Correlations of the AFQT with Other IQ Tests in The NLSY
Sample Correlation with the AFQT
California Test of Mental Maturity 356 .81
Coop School and College Ability Test 121 .90
Differential Atitude Test 443 .81
Henmon Nelson Test of Mental Maturity 152 .71
Kuhlmann-Anderson Intelligence Test 36 .80
Lorge-Thorndike Intelligence Test 170 .72
Otis-Lennnon Mental Ability Test 530 .81
The magnitudes of the correlations between the AFQT (using the age-referenced percentile scores) and classic IQ tests are as high as or higher than the observed correlations of the classic IQ tests with each other. For example, the best-known adult test, the WAIS, is known to correlate (using the median correlation with various studies, and not correcting for restriction of range in the samples) with the Stanford-Binet at .77, with the Ravens Standard Progressive Matrices at .72, the SRA Non-verbal test at .81, the Peabody Picture Vocabulary Test at .83, and the Otis at .78.14 The table below summarizes the intercorrelations of IQ tests, based on the comparisons assembled by Arthur Jensen as of 1980, and adding a line for the AFQT comparisons from the NLSY. The AFQT compares favorably with the other major IQ tests by this measure, which in turn is consistent with the high g-loading of the AFQT.
Correlations of the Major IQ Tests with Other Standardized Mental Tests
Median Correlation with Other Mental Tests
Source: Jensen 1980, Table 8.5, and authors’ analysis of the NLSY.
AFQT (age-referenced, 1989 scoring) .81
Wechsler-Bellevue I .73
Wechsler Adult Intelligence Scale (WAIS) .77
Wechsler Intelligence Scale for Children .64
Stanford-Binet .71
HOW SENSITIVE ARE THE RESULTS TO THE ASSUMPTION THAT IQ IS NORMALLY DISTRIBUTED?
Any good test designed to measure a complex ability (whether a test of cognitive ability or carpentry ability) will have several characteristics that common sense says are desirable: a large number of items, a wide range of difficulty among the items, no marked gaps in the difficulty of the items, a variety of types of items, and items that have some relationship to each other (i.e., are to some degree measuring the same thing).15 Empirically, tests with these characteristics, administered to a representative sample of those for whom the test is intended, will yield scores that are spread out in a fashion resembling a normal distribution, or a bell curve. In this sense, tests of mental ability are not designed to produce normally distributed scores; that’s just what happens, the same way that height is normally distributed without anyone planning it.
It is also true, however, that tests are usually scored and standardized under the assumption that intelligence is normally distributed, and this has led to allegations that psychometricians have bamboozled people into accepting that intelligence is normally distributed, when in fact it may just be an artifact of the way they choose to measure intelligence. For a response to such allegations, Chapter 4 of Arthur Jensen’s Bias in Mental Testing (New York: Free Press, 1980) remains the best discussion we have seen.
For purposes of assessing the analyses in this book, it may help readers to know the extent to which any assumptions about the distribution of AFQT scores might have affected the results, especially since we rescored the AFQT to correct for skew (see Appendix 2). The descriptive statistics showing the breakdown of each variable by cognitive class, presented in each chapter of Part II, address that issue. Assignment to cognitive classes was based on the subject’s rank within the distribution, and these ranks are invariant no matter what the normality of the distribution might be. Ranks were also unaffected by the correction for skew.
The descriptive statistics in the text were bivariate. To examine this issue in a multivariate framework, we replicated the analyses of Part II substituting a set of nominal variables, denoting the cognitive classes, for the continuous AFQT measure. That is, the regression treated “membership in Class I” as a nominal variable, just as it would treat “married” or “Latino” as a nominal characteristic—and similarly for the other four cognitive classes, also entered as nominal variables (See Appendix 4 for a discussion of how to interpret the coefficients for nominal variables as created by the software used in these analyses, JMP 3.0). Below, we show the results for the opening analysis of Part II (Chapter 5), the probability of being in poverty.
Comparison of results when AFQT is treated as a continuous, normally distributed variable and when it is treated as a set of nominal categories based on groupings by centile
Note: For computing the plot, age and SES were set at their mean values.
Whole-Model Test
Source DF -LogLikelihood ChiSquare Prob>ChiSq
Model 3 477222.0 954443.9 0.000000
Error 4488 4587166.7 null
C Total 4491 5064388.7 null
RSquare (U) 0.0942
Observation 4,492
Parameter Estimates
Term Estimate Std Error ChiSquare Prob>Chisq
Intercept -2.6579692 0.0009826 . 0.0000
zAFQT89 -0.8177031 0.0012228 447179 0.0000
zSES -0.2744971 0.0011661 55416 0.0000
zAge -0.0482156 0.0009187 2754.1 0.0000
These are the results using the categorization into cognitive classes by centile:
Whole-Model Test
Source DF -LogLikelihood ChiSquare Prob>ChiSq
Model 6 383494.7 766989.4 0.000000
Error 4485 4680894.0
C Total 4491 5064388.7
RSquare (U) 0.0757
Observations 4,492
Parameter Estimates
Term Estimate Std Error ChiSquare Prob>ChiSq
Intercept -2.5097718 0.0015823 . 0.0000
CogClas.[1-5] -1.0067168 0.0050693 39439 0.0000
CogClas.[2-5] -0.6803606 0.0025486 71265 0.0000
CogClas.[3-5] -0.1905042 0.0018498 10606 0.0000
CogClas.[4-5] 0.64764109 0.0021336 92138 0.0000
zSES -0.3902981 0.0011276 119800 0.0000
zAge -0.1605992 0.000907 31350 0.0000
We repeated these comparisons for a broad sampling of the outcome variables discussed in Part IL The results for poverty were typical When the results for the two expressions of IQ do not correspond (e.g., the relationship of mother’s IQ to low birth weight, as discussed in Chapter 10), the lack of correspondence also showed up in the bivariate table showing the breakdown by cognitive class. Or to put it another way, the results presented in the text using IQ as a continuous, normally distributed variable are produced as well when IQ is treated as a set of categories. Any exceptions to that may be identified through the bivariate tables based on cognitive class.
RELATIONSHIP OF THE AFQT SCORE TO EDUCATION AND PARENTAL SES
The relationship of an IQ test score to education and socioeconomic background is a constant and to some extent unresolvable source of controversy. It is known that the environment (including exposure to education) affects realized cognitive ability. To that extent, it is conceptually appropriate that parental SES and years of education show an independent causal effect on IQ. On the other hand, an IQ test score is supposed to represent cognitive ability and to have an independent reality of its own; in other words, it should not simply be a proxy measure of either parental SES or years of education. The following discussion elaborates on the statistical relationship of both parental SES and years of education to the AFQT score.
The Socioeconomic Status Index and the AFQT Score.
The SES index consists of four indicators as described in Appendix 2: mother’s and father’s years of education, the occupational status of the parent with the higher-status job, and the parents’ total family income in 1979-1980. The correlations of the index and its four constituent variables with the AFQT are in the table below.
Intel-correlations of the AFQT and the Indicators in the Socioeconomic Status Index
AFQT
Mother’s education .43
Father’s education .46
Occupational status .43
Family income .38
SES Index .55
The correlation of AFQT with the SES index itself is .55, consistent with other investigations of this topic.16
There are three broad interpretations of these correlations:
Test bias. IQ tests scores are artificially high for persons from highstatus backgrounds because the tests are biased in favor of people from high-status homes.
Environmental advantage. IQ tends to be genuinely higher for children from high-status homes, because they enjoy a more favorable environment for realizing their cognitive ability than do children from low-status homes.
Genetic advantage. IQ tends to be genuinely higher for children from high-status homes because they enjoy a more favorable genetic background (parental SES is a proxy measure for parental IQ).
The first explanation is discussed in Appendix 5. The other two explanations have been discussed at various points in the text (principally Chapter 4’s discussion of heritability, Chapter 10’s discussion of parenting styles, and Chapter 17’s discussion of adoption). To summarize those discussions, being brought up in a conspicuously high-status or low-status family from birth probably has a significant effect on IQ, independent of the genetic endowment of the parents. The magnitude of this effect is uncertain. Studies of adoption suggest that the average is in the region of six IQ points, given the difference in the environments provided by adopting and natural parents. Outside interventions to augment the environment have had only an inconsistent and uncertain effect, although it remains possible that larger effects might be possible for children from extremely deprived environments. In terms of the topic of this appendix, the flexibility of the AFQT score, the AFQT was given at ages 15-23, when the effect of socioeconomic background on IQ had already played whatever independent role it might have.
Years of Education and the AFQT Score
For the AFQT as for other IQ tests, scores vary directly with educational attainment, leaving aside for the moment the magnitude of reciprocal cause and effect. But to what extent could we expect that, if we managed to keep low-scoring students in school for another year or two, their AFQT scores would have risen appreciably?
Chapter 17 laid out the general answer from a large body of research: Systematic attempts to raise IQ through education (exemplified by the Venezuelan experiment and the analyses of SAT coaching) can indeed have an effect on the order of 2 standard deviation, or three IQ points. As far as anyone can tell, there are diminishing marginal benefits of this kind of coaching (taking three intensive SAT coaching programs in succession will raise a score by less than three times the original increment).
We may explore the issue more directly by making use of the other IQ scores obtained for members of the NLSY. Given scores that were obtained several years earlier than the AFQT score, to what extent do the intervening years of education appear to have elevated the AFQT?
Underlying the discussion is a simple model:
The earlier IQ score affects both years of education and is a measure of the same thing that AFQT measures. Meanwhile, the years of education add something (we hypothesize) to the AFQT score that would not otherwise have been added.
Actually testing the model means bringing in several complications, however. The elapsed time between the earlier IQ test and the AFQT test presumably affects the relationships. So does the age of the subject (a subject who took the test at age 22 had a much different “chance” to add years of education than did a subject who took the test at age 18, for example). The age at which the earlier IQ test was taken is also relevant, since IQ test scores are known to become more stable at around the age of 6. But the main point of the exercise may be illustrated straightforwardly. We will leave the elaboration to our colleagues.
The database consists of all NLSY students who had an earlier IQ test score, as reported in the table on page 596, plus students with valid Stanford-Binet and WISC scores (too few to report separately). We report the results for two models in the table below, with the AFQT score as the dependent variable in both cases. In the first model, the explanatory variables are the earlier IQ score, elapsed years between the two tests, and type of test (entered as a vector of dummy variables). In the second model, we add years of education as an independent variable. An additional year of education is associated with a gain of 3.2 centiles per year, in line with other analyses of the effects of education on IQ.17 What happens if the dependent variable is expressed in standardized scores rather than percentiles? In that case (using the same independent variables), the independent effect of education is to increase the AFQT score by .11 standard deviation—also in line with other analyses.
The Independent Effect of Education on AFQT Scores as Inferred from Earlier IQ Tests
Dependent variable: AFQT percentile score
Independent Variables Model 1 Model 2
Coefficient Std. Error Coefficient Std. Error
Intercept 12.312 1.655 -14.331 2.780
Earlier IQ percentile score .787 .016 .736 .016
Elapsed years between tests -.316 .166 -1.288 .179
Years of education — — 3.185 .273
Type of test (entered as a vector of nominal variables, coefficients not shown.)
No. of observations 1,404 1, 404
R2 (Adjusted) .656 .686
We caution against interpreting these coefficients literally across the entire educational range. Whereas it may be reasonable to think about IQ gains for six additional years of education 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 school, interpreting these coefficients becomes problematic when moving into post-high school education.
The negative coefficient for “elapsed years between tests” in the table above is worth mentioning. Suppose that the true independent relationship between years of education and AFQT is negatively accelerated—that is, the causal importance of the elementary grades in developing a person’s IQ is greater than the causal role of, say, graduate school. If so, then the more years of separation between tests, the lower would be the true value of the dependent variable, AFQT, compared to the predicted value in a linear regression, because people with many years of separation between tests in the sample are, on average, getting less incremental benefit of years of education than the sample with just a few years of separation. The observed results are consistent with this hypothesis.
Appendix 4
Regression Analyses from Part II
This appendix presents the logistic regressions for the figures in Chapters 5 through 12. In the text, the figures are based on regressions that use the entire white sample in the NLSY and are calculated using sample weights. We use the entire sample and weights to take advantage of the NLSY’s supplemental sample of low-income whites; in our judgment, doing so provides the best available estimates of the relationships we discuss. But interpreting standard errors and statistical significance is greatly complicated when using sample weights. In the regression results that follow, we therefore restrict the analyses to the nationally representative cross-sectional sample of whites. This procedure not only enables direct interpretation of the standard errors but also provides the raw material for interested readers to see how much difference there is between the results from the entire white sample and the cross-sectional sample (which you may do by computing the probabilities for the cross-sectional sample and comparing them to the ones shown in the text figures). We have done so ourselves and can report that the differences are so small that they are seldom visually evident.
By “whites,” we mean all NLSY subjects who were identified as “non-black, non-Hispanic” in the NLSY’s racial/ethnic cohort screening (variable R2147, in the NLSY’s documentation), deleting those who identified themselves as being of American Indian, Asian, or Pacific descent in the “first or only racia
l/ethnic origin” item (R96).
In the text, we do not refer to the usual measure of goodness of fit for multiple regressions, R2, but they are presented here for the cross-sectional analyses. As the ratio of the explained sum of squares to the total sum of squares, R2 is in this instance the square of the correlation between the set of independent variables and the dependent variable expressed as the logarithm of the odds ratio. Inasmuch as the values of R2 range widely in the tables to follow, some mention of them is warranted.
The size of R2 tells something about the strength of the logistic relationship between the dependent variable and the set of independent variables, but it also depends on the composition of the sample, as do correlation coefficients in general. Even an inherently strong relationship can result in low values of R2 if the data points are bunched in various ways, and relatively noisy relationships can result in high values if the sample includes disproportionate numbers of outliers. For example, one of the smallest R2 in the following analyses, only .017, is for white men out of the labor force for four weeks or more in 1989. Apart from the distributional properties of the data that produce this low R2, a rough common-sense meaning to keep in mind is that the vast majority of NLSY white men were in the labor force even though they had low IQs or deprived socioeconomic backgrounds. But the parameter for zAFQT in that same equation is significant beyond the .001 level and large enough to make a big difference in the probability that a white male would be out of the labor force. This illustrates why we therefore consider the regression coefficients themselves (and their associated p values) to suit our analytic purposes better than R2, and that is why those are the ones we relied on in the text.
The Bell Curve: Intelligence and Class Structure in American Life Page 70