Empirical tests for compositional epistasis.

In recent papers, Cordell1,2 has argued that statistical tests for interactions are of limited use in elucidating epistasis in a more biological sense of the term in which the effect of a particular genetic variant is masked by a variant at another locus3,4 (“compositional epistasis”5). Suppose two genetic factors, X1 and X2, are binary indicators for genotypes at loci A and B respectively so that X1=0 denotes genotype a/a or a/A and X1=1 genotype A/A and similarly X2=0 denotes genotype b/b or b/B and X2=1 genotype B/B; suppose D is a dichotomous trait. Epistasis in the sense of masking1,3 would be present if there were individuals for whom locus A had no effect without the B/B genotype at locus B, as in Table 1. Table 1 Example of a table of phenotypes for a particular individual for the effects of different genotypes at two loci exhibiting “compositional” epistasis5 in the sense of masking1–3 A statistical model, accommodating statistical interaction, might be formulated as: P(D=1∣X1=x1,X2=x2)=α0+α1x1+α2x2+a3x1x2. Cordell1,2 points out that statistical interaction tests (e.g. α3>0) do not generally allow for conclusions about epistasis in the more biological sense of masking. There are, however, relations between empirical data patterns and compositional epistasis, as in Table 1, that have not been previously noted and that can be used to derive nonstandard interaction tests to empirically test for such epistasis6. Elsewhere I have shown that, provided the effects of the genetic factors are not confounded by stratification or admixture (or alternatively if appropriate control for these has been made7–9), then if α3>2α0 then there must be some individuals with phenotype response pattern of Table 1 i.e. compositional epistasis is present6. If one of the genetic factors is such that its effect is in the same direction for all individuals (i.e. it is not causative for some individuals and preventive for others) then α3>α0 implies compositional epistasis6. Only when both factors are such that their effects are in the same direction for all individuals does the standard interaction test α3>0 imply compositional epistasis6. In most cases, there will be insufficient prior knowledge to make these “monotonicity” assumptions and one would have to test α3>2α0 to detect epistasis in the sense of masking. Suppose instead a log-linear model is used: log{P(D=1∣X1=x1,X2=x2)}=β0+β1x1+β2x2+β3x1x2. It can be shown6, that β3>log(3) implies the existence of individuals with phenotype response pattern of Table 1 (i.e. compositional epistasis) provided β1≥0 and β2≥0. If it can be assumed that the effect of at least one of the factors is monotonic as described above then β3>log(2) with β1≥0 and β2≥0 suffice for compositional epistasis; if it can be assumed the effects of both factors are monotonic then β3>0 suffices. These empirical tests for compositional epistasis presuppose the phenotype probabilities P(D = 1| X1 = x1, X2 = x2) reflect the true effects of the genes; a similar approach to that described above could be used in conjunction with techniques to control for confounding by stratification or admixture7–9. The conditions for compositional epistasis, i.e. epistasis in the sense of masking, are related to but stronger than the notion of “synergism” in Rothman’s sufficient cause framework6,10–12. Compositional epistasis is arguably a more biological notion of epistasis than is “statistical epistasis” but even compositional epistasis need not imply the physical molecular interaction of one protein with another (“functional epistasis”5). The remarks here generalize to settings in which the genetic factors are considered to have three relevant levels. See a fuller report elsewhere6 for additional discussion.