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PAGE 1 1 of 12 Education Policy Analysis Archives Volume 4 Number 20December 24, 1996ISSN 10682341A peerreviewed scholarly electronic journal. Editor: Gene V Glass,Glass@ASU.EDU. College of Educ ation, Arizona State University,Tempe AZ 852872411 Copyright 1996, the EDUCATION POLICY ANALYSIS ARCHIVES.Permission is hereby granted to copy any a rticle provided that EDUCATION POLICY ANALYSIS ARCHIVES is credited and copies are not sold.The Bell Curve : Corrected for Skew Haggai Kupermintz Stanford University Abstract This commentary documents serious pitfalls in the s tatistical analyses and the interpretation of empirical evidence presented in The Bell Curve Most importantly, the role of education is reevaluated and it is shown how, by n eglecting it, The Bell Curve grossly overstates the case for IQ as a dominant determinant of social success. The commentary calls attention to important features of logistic regression coefficie nts, discusses sampling and measurement uncertainties of estimates based on observational s ample data, and points to substantial limitations in interpreting regression coefficients of correlated variables. Introduction The Bell Curve by Richard Herrnstein and Charles Murray (hencefor th H&M) puts forward a strong thesis about the centrality of intelligenc e in determining contemporary American social structure. Following its publication in October 199 4, The Bell Curve sparked an intense public debate over its assertions, methodology and conclus ions. Most of the book's critics, in a flood of newspaper articles, TV talk shows, academic journal articles and a few books, focused on The Bell Curve 's treatment of ethnic and racial group differences in intelligence, the role of heredity in determining these differences, and the social an d political agenda advocated by H&M. The heated debate was clearly another wave of the contr oversies about genes, IQ and public policy (see, e.g., Cronbach, 1975). The Bell Curve is distinguished by its extensive use of statistic al analyses to support a strong social theory. Other authors have provided c ritical examination of some statistical and measurement aspects of The Bell Curve raising concerns about the appropriateness of cau sal inferences, model specification (most notably the a bsence of measures of education from the models), model fit and the validity of IQ and SES m easures, among other issues. Some of these concerns will be echoed here in detail. The current commentary will go beyond delineation of PAGE 2 2 of 12these issues in principle or theory, to reexamine t he statistical evidence and to analyze further the data presented in The Bell Curve H&M explore the relationship between social strati fication and the distribution of cognitive abilities which, according to their thesi s, will inevitably lead to a "world in which cognitive ability is the decisive dividing force" ( p.25). Part I of the book is devoted to an elaborate exposition of the emergence and the incre asing isolation of a "cognitive elite", driven by radical transformations in educational, occupati onal and economic forces in American society throughout the twentieth century. What are the cons equences of this current American landscape that has been stratified so forcefully according to cognitive ability? In part II of the book, H&M launch a series of stat istical analyses to examine the role of intelligence, as measured by an IQ test, in determi ning a myriad of social ailments such as poverty, school dropout, unemployment and labor for ce dropout, welfare dependency and criminal behavior. The analyses of part II use a su bsample of nonLatino white respondents from the National Longitudinal Survey of Youth (NLSY)a nationally representative sample of 12,686 young men and young women who were 14 to 22 years o f age when they were first surveyed in 1979. By focusing on the white subsample, H&M argu e that "cognitive ability affects social behavior without regard to race and ethnicity" (p. 125). Only later, in Part III, when the importance of intelligence as a powerful determinan t of social behavior has been allegedly demonstrated, do H&M turn to examine ethnic and rac ial group differences. An evaluation of the scientific merit of the book will best be served by focusing on how H&M handle and present the less controversial evidence about the role of intel ligence in the lives of young white Americans. As Charles Murray notes, "perhaps the most importan t section of The Bell Curve is Part II" (1995, p. 27). Indeed, many of the arguments and co nclusions to appear later in the book rely heavily on the success of the case made in Part II, which constitutes (together with Appendices 2,3,4) a dense collection of statistics, tables, gr aphs, and technical details. H&M use the case of poverty, presented in Chapter 6, to "set the stage for the social behaviors to follow" (p.125). This chapter provides a basic template for their formula tion of research questions, analysis strategies and use and interpretation of statistical methods. As such, it will be appropriate to focus here in some detail on this chapter. Chapter 6 asks, "What causes poverty?", or more specifically, "If you have to choose, is it better to be born smart or ri ch?" (p.127). Let us examine how H&M arrive at what they claim is an "unequivocal" answer: "smart" Logistic Regression Coefficients The basic analytical tool H&M employ is a set of mu ltiple regression equations. The independent variables are IQ, SES, and age. (Age is included in the models because of the nature of the NLSY sample. It is inconsequential to the ar guments presented here and will not be further discussed.) The IQ test used throughout The Bell Curve is the Armed Forces Qualification Test (AFQT), a subset of the Armed Services Vocational A ptitude Battery (ASVAB). The SES measure is an average of standardized parental educ ation, parental occupation, and family income. The dependent variable is whether a respond ent in the NLSY was below the poverty line in 1989. H&M examine the regression results: they o bserve that the IQ regression coefficient (.84) is much larger than the SES coefficient (.3 3); they then plot a graph showing how the probability of being in poverty is predicted by the model as a function of IQ or SES, holding the other variable constant at its average value. (The regression equation is given in p. 596, and the graph in p. 134.) H&M conclude: "Cognitive ability is more important than parental SES in determining poverty" (p.135), independent of any ro le SES might play in determining the likelihood of poverty. How warranted is this conclu sion? For those not versed in the details of regression a nalysis, H&M provide a primer in Appendix 1 (pp. 553577) entitled: "Statistics for People Who Are Sure They Can't Learn PAGE 3 3 of 12Statistics." After explaining basic statistical con cepts, multiple linear regression is introduced. Logistic regression, the technique employed through out Part II, is presented as a simple adaptation of linear regression to handle binary ou tcomes: "It tells us how much change there is in the probability of being unemployed, married, an d so forth, given a unit change in any given variable, holding all the other variables in the an alysis constant" (p. 567). The unsuspecting reader misses one important point: The value chosen at which to "hold a variable in the analysis constant" has a direct impact on the magnitude of a nticipated change in the probability of the outcome, given a unit change in any other variable. H&M identify the mathematical function responsible for this behavior of the logistic regre ssion, the log odds, or logistic function, later in the introduction to the results in Appendix 4, but they are silent about its consequences. As we shall see, this seemingly insignificant technical p oint has crucial implications for the interpretation of logistic regression results on a probability scale. Let us examine what happens when we use the same r egression coefficients, the same model, but decide to hold SES at other values than its average. Should we expect to see any noticeable difference in the relations between IQ a nd the probability of being in poverty? After all, we are still holding SES constant, and, as H&M assure us, "here is the relationship of IQ to social behavior X after the effects of socioeconomi c background have been extracted" (p. 123). Figure 1 depicts the predicted probabilities of be ing in poverty as a function of IQ at three values of SES: the SES average (the one shown in The Bell Curve ), and 2 standard deviations above and below the SES average. Contrary to what w e might have expected after being told that the effects of SES has been extracted out, the effe ct of IQ on the probability of being in poverty is much stronger when SES level is lower; it is muc h weaker when SES level is higher! This is a necessary consequence of the nature of the logistic regression model. For persons with lower socioeconomic status, the anticipated change in the likelihood of being poor associated with a unit change in IQ, is much larger than for those wi th higher socioeconomic status. This means that the risk of poverty induced by having lower in telligence is far more pronounced under conditions of adverse family environment. On the ot her hand, the privileges of a sound family background seem to mitigate the harsh consequences of lacking in cognitive abilities. Take for example two persons, a "smart" with an IQ of 115 (one standard deviation above the average), and a "dull" with an IQ of 85 (one st andard deviation below the average). How do they compare in their respective risks of being poo r? If they both come from an extremely poor PAGE 4 4 of 12background, the "dull" person is 18% more likely to be in poverty than the "smart"; On the other hand, if they both come from a family of extremely high socioeconomic status, the difference shrinks to only 6%. If we return to H&M original as sertion about the logistic regression coefficient as indicating how much change will occu r in the probability of poverty, given a unit change in IQ, we find that a twounits change (movi ng from 1 to 1 in standard deviations) in IQ, means three times more change in the probability of being poor for those with low SES compared with those with high SES. So much for "hol ding all the other variables in the analysis constant". Clearly, Figure 1 tells a more complicated story th an the one H&M would have the student of their statistics primer believe on the basis of interpreting the logistic regression coefficients a s if they were linear or additive. Even more experien ced researchers, who routinely run linear regression analyses, need more than what H&M are wi lling to provide as a guide to the proper interpretation of their logistic regression results In the authoritative source on Generalized Linear Models, of which logistic regression is a sp ecial case, McCullagh and Nelder (1989) provide such guidance, as well as call attention to the fact that "...statements given on the probability scale are more complicated because the effect on [the probability of an outcome] of a unit change in X2 depends on the values of X1 and X2 (p. 110; italics added). In discussing the "special case of education" (we shall have more to say on this later), H&M quite rightly assert that "...to take education's regression coefficient seriously tacitly assumes that intelligence and education could vary independently and produce simi lar results. No one can believe this to be true in general: indisputably, giving nineteen year s of education to a person with IQ of 75 is not going to have the same impact on life as it would f or a person with an IQ of 125" (p.125). Why should we, then, take the IQ regression coefficient seriously when, as we just saw, having a high (or low) IQ for a person coming from a poor backgro und is not going to have the same impact on life as for a person coming from a wealthy backgrou nd? Let us now review the substantive conclusion H&M d raw from the regression results: "If a white child of the next generation is given a choic e between being disadvantaged in socioeconomic status or disadvantaged in intelligen ce, there is no question about the right choice" (p. 135). Indeed, there is no question: If your parents are rich enough, you can afford to be very dull and still can expect to escape poverty If, on the other hand, you made the poor (literally) choice of being born to a low SES famil y, chances are that intellectual weakness will carry grave consequences for you. This, of course, is a caricature of serious hypothesizing about the dynamics of cognitive abilities and social cond itions, but it brings us to the next issuethe independence (or the lack thereof) of independent v ariables. Independence of Independent Variables H&M point out that "variables that are closely rela ted can in some circumstances produce a technical problem known as multicollinearity whereby the solutions produced by regression equations are unstable and often misleading" (pp.12 4125; italics in original). Attention to potential effects of multicollinearity (meaning simply that the independent variables are correlated with each other), is indeed warranted wh en dealing with an attempt to disentangle via statistical analysis the effects of variables that are highly correlated in nature. Observing correlations of .50 and .64 between education and S ES and IQ, respectively, cause H&M to raise a concern about the interpretation of a regression model that includes all three of them as independent variables. But what about the associati on between SES and IQ? Are they free to vary independently? Are they sufficiently uncorrelated a s not to sound a similar alarm? The correlation between the AFQT scores and parenta l SES in the NLSY data is .55. After reporting this correlation, H&M summarize: "Being b rought up in a conspicuously highstatus or lowstatus family from birth probably has a signifi cant effect on IQ, independent of the genetic PAGE 5 5 of 12endowment of the parent" (p. 589). Although the mag nitude of these effects or their explanation are debatable, the IQ scores used in The Bell Curve to demonstrate the independent role of a cognitive endowment are caused to an important degr ee by parent's SES. This means, to rephrase H&M argument about ignoring years of education in t heir regressions, that when IQ is used as an independent variable, it is to some extent expressi ng the effects of SES in another form. Can this be solved by the machinery of multiple regression? It is too often believed that regression analysis provides the proper statistical control, accounting for" is the usual term, which mathematically remedies the confounding of effects imposed by the realities of the investigated phenomenon or by the study design. The answer is an unequivocal "No." Neter, Wasserman, and Kutner (1990) explain: "Sometimes the standardized regression coefficients b1 and b2, are interpreted as showing that X1 has a greater impact on the [outcom e variable] than X2 because b1 is much larger than b2. However, ...one must be cau tious about interpreting regression coefficients, whether standardized or no t. The reason is that when the independent variables are correlated among themselv es, as here, the regression coefficients are affected by the other independent variables in the model." (By a happy circumstance, the correlation alluded to in t his section is .569, almost exactly the correlation between IQ and SES!) "Hence, it is ordinarily not wise to interpret the magnitudes of standardized regression coefficie nts as reflecting the comparative importance of the independent variables" (p.294). For a detailed discussion of these issues, the rea der is invited to consult Chapter 13 of Mosteller & Tukey's Data Analysis and Regression (1977). They masterfully demonstrate the problems of interpreting regression coefficients, a nd sound very clear warnings concerning the comparison of regression coefficients even for full y deterministic systems under tight experimental control.A Scale is a Scale is a Scale? The correlation between independent variables is no t the only factor affecting the magnitude, and consequently the interpretation, of linear or logistic regression coefficients. It is important to recognize the effects on estimated reg ression parameters due to errors of measurement. H&M go into great detail to document t he superior measurement qualities of their IQ test the AFQT. That the AFQT provides good mea surement of g, general cognitive ability, is demonstrated by high correlations among its four co nstituent tests, by high correlations with other measures of general ability, and by high load ings on the general factor of the ASVAB battery. (The latter is purported to represent g in common psychometric practice. It is interesting to note, however, that Gustafsson and Muthen (1994) show that the ASVAB lacks measures of Fluid Intelligence and its general factor is closer to Crystallized Intelligence, which they interpret as a broad verbal factor, closely associated with a cademic achievement.) The conclusion is that the AFQT is an exceptionally high quality instrumen t. What, then, are the measurement qualities of the me asure of socioeconomic status? Compared with the treatment of the AFQT scale, only meager information is presented to allow evaluating the quality of the SES scale. However, f rom the two pieces of information that are presented, a reliability coefficient of .76 and cor relations among the four indicators comprising the scale ranging from .36 to .63, we can safely co nclude that the SES measure is substantially inferior as a measurement device and is subject to considerable error. Moreover, for more than a quarter of the subjects only three of the indicator s were available, further compromising the reliability of the scale. Therefore, "one must conc lude that as a proxy for 15 years of PAGE 6 6 of 12environment, this is a variable measured with subst antial error" (Delvin et al., 1995, p. 1468). The effect of the SES scale's low reliability on th e regression results is quite clear: an underestimation of the SES effect run in a "horse r ace" against IQ. It is likely that the real differences between the effects of SES and IQ on th e poverty in the population are smaller than what is reflected in H&M's estimates. In addition t o errors of measurement, statistical uncertainties related to sampling are another major source of caution. Uncertainty in Statistical Estimates Based on the logistic regression results, as depic ted by the plots they draw, H&M make two strong predictions to demonstrate the different roles IQ and SES play in determining poverty. Paying attention to the far lefthand side of the p lots on p. 134, we can observe that a white person from an unusually deprived socioeconomic bac kground, with an average IQ, has a probability of about 11% of being in poverty. On th e other hand, an extremely dull person with an average SES, has a probability of about 26% of b eing in poverty more than double. Notice that these prediction use extreme values of IQ and SES to produce dramatic differences. How accurate are these statements? How much confid ence should we have that the real proportions in the population are close to the ones suggested by the statistical model estimated for this particular sample? An appropriate indicato r of statistical uncertainty is the confidence interval of prediction. It informs us about the ran ge of likely values we expect to encounter if we were to sample again from the same population. Conf idence intervals for prediction in logistic regression models are easily obtained by using conv entional methods (see Agresti, 1990, Chapter 12) or alternatively, by utilizing a computer inten sive resampling technique known as bootstrapping (see Efron & Tibshirani, 1993). Using both methods, we may compute confidence inte rvals for the two predictions above (at the 95% confidence level). The range of plausib le values for a person from a deprived socioeconomic background with an average IQ goes fr om 8% to 16%. The range of plausible values for a dull person with average SES goes from 20% to 35%. (Both methods gave similar results.) The confidence interval for the differenc e between the two predictions indicates that this difference can be as small as 6% or as big as 26%. Evidently, The Bell Curve ascribes unwarranted precision to estimates that a re subject to considerable sampling error. The dramatic differenc e between the two estimates becomes much less so when one takes into account the statistical uncertainty associated with them. Thus when H&M declare categorically that the odds of poverty for a person with low IQ and average SES are "more than twice as great as the odds facing th e person from deprived home but with average intelligence" (p.135), one needs to exercise great caution before accepting it on face value. But then, H&M themselves acknowledge (though only in a footnote) the complexities involved in comparing the magnitude of effects in multiple regr ession and promise: "We refrain from precise numerical estimates of how much more important IQ i s than socioeconomic background..." (note 13, p.691). We may also ponder: How valid is a comparison betw een a person with an IQ score of about 70 (two standard deviations below the average ) and a person from a very poor family? That people with very low cognitive capacity face severe limitations in life is hardly a surprising or a fresh finding. For example, Jensen states that "mos t persons with any experience in the matter would agree that those with IQs below 70 or 74 have unusual difficulty in school and in the world of work. Few jobs in a modern industrial soci ety can be entrusted to persons below IQ 70 without making special allowances for their mental disability" (1981, p.12). We should also remember that the youth falling into what H&M call Cognitive Class V, the very dull, are also routinely afflicted by severe socioeconomic conditi onsthey are on average almost an entire standard deviation below the mean in SES. The very dull are also the very poor. Attempts to PAGE 7 7 of 12disentangle the independent effects of cognitive ab ility and harsh environment are doomed, not because of technical complications, but because Ame rican social reality is less than generous towards its weakest citizens. It seems that The Bell Curve has no new story to tell here, but presenting such an extreme situation as an example of the general effect of IQ on social consequences is neither informative nor especially valid. The Special Case of Education The impact of omission of important variables from a regression equation is widely recognized. Not only do the effects of the omitted variables cannot estimated, but other effects in the models might be biased and misinterpreted when an included independent variable is meaningfully correlated with an omitted one. Theref ore, the absence of a measure of educational attainment from regression models set out to explai n the likelihood of poverty, unemployment, welfare dependency and the likes, seems immediately curious. After all, education is the primary social institution responsible for providing the ba sic skills needed for a productive civil participation. The NLSY contains data on years of e ducation respondents completed by 1990, which seems to be a natural scale to capture the ef fects of education. The omission of education from the regression models requires either a compel ling argument for why it should not be included, or strong empirical evidence that educati on does not explain the social behaviors of interest to any meaningful extent. H&M supply four reasons for why "the role of educa tion versus IQ as calculated by a regression equation is tricky to interpret" (p. 124 ). They assert that education is at least partly caused by intelligence 1. effects of education are likely to be discontinuous that is high school or college graduation might be meaningful but not years of education, 2. multicollinearity (that is the degree to which independent variables are correlated) might lead to unstable and misleading regression estimate s, and 3. the effects of education and intelligence are likel y to be complex and require more complicated modeling. 4. Assertions 3 and 4 were treated in some detail earl ier in the sections on the independence of independent variables and logistic regression co efficients. We saw that the same arguments hold when we consider the correlation and complex e ffects of IQ and SESeither the role of SES versus IQ is also "tricky to interpret," which is probably the case, or these two arguments against the inclusion of education should not hold. H&M simply cannot have it both ways. Assertion 2 is nothing more than a technicality eas ily handled by including education in the regressions as a categorical variable with three le vels: less than high school, high school, college or higher education. Moreover, by comparing results from using years of education against results from using this trichotomy, one could direc tly test assertion 2. H&M use this technique successfully to estimate the effects of Cognitive C lasses, rather than a continuous IQ score (see p. 587). Assertion 1 hypothesizes a causal link, whereby IQ determines the number of years of education completed In Appendix 3, H&M present an alternative they entertain the hypothesis that IQ gains are caused by years of education, and note that "it might be reasonable to think about IQ gains for six additional years of educatio n when comparing subjects who had no schooling versus those who reached sixth grade, or even comparing those who dropped out in sixth grade and those who remained through high sch ool" (p. 591). The cause and effect relationship between IQ and education is admittedly complex and open to competing interpretations, but we are not given compelling ar gument or empirical evidence to support the PAGE 8 8 of 12 dismissal of education and the inclusion of IQ in t he regressions because of these complex relationships. We can just as validly argue for the inclusion of education and the dismissal of IQ from the regressions. One last point: if years of e ducation as an independent variable competing with IQ for explanatory power, causes H&M so much c oncern, shouldn't they also worry about the fact that years of education constitute half (a nd sometime more) of the parental SES index? Surely, assertions 24, if valid, pose similar prob lems for the interpretation of the role of IQ versus SES. What about empirical evidence? H&M's solution to th e problems they raise is to run the IQ versus SES regressions separately for those who com pleted 12 years of educationthe high school sampleand those who completed 16 years of educationthe college sample. For college graduates, no matter what their IQ is, the risk of poverty is practically zero. (H&M do not show regression results for the college sample in Append ix 4these are meaningless when only six of these subjects were in poverty, but they still plot the regression lines in p. 136.) For the high school sample, H&M notice similar patterns for IQ a nd SES as were previously observed for the entire subsample. IQ has a strong effect regardles s of SES; SES has much weaker effect. They conclude: "Cognitive ability still has a major effe ct on poverty even within groups with identical education" (p.137). These analyses, however, do not answer the important question about education: What happens to the effect of IQ after accounting for" years of education? Restricting the analysis to a homogenous subgroup in terms of educational attainment provides partial and highly misleading information about this question. When "years of education" is entered into the regression, one finds that it is a highly significa nt predictor of the likelihood of poverty (a regression coefficient of .40), independent of IQ, and, even more importantly, the coefficient for IQ drops from .84 to .63. However, an even better solution exists. Responding to criticisms about the SES scale, Murra y poses a challenge: "Create some other scales and use some other method of combining them.... As scholars are supposed to do, Herrnstein and I check ed out these and many other possibilities the results reported in The Bell Curve were triangulated in numbing detail over the years we worked on the book and w e knew that the critics who bothered to retrace our steps would discover: that there is no way to construct a measure of socioeconomic background using the accep ted constituent variables that makes much difference in the independent role of IQ (1995, p. 29). The following exercise does the obvious. Given the strong correlation between subjects' years of education and parents' SES, and considerin g that doubtless the most direct way in which parental socioeconomic status can be translated int o meaningful advantages for their children is to enable them to get more (and better) education, why not combine these two variables to achieve a better measure of SES? The gains are clea r: we increase the SES index reliability, we avoid having three highly correlated variables in t he same regression, we update the scale to capture directly at least part of the subjects' rea lized potential in socioeconomic status. At the same time we resolve some problems of the special c ase of education. This is achieved simply by averaging the original SES scale with a standardize d variable of the subjects' years of education. Table 1 presents the results of the regression of p overty on IQ and the revised SES index.Table 1 Logistic Regression Results Using Revised SES (cf. The Bell Curve p. 596) EstimateStd. Errort value PAGE 9 9 of 12 (Intercept) 2.695789 0.07884634.1905 IQ 0.652195 0.1062316.1394 Revised SES 0.622218 0.1221955.0920 Age 0.036356 0.0727270.4999 We can now examine how these new results translate to the plots of IQ versus SES in the roles they play in determining whether young white adults are below the poverty line. This simple and straightforward improvement of the SES scale adding the subject's own years of education brings the relative weights of IQ and SES in predicting poverty to a perfect tie. Dominance of IQ? Hardly. A crucial role for SE S? Definitely. Especially if we recall, as H&M themselves acknowledge, that "[SES] has a signi ficant effect on IQ, independent of the genetic endowment of the parent" (p. 589). Moreover this finding has devastating consequences for any argument about the dominance of the inherit ed portion of intelligence, 60 percent is the estimate favored by H&M (see p. 105), over environm ental factors in determining the odds of being poor. Remember the question we started with? "If you have to choose, is it better to be born smart or rich?" (p.127; italics added). The answer is left to the reader. Does the revised SES and IQ model should be consid ered adequate for making sound inferences about the relationships among socioecono mic background, education, intelligence, and social behavior? Certainly not. In reality, the social scientist faces an almost insurmountable task when trying to disentangle and bound causes an d effects that present themselves only indirectly as a complex pattern of things that go t ogether. Rich families provide better home environment and better education for their children children with better home environment and better education do better on IQ tests, students wh o do better on IQ tests are more likely to complete more years of education, they are also mor e likely to come from families who are better off and less likely to end up poor, and so on and s o on. The biggest fallacy behind The Bell Curve PAGE 10 10 of 12statistical analyses in Part II of the book is summ arized by H&M in a single statement: "Regression analysis tells you how much each cause actually affects the result, taking the role of all the other hypothesized causes into account (p. 122; italics in original). If nothing more, t his commentary should provide a demonstration of the da ngers of blindly replacing hard thinking about a problem with an analytical formality, sophi sticated as it may be.Conclusion In a response to The Bell Curve 's critics, Charles Murray repairs to scientific middleoftheroad and claims: "The statistical met hod we use throughout is the basic technique for discussing causation in nonexperimental situati ons: regression analyses, usually with only three independent variables. We interpret the resul ts according to accepted practice" (1995, p. 27). Still, it appears that the analyses of relatio nships among IQ, SES, education, and poverty suffer in The Bell Curve from H&M's quest for simple answers. H&M prefer to ignore important details of their analyses, treat their models and e stimated parameters as if they were accurate and complete descriptions of social reality, and preten d that statistical methods can miraculously unravel or unequivocally differentiate among causes that are inherently confounded The inconsistencies and selectiveness in arguments and analysis choices documented in the current commentary lead one to wonder whether H&M w ere not investing too much of their own IQs to make the case for the dominance of intellige nce stronger than it really is? Otherwise, many of their conclusions, especially the ones they push about the proper policy response to ethnic and racial differences, lose critically in weight and c an hardly be sustained by less extravagant demonstrations of the overarching importance of IQ in the allocation of opportunities in current American society. It is only appropriate to end by rephrasing Murray' s words: "The unfounded criticisms of the statistics in The Bell Curve ... will merely cause embarrassment among a few wh o both understand the issues and have the decency to be em barrassed" (1995, p. 28). It is my hope that the founded criticisms of the statistics in The Bell Curve will not merely cause embarrassment to its author, but will encourage those "who both unde rstand the issues and have the decency" to set the record straight.ReferencesAgresti, A. (1990). Categorical data analysis. New York: Wiley. Cronbach, L.J. (1975). Five decades of public contr oversy over mental testing. American Psychologist, 30, 114. Devlin, B., Fienberg, S.E., Resnick, D., & Roeder, K. (1995). Galton redux: Intelligence, race and society, Journal of the American Statistical Association, 90, pp. 14831488. Efron, B, & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman and Hall.Gustafsson, JE., & Muthen, B.O. (1994). The nature of the general factor in hierarchical models of the structure of cognitive abilities: Alternativ e model tested on data from regular and experimental military enlistment tests. Gothenberg University, Sweden: Unpublished manuscript. Herrnstein, R.J., & Murray, C. (1994). The Bell Curve: Intelligence and class structure in american life. New York: The Free Press. PAGE 11 11 of 12 Jensen, A.R. (1981). Straight talk about mental tests. New York: Free Press. McCullagh, P., & Nelder, J.A. (1989). Generalized linear models New York: Chapman and Hall.Mosteller, F., & Tukey, J.W. (1977). Data analysis and regression Reading, MA: AddisonWesley.Murray, C. (May, 1995). The Bell Curve and its critics. Commentary 2330. Neter, J., Wasserman, W., Kutner, M.H. (1990). Applied linear statistical models. Homewood, IL: Irwin. About the AuthorHaggai KupermintzSchool of Education Stanford University Haggai Kupermintz is a doctoral candidate in Psycho logical Studies in Education, School of Education, Stanford University, Stanford, CA 943 05. His specializations are educational measurement and statistics. haggaik@stanford.edu Home Page Copyright 1996 by the Education Policy Analysis ArchivesEPAA can be accessed either by visiting one of its seve ral archived forms or by subscribing to the LISTSERV known as EPAA at LISTSERV@asu.edu. (To sub scribe, send an email letter to LISTSERV@asu.edu whose sole contents are SUB EPAA y ourname.) As articles are published by the Archives they are sent immediately to the EPAA subscribers and simultaneously archived in three forms. Articles are archived on EPAA as individual files under the name of the author a nd the Volume and article number. 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(6029652692) PAGE 12 12 of 12Editorial Board Greg Camillicamilli@pisces.rutgers.edu John Covaleskiejcovales@nmu.edu Andrew Coulson andrewco@ix.netcom.com Alan Davis adavis@castle.cudenver.edu Sherman Dorn sfxj9x@scfn.thpl.lib.fl.us Mark E. Fetlermfetler@ctc.ca.gov Thomas F. Greentfgreen@mailbox.syr.edu Alison I. Griffithagriffith@edu.yorku.ca Arlen Gullickson gullickson@gw.wmich.edu Ernest R. Houseernie.house@colorado.edu Aimee Howleyess016@marshall.wvnet.edu Craig B. Howley u56e3@wvnvm.bitnet William Hunterhunter@acs.ucalgary.ca Richard M. Jaeger rmjaeger@iris.uncg.edu Benjamin Levinlevin@ccu.umanitoba.ca Thomas MauhsPughthomas.mauhspugh@dartmouth.edu Dewayne Matthewsdm@wiche.edu Mary P. McKeowniadmpm@asuvm.inre.asu.edu Les McLeanlmclean@oise.on.ca Susan Bobbitt Nolensunolen@u.washington.edu Anne L. Pembertonapembert@pen.k12.va.us Hugh G. Petrieprohugh@ubvms.cc.buffalo.edu Richard C. Richardsonrichard.richardson@asu.edu Anthony G. Rud Jr.rud@purdue.edu Dennis Sayersdmsayers@ucdavis.edu Jay Scribnerjayscrib@tenet.edu Robert Stonehillrstonehi@inet.ed.gov Robert T. Stoutstout@asu.edu xml version 1.0 encoding UTF8 standalone no mods:mods xmlns:mods http:www.loc.govmodsv3 xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govmodsv3mods31.xsd mods:relatedItem type host mods:identifier issn 10682341mods:part mods:detail volume mods:number 4issue 20series Year mods:caption 19961996Month December12Day 2424mods:originInfo mods:dateIssued iso8601 19961224 xml version 1.0 encoding UTF8 standalone no record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam a22 u 4500 controlfield tag 008 c19969999azu 000 0 eng d datafield ind1 8 ind2 024 subfield code a E1100069 0 245 Educational policy analysis archives. n Vol. 4, no. 20 (December 24, 1996). 260 Tempe, Ariz. : b Arizona State University ; Tampa, Fla. : University of South Florida. c December 24, 1996 505 The bell curve : corrected for skew / Haggai Kupermintz. 650 Education x Research v Periodicals. 2 710 Arizona State University. University of South Florida. 1 773 t Education Policy Analysis Archives (EPAA) 4 856 u http://digital.lib.usf.edu/?e11.69 