Deriving Epistatic Variance Components

Why Orthogonality Matters for Epistatic Variance

A two-locus interaction describes a genetic value that depends on the combination of genotypes at two loci, beyond the separate contributions assigned to each locus. Its variance is not always a distinct, additive piece of total genetic variance. The key issue is orthogonality: if an interaction contrast is correlated with an additive or dominance contrast, their variances cannot simply be added as though they were independent components.

The distinction depends partly on the population’s allele frequencies and linkage disequilibrium (LD). Under random union of gametes and linkage equilibrium, suitably centered and orthogonal one-locus contrasts yield interaction products that are orthogonal to the single-locus effects. With LD, the loci are statistically dependent, and even a centered product of genotype counts can correlate with additive and dominance contrasts.

Definition: Let $p$ and $q$ be the frequencies of designated alleles $A$ and $B$ at two loci. Let $f_{AB}$ be the frequency of haplotype $AB$. The gametic linkage disequilibrium coefficient is $D=f_{AB}-pq$. Thus $D=0$ means the alleles on a randomly sampled gamete are independent; $D\ne0$ means they are associated on gametes.

The four haplotype frequencies are $pq+D$, $p(1-q)-D$, $(1-p)q-D$, and $(1-p)(1-q)+D$, for $AB$, $Ab$, $aB$, and $ab$, respectively. They must all be nonnegative. Assume throughout that two gametes are sampled independently from this haplotype distribution to form an individual. This gives Hardy-Weinberg proportions at each locus, while allowing LD between loci.

Genotype Contrasts and a Centered Interaction

Write $G_1$ and $G_2$ for the numbers of $A$ and $B$ alleles in an individual, each ranging from 0 to 2. Their means are $2p$ and $2q$. Define centered additive contrasts $U=G_1-2p$ and $V=G_2-2q$. Under the stated gamete-sampling model, their variances are $2p(1-p)$ and $2q(1-q)$, and their covariance is $2D$.

A centered product interaction is $W=UV-E(UV)$. Since $E(UV)=2D$, this is $W=UV-2D$. Centering matters: an interaction term in a variance decomposition should have mean zero, just as the main-effect contrasts do.

Result: Under random union of gametes with allele frequencies $p,q$ and gametic LD $D$, the centered product interaction has variance $$ \operatorname{Var}(W)=4p(1-p)q(1-q)+2(1-2p)(1-2q)D. $$ Its covariances with the additive contrasts are $$ \operatorname{Cov}(W,U)=2(1-2p)D,\qquad \operatorname{Cov}(W,V)=2(1-2q)D. $$

A short derivation makes the LD dependence explicit. For a gamete, let $x=I_A-p$ and $y=I_B-q$, where each indicator is 1 if the named allele is present and 0 otherwise. For the two independently sampled gametes, $U=x_1+x_2$ and $V=y_1+y_2$. On a single gamete, $E(xy)=D$, and the binary-variable identity $x^2=p(1-p)+(1-2p)x$ gives $E(x^2y)=(1-2p)D$ and $E(x^2y^2)=p(1-p)q(1-q)+(1-2p)(1-2q)D$. Expanding $U^2V^2$, terms involving a lone centered factor from one of the independent gametes have expectation zero. The remaining terms give $E(U^2V^2)=2E(x^2y^2)+2p(1-p)q(1-q)+4D^2$. Subtracting $E(UV)^2=(2D)^2$ yields the stated variance. Similarly, expanding $U^2V$ gives $E(U^2V)=2E(x^2y)=2(1-2p)D$, which is $\operatorname{Cov}(W,U)$; exchanging the loci gives the other covariance.

Orthogonal Additive and Dominance Contrasts

At one locus, an additive contrast alone does not describe every possible genotype effect. A standard dominance contrast under Hardy-Weinberg proportions is $Z_1$, with values $-2p^2$, $2p(1-p)$, and $-2(1-p)^2$ for genotypes $AA$, $Aa$, and $aa$, respectively. For the second locus define $Z_2$ in the same way, replacing $p$ by $q$. Each contrast has mean zero and is orthogonal to the additive contrast at its own locus:

\operatorname{Var}(U)=2p(1-p),\quad \operatorname{Var}(Z_1)=4p^2(1-p)^2,\quad \operatorname{Cov}(U,Z_1)=0.

The corresponding statements hold for $V$ and $Z_2$. One useful representation is $Z_1=-2x_1x_2$, where $x_1,x_2$ are the two gametic centered allele indicators at locus 1. This representation also gives $\operatorname{Cov}(W,Z_1)=-4p(1-p)D$ and $\operatorname{Cov}(W,Z_2)=-4q(1-q)D$. Thus LD can make the product interaction overlap with dominance as well as additive effects.

When $D=0$, the genotype contrasts from the two loci are independent. Products $UV$, $UZ_2$, $Z_1V$, and $Z_1Z_2$ are then mutually orthogonal and orthogonal to the four single-locus contrasts $U,Z_1,V,Z_2$. For example, $E[(UV)U]=E(U^2)E(V)=0$, and $E[(UV)(UZ_2)]=E(U^2)E(VZ_2)=0$. These factorizations rely on independence across loci; within each locus, the additive and dominance contrasts are already orthogonal.

Worked Example: Interaction Variance Under Linkage Equilibrium

Suppose $p=0.30$, $q=0.40$, and $D=0$. A genetic value includes the product interaction $\gamma UV$, with $\gamma=1.5$ genetic-value units per product-contrast unit. Find its variance.

Because $D=0$, the genotype counts are independent across loci. Their contrast variances are $\operatorname{Var}(U)=2(0.30)(0.70)=0.42$ and $\operatorname{Var}(V)=2(0.40)(0.60)=0.48$. Therefore,

\operatorname{Var}(\gamma UV) =\gamma^2\operatorname{Var}(U)\operatorname{Var}(V) =(1.5)^2(0.42)(0.48) =2.25(0.2016) =0.4536.

The interaction contributes $0.4536$ squared genetic-value units under this model. It is orthogonal to the single-locus additive and dominance terms, so its variance can be added to their variances without covariance adjustments. This conclusion depends on linkage equilibrium and the stated contrast coding.

What Changes Under Linkage Disequilibrium?

With LD, $U$ and $V$ are correlated, and their product can have nonzero covariance with main effects. The interaction variance formula still applies, but it does not by itself establish an orthogonal variance component. If the genetic value is written as a sum of terms, the variance of that sum includes covariance terms whenever those terms are correlated.

Worked Example: Computing Variance and Covariance With LD

Suppose $p=0.30$, $q=0.40$, and $D=0.05$. The haplotype frequencies are $f_{AB}=0.17$, $f_{Ab}=0.13$, $f_{aB}=0.23$, and $f_{ab}=0.47$. They are nonnegative and sum to 1, so this is a valid haplotype distribution. Calculate the variance of $W$ and its covariance with the two additive contrasts.

Here $p(1-p)=0.21$, $q(1-q)=0.24$, $1-2p=0.40$, and $1-2q=0.20$. Substitution gives

\begin{aligned} \operatorname{Var}(W) &=4(0.21)(0.24)+2(0.40)(0.20)(0.05)\\ &=0.2016+0.0080=0.2096,\\ \operatorname{Cov}(W,U) &=2(0.40)(0.05)=0.0400,\\ \operatorname{Cov}(W,V) &=2(0.20)(0.05)=0.0200. \end{aligned}

The interaction is not orthogonal to either additive contrast. Its covariances with the dominance contrasts are also nonzero: $\operatorname{Cov}(W,Z_1)=-4(0.21)(0.05)=-0.0420$ and $\operatorname{Cov}(W,Z_2)=-4(0.24)(0.05)=-0.0480$. The interaction term has a well-defined variance, $0.2096$ in product-contrast units, but treating it as an independent component alongside all four single-locus contrasts would omit covariance terms.

A useful way to define an interaction that is orthogonal to all single-locus terms under LD is to remove its linear projection onto those terms. Let $C=(U,Z_1,V,Z_2)^\mathsf{T}$, let $\Sigma_C=\operatorname{Cov}(C,C)$, and let $c=\operatorname{Cov}(C,W)$. If $\Sigma_C$ is invertible, define $R=W-c^\mathsf{T}\Sigma_C^{-1}C$. Then $\operatorname{Cov}(C,R)=0$, because $\operatorname{Cov}(C,R)=c-\Sigma_C\Sigma_C^{-1}c=0$. Its variance is $\operatorname{Var}(R)=\operatorname{Var}(W)-c^\mathsf{T}\Sigma_C^{-1}c$. This is a statistical orthogonalization: it assigns the part linearly predictable from single-locus contrasts to those contrasts, leaving a residual interaction. It does not imply that the biological mechanism has changed.

Worked Example: Summing Orthogonal Interaction Components

For a separate hypothetical orthogonal contrast coding, suppose the four product contrasts have variances $0.42(0.48)$, $0.42(0.2496)$, $0.2436(0.48)$, and $0.2436(0.2496)$, respectively. Suppose their genetic-value coefficients are $1.5$, $-0.8$, $0.6$, and $1.0$. The variances are stipulated properties of this coding; they are not being inferred from the allele frequencies in the preceding example. Because the four terms are assumed mutually orthogonal, the total interaction variance is the sum of coefficient-squared times contrast variance.

\begin{aligned} V_E={}&(1.5)^2(0.42)(0.48) +(-0.8)^2(0.42)(0.2496)\\ &+(0.6)^2(0.2436)(0.48) +(1.0)^2(0.2436)(0.2496)\\ ={}&0.4536+0.06709248+0.04209408+0.06080256\\ ={}&0.62358912\approx0.6236. \end{aligned}

The third contribution is $0.36\times0.116928=0.04209408$. The total is $0.62358912$, or $0.6236$ squared genetic-value units to four decimal places. The sum-of-squares formula is justified by the stipulated orthogonality; without it, covariance terms would also be required.

Variance Decomposition and a Common Pitfall

For mutually orthogonal, mean-zero genetic terms $T_1,\ldots,T_k$ with coefficients $\beta_1,\ldots,\beta_k$, the variance of their sum is $\operatorname{Var}(\sum_j\beta_jT_j)=\sum_j\beta_j^2\operatorname{Var}(T_j)$. The reason is that expanding the variance gives this sum plus $2\sum_{j<\ell}\beta_j\beta_\ell\operatorname{Cov}(T_j,T_\ell)$; orthogonality makes every covariance zero. Under LD, product terms generally do not meet that condition automatically. Their covariance structure must be calculated from the population genotype distribution, or the terms must be orthogonalized before they are treated as separate components.

Wrong reasoning: “The interaction is centered, so it is orthogonal to additive and dominance effects.” Centering guarantees a zero mean, not zero covariance.
Correction: Check each covariance. For the centered product $W$, the additive covariances are $2(1-2p)D$ and $2(1-2q)D$; they need not be zero when $D\ne0$.
Wrong reasoning: “The variance of each term can always be added to get total genetic variance.” That ignores covariance between terms.
Correction: Add component variances only after establishing orthogonality, or include all covariance terms in the variance of the sum.

Key takeaway: Allele frequencies determine the marginal contrast variances, while LD affects both the variance of a product interaction and its covariances with main effects. A distinct epistatic variance component requires an explicitly orthogonal decomposition; a centered interaction alone is not enough.

Check Your Understanding

Use the definitions and assumptions stated above to answer the following questions.

For two loci with $p=0.25$, $q=0.50$, and $D=0$, what are the variances of the centered additive contrasts $U$ and $V$?
Why does $D=0$ make cross-locus product contrasts orthogonal to single-locus contrasts under the random-union-of-gametes model?
For $p=0.30$, $q=0.40$, and $D=0.05$, use the covariance formula to find $\operatorname{Cov}(W,U)$. What does its nonzero value imply?
State the distinction between centering an interaction and making it orthogonal to main effects.
When can the variance of a genetic value be calculated as the sum of the variances of its weighted terms without covariance adjustments?