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Genet Epidemiol. Author manuscript; available in PMC 2013 June 25.

Published in final edited form as:

Genet Epidemiol. 2009; 33(0 1): S58–S67.

doi: 10.1002/gepi.20474PMCID: PMC3692280

NIHMSID: NIHMS219662

Ping An,^{1} Odity Mukherjee,^{2} Pritam Chanda,^{3} Li Yao,^{4} Corinne D Engelman,^{4} Chien-Hsun Huang,^{5} Tian Zheng,^{5} Ilija P Kovac,^{6} Marie-Pierre Dubé,^{6} Xueying Liang,^{7} Jia Li,^{8} Mariza de Andrade,^{8} Robert Culverhouse,^{9} Doerthe Malzahn,^{10} Alisa K Manning,^{11} Geraldine M Clarke,^{12} Jeesun Jung,^{13} and Michael A Province^{1}

Corresponding Author: Michael A Province, Division of Statistical Genomics, Box 8506, 4444 Forest Park Blvd, Washington University School of Medicine, St. Louis, Missouri 63108 USA, Email: ude.ltsuw@ecnivorpm

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Interest is increasing in epistasis as a possible source of the unexplained variance missed by genome-wide association studies. The Genetic Analysis Workshop 16 Group 9 participants evaluated a wide variety of classical and novel analytical methods for detecting epistasis, in both the statistical and machine learning paradigms, applied to both real and simulated data. Because the magnitude of epistasis is clearly relative to scale of penetrance, and therefore to some extent, to the choice of model framework, it is not surprising that strong interactions under one model might be minimized or even disappear entirely under a different modeling framework.

The term “epistasis” was first described by the English geneticist William Bateson [1907] to denote the suppression of gene expression at one locus by a gene at another locus. However, modern geneticists more often think of epistasis using Fisher’s [1918] conceptualization, as a departure from additivity in the penetrance for two or more loci, in the same way that dominance is a departure from additivity in the penetrance at one locus. Thus, penetrance models of epistasis require additional interaction terms (each with its own corresponding parameter) in and above the additive “main effect” terms for each locus. In this framework a test of epistasis is a test of whether these gene-gene (G×G) interaction term(s) are zero or not, and lack of epistasis represents a special class of all possible multi-locus penetrance functions. It is of course an empirical question whether epistasis plays a major or minor role for any given trait in any particular population or defined subsample. But interest has lately been increasing in epistasis as one possible source of the so called “dark matter” or “missing *R*^{2}” in genome-wide association scans (GWAS) for complex traits (i.e., the fact that the cumulative main effects from GWAS signals account for far less of the total predictive *R*^{2} than the estimated heritabilities of these traits). Correspondingly, analytical methods to detect and estimate the degree of epistasis are becoming more sophisticated and more numerous. In an attempt to compare and contrast the advantages and disadvantages of various epistatic models and methods of detection, investigators participating in Genetic Analysis Workshop 16 (GAW16) Group 9 applied a number of traditional as well as novel methods to three large, complex trait data sets.

Eight groups analyzed the North American Rheumatoid Arthritis Consortium rheumatoid arthritis (RA) data (GAW16 Problem 1 [Amos et al., 2009]): Chanda et al. [2009], Clarke et al. [2009], Huang et al. [2009], Jung et al. [2009], Li et al. [2009], Liang et al. [2009], Manning et al. [2009], and Mukherjee et al. [2009]. Three groups analyzed the Framingham Heart Study (FHS) data (GAW16 Problem 2 [Cupples et al., 2009]): An et al. [2009], Malzahn et al. [2009], and Yao et al. [2009]. Finally, three groups analyzed the simulated data which was based upon FHS (GAW16 Problem 3 [Kraja et al., 2009]): Culverhouse et al. [2009], Kovac and Dube [2009], and Malzahn et al. [2009]. While most groups assessed discrete traits (RA, coronary heart disease, type 2 diabetes), five groups also focused on quantitative traits including triglyceride/high-density lipoprotein ratio [An et al., 2009], anti-cyclic citrinullated peptide and rheumatoid factor IgM [Mukherjee et al.. 2009], low-density lipoprotein [Kovac and Dube, 2009], coronary artery calcification (CAC) and coronary event [Culverhouse et al., 2009], and CAC and body mass index [Malzahn et al., 2009]. Two groups assessed longitudinal quantitative traits in the FHS data including changes in triglyceride/high-density lipoprotein ratio [An et al., 2009] and changes in CAC and body mass index [Malzahn et al., 2009].

Approaches to detecting epistasis can be classified as either **statistical** or **machine learning** methods. Statistical methods make formal models of stochasticity or randomness, and most propose formal hypothesis tests of epistasis. By contrast, machine learning methods [Alpaydin, 2004] tend to be more heuristic, data-mining techniques that do not necessarily rely on formal statistical tests, but concentrate on efficient algorithms to identify epistatic patterns in high-dimensional spaces, such as the space of all possible G×G interactions among a set of candidate genes, for instance. Some machine learning methods do build their search algorithm around formal statistical models. An early example of such a technique is stepwise regression. Each stage in the “variable selection” model-building process is a formal statistical multiple regression model, but using the stepwise algorithm itself to add and delete new predictors is a heuristic way to reasonably search the space for all possible models, and we typically do not worry about such formal statistical issues as corrections for multiple comparisons between submodels, the possibility of partial null hypotheses, or other probability issues concerning multiple models considered simultaneously. In GAW16 Group 9, both statistical models and machine learning approaches were used to attempt to identify epistasis, as summarized in Table I, and discussed below.

The general approach of statistical epistasis methods is to take the null hypothesis as “no epistasis” (additivity), which is preferred unless there is overwhelming evidence in the data in favor of epistasis. A significance level is provided that quantifies the probability of observing the data (or more extreme data) if the null is true (i.e., no epistasis). Thus, in the typical hypothesis testing paradigm, statistical methods tend to conclude in favor of the null of no epistasis, unless there is strong evidence in favor of epistasis. This is also in keeping with standard epidemiological practice about interactions in general, in which we tend to favor simpler models with no interaction over more complex models requiring interaction, under the principle of Occam’s Razor. If we can reasonably model the data assuming additivity, we prefer this over requiring epistasis to explain the data. In fact, many statistical tests for epistasis are done in the context of specified multivariate penetrance models, which tend to be one form or another of generalized linear models (GLMs).

GLM notation developed by Nelder and Wedderburn [1972] is a general framework for describing multivariable linear models. Given a stochastic random variable phenotype, *Y*, and a set of (fixed, non-random) predictors, *X* (which includes genes *G _{i}* and

$$\mathbf{E}[\mathbf{Y}\mid \mathbf{X}]={\mathbf{l}}^{-\mathbf{1}}(\mathbf{D}[\mathbf{X}]\underset{\_}{\mathbf{\beta}}),$$

where **D**[**X**] denotes the design matrix for **X**, which includes as a submatrix [*G _{i}* ||

Clarke et al. [2009] considered modeling a binary trait as being influenced by two bi-allelic disease susceptibility loci, *F* and *G*, according to a model of joint locus effects. Here, *F* denotes a candidate gene single-nucleotide polymorphism (SNP) and *G* denotes an “equilibrium SNP” (i.e., tag SNPs covering a region which themselves are pairwise in low linkage disequilibrium (LD) *r*^{2}<0.2). They tested for G×G interaction between gene and equilibrium SNPs using GLM tests based on logistic odds, proportional odds, and multinomial link functions. For each model, there are two regressions: first, *F* is modeled as the outcome variable and *G* the predictor, then vice versa. The outcome variable is categorized appropriately according to the relevant model: a binary categorization for the logistic model, an ordinal categorization for the proportional odds model, and a nominal categorization for the multinomial model, which result in three different link functions in the GLM formulation. The predictor variable is categorized as an ordinal variable in all three regressions.

Kovac et al. [2009] and An et al. [2009] used a family-mixed model [Borecki and Province, 2008], which is an extension of the multiple regression model, to deal with association in family data. It can overcome the problem of non-independence of residuals within pedigrees that produces inflation of type I error if one applies regular regression and ignores family relationships. This GLM uses a gaussian probability distribution and an identity link function, just as in linear regression, but includes an additional random effect component predictor for pedigrees.

The underlying principle of this method of Jung et al. [2009] is to identify the association of allelic combination between two unlinked markers with a disease trait so that subjects are assigned an allelic score given their observed genotype information. The score is a conditional probability of obtaining the particular allelic combination given the observed genotypes at the two loci of each subject. A linear trend of proportion of cases over total number of subjects at each allelic combination can be modeled using an extension of the Cochran-Armitage trend regression.

Liang et al. [2009] applied the OT of Chatterjee et al. [2006] to detect epistasis. The omnibus method tests for gene-based effects by considering all SNPs in a given gene or region as a single group and evaluates this gene assuming a second known gene or other risk factor plays a role. Specifically, the method forms *L*(*G*) latent factors from linear combinations of *G* loci, and tests the GLM **E[Y|L(G)] = l**
^{−}** ^{1} (D[L(G)] β)** with

Li et al. [2009] extended the original PC approach to test for association between disease and multiple SNPs in a candidate gene in order to incorporate a test for G×G interaction. The procedure involves the following steps. 1) Let *g _{lk}* be the number of minor alleles at SNP

Li et al. [2009] used PCs that explained at least 80% of the variation as the gene representation to perform a G×G interaction analysis by applying logistic regression to test for interaction between every combination of two PCs. Once significant PC interactions were identified, PC loadings were used to determine the influence of a specific SNP on the PCs because the loading represents the correlation of a SNP with a component.

In a GWAS, or even when there are a large number of candidate genes, there can be too many possible G×G interactions to evaluate exhaustively. Manning et al. [2009] utilized two complementary approaches to reduce the number of possible G×G interactions to test. The first strategy is to use a two-stage approach test for interactions only among genes that show significant main effects. In Stage 1, a set of additive GLMs are fit, one variant at a time, and the susbset of variants that show significance are selected for further consideration. In Stage 2, a series of two-variant GLMs are refit, which include every pairwise combination of the main effect subset as well as their respective interactions. The second approach, interactions among pathway genes, is similar in spirit and design. Again, a subset of genes is selected in Stage 1 and interactions are only evaluated among genes in that subset, but in this strategy, the Stage 1 subset is selected based on external biological knowledge that genes belong to the same relevant pathway, rather than based internal statistical tests from the data itself, interactions among pathway genes.

Li et al. [2009] modified the two-step approach of Murcray et al. [2009] to detect gene-environment interaction to be applied to detect G×G interaction. In the first step, a GLM model is fit predicting *G* from *F* in the combined case-control data, using the approximate method to detect epistasis in PLINK (note that this analysis does not involve phenotype, only the two genotypes). This can be considered as a modified version of the case-only approach for epistasis. Only those SNPs that show significant epistasis in Step 1 are carried forward to Step 2, in which a saturated logistic model (GLM with binomial distribution and logit link) is fit. The test of the G×G product term is the final test of epistasis.

Malzahn et al. [2009] adopted a test statistic from the area of clinical studies [Brunner et al., 2001]. The LNPT tests for association of longitudinal quantitative traits with respect to a set of influencing factors. The latter divide the cohort into subgroups. The LNPT tests the null hypothesis of no difference in trait distribution **F** between these subgroups H_{0}^{F}: **CF**=**0**, where **C** is a contrast matrix and
$\mathbf{F}=\{{F}_{t}^{\mathit{kls}}\}$ is the set of distribution functions
${F}_{t}^{\mathit{kls}}$ ordered by observational time point *t* and the influencing factors (*kls*) of interest (for example, two factors *k*, *l =* 0, 1, 2 for SNP genotype at two bi-allelic loci and additional factors for covariates, e.g., sex *s=*{*m*, *w*}). The LNPT test statistic is invariant with respect to monotone trait transformations. The LNPT is not a GLM because no distributional assumptions are made about **F** and the test is not restricted to contrasts of expected values. However, its test statistic resembles a heteroscedastic repeated measures GLM ANOVA, which is performed on the mid-ranks of the longitudinal traits, estimating longitudinal covariance from the ranks without assuming any structure. The LNPT requires that individuals be followed up at the same time intervals, but individuals with partially missing values for the longitudinal phenotype can be included for computation of the test statistic. The LNPT yields a set of adjusted *p*-values for tests of average effects of the loci, covariates (e.g., sex), number of exam, and for tests of all interactions.

Malzahn et al. [2009] converted longitudinal quantitative traits to event-time data, testing for evidence of G×G interactions with the established semi-parametric Cox model of survival analysis [e.g., Therneau and Grambsch, 2000]. Event time was defined as age at the first exam when the longitudinal trait crossed a predefined threshold. Event times are invariant with respect to monotone transformations of the trait. The Cox model estimates hazard ratios HR(*G _{i}*) to quantify the impact of a genotype

Kovac et al. [2009] utilized a haplotype approach to epistasis, as implemented in the program UNPHASED [Dudbridge, 2008]. It uses a likelihood framework for primarily haplotype-based analysis of data, which can include both familial and unrelated subjects. The test for G×G interaction for a quantitative trait compares the null hypothesis of equal contributions for all gene combinations (in haplotype form) sharing the same alleles at the conditioning marker, versus the alternative hypothesis of differential multiplicative contributions from the test marker. The test uses a likelihood-ratio chi-square statistics to compare models with and without the interaction terms.

Complementary to the statistical methods for epistasis are the machine learning ones, which typically are high-dimensional heuristic search algorithms to detect G×G interactions that mostly rely upon split samples with cross validation to avoid fitting to noise. Some use the basic GLM to evaluate particular interactions, but many of these detection methods do not utilize a formal G×G model per se, and therefore do not provide an estimate of effect size. The emphasis is on efficient search among a large number of possible G×G interactions to determine which are signals and which are noise, rather than on detailed modeling of any particular G×G interaction.

Mukherjee et al. [2009] applied the GMDR method, a score-based extension of the MDR [Ritchie et al., 2001]. In MDR, multi-locus genotype combinations are classified as high-risk or low-risk genotype combinations using a threshold on the ratios of cases to controls in each combination. The best model is selected as the combination of marker with maximum cross-validation consistency and minimum prediction error. GMDR generalizes this framework, replacing the ratio of cases to controls by scores in each cell to discriminate between high risk and low risk. The GMDR algorithm allows for increased flexibility to use covariates, handling both dichotomous and continuous phenotypes, and a variety of population-based study designs such as using unbalanced case control samples.

Culverhouse et al. [2009] applied the RPM [Culverhouse et al., 2004], which reduces the dimensionality of genotype comparisons by using a multiple comparisons ANOVA to evaluate whether the phenotypes associated with each (multilocus) genotype in a particular model (e.g., a two-SNP model consisting of nine genotypes) come from the same distribution. If the answer is no, the method proposes a partition of the genotypes. The test statistic is the proportion of the trait variation explained by the partition. Statistical significance is determined by permutation testing. The RPM was developed as a method for analyzing data sets consisting of unrelated subjects, and hence can be considered only an approximately correct screening tool when applied to pedigree data, such as the FHS.

Yao et al. [2009] utilized GUIDE [Loh, 2002], a tree building software package. GUIDE develops classification trees using three steps: 1) perform a χ^{2} test to select the most significant variable to split a node; 2) select the split threshold that minimizes the node impurity measure; 3) recursively repeat Steps 1 and 2 until there are too few observations in each node. After building a complete tree, three methods are used to decide how much of the tree to retain: cross-validation pruning, test-sample pruning, and no pruning, where the criterion for judging the correct amount of pruning is that which minimizes the unbiased estimate of misclassification cost. GUIDE allows fast computation, provides a natural extension to data sets with categorical variables, and direct detection of local two-variable interactions. It has four useful properties: 1) negligible selection bias; 2) sensitivity to curvature and local pairwise interactions between regressor variables; 3) inclusion of categorical predictor variables; and 4) choice of three roles for each ordered predictor variable: split selection only, regression modeling only, or both. GUIDE can process a large number of SNPs in one run. However, it is still not feasible to run the entire Problem 2 FHS data set at one time due to computation limitations (350k SNPs after filtering for lack of Hardy-Weinberg equilibrium (*p*<0.001) and low minor allele frequency < 5%, which results in a file over 10 GB in size).

Liang et al. [2009] utilized the RF approach of Breiman [2001], which involves tree models fit to bootstrapped samples of subjects and predictors. Each bootstrap tree provides a classification, and these are aggregated as votes to form a final classification. RFs are less likely to fit to noise than are simple trees.

Chanda et al. [2009] developed an entropy-based method for detecting epistasis called KWII. KWII is defined as the amount of information (redundancy or synergy) present in the set of variables that is not present in any subset of these variables. Formally, if *S* is a set of variables that includes both predictors (including genes) as well as the response phenotype, then KWII (*S*) = Σ_{t}_{}* _{S}* (−1)

The GTD method of Huang et al. [2009] is a variation on the backward haplotype transmission association [Lo and Zheng, 2002] and backward genotype-trait association methods of Zheng et al. [2006]. Given *k* SNP markers, there are 3*k* possible unphased genotypes. The GTD statistic, *V*, is defined on the sum of squared difference between genotypes’ relative frequency among the cases and controls and measures the joint effects of these *k* SNPs on the disease status. Specifically *V* = Σ_{g}_{}* _{G}* (

In Figure 1, we show the epistasis detected by various methods for the GAW16 Problem 1 (RA data set), which was the data that was analyzed by most groups for epistasis. The most striking observation is the lack of consistency of results. There are a few genes that show consistent epistasis by multiple methods, such as *TRAF1-C5* and *PTPN22* (three methods: MDR, RF, and OT); *HLA-DRB1* and *PTPN22* (two methods: OT and RF), *HLA-DRB1* and *TRAF1-C5* (two methods: RF and OT); and *HLA-B* and *HLA-C* (two methods: GTD and KWII). Some methods found many G×G interactions that few (if any) other methods found. For instance, GMDR found 18 interactions, not one of which any other method found. GTD found 17 interactions, only one of which one other method found. Some methods found only a small number of interactions. For example, MDR identified two: *HLA-C* and *PTPN22* as well as *PTPN22* and *TRAF1-C5*. Either some of these methods are homing in on information not being used by other methods, or some are more powerful than others, or some are more prone to fitting to noise than others. It is difficult to reach a definitive conclusion because this is a real data example, and we therefore do not know the truth. Hence, we do not know when to praise or scold a method for either finding or not finding what it “should have.” But part of the difficulty in comparing methods may arise from the relativity of epistasis to scale of penetrance.

Because epistasis is simply a departure from additivity in multi-locus penetrance, it has been known for some time that such statistical interactions are scale dependent [Greenland et al., 2008]. Recently, several authors [Cordell, 2002; Frankel and Schork, 1996; Greenland and Rothman, 1998] have emphasized that the choice of how one models epistasis and in particular, the scale upon which penetrance is measured, will greatly affect whether additivity is maintained and therefore whether there “is” or “is not” epistasis. In particular, by simply rescaling the problem we can “create” or “remove” epistasis. This is illustrated in Figure 2, for two hypothetical two-locus examples. In Figures 2a and 2b, we show the interaction between two genes *G* and *F* in which the probability of disease (penetrance) for the baseline genotype group is *P*[disease|*G*=*aa*, *F*=*bb*]=0.001, and each *A* allele dose for gene *G* increases the conditional probability risk by six-fold, while each *B* allele for gene *F* increases the conditional probability risk by five-fold. In Figure 2a, both genes have no dominance and no interaction when modeled on the multiplicative scale (i.e., all three *F* genotype lines when plotted against the *G* genotype on the x-axis and the log probability of disease given genotype along the y-axis, are linear and parallel). This would utilize a log(*P*) link function in the GLM, which corresponds to a multiplicative risk model. However, these same data are shown in Figure 2b on the log(odds) scale as is relevant when analyzing by logistic regression (using a logit link = log(*P*/(1−*P*)) in the GLM). Here, both genes show strong dominance (non-linear response to the *G* genotype by the *F* genotype) as well as strong G×G interaction (non-parallel lines). We have not changed the data, just the scale of the y-axis, and we have created epistasis. By contrast, in Figures 2c and 2d, consider two other genes *J* and *K* in which the baseline genotype group penetrance is *P*[disease | *J*=*cc*, *K*=*dd*]=0.5, (which corresponds to odds(*P*)=1), and each *C* allele dose for gene *J* increases the odds four-fold, while each *D* allele dose for gene *K* increases the odds two-fold. Both genes show strong dominance as well as G×G interaction on the log(*P*) scale (Figure 2c), which would be the conclusion according to a multiplicative model, but these same data show no dominance and no interaction on the log(odds) scale for logistic regression (Figure 2d). By simply rescaling the y-axis, we have removed epistasis.

Hence, for most methods, the existence of epistasis and/or dominance is dependent upon the scale of the response and therefore also on the choice of model or link function. It would therefore not be surprising that some models might find strong epistasis, while others applied to the same data might find little or no epistasis, just as we observed in GAW16. It is important to realize that this is a deeper issue than just that the “power to detect epistasis” differs by method. In Figures 2a and 2d, there is metaphysically no epistasis (not just close to zero epistasis, which might possibly be detected by some more powerful methods). We have found a scale in which it is impossible to detect epistasis, because by simply changing scale, we have flipped from the alternative to the null hypothesis (where issues of power are moot). Evidently, the term “epistasis” in the sense of Fisher is non-additivity, not some objective biological condition that exists in and of itself, outside of the way in which we model it. Rather, “epistasis” versus “additivity” are relative concepts for which we must specify a particular penetrance scale, much like in physics where the ideas of “motion” and “rest” only make sense with respect to a particular frame of reference.

Just because the concept of epistasis requires a scale or frame of reference to makes sense, it does not mean that it is an imaginary or unimportant phenomenon. No one would suggest that the concept of motion is illusory just because it is relative, nor that all things are really standing still. In fact, much of the success of classical Newtonian physics centers around embracing the frame of reference concepts in order to form strong and accurate models of bodies in motion. Indeed, the relativity of epistasis means that essentially, for any pair of genes there is **at most** one frame of reference, one scale, upon which additivity holds, and on all other scales there is non-additivity or epistasis, just as in our examples in Figure 2. Moreover, it is **not** true that there is always some monotonic rescaling of the penetrance function that will reduce epistasis to zero. Whenever different genotypes at one locus cause the order of the penetrances by genotype at the other locus to reverse, there can be no monotonic transformation that will “remove” epistasis, as illustrated in Figure 3. Here we reproduce two real examples of epistasis from model organisms. Carlborg et al. [2004] found this type of persistent epistasis in chicken growth (their Figure 3), and Leamy et al. [2005] found it in mice for both molar size and shape (their Figure 2). This pattern of epistasis will persist, regardless of how we monotonically transform the penetrance function, and we can never find a scale on which the two genes act additively. Therefore, if anything, it might be more rightly emphasized that every pair of genes will show some degree of epistasis on almost every scale of reference (all but save at most one scale) and therefore, we should be cautious about making untested assumptions that there is no epistasis on the particular scale on which we model our data.

The Genetic Analysis Workshops are supported by NIH grant R01 GM031575 from the National Institute of General Medical Sciences. The authors acknowledge the investigators that contributed the phenotype, genotype, and simulated data. The Framingham Heart Study is conducted and supported by the National Heart, Lung, and Blood Institute (NHLBI) in collaboration with Boston University (N01 HC25195). Creation of the simulated Framingham Heart Study data was supported by the Washington University Institute of Clinical and Translational Sciences, NIH grant 1U54RR023496. The GAW16 Framingham Heart Study and simulated data were obtained through dbGaP (accession number phs000128.v1.p1). The RA data was supported by NIH grant AR44422 and NIH contract N01-AR-7-2232. This work was partially supported by USA National Institute of Health grants AA01572, DA023166, GM070789, GM69590, HL087700, HL088215, HL088655, HL87660; USA National Science Foundation grant DMS 0714669; American Cancer Society grant IRG-58-010-50; a grant from the Urological Research Foundation, the German Federal Ministry of Education and Research BMBF (01GS0837); Fonds de la recherche en santé du Québec (FRSQ); and Fogarty International Postdoctoral Fellowship TW0511-05. The authors thank the GAW16 Group 9 participants for their stimulating discussion and for sharing their insights into the epistasis problem.

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