Variance Decomposition
Fisher’s partition of genetic variance decomposes the genotypic value $G_{ij}$ at two loci into orthogonal components:
\[G_{ij} = \mu + \alpha_i + \alpha_j + \delta_{ij} + (\alpha\alpha)_{ij} + (\alpha\delta)_{ij} + (\delta\delta)_{ij}\]where $\alpha$ are additive effects, $\delta$ dominance deviations, and the interaction terms constitute epistatic variance $V_I$:
\[V_G = V_A + V_D + V_{AA} + V_{AD} + V_{DD} + \cdots\]In large outbred populations, $V_{AA}$ (additive-by-additive epistasis) dominates because allele frequencies weight interaction terms differently.
Statistical Epistasis in Linear Models
A two-SNP interaction test adds a product term:
\[y_i = \mu + \beta_1 g_{i1} + \beta_2 g_{i2} + \beta_{12}\,(g_{i1} \cdot g_{i2}) + \varepsilon_i\]Significance of $\hat{\beta}_{12}$ indicates statistical epistasis. The multiple testing burden is severe: with $M = 10^6$ SNPs, all pairwise tests number $\binom{M}{2} \approx 5 \times 10^{11}$, requiring thresholds of order $p < 10^{-13}$ for FWER control.
Biological vs. Statistical Epistasis
| Aspect | Biological epistasis | Statistical epistasis |
|---|---|---|
| Definition | Physical interaction in a pathway | Non-additive term in a statistical model |
| Allele-frequency dependence | No | Yes |
| Detectable by GWAS interaction test | Not necessarily | By definition |
| Contributes to $V_I$ | Partly | Depends on model |
A biological interaction can appear additive at the population level if interacting alleles are rare or at extreme frequencies.
FWER Inflation and Power
Exhaustive epistasis searches inflate type-I error dramatically. Common strategies include:
- Restricting search to cis-pairs within LD blocks
- Testing known pathway gene pairs (biologically informed)
- Using LASSO-penalised regression to select interaction candidates
- Bayesian approaches (BSLMM) that shrink interaction effects genome-wide