The Overprediction of NAM Achievement

This graph shows three hypothetical regression lines relating one measurement of academic achievement (e.g., test score) and an outcome (e.g., grade point average, another test). The center line represents a regression for the entire population, the top line the regression line for the higher achieving population where the test underpredicts the outcome, and the bottom for a low achieving population where the test overpredicts the outcome. For example, the relationship between SAT scores and first year college GPA for NAMS and the Whites and Asians majority (WHAM). The SAT is said to overpredict minority first year grades compared to the entire population and underpredict WHAM grades. For example, a NAM with a SAT of 2100 will on average have a lower GPA than a WHAM with the same score.

Many explanations are offered for this phenomenon. When the y-axis represents the California High School Exit Exam ( CASHEE) and the x-axis the California Standardized Test ( CST ) Sean Reardon, professor of Education at Stanford University, speculates that the phenomenon is caused by stereotype threat. As Steve Sailer ridiculed, Reardon’s theory is that the NAMs are afraid of being pigeonholed as high school drop outs so they choke on the exit exam and become high school drop outs. When the y-axis represents bar exam scores and the x-axis and LSAT scores, Guy White, noted HBD blogger, speculates that the ETS has modified the LSATs to be biased against the WHAMs.

I cannot dismiss Guy White’s or Prof. Reardon’s explanations out of hand, but I can offer a simpler one: measurement error. Assume that tests of academic achievement are actually imperfect but unbiased measures of intelligence, and suppose, just for fun, that this intelligence is distributed differently between the two different populations. This would imply that a NAM student who obtains a score higher than their group mean on a test is likely to score closer to their group mean on their next test of academic achievement. The student, like us all, regresses to the mean. When this occurs you will observe the graph above without test bias and no difference in test anxiety. I show a simple example below.

Others have pointed this out. La Griffe du Lion discuss this phenomenon and coldly states that if a college wants to maximize first year GPA it should require that NAMs have higher SAT scores than WHAMs. On the other side of the debate, anti-HBD blogger Three Toed Sloth complains about the failure of Herrnstein and Murray to account for this type of error in their analyses. Sloth links to the work of psychometrician Roger Millsap who says

The direction of the intercept difference [in the graph above] is determined by the factor mean difference: the group with the larger factor mean will have the larger intercept. If a common regression line is imposed on the two groups, the group with the higher intercept will show systematic underprediction via the common line. In many applications, this group is the majority or reference group.

Millsap, the current editor of Psychomterika, gives conditions where the test is “measurement invariant” across population but it is not “prediction invariant” across populations. That is, the test is “fair” but you still overpredict NAM performance.

Here is a simple example to illustrate how over or underprediction can occur even on a fair test. Define M_1N = mean NAM score on test 1 and and M_1W = mean WHAM score on test 1. Define M_2N and M_2W similarly, assume that test 1 and test 2 are distributed bivariate normal in both populations with a standard deviations and a correlation of R. Finally, assume that both means are one standard deviation apart (i.e., M_1W – M_1N = S = M_2W – M_2N).

Suppose a NAM and a WHAM both receive a score of X on this test 1. Who will score higher on test 2? For those of you lucky enough to have avoided an introduction to mathematical statistics class, the expected scores for the NAM and WHAM are

(1) E(Test 2 NAM) = M_2N – R*(M_1N – X) and
(2) E(Test 2 WHAM) = M_2W – R*(M_1W – X)

respectively. Under the one standard deviation assumption the expected difference in the scores will be S(1 – R) in favor of the WHAMS. As noted more generally by Robert Miller, unless the tests are perfectly correlated (i.e., R=1), you cannot be “color blind” if you want to calculate the expectation.