability in school size on educational achievement and a variety of re lated outcomes remains a subject of sometimes intense, sometimes dilatory, inquiry and debate. In the study reported here, we use a Texas data se t representing 1,001 high schools to build on previous research, completed fi rst in California and then replicated in six very different states. (The data set is availab le here for researchers who wish to replicate or extend our analyses.) This line of res earch has, with unusual consistency, found an interesting interaction effect between soc ioeconomic status (SES) and school size in the production of achievement: as school si ze increases, school performance (aggregate achievement at the school level) decreases for economically disadvantaged students In short, as schools get larger, those with poor children as students perform increasingly less well when achievement is the outc ome measure. School size imposes increasing "achievement costs" in schools serving i mpoverished communities.Research Questions Continuing this line of research, we address two s pecific questions. First, will a replication that deploys a more fully specified reg ression model find the same size-by-SES interaction effect among the high schoo ls in our Texas data set as was previously found (Bickel, 1999b)? Second, whatever the merits of small schools, are large high schools with conventionally narrow grade ranges necessary to minimize expenditures per pupil ("save money"), or can "savi ngs" occur without increased size?Replication Through Re-Specification
3 of 32 In previous analyses, the independent variables in cluded in regression equations were limited to a measure of school size, either to tal number of students or number of students per grade level; a measure of SES, most of ten percent of students eligible for free or reduced cost lunch; and the multiplicative interaction term. Some analyses included student-teacher ratio (Howley, 1999a, 1999 b) or ethnicity variables (Bickel, 1999b). The most notable exception, however, is a m ulti-level analysis of Georgia data, which incorporated both ethnic composition and stud ent/teacher ratio (Bickel & Howley, 2000). To improve on past research, the primary dif ference between the work reported in the present study and the previous replications is a more fully specified regression model. Therefore, we are now asking if the size-by-SES in teraction effect will prove unduly sensitive to better-informed regression mode l specification, diminishing the credibility of the consistent results reported from previous research. In other words, does the interaction effect merely mask the influence of "the usual suspects" through inadequate model specification? (Note 1)Fiscal Practicality In addition, we examine the claim that large schoo ls with a narrow range of grades are a necessary organizational consequence o f the modern need to minimize expenditures (fiscal efficiency). Many policy maker s and administrators who have persisted in off-handedly dismissing the small-is-b etter research have done so in the name of fiscal practicality. Large consolidated sch ools, specializing in just a few grade levels, are viewed as essential to achieve "economi es of scale" and to meet the supposedly critical developmental needs of students of differing ages. Those who hold contrary views are dismissed as romantics. (For a m ore balanced view, however, see Boex & Martinez-Vasquez, 1998). School size is negatively related to expenditure per pupil in zer o-order correlational analysis. However, our analyses of th e link between school size and expenditure per pupil go beyond the usual simplicit ies to include the under-researched concept of grade span configuration (Note 2) Specifically, 116 of the high schools in our Texas data set are single-unit schools: the only school in a typically small, typically rur al district, containing all elementary and secondary grades under a single roof. (Note 3) With expenditure per pupil as the outcome measure, multiple regression analysis shows that single-unit schools, on average, correspond to a reduction in expenditure of $1,017 per pupil a substantial efficiency, when compared with conventionally grade -specialized high schools. (See Table 6.) The "savings" can be statistically attributed to t wo distinctive characteristics of single-unit schools in Texas: each is the only scho ol in its district, and each has an unusually broad grade configuration, K-12, PreK-12, or early childhood-12 (see Table 7). We find, however, that the savings decline as such schools become larger In other words, in Texas, small K-12 unit schools are cost effective, all else equ al. They are also, as we shall see, educationally effective because, o verall, such schools do tend to be small.School Size: A Timely Issue Writing on the role of school size as a determinan t of school performance has a long history and is embedded in a voluminous litera ture (see, for example, Barker &
4 of 32Gump, 1964; Fowler, 1991; Guthrie, 1979; Khattari, Riley, & Kane, 1997; McDill, Natriello, & Pallas, 1986; Smith & DeYoung, 1988; W alberg & Walberg, 1994). As with so many commonly invoked explanatory factors i n the social and behavioral sciences, reports about the effects of school size have been contradictory (Caldas, 1993; Lamdin, 1995; Rivkin, Hanushek, & Kain, 1998; Rossm iller, 1987). Part of the problem is that findings about size have often been a footn ote in research focused on "effective schools," "school restructuring," or other species of broad-based reform efforts. As a consequence, school size sometimes has been relegat ed to the status of an obligatory but uninteresting control variable. Not infrequently, i t simply has been ignored (Barr & Dreeben, 1983; Gamoran & Dreeben, 1986; Farkas, 199 6; Hanushek, 1997, 1998; Wyatt, 1996). Uncertainty as to the import of school size has yi elded state-of-the-art school effectiveness research that fails to designate size a "resource," much less a resource worthy of investigation. A recent school effectiven ess review by eleven production function virtuosos, for example, devoted four of it s three hundred ninety-six pages to school size (Hodges & Greenwald, 1996, p. 81; Betts 1996, pp. 166-168). Consequences of variability in school size were, in sum, judged to be uncertain. This assessment is simplistic and wrong according to recent studies. In fact, the Education Commission of the State (ECS) has for som e time recommend smaller school size as one of the "best investments" policy makers could sponsor (Fulton, 1996). The research base on the influence of size per se (rath er than as a feature of reformed or restructured schools) is developing quite rapidly, and may be said to have spawned a "movement" (Fine & Somerville, 2000).One Size Fits All One important limitation of most literature coveri ng school size has been failure to examine the interaction of school size with othe r variables (Howley, 1989; Lee & Smith, 1995; Mik & Flynn, 1996; Riordan, 1997). Thi s deficiency tends to give rise to a one-size-fits-all point of view. Within any school, it may seem, size-related benefits accrue and size-related costs are borne equally by all students (Conant, 1959; Haller, 1992; Haller, Monk, & Tien, 1993; Hemmings, 1996). This turns out to be a dubious assumption (Bickel & Howley, 2000).Discounting Equity In an era of cult-of-efficiency institutional rest ructuring, moreover, questions as to the "best" size for any school are often express ed in the scientific management terms of organizational efficiency. In economists' termin ology, presumed economies of scale frequently have been given pride of place (Haller, Monk, Bear, Griffith, & Moss, 1990; Purdy, 1997; Tholkes & Sederberg, 1990). As with mu ch contemporary educational research, equity questions are usually dismissed as irrelevant to the school size discussion, at least when fiscal efficiency is at s take. For many, this has simply come to mean that bigger is better, inevitably and always ( Stevenson, 1996), when choices about school construction are made.Small is Better? Recently, nevertheless, attention has been drawn t o a growing body of empirical
5 of 32research that holds that school size is negatively associated with conventional measures of educational productivity. This includes measured achievement levels, dropout rates, grade retention rates, and college enrollment rates (see, for example, Bickel & McDonough, 1997; Fowler, 1995; Fulton, 1996; Mik & Flynn, 1996; Stevens & Peltier, 1995; Walberg & Walberg, 1994).Size-by-SES Interaction Effects In part, renewed interest in smaller schools is du e to research concerning the joint or interactive, rather than independent or main, ef fects of school size and SES. Specifically, interaction effects have been identif ied which suggest that the well known adverse consequences of socioeconomic disadvantage are tied to school size in substantively important ways. In brief, as school size increases, the mean measu red achievement of schools with less-advantaged students declines. The larger the n umber of less-advantaged students attending a school, the greater the decline (Bickel & Howley, 2000; Friedkin & Necochea, 1988; Howley, 1995, 1996; Howley & Bickel 1999; Huang & Howley, 1993). In addition to helping revive interest in school s ize as a variable of importance in educational research, this work has begun to sensit ize researchers and policy makers to equity concerns associated with school size. One-si ze-fits-all is no longer a unanimous judgment. Some researchers and policy makers are no w asking, "Best-size-for-whom?" (Devine, 1996; Henderson & Raywid, 1994).Reproducible Findings: A Research Agenda Research on size-by-SES interactions, moreover, ha s substantial geographic scope. The same school-level interactions have been found in California (Friedkin & Necochea, 1988), West Virginia (Howley, 1995, 1996) Alaska (Huang & Howley, 1993); Montana (Howley, 1999a), Ohio (Howley, 1999b ); Georgia (Bickel, 1999a; Bickel & Howley, 2000), and Texas (Bickel, 1999b). In contrast to so much research which has yielded initially interesting findings, t he likelihood that additional replications will yield sharply conflicting results has been sub stantially reduced by the findings from these studies.Texas High School Data for 1996-97 By way of continuing this line of investigation, w e use a data set consisting of 1,001 Texas high schools. This represents 83.6 perc ent of all high schools in the state for academic year 1996-97. The 196 excluded schools are those for which values were not available for one or more of the variables used in our analyses (Bickel, 1999b). Independent Variables As already explained, previous research on this is sue has been marked by simplified regression model specification. In large part, this parsimonious approach derives from the fact that proper specification for research on school size or any other correlate of achievement is substantively uncertain and theoretically thin. The usual suspects are SES and ethnicity variables, but a hos t of other variables has often been included in production function models. The debate continues (cf. Greenwald, Hedges,
6 of 32 & Laine, 1996; Hanushek, 1996; Hedges, 1996). Pending a resolution to this debate, the independe nt variables included in Table 1 seem appropriate. (Note 4) They reflect the ethnic, linguistic, and socioeconomic diversity of the state's high school students (PCTB LACK, PCTHISP, PCTLEP, PCTPOOR); they show substantial variability in Texa s high schools' organizational characteristics and resources, including size (SIZE S/TRATIO, EPP, PCTINST, UNIT, LEVELS, HIGHSKLS); and they manifest pertinent vari ability in curricular composition (PCTTECH, PCTSPECL, PCTGIFT). (Note 5) Inclusion of student/teacher ratio (S/TRATIO), a u seful proxy for class size among the additional independent variables, enables us to address questions as to whether small classes in large schools diminish the adverse consequences of increased size. As it turns out, they do not. This result is consistent with tests of the hypothesis in ancillary analyses provided in two of the previous studies (Howley, 1999a, 1999b).Table 1 Definitions of VariablesSIZE Number of students. (Expressed in thousandstudent units in Tables 3 through 5; expressed in natural logarit hms of single-student units in Tables 6 and 7.) PCTPOOR Percentage of students eligible for free or reduced -cost lunch. PCTBLACK Percentage of students who are Black. (Expressed innatural logarithms.) PCTHISPPercentage of students who are Hispanic.PCTLEP Percentage of students classified as limited Englis h proficient. (Expressed in natural logarithms.) S/TRATIOStudent/teacher ratio.EPP Expenditure per pupil. (Expressed in thousanddoll ar units in Tables 3 through 5.) PCTINSTPercentage of total budget allotted for inst ruction. PCTTECH Percentage of students enrolled in a fulltime car eer and technical education curriculum. PCTSPECL Percentage of students enrolled in a fulltime spe cial education program. PCTGIFTPercentage of students classified as gifted.UNITCoded 1 for single-unit schools, and 0 otherwis e.
7 of 32 HIGHSKLS Number of high schools in a district. A high school is any school which includes grade 12. (Expressed in natur al logarithms.) LEVELSNumber of grade levels.R10 Texas Assessment of Academic Skills tenth grade rea ding test. M10 Texas Assessment of Academic Skills tenth grade mat h test. W10 Texas Assessment of Academic Skills tenth grade wri ting rest. Dependent Variables: Measures of Achievement In Tables 3, 4, and 5, the dependent variables are taken from the mandatory Texas Assessment of Academic Skills (TAAS) end-of-grade b attery, used on a limited basis since the Fall of 1990, and fully implemented in 19 94. The tests are criterion-referenced measures of attainment in reading, math, and writin g, administered to tenth graders throughout the state, and used to evaluate the perf ormance of students and, by implication, the effectiveness of schools and schoo l districts in promoting measured achievement. Measures of internal consistency for t he TAAS are reported to range from .80 to .90 (Texas Education Agency, 2000). (For cri tical discussions of the use and interpretation of TAAS, see Clopton, Bishop, & Klei n ,1997; Haney, 2000; and Klein, Hamilton, McMaffery, & Stecher, 2000).Dependent Variables: Expenditure Per Pupil In Tables 6 and 7, expenditure per pupil is the de pendent variable, and measured achievement is used for purposes of statistical con trol rather than as an outcome measure. Since scores for R10, M10, and W10 are clo sely correlated, use of all three in the same equation produces multicollinearity, with Condition Indices well above thirty (Gujurati, 1995, p. 338). To eliminate this threat, we have created a summar y achievement measure, COMPOSITE, which is the sum of the Z scores of R10, M10, and W10. All bivariate correlations between COMPOSITE and its three consti tuents exceed .935. We have also found that the relationship between S IZE and EPP is curvilinear, but that the relationship can be linearized using n atural logarithms of SIZE. Use of this transformation is discussed further in the next sec tion. Descriptive Statistics Table 2 shows that the mean value for SIZE, total number of students enrolled, is 877.19. The size of the standard deviation, 849.88, indicates that SIZE manifests a good deal of variability, with a coefficient of variatio n of 103.2 percent. While SIZE has a positive skew, the skew is not so extreme that the variable warrants logarithmic or other transformation (Fox, 1997, pp. 64-68). In fact, using the
8 of 32 Studentized range test for normality, SIZE more clo sely approximates a normal distribution when non-transformed values are used ( see Kanji, 1993, p. 65). Therefore, actual SIZE values are used in the analyses with ac hievement tests as outcome measures, reported in Tables 3, 4, and 5. The relationship between SIZE and EPP is curviline ar: concave and sloping downward for the smallest values of school size; al most perfectly straight with a modest downward slope for SIZE values between 220 and 550; almost perfectly straight with a diminished downward slope between size values 550 a nd 1800; then sloping still less, and eventually becoming level for SIZE values of mo re than 3200 students. This is similar to the curvilinear relationship between hig h school size and cost discovered by Stiefel, Berne, Iatarola, & Fruchter (2000) in thei r New York City data. We have linearized the relationship between SIZE a nd EPP in our Texas data by taking natural logarithms of SIZE for the analyses reported in Tables 6 and 7 (where EPP is the dependent variable)..Table 2 Descriptive Statistics Means and (Standard Deviations) N=1001SIZE 877.19 (849.88) PCTPOOR 36.51 (30.93) PCTBLACK 11.07 (17.34) PCTHISP 27.73 (27.78) PCTLEP 4.95 (8.99) S/TRATIO 13.24 (3.15) EPP 4745.67 (1318.94) PCTINST 69.92 (7.34) PCTTECH 56.12 (20.59) PCTSPECL 13.54 (6.08) PCTGIFT 9.02 (7.07)
9 of 32 UNIT 0.12 (0.32) HIGHSKLS 2.96 (5.12) LEVELS 5.34 (3.11) R10 39.17 (2.30) M10 45.51 (4.08) W10 32.88 (1.80) Means and standard deviations for PCTBLACK, PCTLEP and HIGHSKLS are reported in Table 2 before the variables were logge d. Since, however, each has a sharp positive skew, with most of the observations confin ed to a very narrow range of data on the left side of the distribution, the variability of each is tightly constrained. Taking natural logarithms spreads each distribution, makin g it more informative (Fox, 1997, pp. 64-68). It is also worth noting that the standard deviatio ns for the R10, M10, and W10 achievement tests are small: 2.30, 4.08, 1.80. Coef ficients of variation are similarly small, 5.9 percent, 9.0 percent, and 5.5 percent. Routine tests for violations of assumptions of the classical normal linear regression model, and for the presence of influenti al observations ("outliers"), were conducted. No assumptions were violated, and there were no speciously influential observations.Regression Results: A Robust Interaction Effect Tables 3, 4, and 5 provide results of regression a nalyses using TAAS reading, math, and writing scores as dependent variables. Th e most interesting finding for present purposes is that the size-by-SES interaction effect is statistically significant and negative in each instance. This interaction, in each case, i s, in fact, the most influential variable after SES and the ethnicity variables (the influenc e of which varies across subject areas). As school size increases, the cost to school perfor mance of schools serving economically less-advantaged students increases, as well. This, of course, was the finding in all previous replications.Table 3 TAAS Reading Achievement Unstandardized and (Standardized) Coefficients N=1001SIZE0.177 (.065)
10 of 32 PCTPOOR -0.040*** (-.367)PCTBLACK! -0.253*** (-.142)PCTHISP -0.010** (-.123)PCTLEP! -0.268** (-.117)S/TRATIO 0.008 (-.011)EPP 0.027 (.015)PCTINST 0.007 (.022)UNIT 0.733** (.102)PCTTECH 0.004 (.040)PCTSPECL 0.047** (-.123)PCTGIFT 0.038** (.118)SIZE-by-SES -0.035** (-.143) Adjusted R-Squared = 40.3% *** <.001** <.01* <.05 Expressed as Natural Logarithms Partial Derivative = -0.035(PCTPOOR) Effect SizePoints (S.D. Units) PCTPOOR(Quartiles) -0.76 (-0.33)21.6%-1.14(-0.50)32.5%-1.73 (-0.75)49.5%-3.50 (-1.52)100.0%Table 4 TAAS Math Achievement Unstandardized and (Standardized) Coefficients N=1001SIZE0.019 (.040)PCTPOOR-0.062*** (-.318)PCTBLACK!-0.631*** (-.200)PCTHISP-0.022** (-.152)PCTLEP!0.010 (.002)
11 of 32 S/TRATIO-0.146** (-.113)EPP-0.149 (-.048)PCTINST0.007 (.013)UNIT0.611 (.048)PCTTECH0.005 (.024)PCTSPECL-0.064** (-.095)PCTGIFT0.052** (.090)SIZE-by-SES-0.060** (-.144) Adjusted R-Squared = 30.5% *** <.001 ** <.01* <.05 Expressed as Natural Logarithms. Partial Derivative = -0.060(PCTPOOR) Effect Size Points (S.D. Units) PCTPOOR(Quartiles) -1.30 (-0.32)21.6%-1.95 (-0.48)32.5%-2.97 (-0.73)49.5%-6.00 (-1.47)100.0%Table 5 TAAS Writing Achievement Unstandardized and (Standardized) Coefficients N=1001SIZE 0.052 (.025)PCTPOOR -0.031*** (-.366)PCTBLACK!-0.183*** (-.132)PCTHISP -0.002 (-.037)PCTLEP!-0.310*** (-.173)S/TRATIO -0.041 (-.072)EPP-0.007 (-.006)PCTINST0.007 (.027)UNIT 0.505** (.090)
12 of 32 PCTTECH -0.001 (-.010)PCTSPECL -0.036*** (-.123)PCTGIFT0.027** (.105)SIZE-by-SES-0.033*** (-.171) Adjusted R-Squared = 40.3% *** <.001** <.01* <.05 Expressed as Natural Logarithms. Partial Derivative = -0.033(PCTPOOR) Effect Size Points (S.D. Units) PCTPOOR(Quartiles) -0.71(-0.40)21.6%-1.07 (-0.60)32.5%-1.63 (-0.91)49.5%-3.30 (-1.84)100.0% Clearly, the interaction effect involving school s ize and the percentage of students who are poor is robust and strong in the presence o f regression model re-specification. This result adds credibility to the repeatedly repl icated finding that smaller schools diminish the achievement disadvantages associated w ith being poor. Larger schools, by contrast, exaggerate these disadvantages.Effect Size As with previous research on size-by-SES interacti ons, we have computed illustrative effect sizes by using partial derivati ves. This is done by differentiating the regression equations in Tables 3 through 5 with res pect to SIZE (expressed in thousand-student units), while treating the other i ndependent variables as constants (Purcell and Varberg, 1984, pp. 308-309, 636-639). Statistically nonsignificant coefficients are set equal to zero. (Note 6) The results, reported at the bottom of each table, are the average achievement decrements, in test score points and standard devia tion (S.D.) units, which come with each quartile increment in PCTPOOR. In each instanc e, we see that there are mean achievement test score costs associated with econom ically disadvantaged students, and these costs increase as the percentage of less-adva ntaged students increases. The substantial nature of the achievement costs be comes clearer when we recall that the standard deviations and coefficients of va riation for R10, M10, and W10 are small. This replication, based on informed regression mode l re-specification, makes clear that the size-by-SES interactions are robust and strong .Can Costs Decline Without Increasing Size?
13 of 32 In spite of the consistently strong findings about school performance, small schools with a broad range of grade levels seem to many--if not most--observers singularly anachronistic. The move toward ever-larg er, ever-more grade-specialized schools, is proceeding apace (Lyons, 1999; Funk & B ailey, 1999; Boex & Martinez-Vasquez, 1998). One of the coauthors recen tly received a query from a former student about whether any research addressed the gr eater effectiveness of a K-3 versus a K-5 school. The answer, not surprisingly, is "no." This study, however, together with several other s tudies (e.g., Franklin & Glascock, 1998; Howley & Harmon, 2000a; Wihry, Cola darci, & Meadow, 1992; DeYoung, Howley, & Theobald, 1995), attempts to rai se the issue of grade span configuration more systematically In general, the present analysis finds that restricting the grade span of a school increases costs. That is, given a level of school performanc e, the school with a broader grade span will provide t hat level of performance at lower cost (all else equal). A critical problem for such an analysis is differe nces in grade level expenditures per pupil, which are higher for secondary than for elementary grades. Without controlling for this difference, we bias the analys is to favor cost reductions for schools with the broadest range of grades. Therefore, we cr eated a weighting variable for the EPP at each grade level to control for such grade l evel differences in expenditure per pupil. This additional variable (which does not app ear in the tables) was created by multiplying the number of students at each grade le vel by the mean EPP at each grade level, summing across the grades included in a scho ol, then dividing by school size. (Note 7) We make these analyses because, administrators and policy makers deal with fiscal constraints that render findings about the e ducational benefits of small size seem impractical to them. For them, cost remains a prima ry consideration. For instance, rural superintendents who operated small rural high schoo ls (enrolling fewer than 400 students) recently cited fiscal constraints as the primary threat to the continued existence of such schools (Howley & Harmon, 2000b). Departure from the large, grade-specialized mode in pursuit of equity appears to many administr ators to be a luxury they cannot afford (Keller, 2000). Findings reported here shoul d help administrators and policy makers revise commonly held views about the fiscal practicality of operating small high schools in the 7-12 and K-12 configurations. (Note 8) Multiple Regression Analysis: Expenditure Per Pupil In the regression analysis reported in Table 6, th e dependent variable is expenditure per pupil. The independent variables ar e otherwise the same as with Tables 3, 4, and 5, except that the size-by-SES interactio n term has been deleted as irrelevant to this analysis (since the theory links the interacti on to school performance, which is not the dependent variable in these analyses), and the three achievement test scores are now used as independent variables for purposes of stati stical control, appearing jointly in the COMPOSITE variable. Finally, a multiplicative interaction term created using UNIT and SIZE (with SIZE logged and centered, see Cronbach, 1987) has b een added. Given statistically significant coefficients for these two variables, a UNIT-by-SIZE interaction term wiIl enable us to determine if the relationship between SIZE and EPP varies between single-unit schools and conventional high schools.
14 of 32 Table 6 Unit Schools and Expenditure Per Pupil Unstandardized and (Standardized) Coefficients N=1001SIZE!-254.415*** (-.199)PCTPOOR-4.158 (-.066)PCTBLACK! 81.239** (.080)PCTHISP5.668** (.119)PCTLEP!37.920 (.029)S/TRATIO -284.614*** (-.680)PCTINST-35.422*** (-.199)PCTTECH-2.923 (-.046)PCTSPECL 1.291 (.006)PCTGIFT 4.823 (.026)COMPOSITE-3.551 (-.008)UNIT -1017.607*** (-.247)UNIT-by-SIZE-730.195*** (-.172) Adjusted R-Squared = 51.4% *** <.001 ** <.01* <.05 Expressed as Natural Logarithms.!! Weighted for differences in mean EPP by grade le vel. Partial Derivative = -254.415(1/SIZE) 730.195(UNI T)(1/SIZE) Effect Size (Dollars) SIZE(Quartiles) UNIT=1 UNIT=0 -4.48 -1.16220-2.20 -0.57447-0.67 -0.171459-0.22 -0.064434 Regression Results: Anticipated and Unanticipated F indings Not surprisingly, school size (SIZE) has a statist ically significant and negative
15 of 32relationship to expenditure per pupil. The same is true of student-teacher ratio (S/TRATIO), the variable exercising the greatest in fluence on expenditure per pupil. Smaller schools and smaller classes are associated with higher expenditures overall (but not with all else equal). Less predictably, the statistically significant re gression coefficient corresponding to UNIT is notable: Being a single-unit school is a ssociated with an average reduction in expenditure-per-pupil of just over $1,017. Other th ings being equal (that is, with the full complement of controls in place, including achievem ent level, class size, and grade-level differences in EPP), having only one school, coveri ng all grades in a district, represents substantial dollar savings. The multiplicative interaction term, UNIT-by-SIZE, however, also has a negative and statistically significant coefficient. This int eraction indicates that the net influence of increases in school size provides more substantial cost reductions for single-unit schools than for conventional schools.Reduced Costs Without Increased Size? The results reported in Table 6 affirm the convent ional wisdom that size is negatively related to expenditure per pupil, for bo th single-unit schools and conventional high schools. Table 6 also shows, however, that the relationship is more complex than commonly acknowledged. After controlling for size a nd a reasonable complement of other factors, single-unit schools are associated w ith substantial savings in expenditure per pupil, and increases in size yield greater cost reductions for single-unit schools than for conventional grade-specialized schools. What ca n explain such unexpected findings? We seek possible answers to such questions in the o rganizational distinctiveness of single-unit schools as defined in this study.Single-Unit Schools: Organizational Distinctiveness Organizationally, the characteristics that conspic uously set these single-unit schools apart are number of grade levels, and the f act that, in this data set, each is the only school in its district. (Note 9) Seventy-five percent of the high schools in our data set have four or fewer grades (LEVELS). Single-unit schools, however, with K-12, PreK-12, or early childhood-12 configurations, have thirteen, fourteen, or fifteen grade levels. Similarly, the mean of the variable HIGHSKL S (before logging) tells us that the average number of high schools per district is near ly three, while a single-unit school is the only school of any kind in its district. Single-Unit Distinctiveness and Expenditure Per Pup il In an effort to explain cost savings associated wi th single-unit schools, therefore, in Table 7 we have added two additional independent variables, representing the distinctive characteristics of single-unit schools. Since LEVELS is very closely correlated with UNIT (r=.965), the UNIT variable ha s been deleted, replaced by the organizational components of the Texas single-unit school phenomenon (i.e., LEVELS and HIGHSKLS). We construe the new independent vari ables as essential components of the global, complex variable UNIT (Rosenberg, 19 68, pp. 40-52). In effect, we are trying to identify the specific characteristics of UNIT that may account for its unexpected relationship with expenditure-per-pupil (EPP). Thes e characteristics, of course, may also
16 of 32 be associated with reduced costs in conventional hi gh schools. We have also created a multiplicative interaction term with SIZE and each of the components of UNIT. Thus, we are also adding to the regression equation LEVELS-by-SIZE and HIGHSKLS-by-SIZE, with all varia bles used in creating the interaction terms centered (Cronbach, 1987).Table 7 One High School, Grade Levels, and Expenditure Per Pupil Unstandardized and (Standardized) Coefficients N=1001SIZE!-290.519*** (-.227)PCTPOOR-2.927 (-.046)PCTBLACK35.476 (.035)PCTHISP4.160* (.088)PCTLEP!23.216 (.018)S/TRATIO-314.462*** (-.751)PCTINST -34.101*** (-.191)PCTTECH -3.365 (-.053)PCTSPECL 1.318 (.006)PCTGIFT0.646 (.003)COMPOSITE 8.725 (.019)HIGHSKLS332.023*** (.223)LEVELS-98.358** (-.232)HIGHSKLS-by-SIZE -114.038* (-.076)LEVELS-by-SIZE-48.445** (-.108) Adjusted R-Squared = 52.8% *** <.001 ** <.01* <.05 Expressed as Natural Logarithms.!! Weighted for differences in mean EPP by grade le vel. Partial Derivative = 290.519(1/SIZE) 114.038(HIGHSKLS)(1/SIZE) 48.445(LEVELS)(1/SIZE) Effect Size (Dollars) SIZE(Quartiles) HIGHSKLS(Quartiles) LEVELS(Quartiles) -2.2022004
17 of 32 -1.08447 04-0.4114590.696-0.314434 3.26 15 LEVELS, HIGHSKLS, and Expenditure Per Pupil The results are instructive. Predictably, as with Table 6, the coefficients corresponding to SIZE and S/TRATIO are negative and statistically significant. This holds in spite of the fact that SIZE and S/TRATIO a re substantially correlated (r=.736), thereby reducing statistical power. However, the va riance inflation factors for each, though the largest for the equation, are well withi n acceptable limits, 4.870 and 4.131 (Chatterjee, Hadi, & Price, 2000, pp. 240-241). (No te 10) Furthermore, given that LEVELS and HIGHSKLS are co nstrued as effective components of UNIT, the following results are not s urprising: as the number of high schools in a district increases, expenditure per pu pil also increases averaging just over 332 dollars per school. In addition, each grade lev el added to a high school is associated with an average expenditure per pupil decrease of just over 98 dollars. (The distribution of high schools per district and by grade levels is reported in Table 8 and Table 9.)Table 8 High Schools Per District12345681012212692.49%(727) 3.69%(29) 1.14%(9) 1.27%(10) 0.76%(6) 0.51%(4) 0.25%(2) 0.25%(2) 0.13%(1) 0.13%(1) 0.13%(1)Table 9 Grade Levels Per High School23 45679121314 150.50%(5) 1.80%(18) 72.43%(725) 0.80%(8) 11.09%(111) 1.70%(17) 0.09%(1) 0.09%(1) 5.79%(58) 3.70%(37) 2.00%(20) Finally, the statistically significant interaction terms make clear that as SIZE increases, the increased costs associated with havi ng more than one high school in a district are diminished; while the reduced costs as sociated with having more grade levels are reduced still more.What Is To Be Made of All This?School Size and Expenditure Per Pupil: Diminishing Returns One way to summarize these complex results is to r efer to the illustrative effect sizes reported (by quartiles of significant variabl es) on Tables 6 and 7. For each analysis, as school size increases, the partial derivatives s how savings, but progressively
18 of 32diminished savings. (Note 11) School size is negati vely related to expenditure per pupil, but savings diminish with each increment in size (s ee the following discussion of diseconomies of scale for our interpretation of thi s finding.) School Size and Expenditure Per Pupil: Single-Unit Schools Furthermore, with a judiciously selected complemen t of controls in place, single-unit schools and their defining characterist ics--number of grade levels and uniqueness in their district--are associated with s ubstantial savings in expenditure per pupil. For these organizationally distinctive schoo ls, moreover, size contributes more to reducing costs than in conventional high schools. One related observation needs still to be undersco red. Despite the comparative cost-advantages of increased size for single-unit s chools, change in the rate of reduction in EPP as size increases slows for both K-12 schools and other schools--the slowing is simply less dramatic for other schools. See the effect sizes given in T ables 6 and 7 to gauge this difference. (Note 12)School Size and Expenditure Per Pupil: HIGHSKLS and LEVELS Not surprisingly, given the savings associated wit h single-unit schools, as the number of schools in a district increases, so does expenditure per pupil, though this additional cost is less for larger schools than for smaller schools. While this finding might suggest that building additional large, as co mpared to small, schools is cost-effective, readers need to recall two other fa cts. First, the law of diminishing returns to investment is definitely applicable: Ever-larger size assures ever-diminishing returns with regard to expenditure per pupil. Second, large r consolidated schools typically do have conventionally narrow grade spans, and, as the number of grade levels in a school decreases, expenditure per pupil is again increased So far as expenditure-per-pupil goes, size (total enrollment), grade span configuration, and district organization structure a quite complex playing field for the game of minimiz ing costs. "Larger schools cost less to operate" is not even a close approximation of su ch complexity. Most succinctly: Bigger is not always or even usua lly cheaper. The questions to be answered locally are: (1) how big (when do the r eturns to increased size yield negligible savings)? (2) bigger for whom (poverty, ethnicity--poorer communities require smaller schools to maximize achievement)? a nd (3) bigger under what circumstances (district organization and grade span configuration)? The analyses presented so far show that answers to questions 2 a nd 3 constrain the answer to question 1. Those who govern school funding and school const ruction have not, to our knowledge, even begun to recognize the real constra ints to large size. Diseconomies of Scale Typically, economists attribute diseconomies of sc ale to problems posed by the need for coordination and control (Bidwell & Kasard a, 1975; Boex & Martinez-Vasquez, 1998; Friedman, 1990). This obser vation follows from different interests among organizational participants, includ ing lack of consensus with regard to organizational objectives. The usual response is a system of personnel and procedures for supervision and monitoring: bureaucratic organi zation. Supervision and monitoring are costly additions to an organization, but in inc reasingly large organizations these
19 of 32additional costs are (ironically) increased by the need to coordinate and control those who supervise and monitor. Bureaucratic organizatio n, a feature of increased organizational scale, inevitably has the effect of complicating organization itself. This is a concern faced by any large organization, not just schools. As organizations become larger and more complicate d, with ever-greater specialization among employees, departments, and le vels, threats of organizational anomie and anarchy not only come into play, but are often realized and disorganization begins to prevail (Shedd & Bachrach, 1991). In dyna mic fashion, additional "negative feedback loops" necessary to maintain stability inc rease supervision and monitoring costs to unacceptably high--and ultimately counterp roductive-levels. Change and adaptation become so costly that they are sacrifice d to the imperative of sheer survival. A school enrolling 750 students can easily offer all the curricular and co-curricular "iconography" that characterizes the American compr ehensive high school (Haller et al., 1990), and increases in size beyond some hypothetic al level of what might be called "programatic surfeit" come at a cost to efficiency, recognizable as diminishing returns to size if not as absolute diseconomies of scale. For instance, high schools enrolling 3,500 as compared to 750 students would (hypothetically) realize little or no economic advantage to their increased size, they might encou nter diseconomies of scale that counterbalance and, beyond some hypothetical thresh hold, overwhelm the accumulated advantages of economies of scale (see, e.g., Bidwel l & Kasarda, 1975; Friedkin & Necochea, 1988). This description will sound familiar to many reade rs in big-city mega-districts as well as to readers of very large districts in rural and suburban locales. The so-called "small schools movement" is a reform strategy to ad dress this dilemma in metropolitan districts. Elsewhere, in rural areas and small town s, extant small high schools are often regarded as too expensive to exist, in part because analyses with adequate controls (such as appear in the present study) are so seldom under taken or even understood as necessary. According to some observers, both policy analysts and policy makers have tended to ignore the issue of organizational scale as an influence on school performance (Guthrie, 1979; Howley, 2000; Wasley, Fine, Gladden Holland, King, Mosak, & Powell, 2000) The results of this study may indicate that inclus ion of all grade levels in the same setting fosters a common, perhaps strongly tacit, u nderstanding of organizational purpose. A K-12 school, for example, includes all p ersonnel who teach and administer in all grades in the same location. This may foil deve lopment of the usual articulation problems that characterize relationships among elem entary schools, middle schools, and high schools, diminishing the need for costly monit oring and supervision. Similarly, if a school is the only one in its dist rict, between-school differences in purpose and procedure cannot occur, further reducin g the need for coordination and control through monitoring and supervision. When a single school with a broad range of grade levels is also small the seemingly antithetical goals of saving money and promoting equity in achievement may well be attaine d simultaneously; the odds of doing so are at any rate increased, according to the anal yses in this article. This tentative account, of course, shifts our focu s from schools to school districts. This is consistent with earlier Georgia research, i n which we found that the achievement of less-advantaged students in larger schools was d iminished less if the schools were located in smaller districts. In addition, we found that the expected achievement gains of less-advantaged students in small schools were unde rcut in large districts (Bickel & Howley, 2000; Howley, 2000).
20 of 32 Size-by-SES and Cost Table 10 joins the size-by-SES and cost issues sti ll more closely together. We use the same regression model specification employed in Table 7. Our achievement composite is now the outcome measure, and we reintr oduce the size-by-SES interaction term.Table 10 Composite Achievement Unstandardized and (Standardized) Coefficients N=1001SIZE!0.218 (.079)PCTPOOR -0.054*** (-.403)PCTBLACK!-0.270*** (-.123)PCTHISP -0.008 (-.081)PCTLEP! -0.255* (-.090)S/TRATIO 0.017 (.019)PCTINST 0.004 (.011)PCTTECH 0.001 (.009)PCTSPECL-0.056*** (-.121)PCTGIFT 0.051*** (.130)HIGHSKLS! -0.946*** (-.297)LEVELS 0.130** (.142)HIGHSKLS-by-SIZE 0.534*** (.166)LEVELS-by-SIZE 0.050 (.051)SIZE-by-SES -0.034* (-.116) Adjusted R-Squared = 42.7% *** <.001 ** <.01* <.05 Expressed as Natural Logarithms. Partial Derivative = 0.534(HIGHSKLS)(1/SIZE) 0.034PCTPOOR) Effect SizePoints (S.D. Units) SIZE(Quartiles) HIGHSKLS (Quartiles) PCTPOOR(Quartiles)
21 of 32 -0.73 (-0.26) 220021.6-1.10 (-0.39) 447 0 32.5-1.68 (-0.59)1459 0.6949.5-3.39 (-1.20) 44343.26100.0 Interestingly, LEVELS, the component of UNIT which was associated with reduced expenditures, is now associated with increased achievement. HIGHSKLS, the component of UNIT which was associated with increas ed expenditures, is now associated with decreased achievement. In most other respects the results in Table 10 are like the results reported in Tables 3, 4, and 5. Once again, the size-by-SES int eraction term is statistically significant and negative (equal in magnitude to PCT BLACK! and PCTSPECL, i.e., = -.116, p<.05), and the illustrative effect sizes de monstrate that as school size increases, the presence of economically disadvantaged students is associated with diminished average achievement. Most significantly, perhaps, the influence of size across the SES spectrum, from relatively affluent to impoverished, is negative, t hough the influence of size is most harmful in larger districts serving many poor stude nts (effect size = -1.20, see Table 10). Even in small unit schools serving a relatively aff luent community, however, data in Table 10 show that a one standard-deviation-unit in crease in size (850 students, see Table 2), would depress school performance by about one-fourth of a standard deviation.Cautions Our data set contains a large number of cases and a broad range of pertinent variables. Nevertheless, it is useful to bear in mi nd that Texas is a distinctive state. For this reason, our analysis is limited in specific wa ys (to be considered shortly). We do not claim that these results necessarily apply in other states; indeed, many states retain no single-unit schools, and the present analysis can n ot be completed in them. However, the Texas case also shares certain featur es of policy context with other states. First, most states continue to make changes to their accountability schemes, and these changes notably include changes to assessment instruments. Texas (and many other states) claim, for instance, to be creating tougher" tests all the time. Such changes are usually more cosmetic than substantive, and the re is little reason to suspect that even substantive changes would dramatically alter relati onships that prevail among influences in the present study. The fact also remains that th e previous studies in this line of research have analyzed data from different states a nd employed different sorts of achievement measures (both normand criterion-refe renced standardized tests) with rather consistent results. We would predict that th e realtionships apparent here would persist with marginally different sorts of tests--s omewhat "tougher," "more authentic," or measuring incrementally different achievement const ructs. Second, in Texas, as in other states, school finan ce litigation continually produces marginal changes in how schools are funded. Nonethe less, it remains an American principle that schools in wealthy communities susta in their funding advantages through all such changes. Political contest, after all, rev olves around the way money is deployed by powerful interests for public and private purposes, and poor people are not well positioned to prevail in such contests. For instanc e, power equalization school finance schemes may, in Texas and elsewhere, mute the relat ionships reported here, but they are
22 of 32unlikely to substantially obscure them. Our use of tests of statistical significance serves as a modest hedge against the effect of incremental policy movement, such as changes in assessment and finance systems may entail.Model Specification Misleading results due to specification error are a good deal less threatening in our achievement analyses than in our analyses of ex penditure per pupil. The size-by-SES interaction effect has proven robust across seven v ery different states, and for at least four different regression model specifications, two in this paper alone. (Compare Tables 3, 4, and 5 with Table 10. Also see Bickel & Howley 2000; Friedkin & Necochea, 1988; Howley, 1995; Howley & Bickel, 1999; Huang & Howley 1993). Misleading results due to specification error are more likely in our analyses of expenditure per pupil because the variables we have found to be especially interesting, UNIT, LEVELS, and HIGHSKLS, as well as the interact ion effects created with SIZE, have not been adequately researched. The research that has been done on these issues, m oreover, does not address relationships between expenditure and variables suc h as UNIT, LEVELS, and HIGHSKLS (see Wihry, Coladarci, and Meadow, 1992; A lspaugh, 1996; Howley & Harmon, 2000a; Franklin & Glascock, 1998). Therefor e, though our choice of independent variables and functional forms seems re asonable, our regression model specification is necessarily tentative, and we read ily acknowledge that a better-informed alternative might yield different results.Concepts: Single-Unit School We have defined single-unit schools as the only sc hool in a district, including all grade levels. The performance of the component vari ables LEVELS and HIGHSKLS, along with interaction effects created with these v ariables and SIZE, suggests that there is merit to this way of construing the single-unit school and its distinctive components. However, in the only national survey of single-uni t schools, Howley & Harmon (2000a) suggest that the single-unit designation be applied to any K-12 school, whether or not it is the only school in its district. In Te xas, however, each such school is, in fact, the only school in its district. In a real sense, a s we have seen, Texas single-unit schools are districts as well as schools. This account of the simultaneous realization of th e supposedly competing objectives of equity and cost efficiency suggests t hat having more than one single-unit school in a district would diminish its attractiven ess. The uniqueness-in-district that is a defining characteristic of single-unit schools in T exas is a common feature of many K-12 schools (Howley & Harmon, 2000). But it does n ot characterize all single-unit schools still in existence.Concepts: Expenditure Per Pupil We have measured cost in terms of expenditure per pupil. Funk and Bailey (1999), however, in their Nebraska research, judged cost per graduate to be a superior measure of cost efficiency. After all, one virtue o f smaller school size is a lower dropout rate. Similarly, Stiefel, Berne, Iatarola, and Fruchter (2000) measured cost in terms of
23 of 32total budget per pupil and total budget per graduat e. Neither measure revealed the cost inefficiencies commonly attributed to small schools Whatever the virtues of per-graduate measures, the ir calculation requires dropout data which covers all grades in the schools being a nalyzed (Stiefel, Berne, Iatrola, & Fruchter, 2000, p. 33). Twenty-five percent of our Texas high schools, however, have five or more grades, and information on dropouts is often not reported for lower grades. Our choice of the traditional expenditure per pupil measure, therefore, was dictated by the information available in our Texas data set. It s use, together with use of our grade-level-expense weighting variable (described p reviously), nonetheless means that the findings reported here probably represent conservative estimates of cost efficiency. Multi-Level Analysis? With the individual high school as the unit of ana lysis, an obvious strategy would be to conduct a multi-level analysis, with school d istricts constituting the second level (schools within districts). As it turns out, howeve r, while only 11.6 percent of the schools are of the single-unit variety, 72.6 percen t of the districts operate just one high school. This yields an average within-group sample size of 1.27. High schools and districts are thoroughly confounded in the organiza tional structure of public secondary education in Texas, a situation common to many stat es. In short, for this analysis, the multi-level approach is simply inapt. (Note 13) In addition, Singer (1987) has shown that with sma ll within-group sample sizes, and small residual intra-class correlations, standa rd errors of regression coefficients are diminished very little by intra-class correlation, and tests of significance are reliable (Note 14). In all our analyses, deflation of standa rd errors due to intra-class correlation is less than two percent (Singer, 1987, pp. 224-226).Conclusions As with seven previous analyses, we have found tha t as school size increases, achievement test score costs associated with having economically disadvantaged students in schools increase, as well. This finding has now proven robust across seven states and at least four different regression model specifications. This degree of consistency is rare, indeed, in educational researc h. We have also found that, while administrators and policy makers are correct in their judgment that school size is negatively relat ed to costs, that is far from the whole story, at least with regard to expenditure per pupi l. The negative relationship between size and expenditure per pupil becomes increasingly tenuous as school size increases, and eventually savings become negligible. In addition, organizational factors, especially as manifest in the distinctive components of the single-unit school, reveal unanti cipated relationships to cost reduction If we were designing schools solely to minimize expenditure per pupil (an educationally counterproductive goal in the view of the authors), the best configuration might very well be a large single-unit school. However, if we were also interested in balancing e xpenditure per pupil with achievement-based equity, the best configuration se ems to be a small single-unit school. While decreased size would increase costs, a (logge d) value of 1 on HIGHSKLS (equaivalent to approximately 3 high schools in a d istrict) and a value of 13 to 15 on LEVELS would substantially diminish costs (Note 15). This makes the achieveme nt
24 of 32advantage of small schools (where they are most nee ded, that is, in impoverished communities) more affordable than previously expect ed. This study once again corroborates the manner in w hich SES regulates the relationship of school size to school performance. The findings have proven to be unusually robust, which makes them difficult to dis miss. This study's findings with regard to ways to reduce school costs without incre asing size are more tentative, and our explanations of them are more tentative as well. Ne vertheless, in the effort to resolve the aim of achievement equity within manifest fiscal co nstraints, it seems time to consider the issue of district organization and school grade span configuration .AcknowledgementThe findings reported here are perhaps surprising, but not miraculous. Support for this study was provided, in part, through a contract fro m the Policy Program of the Rural School and Community Trust during the academic year s 1997-8 and 1998-9. We thank Marty Strange and his staff for their continuing in terest in the research itself and for their commitment to interpret the findings to a wide audi ence.NotesEvidence from a related study conducted with Alaska data (Huang & Howley, 1993) which included several blocks of contextual student background, and school-level process variables, suggests that the i nteraction effect may be robust. Using individual-level data, the interaction term r emained significant after entry of all blocks of relevant data. 1. The controlled vocabulary of the ERIC database incl udes "grade span configuration" as an "identifier," but not as a "de scriptor." Descriptors are main indexing terms and are adopted after a lengthy and formal deliberation; identifiers may be coined by any ERIC clearinghouse at any time and serve as proto-descriptors. As of this writing, "grade span configuration," added in the early 1990s, had been used to index just 4 items. 2. These are sometimes referred to as "union schools" (e.g., in the Southeast) or "unit schools" (e.g., in the West) schools. 3. All independent variables originate with the Texas Department of Education. In particular, PCTINST is computed by dividing the DOE 's dollar value for instruction by "total campus budget." Approximately 80 percent of PCTINST, which varies among schools, is accounted for by tea cher salaries. 4. These three programmatic terms are included for the sake of model specification as control variables hypothetically associated with increased EPP. Our analytical focus, however, remains organizational rather than programmatic. One anonymous reviewer of an earlier draft of this arti cle observed that most children eligible for special education services are not in full-time programs (PCTSPECL). We recognize this fact, of course, but for our purp oses, PCTSPECL is a proxy for the additional cost of providing special services i n a school. The correlation of PCTSPECL and EPP is positive, as expected r = +.53) ; S/TRATIO, however, predictably covaries with PCTSPECL r = -.44) and th e net influence of PCTSPECL (in the multivariate analyses) becomes sta tistically nonsignificant when both independent variables appear in our equat ions. 5. A full discussion of the use of partial derivatives in this fashion appears in Bickel and Howley (2000) and in Howley (1995). 6.
25 of 32This is not readers unfamiliar with economic analysis should note, case weighting as used in in analyses of data produced by oversamp ling, but an application of a simple weighted average serving as a proxy for average differences in cost, statewide, by educational level. The variable incor porates these norms into a single, school-wide metric as a control variable, o nce again for the sake of model specification. 7. The Texas case, we think, is illustrative of the la rger policy issue of size and grade span configuration. One of the authors (Howley) has consistently argued that the ratio of total school enrollment to grade span is t he most proper metric of school size. That metric, however, makes it impossible to treat the influence of grade span configuration separately from school size. Sep arating the two issues allows for grade span configurations other than the domina nt 9-12 arrangement (10-12, 7-12, 5-12, or, indeed, K-12). 8. In a survey of all unit schools nationally, two-thi rds of responding superintendents indicated that their school district operated a sin gle school--the K-12 unit school in question. All responding Texas superintendents i ndicated their schools were in this category. Among the other states, most seemed to maintain unit schools principally on this model. States where unit school s were more frequently part of multi-school districts included Alabama, Alaska, Lo uisiana, and Mississippi (Howley & Harmon, 2000a). 9. That is, the moderately strong correlation did not introduce multi-collinearity problem, which means we can affirm that as SIZE inc reases, EPP declines, and as S/TRATIO increases, EPP declines. 10. The possible value combinations of the independent variables in the partial derivative are considerable, and so are the possibl e effect sizes that are the function of such values. In Table 7, then, The 12 v alues of the relevant independent variables chosen to illustrate the rang e of effect size variation, then, are merely illustrative. For another application of this sort of illustration, see Bickel & Howley, 2000; see also note six for refere nce to the use of partial derivatives to estimate effect sizes across this se ries of studies. 11. In Table 7, recall that HIGHSKLS is logged, so that a value of 0 (ln=0) is equivalent to an unlogged value of 1, indicating a single high school, the category to which all single-unit schools belong. 12. Because we were more interested in policy matters t han in the conditions of instruction, we did not plan for a multi-level anal ysis of students within schools; individual-level information was not part of our da ta set. 13. We provide significance levels on the assumption th at "A population...in a given time interval includes not only the actual history represented by the values that were in fact observed but also the potential histor y consisting of all the values that might have occurred but did not. The population so defined is obviously an infinite one....This view underlies virtually all p olicy-oriented research in economics and econometrics" (Kmenta, 1997, p. 4). T he use of significance levels also provides one rubric, when a study population s o closely represents the universe, for judging practical significance. In th is study, we dismiss as practically insignificant influences that do not attain levels of statistical significance. 14. Recall that e1 2.72; that is, the unlogged value of ln=1 is e, or approximately 2.72. 15.References
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30 of 32Singer, J. (1987). An intraclass correlation model for analyzing multilevel data. Journal of Experimental Education, 18 219-228. Smith, D., & DeYoung, A. (1988). Big school vs. sma ll school: Conceptual, empirical, and political perspectives on the re-emerging debat e. Journal of Rural and Small Schools, 2 2-11. Shedd, J., & Bachrach, S. (1991). Tangled hierarchies San Francisco, CA: Jossey-Bass. Stevens, N., & Peltier, G. (1994). A review of rese arch on small-school participation in extracurricular activities. Journal of Research in Rural Education, 10 116-120. Stevenson, K. (1996). Elementary school capacity: W hat size is the right size? The Educational Facility Planner, 33 ,10-14. Stiefel, L., Berne, R., Iatarola, P., & Fruchter, N (2000). High school size: Effects on budgets and performance in New York City. Educational Evaluation and Policy Analysis, 22, 27-39. Texas Education Agency. (2000). Welcome Researchers Austin, TX: Author, Student Assessment Division. World Wide Web Site: http://ww w.tea.state. txt.us/student assessment/researchers.htm.Tholkes, R., & Sederberg, C. (1990). Economies of s cale and rural schools. Journal of Research in Rural Education, 7 9-15. Walberg, H., & Walberg, H. (1994). Losing local con trol of schools. Educational Researcher, 23 19-26. Wilson, B., & Corcoran, C. (1988). Successful secondary schools New York: Falmer. Wyatt, T. (1996). School effectiveness research: De ad end, damp squib, or smoldering fuse. Issues in Educational Research, 6 79-112.About the AuthorsRobert Bickel is Professor of Advanced Educational Studies at Ma rshall University. His recent research has dealt with crime on school prop erty, the nature of rural neighborhoods, and consequences of geographical mob ility among high school students. Craig Howley directs the ERIC Clearinghouse on Rural Education and Small Schools at AEL, Inc., and is adjunct associate professor in th e Educational Studies Department at Ohio University. His recent research concerns small rural high schools, rural school busing, and principals' perspectives on planning.Tony Williams is Professor of Advanced Educational Studies at Ma rshall University. His recent publications have concerned teen pregnan cy and early teen pregnancy. He is also author of a widely used textbook on the histor y of West Virginia. Catherine H. Glascock is Assistant Professor in the educational studies department at Ohio Univeristy. She holds an MBA in Finance and Ph .D. in Educational
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