How heritability can mislead, Peter Taylor, 30 Oct. '97 (revised 6 Oct. '04)

Imagine a world of only 4 distinct varieties and 4 distinct environments, and that for each variety-environment combination the almighty has established the ideal value for the attribute in question:

Variety 1Variety 2Variety 3Variety 4
Environment A1.55-0.15-2.550.85
Environment B0.55-1.15-1.55-0.15
Environment C-2.15-0.45-1.150.55
Environment D-3.15-1.45--2.15-0.45

Suppose, however, that social conditions meant that not all the combinations are realized. Inspection of the full data set indicates that if, say, varieties 1&2 experienced environments A&B they would look much better than varieties 3&4 if these experienced environments C&D. Yet, if the environments were switched, the distinction would effectively disappear.

Discounting some technical issues, one can use this data to look at heritability. The figures come out, I think, as follows -- considering first the whole set of possibilities then those for various subsets.

Varieties 1-4 experiencing environments A-D heritability = 11%
Varieties 1&2 experiencing environments A&B heritability = 74%
Varieties 3&4 experiencing environments C&D heritability = 74%
Varieties 2&3 experiencing environments B&C heritability = 20%

Clearly heritability changes according to the subset available for analysis by not-all-seeing behavioral geneticists. Heritability is a statistic that is "local" to the particular data subset, i.e., to the particular set of variety-environment combinations.

The term heritability suggests that it refers to something about actual causal relations, but it does not. High heritability indicates only that, within the particular population of individuals and range of environments surveyed, differences in variety-type for individuals are statistically associated with larger changes in the attribute than are differences among environments for individuals. Perhaps we should call these the "variety differences' contribution to variation among individuals" vs. the "environmental differences' contribution for this particular set of..." Or, the "contribution of differences in variety means (when averaged over observed environments) to variation among individuals etc..." These expressions are long-winded, but using them would avoid the confusion that is endemic because the terms genetic and environmental are also used to think about causation of any individual's IQ, etc.

Notice also that the heritabilty happens to be high for varieties 1&2 experiencing environments A&B, and ditto for 3&4 in C&D. Imagine the attribute was IQ, and the environments were those in the USA experienced by people labeled "white" and "black" respectively.* High heritability within the two groups in their environments does not mean that their difference on average must also be associated with so called "genetic" differences (the differences among the average variety-type for the two groups), but Arthur Jensen and others have argued that it is highly probable. Their argument is that the only alternative is improbable, namely, that there is a "factor X" which affects all individuals in one group differently from all individuals in the other group, and within each group all individuals are affected equally. However, the table above shows that this is not the only alternative. The difference in the averages between the two groups of varieties in their environments is large, but in each variety group there is variation across the different environments in their environment group.
The Jensenian position about the likelihood of "genetic" causes for between groups average differences gets into more trouble once you note that in this example the difference in the averages between the two groups of varieties in their environments consists of a negative effect on average of moving from A&B to C&D (if all varieties and environments are considered, i.e., the whole table) and a positive effect of moving from varieties 1&2 to 3&4. If this tempts you to conclude that, taking the whole world into consideration, "blacks" are genetically better than "whites," remember there is plenty of overlap (one "black" in the favorable environments A&B is lower than three of the "whites" in the unfavorable environments C&D, etc.).
As a take home issue to chew on, notice that the following table preserves the heritability figures for 1&2 in A&B and for 3&4 in C&D as well as the difference between 1&2 and 3&4 in their usual environments, yet the difference in the averages between the two groups of varieties in their environments consists of a zero effect on average of moving from A&B to C&D and a negative effect of moving from varieties 1&2 to 3&4. The world set up by the almighty is quite different here, but the observed world is identical!

Variety 1Variety 2Variety 3Variety 4
Environment A1.55-0.150.55-1.15
Environment B0.55-1.15-0.45-2.15
Environment C-0.151.55-1.150.55
Environment D-1.150.55--2.15-0.45

[*] The figures above would look like IQ scores if you add .3 to each, multiply by 15, then add 100.