Statistical Epistasis Versus Biological Interaction

Why “Interaction” Needs a Model

When two loci are associated with a phenotype, researchers may ask whether their joint pattern departs from what an additive model predicts. That question is statistical: its answer depends on how the phenotype is represented and what “additive” means in the chosen model. It is not, by itself, a claim that the two gene products physically interact or jointly cause the phenotype.

The distinction matters because the same two-locus pattern can show no departure from additivity on one scale and a departure on another. A precise report therefore names the phenotype, scale, genotype comparison, and model. Without those details, “epistasis” can sound more definitive than the analysis warrants.

Two-Locus Additivity as a Statistical Statement

To make the idea concrete, consider two loci, each represented here by two states: 0 and 1. Let \(m_{ab}\) denote the expected phenotype for individuals in state \(a\) at the first locus and state \(b\) at the second, with \(a,b\in\{0,1\}\). The expectations might be cell risks, mean trait values, or another outcome summary. The meaning of \(m_{ab}\) depends on the chosen phenotype scale.

On a scale where additivity means that the change associated with the first locus is the same in both states of the second, the two-locus model has the form \(m_{ab}=\mu+\alpha a+\beta b\). The model predicts that the joint state combines the two single-locus changes without an additional cross-term. For these two binary states, its implication can be checked with one difference-in-differences contrast.

The first expression compares the change from locus 0 to locus 1 when the second locus is in state 1 with that change when it is in state 0. Under the additive model, both changes equal \(\alpha\), so \(\Delta=0\). A nonzero value is a departure from this particular additive model on this particular scale. It does not establish that the departure is statistically distinguishable from sampling variation; that requires uncertainty estimates and an appropriate inferential procedure.

Worked Examples: The Scale Changes the Answer

Statistical and Biological Interaction Are Different Claims

A statistical interaction describes how a chosen model summarizes observed or expected outcomes. Biological interaction concerns processes: for example, whether one gene product alters another product's activity, whether two pathways converge, or whether one locus changes the functional consequence of another. Such mechanisms can be investigated with molecular and experimental evidence, but a nonzero statistical contrast alone does not demonstrate them.

The distinction also works in the other direction. A biological dependence may be present while a particular statistical contrast is zero. Effects can be masked by the chosen outcome, offset by other processes, averaged over different biological contexts, or represented additively on the selected scale. A statistical result is therefore not a universal verdict about whether two loci interact in nature.

In genetic data, genotype groups are generally observed rather than assigned at random. Differences between groups can reflect ancestry, environment, selection, or other sources of structure as well as genetic associations. Consequently, statistical epistasis in an observational analysis is not automatically a causal effect of jointly changing two loci. The population, conditioning variables, genotype comparison, and assumptions behind the analysis all matter.

A Short Justification of the Additive Contrast

The contrast has a direct model-based justification. If the expected outcome follows the additive model \(m_{ab}=\mu+\alpha a+\beta b\), then the four expected values are \(m_{00}=\mu\), \(m_{10}=\mu+\alpha\), \(m_{01}=\mu+\beta\), and \(m_{11}=\mu+\alpha+\beta\). Substitution gives

Conversely, with these four cell expectations, if the contrast is zero, then \(m_{11}=m_{10}+m_{01}-m_{00}\). Setting \(\mu=m_{00}\), \(\alpha=m_{10}-m_{00}\), and \(\beta=m_{01}-m_{00}\) gives the additive model's prediction for all four cells, including \(m_{11}\). Thus, for these two binary states, the zero contrast is exactly the restriction that the four cell expectations can be represented by this additive model. This justification applies to whichever outcome scale defines \(m_{ab}\); changing that scale changes the cell expectations and can change the conclusion.

What to Specify When Reporting Epistasis

A useful report makes the statistical claim reproducible and limits its interpretation to what the analysis supports. State:

• The outcome and scale: for example, mean biomarker concentration, risk, or log odds.
• The genotype comparison and reference states: define which two-locus groups the contrast compares.
• The null model: say what additivity means on that scale, rather than using “additive” without qualification.
• The estimate and uncertainty: provide the interaction contrast or model-based effect, with its standard error or interval and the inferential method when making a population-level claim.
• The scope of interpretation: distinguish a statistical association from a causal or mechanistic conclusion.

A frequent mistake is to reason, “The interaction test was not significant, so the two genes do not interact biologically.” The correction is that a test concerns evidence against a specified statistical restriction, given the data and analysis. A nonsignificant result does not prove that restriction true, and neither a nonsignificant nor a significant result alone settles a biological mechanism. Another mistake is to report “epistasis is present” without naming the scale. A more precise statement is that the data show evidence of departure from a specified additive model on a named outcome scale.