Number 4 in the social science’s top 10 list of “grand challenge questions that are both foundational and transformative” (Giles, 2010) is: “How do we reduce the ‘skill gap’ between black and white people in America?” Presumably, figuring out the cause of this psychometric intelligence differential would help when it comes to deciding how best to minimize it. If so, we can thank Meng Hu for his recent efforts focused on determining the cause. This includes his recent extensive exploration of differential regression.
At the finish of his lengthily post on differential regression, Hu (2013, April, 18) concluded:
Discussion. If, for reasons mentioned above, the BW sibling regression gap cannot be fully interpreted in terms of environments, we may think of a combination of genetic and shared environmental differences…One cannot even begin to explain why blacks should be more environmentally depressed relative to whites at higher levels of IQ.
I wish here to comment on this point and to offer my own deliberations with regards to the interpretation of the results found.
As Hu (2013, April, 18) has noted, the meaning of the differential regression results has been subject to continual debate. Mackenzie (1980), commenting on Jensen (1973), argued that the results were a statistical artifact. Brody (2002), commenting on Jensen (1998), granted their statistical reality but argued that they were consistent with “virtually any” environmental hypothesis. Kaplan (2001), citing Fynn (1980), seemed to concur with Brody (2002). Murray (1999) argued that they were either consistent with a genetic hypothesis or a non shared environmental hypotheses – and he considered the latter to be implausible. Jensen and Rushton (2010), criticizing Nisbett (2009), argued that that these results were consistent with a genetic hypothesis and not readily consistent with a culture only hypothesis. Pinker (2012, August 6) indicated that they provided support for a genetic hypothesis of group differences.
There seems to be much confusion here. Let us try to shed some light on the issue.
Regression to the Mean and Inheritance of Deviance
In context to behavior genetics, people not infrequently discuss regression to the mean. Regression to the mean simply results, when it does, from deviance not being completely inherited. The inherited portion of a trait deviation from the mean is the portion conditioned by additive genetics and shared environment. It is the portion of a trait deviance that biological families (e.g., full siblings or biological parents and biological children) share. Regression to the mean is simply the non-transmission of trait deviance. It occurs, for example, when very smart parents have only somewhat smart children — because intelligence is only partially environmentally and genetically inherited. Generally speaking, we can define:
Regression to the Mean (R) = 1 – Inheritance of Deviance (I),
where, (I) =~ shared environmental and additive genetic effect.
Mostly people discuss R/I within populations but one can just as well discuss this dual phenomena as it occurs between populations. If two populations differ in terms of the inheritance of a trait, either due to differences in shared environment or to differences in additive genetics, they will show “differential regression”.
Measure and Meaning of Differential Regression
Differential regression is frequently measured by matching individuals on a trait between populations and then comparing the traits of their full siblings. Typically, graphs of the sibling regression lines are presented. A between population deviation in sibling regression lines results from some set of factors causing a population deviance. This deviation, of course, in itself, does not tell one anything more than what one already knows: that there is a population mean difference in the trait being investigated. This is the point that environmentalists such as Flynn (1980) have tried to make. However, the point that hereditarians, such as Jensen (1973), seemed to have tried to make is that the slopes of the regression lines do tell one something. Unfortunately for this debate, hereditarians have not spelled out, precisely, what they mean. We will do that below:
Differential regression tells one something about the within population distribution of a between population trait difference. To illustrate, using Meng Hu’s NLSY 1979 White sibling scores and the scores of created pseudo White siblings, I modeled four different environmental effects:
(1) The first graph shows the sibling regression lines produced by a uniformly distributed non-shared environmental difference which induces 1.2 SD of between population difference. Here, the regression line for White siblings is shown in blue. In purple is shown the regression line for pseudo White siblings. Since the between population effect is non shared, 2.4 standardized units was randomly subtracted from one of the pseudo White siblings per pseudo White pair (2.4/2 =1.2). The scores of the first White and first pseudo White siblings were then matched and averaged per tenth of a standard deviation and the scores of the second siblings were compared. As can be seen, the result is a non linear regression for the pseudo White group.
Now these are odd results, so let me explain what’s happening. Imagine that you had a set of sibling pair scores, sibling1 and sibling2 and that you created a new set of scores, pseudosibling1 and pseudosibling2, by taking the old set and then randomly subtracting 30 points from one of the two siblings per pair. These 30 points would, on average, represent a 15 point difference per individual. And your second set of siblings would, on average, be depressed 15 points relative to your first set. Since only one sibling per pseudo pair would be affected, this would constitute an unshared family effect. An example of this is shown below:
Now imagine that you went ahead and matched sibling1 and pseudosibling1 on IQ. These scores would give you a sibling1-pseudosibling1 column. If you went ahead and plotted sibling2 and pseudosibling2 scores on sibling1-pseudosibling1 scores, you would get something similar to the graph above, non-linear curve and all. This is because you end up with a large number of low IQ pseudosibling1 scores (e.g., 100-30=70) that end up showing a large regression upwards. These are individuals that would have had IQs of e.g., 100 and that have siblings with IQs of around 100 but were depressed 30 points. You might think that these scores would average out with those in the case where the other sibling was depressed. That is, for every 100-30 =70 pseudosibling1 (pseudosibling2 IQ = 100) you should have a 70 IQ pseudosibling (pseudosibling2 IQ = 40) — but you don’t because we are dealing with a normal distribution. More 100 IQ individuals will be depressed than 70 IQ individuals because there are more of the former. Now, we might suppose that the depressive effect is somehow systematic such that it depresses the IQs of only our second sibling. Maybe every second sibling born is depressed 30 points. In that case, we get a sibling depression that looks as follows:
The regression lines are parallel but now there’s a two standard deviation difference — almost three times as large as the difference found. Moreover, the intraclass sibling correlations come out as negative. This is because these correlations take into account the magnitude of the absolute sibling differences. To put the above point into perspective, the figure immediately below shows the full sibling intraclass correlations for Blacks and White for the NLSY97 and NLSY79. And the figure immediately after this shows the intraclass correlations that would result were the second of the White NLSY79 siblings depressed various amounts. As can be seen, only a “systematic” unshared depressive effect of 0.6 per sib pair — and therefore 0.3 SD overall — is somewhat consistent with the actual correlations found.
In the more realistic situation were the 0.6 per sib pair difference is normally distributed (SD=.3), the intraclass correlation drops to 0.405. Generally speaking, the Black intraclass full sibling correlations places a limit on the amount of between population variance that could possibly be explained by unshared environmental effects.
(2) The second graph shows the sibling regression lines produced by a partially uniformly distributed shared environmental difference which induces 1.2 SD of between population difference, where 10% of the depressed population is unaffected. Here we see that the regression line for the pseudo Whites is linear and that it converges with that of the Whites as IQ increases. This is because the 10% percent of unaffected sib pairs have higher IQs, on average, being unaffected, and because unaffected individuals will show no differential regression.
(3) The third graph shows the sibling regression lines produced by a normally distributed shared environmental difference which induces 1.2 SD of between population difference, where the standard deviation of depressive effect is 0.6. In this case, ~2.2% of the pseudo White groups is completely unaffected. Here, again, we see that the regression line for the pseudo Whites is linear and that it converges with that of the Whites as IQ increases. This is, again, because the less affected sib pairs have higher IQs and so higher IQ individuals tend to be less affected and so show less differential regression.
(4) The fourth graph shows the sibling regression lines produced by a normally distributed shared environmental difference which induces 1.2 SD of between population difference, where the standard deviation of depressive effect is 0.3. In this case, ~0.15% of the pseudo White groups is completely unaffected. Here, again, we see that the regression line for the pseudo Whites is linear but we also see that it shows little convergence. This is because the variability in depressive effect is minimal.
We can now compare the above theoretical results to the actual results presented by Meng Hu, which look like:
Clearly (1) and non shared environmental effect models, in general, are untenable. The Black regression line is linear — and, moreover, the magnitude of the found difference in regression lines at its maximum is not in line with this type of model. Models (2) and (3), which represent shared environmental effects, are also untenable, since the results of Hu (2013, April, 18) and Murray (1999) show that the differential regression lines do not converge or even narrow with increasing IQ. On the other hand, the found results are somewhat consistent with model (4), that is, with a shared environmental model which proposes that the standard deviation of the depressive effect is less than 0.6. Other considerations, particularly concerning measurement invariance, imply that the SD of the depressive effect can not be zero or near zero — as this is equivalent to having an x-factor. If so, this would violate MI, but MI has repeatedly been found to hold in the case of the Black-White differential. As such, any tenable environmental hypothesis must be a mostly shared environmental hypothesis which proposes that the effect depressing the Black mean is narrowly distributed in the Black population i.e., 0 > SD < 0.6.
Now, to note, for comparison, the standard deviation of depressive effect due to shared environment within populations is about 0.45 SD (note 1). So this hypothesized narrow variability in the between group difference (i.e., the variance in how much Blacks would be adversely impacted, on the account of the hypothesized shared environmental difference, relative to Whites) is not overly inconsistent with the variability in the with group differences within the Black population (i.e., the variance in how much Blacks are adversely impacted, on the account of shared environment, relative to Blacks). Readers familiar with this debate will spot the problem, though: the amount of total shared environmental difference between populations that would be needed to account for the (1.2 SD) difference would be 2.7 SD (note 2, 3). This simply does not exist. There have been attempts to deconstruct the estimates of shared environmentality, but these are of no help, in this case, because if the variance due to shared environmentality is increased within population it will likely be so between populations leading to differential regression results that match with models (2-3). To put this point another way, you can shared environmentally explain the differential regression results by positing that there is not much variance in depressive effect and you can explain the dearth of this variance by pointing out that there is not much within population variance on the account shared environment, but then this leads, inevitably, to the question of why there are such large differences between populations in the first place given the impotency of shared environments. If you try to explain this by shattering heritability (or shared environmentality) estimates, you are led back to the problem of non-convergence of differential regression lines. The only apparent escape from this catch 22 is simply to not deal with the totality of the evidence — and this generally seems to be the strategy employed.
Meng Hu claimed:
One cannot even begin to explain why blacks should be more environmentally depressed relative to whites at higher levels of IQ
I agree with Meng Hu that this situation is curious. Nonetheless, it seems to me that the general regression to the mean results, while consistent with a additive genetic model of group differences, are also, at least when taken in isolation, consistent with some shared environmental models. Another possibility is that the found differential regression slopes could be due to some combination of shared environmental and unshared environmental effect. It might be worthwhile to explore these models. With regards to shared environmental models, no models by which an appreciable portion (e.g., more than 2.5%) of the Black population is unaffected are tenable. Likewise, by all tenable models, more than 85% of the Black population must be depressed by at least 0.6 SD (using a conservative estimate).
Are the remaining environmental models plausible? In my estimation, no — when taking into account the totality of the evidence. But to answer this query properly, we would have to look at specific models and explanations or classes of them and evaluate them in particular. In general, it is curious, though, that between the time of Jensen's early work and the NLSY 97 the regression lines have not begun to converge — as would have happened if a non-trivial portion of the Black population (e.g., 10%) managed to escape the cumulative effect depressing the Black mean. Eventually, if the gap is to close, subsections of the African-American population will need to escape the mysterious cognitive depressing effect, an event which will result in a convergence of the sibling regression lines at the right end of the spectrum (model 2, 3). Insofar as there is no convergence, again, the results are consistent with an additive genetic model.
(1) It would be: SQRT((SD^2)*C^2)), where SD is the variance in the trait and C^2 is the portion of variance due to shared environment. The C^2 found for IQ is typically around 0.2, age depending.
(2) The difference in g-scores is ~1.2 SD. The shared envrionmentality is about 0.2– and so the correlation between cumulative shared environmental effect and g-scores is about 0.4 (e.g., SQRT(0.2). The amount of cumulative shared environmental effect, then, needed to explain a ~1.2 SD difference would then be ~1.2/0.4 or 2.7 SD.
(3) Some have argued that there are 2.7 or so standard deviations of shared environmental effects between contemporaneous Black and White Americans on the account that there are 2.7 or so standard deviations in cumulative “environmental” differences (e.g., Fryer and Levitt). But those who have have failed to grasp the distinction, among other things, between environmental factors and external factors. The shared envrionmentality of external factors (e.g., parental income, number of book read to by parents, peer groups, home “environment”) is only about 0.5 (Vinkhuyzen et al., 2009; Plomin and Bergeman, 1991). As such, the total amount of difference in cognitive affecting external factors needed to account for the difference would have to be 2.7 SD/ SQRT(0.5). or 3.6 SD. There would have to be almost no overlap. Also, the relevant amount of cumulative external factor difference would not be the sum of the effects but the sum of the independent effects of external factors determined with multiple regressions. Interested readers can explore the possibility that there is such a difference using the publicly available NLSY79 Children and Young Adults survey.
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