Two-Locus Epistasis And Its Statistical Scales

Why the Scale of an Interaction Matters

At two loci, the effect of a genotype at one locus may depend on the genotype at the other. This dependence is called epistasis, or statistical interaction between loci. But an interaction is not a scale-free feature of a set of genotype means: it is defined relative to a chosen genotype coding and a chosen scale for the phenotype. A pair of loci can have no interaction on one scale and show an interaction on another.

To describe the possibilities precisely, represent each locus with two genotype contrasts: one for additive dosage and one for dominance. Products of those contrasts give four types of two-locus terms. Their names identify which contrast is used at each locus; they do not, by themselves, imply a particular biological mechanism.

Definition: Let \(A_1\) and \(D_1\) be additive and dominance contrasts for the genotype at locus 1, and let \(A_2\) and \(D_2\) be the corresponding contrasts at locus 2. The two-locus products are \(A_1A_2\) (additive-by-additive, or AA), \(A_1D_2\) (additive-by-dominance, or AD), \(D_1A_2\) (dominance-by-additive, or DA), and \(D_1D_2\) (dominance-by-dominance, or DD). Each coefficient describes a term in a model for the phenotype on a specified scale and under the stated genotype coding.

A Two-Locus Mean Model

For a quantitative phenotype, one way to describe the genotype means is a linear model for the conditional mean \(E[Y\mid G_1,G_2]\), where \(G_1\) and \(G_2\) are the genotypes at the two loci. With contrasts that distinguish all three genotypes at each locus, a complete two-locus mean model is

E[Y\mid G_1,G_2] =\mu+a_1A_1+d_1D_1+a_2A_2+d_2D_2 +aa\,A_1A_2+ad\,A_1D_2+da\,D_1A_2+dd\,D_1D_2.

Here \(\mu\) is an intercept; \(a_1,d_1,a_2,d_2\) are single-locus coefficients; and \(aa,ad,da,dd\) are the two-locus coefficients. This parameterization has nine terms, matching the nine possible pairs of genotypes for two three-genotype loci. When the contrast columns are linearly independent across those genotype combinations, the model can represent any set of nine genotype means. In that saturated setting, the product terms describe departures from the lower-order additive-and-dominance model.

The word “additive” refers to the contrast used for a locus, not to a claim that its effect is constant across every possible scale or model. A common coding, used in the first example, assigns \(A=-1,0,1\) to the reference homozygote, heterozygote, and alternate homozygote, respectively. At allele frequency one half, a corresponding dominance contrast can be coded \(D=-0.5,0.5,-0.5\). These are one convenient choice, not a universal unit system. The values and orthogonality properties of contrasts depend on coding and, for some choices, allele frequency.

Worked Example: Reading an Additive-by-Dominance Term

Suppose hypothetical genotype means, measured in a quantitative trait’s units, follow the model above with \(\mu=10\), \(a_1=2\), \(d_1=1\), \(a_2=-1\), \(d_2=0.5\), \(aa=0.8\), \(ad=0.6\), \(da=-0.4\), and \(dd=1.2\). Use the stated contrast coding. Find the predicted mean for a heterozygote at locus 1 and an alternate homozygote at locus 2.

For this genotype pair, \(A_1=0\), \(D_1=0.5\), \(A_2=1\), and \(D_2=-0.5\). Substitution gives

\begin{aligned} E[Y\mid G_1,G_2] &=10+2(0)+1(0.5)+(-1)(1)+0.5(-0.5)\\ &\quad+0.8(0)(1)+0.6(0)(-0.5)+(-0.4)(0.5)(1)+1.2(0.5)(-0.5)\\ &=10+0+0.5-1-0.25+0+0-0.2-0.3\\ &=8.75. \end{aligned}

The additive-by-dominance contribution is \(ad A_1D_2=0.6(0)(-0.5)=0\) for this particular genotype because the additive contrast at locus 1 is zero. That does not mean the AD coefficient is zero: it means this genotype pair does not activate that product term. The predicted mean, 8.75 trait units, combines the intercept, both single-locus terms, and the active interaction terms.

Interaction as a Contrast Among Genotype Means

A useful way to see what an interaction coefficient means is to compare genotype means across the two loci. First consider only two genotype levels per locus, coded \(x_1,x_2\in\{0,1\}\), and the model \(E[Y\mid x_1,x_2]=\beta_0+\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2\). The cross-difference of its four means is

\begin{aligned} \Delta &=M_{11}-M_{10}-M_{01}+M_{00}\\ &=(\beta_0+\beta_1+\beta_2+\beta_{12}) -(\beta_0+\beta_1)-(\beta_0+\beta_2)+\beta_0\\ &=\beta_{12}. \end{aligned}

Thus, for 0/1 coding, the interaction coefficient is the difference between the locus-1 contrast at \(x_2=1\) and the locus-1 contrast at \(x_2=0\). With three genotype levels per locus, the same idea applies to contrasts among genotype means, though there are multiple additive and dominance contrasts and therefore multiple interaction coefficients.

Worked Example: Calculating an Interaction Contrast

Suppose a hypothetical study has the following four conditional mean phenotypes, in trait units, for two selected genotype levels at each of two loci. The levels are coded 0 and 1.

	\(x_2=0\)	\(x_2=1\)
\(x_1=0\)	10	12
\(x_1=1\)	13	17

The difference between the locus-1 means at \(x_2=0\) is \(13-10=3\) trait units. At \(x_2=1\), it is \(17-12=5\) trait units. Their difference is \(5-3=2\) trait units. Equivalently, the cross-difference is

\Delta=M_{11}-M_{10}-M_{01}+M_{00} =17-13-12+10=2.

Under the 0/1 model, \(\beta_{12}=2\) trait units. The locus-1 contrast is two units larger at \(x_2=1\) than at \(x_2=0\). This is a statistical description of these conditional means; it does not establish that either locus causally changes the effect of the other.

Why Genotype Coding Changes Coefficients

The previous example used 0/1 coding. Now recode each locus as \(z=2x-1\), so the two levels are \(-1\) and \(+1\). The same four means can be written as

E[Y\mid z_1,z_2]=\gamma_0+\gamma_1z_1+\gamma_2z_2+\gamma_{12}z_1z_2.

Because changing from \(x\) to \(z\) changes the numerical units of the contrasts, the interaction coefficient changes even though the four fitted means do not. The cross-difference is now \(4\gamma_{12}\): changing either \(z\) from \(-1\) to \(+1\) spans two coded units, so the product contrast spans four. This distinction matters whenever coefficients are compared across analyses that use different genotype codings.

Worked Example: The Same Means Under a New Coding

Use the four means from the previous example, but code each genotype level as \(z=-1\) or \(z=+1\). The cross-difference remains \(2\) trait units. Since it equals \(4\gamma_{12}\) under this coding,

\gamma_{12}=\frac{2}{4}=0.5\text{ trait units per coded product unit}.

For completeness, the intercept is the average of the four means: \(\gamma_0=(10+12+13+17)/4=13\) trait units. The \(z_1\) coefficient is the average mean at \(z_1=+1\) minus the average at \(z_1=-1\), divided by two: \(\gamma_1=[(13+17)-(10+12)]/4=2\) trait units. Similarly, \(\gamma_2=[(12+17)-(10+13)]/4=1.5\) trait units. At \(z_1=z_2=+1\), for example, the model predicts \(13+2+1.5+0.5=17\), matching the table.

The 0/1 interaction coefficient was \(2\), whereas the \(-1/+1\) coefficient is \(0.5\). Both models encode the same four means; the coefficient difference is a coding-scale difference, not a disagreement about the data. A coefficient cannot be compared meaningfully without specifying the contrast coding and its units.

For three genotypes per locus, additive and dominance contrasts provide two dimensions for describing each locus. Their four products distinguish AA, AD, DA, and DD components. If contrasts are rescaled or redefined, coefficients can change, and when a new contrast mixes additive and dominance directions, the interaction components can be redistributed among the four product terms. With a complete model, the fitted genotype means can remain identical. It is therefore better to report the coding and, where useful, the genotype means or contrasts themselves rather than treating a coefficient as an invariant property of a locus pair.

Phenotype Scale Can Create or Remove Interaction

The scale on which means are modeled is another choice. A model may describe \(E[Y\mid G_1,G_2]\) directly, or it may model a transformed mean, such as its logarithm. Interaction on one scale need not imply interaction on another. This is a consequence of nonlinear transformations: differences on the transformed scale generally do not translate into equal differences on the original scale.

Worked Example: No Interaction on the Log Scale, Interaction on the Raw Scale

Consider hypothetical positive mean outcomes for two binary genotype indicators \(x_1,x_2\in\{0,1\}\). Suppose the log of the mean is additive: \(\log M_{x_1x_2}=0.2x_1+0.3x_2\), with no interaction term. The four log means are \(0\), \(0.2\), \(0.3\), and \(0.5\), so their cross-difference is \(0.5-0.2-0.3+0=0\). On the log-mean scale, there is no interaction.

The corresponding raw-scale means are \(M_{00}=e^0=1\), \(M_{10}=e^{0.2}\), \(M_{01}=e^{0.3}\), and \(M_{11}=e^{0.5}\). The raw-scale difference-in-differences is

\begin{aligned} M_{11}-M_{10}-M_{01}+M_{00} &=e^{0.5}-e^{0.2}-e^{0.3}+1\\ &\approx 1.64872-1.22140-1.34986+1\\ &\approx 0.07746. \end{aligned}

Thus the raw-scale interaction contrast is about \(0.07746\) mean-outcome units, or \(0.0775\) to four decimal places. The rounding is applied after evaluating the full expression; subtracting rounded intermediate values can give a different last digit. This example has no interaction on the log-mean scale but a nonzero interaction contrast on the raw scale. The conclusion is about the mathematical scale of the means, not evidence of a biological interaction.

Interpreting and Reporting Epistasis

An interaction term is meaningful only together with the model that defines it. For a quantitative phenotype, state whether the model is for the raw conditional mean or a transformed mean, name the genotype contrasts, and give the coefficient units. If the outcome is modeled on a log scale, the coefficient describes a departure from additivity on that log scale; it is not directly a raw-unit difference-in-differences.

The four product types distinguish which contrast is used at each locus. For example, \(ad A_1D_2\) uses an additive contrast at locus 1 and a dominance contrast at locus 2, while \(da D_1A_2\) reverses those roles. They are not interchangeable labels. If the chosen contrasts and the analysis design make the two loci symmetric, some parameter values may coincide, but the terms still refer to different contrast directions.

These statistical terms describe patterns among genotype-specific means. They do not, on their own, identify molecular mechanisms, establish a causal interaction, or quantify an epistatic variance component. Variance components require additional population and covariance calculations. For an interaction analysis, first clarify the mean model and scale; interpretation of the coefficient follows from those choices.

Key takeaway: Two-locus interaction is a departure from additivity defined on a particular phenotype scale and with particular genotype contrasts. Always report both choices; otherwise, an interaction coefficient is not fully interpretable.

Common Mistakes

Comparing coefficients coded on different scales. A coefficient of 2 under 0/1 coding and 0.5 under \(-1/+1\) coding can describe the same genotype means. Compare the model coding or the implied means, not the raw numbers alone.
Calling an interaction biological by definition. A product term represents statistical nonadditivity under a model. It does not prove a molecular mechanism or causation.
Failing to name the phenotype scale. No interaction on a log-mean scale does not imply no interaction on the raw-mean scale. State which scale the model uses.
Treating a zero product for one genotype as a zero coefficient. A product can be zero for a particular genotype even when its coefficient is nonzero. Evaluate the term and the coefficient separately.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In the two-locus mean model, which contrast product corresponds to dominance-by-dominance interaction?
For 0/1 coding at both loci, what genotype-mean contrast equals the interaction coefficient?
Why can two analyses of the same genotype means report different interaction coefficients?
In the log-mean example, is the interaction contrast zero on the raw scale? Explain what the calculation shows.
What coding and scale information should accompany a reported two-locus interaction coefficient?