Reports of the Death of the Epistasis Model Are Greatly Exaggerated

To the Editor:

I was surprised by the conclusions drawn by Vieland and Huang (2003) that linkage studies of affected sibling pairs (ASPs) cannot, in general, as a matter of mathematical principle, be used to distinguish heterogeneity from epistatic models. A glance at the citation list suggests that the authors have overlooked a critical body of scholarly work that is directly relevant to this issue and that flatly contradicts their conclusions.

Epistasis (interaction) between genes influencing inherited traits has been recognized since 1865, when the results of Gregor Mendel's hybridization experiments were published. Fisher (1918) was the first to partition genetic variance into a series of additive components corresponding to the “main effects” (additive and dominance components) attributable to individual genotypes and “interactions” (epistatic components) determined by combinations of genotypes. Cockerham (1954) used orthogonal contrasts to decompose the epistatic variance into several components; for a two-locus example, under the assumption of linkage equilibrium, the total genetic variance can be written as (V_G)=V_A1+V_A2+V_D1+V_D2+V_A1A2+V_A1D2+V_D1A2+V_D2D2, where V_A1 and V_A2 are additive components for the first and second loci; V_D1 and V_D2 are dominance components; and V_A1A2, V_A1D2, V_D1A2, and V_D2D2 are additive-additive, additive-dominance, dominance-additive, and dominance-dominance components, respectively, for the two loci. Epistasis is present in the model when one or more of the V_A1A2, V_A1D2, V_D1A2, and V_D2D2 components are >0. In experimental intercrosses, analysis-of-variance (ANOVA) techniques are traditionally used to assess the significance of each component, and the classic two-locus statistical test for epistasis compares the fit of the general epistasis model (eight components) to a nested (hierarchical) model with four main effects (i.e., V_A1A2=V_A1D2=V_D1A2=V_D2D2=0). More elaborate methods using models based on the variance-components framework have been developed (e.g., Kao and Zeng 2002). It is commonplace to colloquially refer to the main-effects model as the “additive” model. The real-world meaning of the additive model is crystal clear: the effects of each locus on the phenotype are independent of each other—the very same definition of “genetic heterogeneity” used by Vieland and Huang (2003). Or, to put it another way, it doesn’t matter on what genetic background you choose to estimate the effects of a locus, you will measure the same effect.

Risch (1990) introduced an elegant mathematical approach to the generalized study of complex human diseases. Identity-by-descent (IBD) vectors in ASPs could be modeled “on the back of an envelope” using mathematically simple models of gene interaction. His “additive,” two-locus model carefully defines the joint penetrance (the probability that an individual with a particular multilocus genotype is affected) as a sum of “penetrance summands,” one for each locus. The critical issue here is that the “penetrance summands” are deliberate abstractions—they are distinct from marginal, locus-specific penetrances. This is because the sole purpose of the “penetrance summands” is to specify the joint penetrances and thus specify the joint IBD probability vector (an analogous trick was used by Risch et al. [1993] and extended by Bonyadi et al. [1997] to analyze affected animals in backcrosses and intercrosses). The marginal IBD probability vector (IBD observed at each constituent locus) can then be easily solved but not some marginal penetrance vector. If I understand them correctly, it is these marginal penetrance vectors (one for each locus) that Vieland and Huang (2003) seek to estimate. The reason why this search is pointless in the context of ASP linkage studies can be understood by reference to the work of James (1971) and Suarez et al. (1978). First, the expected probabilities of the three IBD configurations in an ASP can be calculated from a set of allele frequencies and single-locus penetrances (for any number of alleles; a minimum of four parameters for a two-allele model). However, there is no inverse solution, since the penetrances and allele frequencies cannot be identified starting from a set of IBD probabilities. This was confirmed for ASPs by Whittemore et al. (1991), who pointed out that the inverse solution can be solved in larger families. It is this unique statistical property of ASPs that validates the term “non-parametric” to test statistics based on IBDs and ASPs. For aficionados of the ASP paradigm, this is valued as a strength (Farrall 1997b); for detractors, however, it is apparently perceived to be a weakness (Greenberg et al. 1996; Spence et al. 2003). The point here is that ASPs are good for detecting linkage (via IBD distortion), but they are hopeless for measuring allele-specific or genotype-specific parameters. This latter objective is of great interest to both “earlier generations” and the “next generation” of gene mappers, but I suspect that more insights will be gained through genotype/haplotype association techniques than by pure linkage tests.

Anyway, Cordell and colleagues (1995) built on the findings of Risch (1990) to expand and generalize the variance-components model for two-locus disease models; in effect, they implemented the ASP equivalent of Cockerham’s variance-components model. This was informative, since it led to the conclusion that the main-effects model (see above) was equivalent to Risch’s additive model; Risch had chosen this moniker well. Consequently, for ASPs, the classic linkage test for epistasis was immediately obvious: use likelihood-ratio tests to compare the general epistasis model with the additive model (or GEN-ADD in Cordell et al. 1995). This test has been successfully applied (Cordell et al. 1995) and theoretically extended to the case of linked susceptibility genes (Farrall 1997a). The existence and mathematical justification of this linkage test directly contradicts the main conclusion of Vieland and Huang (2003).

Vieland and Huang (2003) comment on their surprise on reaching their conclusions. They had counted the number of degrees of freedom in a two-locus IBD matrix (eight) and were suspicious that there might be eight underlying parameters to describe a saturated model. Of course, the eight degrees of freedom are mirrored by the eight variance components in the general epistasis model of Risch (1990) and Cordell (1995). It seems that it will be impossible to reconcile the variance-components epistasis ASP model with the conclusions of Vieland and Huang (2003). I look forward to Vieland and Huang’s critique of the variance-components epistasis model and its application to ASP data and also to their re-evaluation of their findings.