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Stat Appl Genet Mol Biol. 2010 January 1; 9(1): 1.
Published online 2010 January 6. doi:  10.2202/1544-6115.1517
PMCID: PMC2861312

Epistatic Interactions*

Abstract

The term “epistasis” is sometimes used to describe some form of statistical interaction between genetic factors and is alternatively sometimes used to describe instances in which the effect of a particular genetic variant is masked by a variant at another locus. In general statistical tests for interaction are of limited use in detecting “epistasis” in the sense of masking. It is, however, shown that there are relations between empirical data patterns and epistasis that have not been previously noted. These relations can sometimes be exploited to empirically test for “epistatic interactions” in the sense of the masking of the effect of a particular genetic variant by a variant at another locus.

Keywords: epistasis, gene-gene interaction, synergism

Introduction

Writing in 1909, Bateson used the term “epistasis” to describe instances in which the effect of a particular genetic variant was masked by a variant at another locus so that variation of phenotype with genotype at one locus was only apparent amongst those with certain genotypes at the second locus (Bateson, 1909). In recent papers, Cordell (2002, 2009) has argued that the statistical tests that are often used to assess interactions (Ritchie et al., 2001; Hahn et al., 2003; Moore, 2004; Chung et al., 2007; Purcell et al., 2007; Zhang and Liu, 2007; Ferreira et al., 2007; Gavan et al., 2008) are of limited use in elucidating the type of biologic interaction that Bateson had originally conceived. Recent developments have extended interaction tests for case-control design to settings of case-only designs (Piegorsch et al., 1994; Khoury and Flanders, 1996; Yang et al., 1999; Weinberg and Umback, 2000) and to family-based association studies (Cordell and Clayton, 2002; Cordell et al., 2004; Laird and Lange, 2006; Martin et al., 2006; Kotti et al., 2007; Lou et al., 2008; Hoffmann et al., 2009); however, these developments are arguably also subject to Cordell’s critique (2002). In this paper, it is argued that there are relations between empirical data patterns and epistasis in the sense of masking that have not been previously noted and that can sometimes be exploited to empirically test for epistasis as originally conceived by Bateson.

Empirical Tests for Epistasis

Under Bateson’s original conception, epistasis would be said to be present if variation of phenotype with genotype at one locus was only apparent amongst those with certain genotypes at the second locus. Those with other genotypes at the second locus would show no effect at the first. Consider first a setting in which genotypes at both loci can effectively be considered binary as in Table 1; below we will consider more general settings.

Table 1.
Example of a table of phenotypes for a particular individual for the effects of different genotypes at two loci exhibiting epistasis under Bateson’s (1909) original definition

Table 1 describes a potential phenotype pattern for a particular individual such that the effect of genotype at locus A is only present for the B/B variant; if the genotype at locus B is not B/B then the effect of variation at locus A is not apparent. The effect of genetic variation at locus A can be masked by that at locus B and we might say that that locus B is epistatic to locus A. By symmetry in this example, it is also the case the effect of genotype at locus B is only present for the A/A variant so the effect at locus B can be masked by that at locus A and we might thus also say that that locus A is epistatic to locus B (Cordell, 2002).

In this simple setting in which the genotype at these two loci can effectively be considered binary, one way to conceive of epistasis then is whether there are any individuals for whom the response pattern follows that in Table 1. In populations with heterogeneity and for complex traits with non-Mendelian inheritance, the response patterns may vary between individuals but we may be interested whether there are any individuals whose phenotype response patterns manifest such epistasis. Let X1 be a binary indicator for genotype at locus A (in the example, X1=0 for genotype a/a or a/A and X1=1 for genotype A/A); let X2 be a binary indicator for genotype at locus B (in the example, X2=0 for genotype b/b or b/B and X2=1 for genotype B/B). Let D be a binary indicator of phenotype, indicating the presence of some dichotomous trait. For each individual in the population let Dij denote what the trait would have been if X1 were i and if X2 were j. For each individual we could thus consider D11, D10, D01 and D00 i.e. what would have happened to the individual under the presence or absence of each of the two factors. An epistatic interaction, in Bateson’s original sense of masking, would be present if there were an individual for whom Table 1 describes the phenotype response pattern for that individual. In others words, an epistatic interaction would be present if there were an individual for whom:

D11=1butD10=0,D01=0,D00=0.
(1)

Another way of asking whether there are individuals for whom relation (1) is satisfied is to ask whether there are individuals for whom

D11D10D01D00>0.
(2)

In general, we may not be able to infer what all of D11, D10, D01, D00 would be for a particular individual. However, in a genetic association study we might hope to be able to estimate the values of D11, D10, D01, D00 on average for a population by careful control for confounding by stratification and admixture. If we let C denote a genetic marker for population substructure based on many loci (Pritchard and Rosenberg, 1999; Pritchard, 2000; Satten et al., 2001; Hoggart et al., 2003; Price et al., 2006) then if control for the marker suffices to control for confounding then we can estimate the average likelihood of the outcome D when X1=i and X2=j for those with genetic marker C=c by:

P(Dij=1|C=c)P(D=1|X1=i,X2=j,C=c).
(3)

If marker C suffices to control for confounding, then the average effects of genetic factors X1 and X2 on D can be estimated with data and (3) will hold (Hernán, 2004). We could thus test whether there are any individuals with C=c whose response patterns satisfy (2) (i.e. for whom the response pattern is that in Table 1) by testing:

p11cp10cp01cp00c>0
(4)

where pijc = P(D=1|X1=i,X2=j,C=c). Provided the genetic marker C suffices to control for confounding by stratification and admixture so that relation (3) holds, then if for some value of the genetic marker we find that p11c – p10c – p01c – p00c > 0 then there must be some individuals with C=c for whom the response pattern is given by Table 1 i.e. for whom an epistatic interaction, in the sense of Bateson, is present. Condition (4) is not the usual statistical test for interaction but it can be tested empirically from data to draw conclusions about whether there are at least some individuals with an epistatic response pattern. We will discuss below the relationship between condition (4) and the usual statistical tests for interactions. Note that the implication described above is one-way; if condition (4) is satisfied then there are individuals for whom D11=1 but D10=0, D01=0, D00=0; however, if condition (4) is not satisfied we cannot necessarily conclude that there are no individuals for whom D11=1 but D10=0, D01=0, D00=0. Condition (4) is a sufficient condition for an epistatic interaction but not a necessary condition.

Using the same logic as that given above, we could empirically test for other epistatic response patterns. We could test whether there are individuals for whom D00=1 but D11=D10=D01=0 by testing p00c – p11c – p10c – p01c > 0; we could test whether there are individuals for whom D10=1 but D11=D01=D00=0 by testing p10c – p11c – p01c – p00c > 0; finally, we could test whether there are individuals for whom D01=1 but D11=D10=D00= 0 by testing p01c – p11c – p10c – p00c > 0.

Tests under Additional Assumptions

In some cases, we might be willing to assume that genotype X1=1 (as compared with X1=0) never prevents the outcome so that if D00=1 then it is also the case that D10=1 and if D10=0 then it must also the case that D00=0 and similarly, if D01=1 then it is also the case that D11=1 and if D11=0 then it must also the case that D01=0. In such cases, X1=1 (as compared with X1=0) is never preventive in that it has either a neutral or causative effect on all individuals. Such cases of no preventive effects are sometimes referred to as monotonicity relationships. Stated more succinctly, we may say that X1 has a monotonic effect on D if for all individuals in the population, D1j≥ D0j for j=0,1. Similarly, we say that X2 has a monotonic effect on D if for all individuals in the population, Di1≥ Di0 for i=0,1. Monotonicity is a strong assumption and will often not hold. For X1 to have a monotonic effect on D, it must be the case that X1=1 (as compared with X1=0) is either neutral or causative of the outcome D=1 for all individuals in the population i.e. X1=1 (as compared with X1=0) never prevents the outcome. For X2 to have a monotonic effect on D, it must be the case that X2=1 (as compared with X2=0) is either neutral or causative of the outcome D=1 for all individuals in the population i.e. X2=1 never prevents the outcome. Whenever a particular genetic variant is such that it makes the outcome more likely in some populations but less likely in others, this monotonicity relation will not hold. In some cases, monotonicity of X1 or X2 might hold within certain strata of a genetic marker C but not in others.

When monotonicity assumptions do hold, we can test for epistatic interactions by testing a condition weaker than that given in (4) above. Suppose for example that X1 had a monotonic effect on D then if it were the case that D10 were 0 then it must also be the case that D00 is 0. Thus if X1 had a monotonic effect on D and there were individuals for whom D11=1 and D10=D01=0 then we could also conclude for such individuals that D00=0 by monotonicity and thus that relation (1) held for such individuals, i.e. that an epistatic interaction as given in Table 1 was present. If some genetic marker C suffices to control for confounding by stratification and admixture then we could test whether there were individuals for whom D11=1 and D10=D01=0 within stratum C=c by testing

p11cp10cp01c>0
(5)

Note that condition (5) is a weaker condition than condition (4); condition (5) does not require subtracting p00c. When we can assume that the effect of X1 on D is monotonic (i.e. never preventive) then we can test this weaker condition instead. By symmetry, it is also the case that if X2, rather than X1, has a monotonic effect on D, then condition (5) can be used to test whether there are individuals with a response pattern like that given in Table 1. We seen then that if either X1 or X2 has a monotonic effect on D then we can use the weaker condition (5), rather than condition (4), to test for epistasis in the sense of Bateson (1909). Once again, however, condition (5) does not correspond to a standard statistical test for interaction.

Finally suppose that both X1 and X2 have monotonic effects on D. Suppose X1=1 (as compared with X1=0) never prevented the outcome for any individual and X2=1 (as compared with X2=0) never prevented the outcome for any individual. Stated another way, we are supposing that Dij is non-decreasing in i and j. In the appendix we show that if some genetic marker C suffices to control for confounding by stratification and admixture then we could test whether there were individuals for whom D11=1 and D10=D01=D00=0 within stratum C=c by testing

p11cp10cp01c+p00c>0
(6)

Note that condition (6) is weaker than both conditions (4) or (5) because now we are adding back the term p00c. Condition (6) is how interaction is often ordinarily assessed in statistical models; condition (6) essentially examines whether the effects of X1 and X2 combined are greater than the sum of the effects of X1 and X2 considered separately. However, condition (6) will only imply individuals with the epistatic response pattern in Table 1 if it can be assumed that both X1 and X2 have monotonic effects on D. In other words, if there are any individuals with C=c for whom the outcome would be present if X1=0 but for whom it would not be present if X1=1 (or similarly for whom the outcome would be present if X2=0 but for whom it would not be present if X2=1) then the monotonicity conditions would be violated and one could not use condition (6) to test for epistatic response patterns. We have seen above that even if these monotonicity assumptions are violated then condition (4) or (5) could be used to test for epistasis in the sense of masking; however, condition (6) which is the usual test for interaction, only gives a test for epistasis, in the sense of masking, under strong monotonicity assumptions for both genetic factors.

More General Settings

Consider now the more setting in which at loci A and B there are three distinct relevant genotypes: a/a, a/A and A/A at locus A and b/b, b/B and B/B at locus B. Now let V1 and V2 be variables with three levels indicating the genotype at loci A and B respectively (e.g. V1=0 for a/a, V1=1 for a/A, V1=2 for A/A and V2=0 for b/b, V2=1 for b/B, V2=2 for B/B). Once again, let D be a binary indicator of phenotype, indicating the presence of some dichotomous trait. For each individual in the population let Dij denote what the trait would be if V1 were i and if V2 were j. Again let C denote a genetic marker for population substructure and suppose that the marker suffices to control for confounding by stratification and admixture so that P(Dij=1|C=c) ≈ P(D=1|V1=i,V2=j,C=c).

As before we will let pijc = P(D=1|V1=i,V2=j,C=c). We can then consider a variety of response patterns that would constitute instances of epistasis. For simplicity now assume that the effects of V1 and V2 on D are monotonic so that whenever i≥i’ we have Dij≥ Di’j and whenever j≥j’ we have Dij≥ Dij’. Consider the response pattern in Table 2.

Table 2.
Example of a table of phenotypes for the effects of genotypes at two loci exhibiting epistasis, with three relevant genetic variants at each locus

By arguments similar to those given above, there must be individuals with genetic marker C=c who have response patterns given by Table 2 if it is the case that

p22cp21cp12c+p11c>0
(7)

Note that for the response pattern given in Table 2, the effect of genetic variation at locus A is only apparent when the genotype is B/B at locus B (similarly, the effect of genetic variation at locus B is only apparent when the genotype is A/A at locus A). Thus the response pattern in Table 2 would be another instance which would be considered epistasis under Bateson’s original conception. If control for genetic marker C suffices to control for confounding by stratification and admixture and if condition (7) holds then there must be some individuals with genetic marker C=c who have response patterns given by Table 2. We can once again test for epistasis empirically.

A number of other response patterns constituting instances of epistasis are also possible. Consider, for example, the response pattern in Table 3.

Table 3.
Example of a table of phenotypes for the effects of genotypes at two loci exhibiting epistasis, with three relevant genetic variants at each locus

There must be individuals amongst those with genetic marker C=c who have response patterns given by Table 3 if it is the case that

p12cp21cp02c+p01c>0.
(8)

There must be individuals amongst those with genetic marker C=c who have response patterns given by Table 4 if it is the case that

p21cp12cp20c+p10c>0.
(9)

There must be individuals amongst those with genetic marker C=c who have response patterns given by Table 5 if it is the case that

p11cp21cp12c+p00c>0.
(10)
Table 4.
Example of a table of phenotypes for the effects of genotypes at two loci exhibiting epistasis, with three relevant genetic variants at each locus
Table 5.
Example of a table of phenotypes for the effects of genotypes at two loci exhibiting epistasis, with three relevant genetic variants at each locus

The tests represented by conditions (7)–(10) presupposed that the effects of V1 and V2 on D were monotonic (i.e. never preventive). In the appendix, we consider tests for the epistatic interactions given in Tables 25 without monotonicity assumptions or when only one of V1 or V2 have a monotonic effect on D. We also consider tests when one factor has two levels and the other has three levels. The basic point here is that the approach described in the previous section to empirically test for epistasis can be employed even when genotypes are considered to have three possible relevant variants rather than two.

Relation to Statistical Models

In this section we will briefly relate the empirical tests for epistasis to standard tests for interactions in statistical models. For simplicity we will again return to the setting in which the relevant genotypes can effectively be considered binary as in Table 1. Similar remarks apply to the more general settings described in the previous section. When two binary genetic variants are considered, a statistical model of the following form is sometimes used to test for a statistical interaction:

P(D=1|X1=x1,X2=x2)=α0+α1x1+α2x2+α3x1x2
(11)

To control for confounding by stratification and admixture, one can fit a separate model like (11) within each stratum C=c of some genetic marker. Statistical interaction is then often assessed by testing whether α3>0. Testing whether α3>0 corresponds to a test of condition (6). We saw above that condition (6) can be used to test for epistasis in the sense of masking only under the strong assumption that both X1 and X2 have monotonic effects on the outcome. However, we also saw that even if these monotonicity assumptions do not hold, we could still test for epistatic response patterns; however, we would have to use more stringent conditions like (4) or (5).

We can also express conditions (4) and (5) in terms of the coefficients of the statistical model in (11). We saw above that we could use condition (5) to test for epistasis in the sense of masking if at least one of X1 or X2 has a monotonic effect on the outcome. Condition (5) can expressed in terms of the coefficients of statistical model (11) as α3> α0. Thus if at least one of X1 and X2 had monotonic effects on the outcome then we could test for such epistasis by testing whether α3> α0. Even if neither X1 nor X2 had monotonic effects (i.e. we make no assumptions about monotonicity) we could still test for epistasis in the sense of masking by testing condition (4). Condition (4) can be expressed in terms of the coefficients of statistical model (11) as α3> 2α0. Thus even without making any assumptions about monotonicity we could test for such epistasis by testing whether α3> 2α0. These are non-standard tests for interaction but, when satisfied, allow for conclusions to be drawn not just about statistical interaction but about epistatic response patterns. As noted above, these tests are sufficient conditions for epistasis in the sense of masking, but not necessary. If the conditions are satisfied then there are at least some individuals with response patterns manifesting epistasis in Table 1. If the conditions are not satisfied then there may or may not be individuals with response patterns exhibiting epistasis; we cannot tell from the data.

Multiple Uses of the Word “Epistasis”

Cordell and Clayton (2005) note that although Bateson conceived of epistasis in terms of the masking of the effect of one genetic factor by another, the term “epistasis” soon began to take on a variety of meanings. They note that not long after Bateson, Fisher (1918) used the term “epistacy” to refer to a statistical interaction in the sense of deviation from additive effects such as α3>0 in the statistical linear model given above. Cordell and Clayton further argue that Fisher’s “epistacy” quickly evolved into “epistasis” so that in the modern genetics literature the two uses of the word coexist creating ambiguity. Cordell (2002, 2009) argues that epistasis in the statistical sense does not in general imply epistasis in original sense of Bateson, the sense of the masking of the effect of one genetic factor by another.

We have seen here, however, that the two uses of the word “epistasis” are not entirely unrelated. In particular, in some very special circumstances epistasis in the statistical sense (α3>0) implies epistasis in the sense of the masking the effect of one genetic factor by another (D11=1 but D10=D01=D00=0). More specifically, epistasis in the statistical sense (α3>0) implies epistasis in the sense of the masking the effect of one genetic factor by another only when it can be assumed that both X1 and X2 had monotonic effects on the outcome. This is a very strong assumption and one which in many contexts will not hold. When it does not hold, statistical epistasis does not imply epistasis in the sense of masking. However, we have also seen in this section, that there are further relationships between statistical models and data patterns on the one hand and epistasis in the sense of masking on the other; these relations have been previously unrecognized. We have seen that if at least one of X1 or X2 have a monotonic effect on the outcome then we can test for epistasis in the masking sense (D11=1 but D10=D01=D00=0) by testing whether α3> α0. Even without any monotonicity assumptions, we can test for epistasis in the masking sense by testing whether α3> 2α0. Again, these are stronger conditions than regular tests for statistical interactions.

In a recent review article, Phillips (2008) also discusses the ambiguity in the term “epistasis” and he distinguishes what he considers as three distinct forms of epistasis. Phillips used “statistical epistasis” to refer to a departure from additive effects in a statistical model (or more generally a departure from independent effects on some scale of measurement). Phillips introduced the term “compositional epistasis” to refer to epistasis in Bateson’s original sense of the term, i.e. the masking of the effect of an allele at one locus by an allele at another locus. Finally, Phillips used “functional epistasis” to describe the physical molecular interactions between various proteins (and other genetic elements). Compositional epistasis, as defined by Phillips, need not necessarily imply “functional epistasis” but compositional epistasis is nevertheless arguably a more biological form of interaction than mere “statistical epistasis.” The tests described in the previous sections constitute empirical tests for what Phillips referred to as “compositional epistasis.”

Testing for Epistatic Interactions in Case-Control Studies

Many analyses of interaction use data from a case-control study. In such case-control studies risks like p11c, p10c, p01c, and p00c cannot in general be estimated but odds ratios for the effects of genetic factors can be estimated. Thus in such studies, logistic regression is often used which for interaction analyses may take the form of:

logit{P(D=1|X1=x1,X2=x2)}=β0+β1x1+β2x2+β3x1x2
(12)

Model (12) to can be used to calculate odds ratios comparing the odds of the outcome when both X1 and X2 are present to when both are absent (denoted by OR11), the odds when X1=1 and X2=0 to when both are absent (denoted by OR10) and the odds when X1=0 and X2=1 to when both are absent (denoted by OR01). When the outcome is rare these odds ratios approximate the corresponding relative risks, denoted by RR11, RR10, RR01. Although we cannot test conditions (4) or (5) or (6) above directly using risks we could divide these conditions by p00c. Condition (4) becomes

RR11cRR10cRR01c1>0.
(13)

Condition (5) becomes

RR11cRR10cRR01c>0.
(14)

Condition (6) becomes

RR11cRR10cRR01c+1>0.
(15)

Under the assumption that the outcome is rare, these conditions could be tested using the odds ratios from a logistic regression. Thus even in a case-control study one can potentially test for epistasis in settings in which the outcome is rare. The quantity RR11c – RR10c – RR01c + 1 is sometimes described as the “relative excess risk due to interaction” or RERI (Rothman, 1986). The three conditions given above could thus be written respectively as RERI>2, RERI>1, RERI>0. Statistical tests and confidence intervals for this quantity, RERI, are given elsewhere (Richardson and Kaufman, 2009). In a case-control study with a rare outcome, epistasis could be tested by testing RERI>0 if both X1 and X2 can be assumed to have monotonic effects, by testing RERI>1 if one of X1 or X2 can be assumed to have a monotonic effect, and by testing RERI>2 if no monotonicity assumptions are made. Alternatively, it can also be shown (see the appendix) that if the outcome is rare then (4), (5) or (6) will be satisfied, respectively, if for the coefficients in logistic model (12), we have β3>log(3), β3>log(2), or β3>0 provided that the main effects β1 and β2 are non-negative. Similar results hold when one or both factors have three levels; see the appendix for additional discussion.

Relation to Sufficient Causation

The tests described above for epistatic response patterns bear a certain relation to tests for synergism in the sense conceived of by Rothman (1976). Rothman conceptualized causation as a series of mechanisms for the outcome each of which involved the conjunction of various factors such that whenever all the factors for a particular mechanism were present the outcome would occur. Such mechanisms or “sufficient causes” might require the absence of presence of two or more particular factors of interest, X1 and X2, along with other possibly unknown factors. Rothman conceived of synergism being present whenever there was a sufficient cause that required both X1 and X2 to operate. VanderWeele and Robins (2007, 2008) formalized Rothman’s sufficient cause framework and introduced the notion of a sufficient cause interaction. A sufficient cause interaction is present if there are individuals who have a response pattern such as that given in Table 6 i.e. if there are individuals for whom D11=1 and D10=D01=0; D00 can be either 1 or 0. VanderWeele and Robins (2008) showed that a sufficient cause interaction implied synergism as conceived of by Rothman.

Table 6.
Example of a table of phenotypes for two factors, X1 and X2, exhibiting a sufficient cause interaction, implying synergism as conceived by Rothman (1976)

The notion of an epistatic response pattern such as that given in Table 1 is stronger than that of a sufficient cause interaction because the response pattern in Table 1 requires that D00=0. If at least one of the two genetic factors has a monotonic effect on the outcome D then the concepts of an epistatic interaction (“compositional epistasis”) and a sufficient cause interaction between two factors coincide. If neither of the two factors has a monotonic effect on the outcome, then an epistatic interaction is a stronger condition than a sufficient cause interaction. Statistical tests for sufficient cause interactions have been described elsewhere (VanderWeele and Robins, 2007; Vansteelandt et al., 2008; VanderWeele, 2009, VanderWeele et al., in press) and these statistical tests could also be used for epistatic interactions if at least one of the two factors has a monotonic effect on the outcome.

Discussion

Bateson’s original conception of epistasis was that the effect of a gene at one locus would be masked for certain values of the genotype at a second locus. In this paper we have derived conditions that can be tested empirically for detecting whether there are individuals whose response patterns manifest epistasis in the sense of masking originally conceived by Bateson. It was shown that only under some very strong assumptions would tests for regular statistical interactions correspond to epistasis in the masking sense of the term. We have, however, further seen that even without such strong assumptions one can still test whether there are individuals for whom the effect of a gene at one locus would be masked for certain values of the genotype at a second locus. The empirical conditions described above for detecting epistasis are quite strong but the conclusions which tests of these conditions allow (conclusions concerning “compositional epistasis”) may be of interest in a wide range of studies.

The tests derived required control for a genetic marker, denoted in this paper by C, to control for control for confounding by stratification and admixture so that the associations observed between the genes of interest and the outcome at least approximately correspond to the true effects of these genes. The tests will be valid only to the extent that this approximation holds. Other genetic or environmental factors could be included in C to attempt to better control for confounding. When C contains multiple factors, more sophisticated statistical techniques may be desirable to allow for multivariate control.

In many studies, identified genetic risk factors will be in linkage disequilibrium with the true causal genetic factor; in such cases of linkage disequilibrium the genetic risk factor in the study might be conceptualized as a misclassified version of the true causal factor. Future research will examine the extent to which conclusions about epistasis concerning the true causal genetic factors can be drawn from identified genetic risk factors in linkage disequilibrium with the true causal factors.

This paper has focused on epistasis for two genetic factors by considering the response pattern tables that exhibit epistasis as conceived of by Bateson. The tests described in this paper may also be of interest, however, in assessing gene-environment interactions. In particular, the tests that have been described could also be used to detect individuals with particular gene-environment interaction response patterns corresponding to Tables 15; we might refer to such response patterns as instances of “compositional” gene-environment interaction. As described in this commentary, these will only correspond to the ordinary tests for statistical interactions between genetic and environmental factors in very special cases.

It is hoped that the contributions in this commentary have clarified some of the conceptual relationships between epistasis as conceived of by Bateson and statistical tests using data. It is further hoped that the empirical tests for epistasis derived in this paper will be employed in future analyses of genetic data.

Appendix

Proof that relation (6) suffices for epistasis under monotonicity

Here we prove that under the assumption that both X1 and X2 have monotonic effects on D then p11c – p10c – p01c + p00c > 0 implies there is an individual for whom D11=1 and D10=D01= D00=0. Suppose that both X1 and X2 have monotonic effects on D so that Dij is non-decreasing in i and j i.e. for no individual does X1=1 (as compared with X1=0) ever prevent the outcome and for no individual does X2=1 (as compared with X2=0) ever prevent the outcome. Under these monotonicity assumptions if there is an individual for whom D11=1 and D10=D01=0 then it is also the case for that individual that D00=0. Suppose there were no individual for whom D11=1 and D10=D01=0; then whenever D11=1 then we must have that either D10=1 or D01=1. We also have that D10≥ D00 and that D01≥ D00. Thus if for some individual it is not the case that D11=1 and D10=D01=0 then we must have that D11 – D10 – D01 + D00 ≤ 0. Thus if there is no individual in some subpopulation with C=c such that D11=1 and D10=D01=0 then it must be the case that for all individuals with C=c that D11 – D10 – D01 + D00 ≤ 0. From this it would follow that if some genetic marker C suffices to control for confounding by stratification and admixture then if there were no individuals with C=c and with D11=1 and D10=D01=0 then we would have

p11cp10cp01c+p00c0.

From this it follows that if we were to find

p11cp10cp01c+p00c>0

then there must be some individuals with C=c such that D11=1 and D10=D01=0 and by monotonicity it would also be the case for these individuals that D00=0. Thus these individuals would have an epistatic response pattern.

Further tests for epistasis when the genetic factors have three levels

Here we will consider further tests for epistatic response patterns in which, at each locus of interest, there are three distinct relevant genotypes. As in the paper we will let V1 and V2 denote variables with three levels indicating the genotype at loci A and B respectively (e.g. V1=0 for a/a, V1=1 for a/A, V1=2 for A/A and V2=0 for b/b, V2=1 for b/B, V2=2 for B/B). Once again, let D be a binary indicator of phenotype, indicating the presence of some dichotomous trait. For each individual in the population, Dij denotes what the trait would be for that individual if V1 were i and if V2 were j. We again let C denote a genetic marker for population substructure and suppose that control for the marker suffices to control for confounding by stratification and admixture so that P(Dij=1|C=c) ≈ P(D=1|V1=i,V2=j,C=c). As in the text, we will let pijc = P(D=1|V1=i,V2=j,C=c).

In the main text we considered tests for response patterns exhibiting epistasis such as those in Tables 25; tests were derived that assumed that both V1 and V2 had monotonic effects on D i.e. that Dij was non-decreasing in i and j. Here we describe tests for such epistatic response patterns when only one or neither of the genetic factors has a monotonic effect on D.

Suppose first that only V1 has a monotonic effect on D so that Dij is non-decreasing in i (but possibly not j) for all individuals. Using arguments similar to those in the text, it can be shown that there will be individuals with C=c and with response patterns like those in Table 2 if it is the case that:

p22cp21cp20cp12c>0.

Essentially, if p22c – p21c – p20c – p12c – p02c > 0 then there must be individuals for whom D22=1 but for whom D21=D20=D12=0. But for such individuals if D21=0 then it must also be the case that D11=0 and D01=0 by the monotonicity of V1; similarly by the monotonicity of V1, if D20=0 then it must also be the case that D10=0 and D00=0; finally, by the monotonicity of V1, if D12=0 then it must also be the case that D02=0 We thus have that if p22c – p21c – p20c – p12c > 0 then there must be individuals for whom D22=1 but for whom D21=D11=D01=D20=D10=D00=D12=D02=0 i.e. for whom the phenotype response pattern is given by that in Table 2.

By similar reasoning it can be shown that if V1 has a monotonic effect on D then there are individuals with response patterns like those in Table 3 if it is the case that:

p12cp21cp20cp02c>0.

It is not in general possible to test for epistatic response patterns like those in Tables 4 and and55 if it can only be assumed that V1 has a monotonic effect on D. This is because it is not possible to test for individuals for whom it is the case that both D22=1 and D21=1 without making monotonicity assumptions about V2.

Now consider the case in which it can be assumed that V2 has a monotonic effect on D but for which it may not be reasonable to suppose that V1 has a monotonic effect on D. By similar reasoning to that above it can be shown that if V2 has a monotonic effect on D then there are individual with response patterns like those in Table 2 if it is the case that:

p22cp12cp02cp21c>0.

Likewise, if V1 has a monotonic effect on D then there are individual with response patterns like those in Table 4 if it is the case that:

p21cp12cp02cp20c>0.

It is not in general possible to test for epistatic response patterns like those in Tables 3 and and55 if it can only be assumed that V2 has a monotonic effect on D. This is because it is not possible to test for individuals for whom it is the case that both D22=1 and D12=1 without making monotonicity assumptions about V1.

Now consider the case in which no monotonicity assumptions are made. In such settings, it is not in general possible to test for response patterns like those in Tables 35. One can still test for response patterns like that given in Table 2. However, to conclude that there are individuals with response patterns like those in Table 2 without making any monotonicity assumptions one would need to test:

p22cp21cp20cp12cp11cp10cp02cp01p00c>0.

If this condition were satisfied and if it could be assumed that genetic marker C suffices to control for confounding by stratification and admixture then it could be concluded that there were individuals with response patterns like those in Table 2 even without making any monotonicity assumptions.

Tests for epistasis when one factor has two levels and one factor has three

Suppose now that V1 has two levels and V2 has three levels indicating the genotype at loci A and B respectively (e.g., V1=0 for genotype a/a or a/A and V1=1 for genotype A/A and V2=0 for b/b, V2=1 for b/B, V2=2 for B/B). Let D denote an indicator for dichotomous trait, Dij denote what the trait would be for an individual if V1 were i and if V2 were j, and C denote a genetic marker for population substructure. The effect of V1 or V2 on D is said to be monotonic if Dij is non-decreasing in i or j respectively. We suppose that control for the marker suffices to control for confounding by stratification and admixture so that P(Dij=1|C=c) ≈ P(D=1|V1=i,V2=j,C=c). We again let pijc = P(D=1|V1=i,V2=j,C=c).

Epistasis, in the sense of masking, would be present if there were individuals for whom

D12=1butD11=0,D10=0,D02=0,D01=0,D00=0.

Using arguments similar to those above, if the effects of V1 and V2 on D are both monotonic then there are individuals with the epistatic response pattern above if

p12cp11cp02c+p01c>0.

If only the effect V1 on D can be assumed to be monotonic then there are individuals with the epistatic response pattern above if

p12cp11cp10cp02c>0.

If only the effect V2 on D can be assumed to be monotonic then there are individuals with the epistatic response pattern above if

p12cp11cp02c>0.

If neither the effect of V1 or V2 can be assumed to be monotonic then there are individuals with the epistatic response pattern above if

p12cp11cp10cp02cp01cp00c>0.

Epistasis, in the sense of masking, would also be present if there were individuals for whom

D12=D11=1butD10=0,D02=0,D01=0,D00=0.

Using arguments similar to those above, if the effects of V1 and V2 on D are both monotonic then there are individuals with the epistatic response pattern above if

p11cp10cp02c>0.

If only the effect V1 on D or if neither the effect of V1 or V2 can be assumed to be monotonic then it is not possible to detect this second epistatic response pattern simply using observed outcome probabilities.

Tests for epistasis in case-control studies using logistic models

Suppose that the outcome is rare so that odds ratios approximate risk ratios. Consider model (12) in the text:

logit{P(D=1|X1=x1,X2=x2)}=β0+β1x1+β2x2+β3x1x2

Suppose it is known a priori that the main effects β1 and β2 are non-negative. The fact that β3>0 implies condition (6) holds at least approximately and that β3>log(2) implies condition (5) holds at least approximately have been shown elsewhere (VanderWeele, 2009). To see that β3>log(3) implies condition (13), RR11c – RR10c – RR01c – 1 > 0, and hence condition (4), p11 – p10 – p01 – p00 > 0, note that:

RR11cRR10cRR01c1OR11cOR10cOR01c1=exp(β1+β2+β3)exp(β1)exp(β2)1=(1/3)exp(β1+β2+β3)exp(β1)+(1/3)exp(β1+β2+β3)exp(β2)+(1/3)exp(β1+β2+β3)1=exp(β1)((1/3)exp(β2+β3)1)+exp(β2)((1/3)exp(β1+β3)1)+((1/3)exp(β1+β2+β3)1)

If β3>log(3) and also the main effects β1 and β2 are non-negative, then each of the terms, ((1/3)exp(β2 + β3) – 1) and ((1/3)exp(β1 + β3) – 1) and ((1/3)exp(β1 + β2 + β3) – 1) will be positive and thus we will have that RR11c – RR10c – RR01c – 1 > 0 and consequently also p11 – p10 – p01 – p00 > 0 i.e. condition (4) will be satisfied.

Using similar arguments and the results of VanderWeele (in press), similar relations can be obtained when one or both of the two exposures have three levels. In such cases, what VanderWeele (in press) defined as “definite interdependence” will correspond to an epistatic response pattern if at least one of the two factors have a monotonic effect on the outcome; otherwise definite interdependence is a weaker condition than an epistatic response pattern.

Suppose V1 has two levels and V2 has three levels and we use the regression model:

logit{P(D=1|V1=v1,V2=v2)}=β0+β1x1+β2x2+β3x3+β4x1x2+β5x1x3

where X1=1 if V1= 1 and 0 otherwise, X2=1 if V2[set membership]{1,2} and 0 otherwise and X3=1 if V2=2 and 0 otherwise. Suppose further that the outcome is rare and that P(D=1|V1= v1,V2= v2) is non-decreasing in v1 and v2. Then there are individuals with epistatic response pattern D12=1 and D11=0, D10=0, D02=0, D01=0, D00=0 if (i) β5>0 and both V1 and V2 have monotonic effects on D or if (ii) β5>log(2) and just V2 (the factor with three levels) has a monotonic effect on D or if (iii) β5>log(3) and just V1 (the variable with two levels) has a monotonic effect on D or if (iv) β5>log(5) and it is not assumed that either V1 and V2 have monotonic effects on D.

Similarly, it is the case that there are individuals with epistatic response pattern D12= D11=1 and D10=0, D02=0, D01=0, D00=0 if (i) β4–β3>0 and both V1 and V2 have monotonic effects on D or if (ii) β4–β3>log(2) and just V2 has a monotonic effect on D. If only V1 has a monotonic effect on D or neither V1 nor V2 have a monotonic effect on D then it is not in general possible to detect this second type of epistatic response pattern.

Suppose V1 and V2 have three levels and we use the regression model:

logit{P(D=1|V1=v1,V2=v2)}=β0+β1x1+β2x2+β3x3+β4x4+β5x1x3+β6x1x4+β7x2x3+β8x2x4

where X1=1 if V1[set membership]{1,2} and 0 otherwise and X2=1 if V1=2 and 0 otherwise and similarly X3=1 if V2[set membership]{1,2} and 0 otherwise and X4=1 if V2=2 and 0 otherwise. Suppose further that the outcome is rare and that P(D=1|V1= v1,V2= v2) is non-decreasing in v1 and v2. Then there are individuals with the epistatic response pattern of that in Table 2 if (i) β8>0 and both V1 and V2 have monotonic effects on D or if (ii) β8>log(3) and either V1 or V2 have a monotonic effect on D or if (iii) β8>log(8) and it is not assumed that either V1 and V2 have monotonic effects on D.

There are individuals with epistatic response pattern of that in Table 3 if (i) β6–β7–β2>0 and both V1 and V2 have monotonic effects on D or if (ii) β6–β7–β2>log(3) and V1 has a monotonic effect on D. There are individuals with epistatic response pattern of that in Table 4 if (i) β7–β6–β4>0 and both V1 and V2 have monotonic effects on D or if (ii) β7–β6–β4>log(3) and V2 has a monotonic effect on D. There are individuals with epistatic response pattern of that in Table 5 if β5–β4–β2>0 and both V1 and V2 have monotonic effects on D.

Alternatively, with case-control data with a rare outcome, one could test the conditions in the previous two sections of the appendix somewhat more directly. Each of the conditions in the previous two sections of the appendix could be divided by p00 to express the conditions in terms of risk ratios; because the outcome is rare, the risk ratios will be approximated by odds ratios which can be obtained from the logistic regression models. This approach was described in the text for two binary genetic factors but applies also to settings in which one factor has two levels and the other three or to settings in which both factors have three levels.

Footnotes

*The author thanks David Clayton for a presentation at the Channel Network Conference of the International Biometric Society that in part prompted the development of these results. The author also thanks Jonathan Pritchard, Nan Laird, the editor and two anonymous referees for helpful comments on this paper. This research was supported by NIH grant R01 ES017876.

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