Power and Sample Size for Gene–Environment Interaction

Why Interaction Studies Need Deliberate Design

A study can have a large overall sample and still provide little evidence about a gene–environment interaction. The interaction is informed by how the outcome changes across combinations of genotype and exposure. If one combination is rare, its outcome mean or event rate is estimated imprecisely, and that uncertainty can dominate the interaction test. Planning therefore requires more than specifying a total sample size: it requires the genotype and exposure distributions, the outcome model and scale, and a scientifically meaningful interaction effect.

Power is always tied to a particular test and alternative. The same data can have different power for an additive interaction in mean outcome and a multiplicative interaction in odds, because those are different parameters with different standard errors. The calculations below use standard large-sample approximations. They are useful for planning and for understanding which design features matter; when cell probabilities, covariate adjustment, or sampling are complex, simulation or model-specific software is preferable.

Binary Genotype and Binary Exposure: Mean-Scale Interaction

Let \(G\) indicate genotype group (0 or 1) and \(E\) indicate exposure status (0 or 1). Write \(n_{ge}\) for the number of participants in joint group \((G=g,E=e)\), and let \(\mu_{ge}\) be that group's mean quantitative outcome. The mean-scale interaction is the difference in exposure effects between genotype groups:

This is also the coefficient of \(GE\) in a linear model with an intercept, \(G\), \(E\), and \(GE\), when \(G\) and \(E\) are coded 0/1. Suppose outcomes have independent, normally distributed errors with common variance \(\sigma^2\), and observations are independent across participants. With fixed group sizes, the four cell means are independent and have variances \(\sigma^2/n_{ge}\). Therefore:

The variance follows because each cell mean contributes once to the contrast, with coefficient either \(+1\) or \(-1\); squaring those coefficients gives 1 in every variance term. If genotype and exposure are independent in the target population, with \(P(G=1)=q\) and \(P(E=1)=r\), expected cell sizes are \(Nqr\), \(Nq(1-r)\), \(N(1-q)r\), and \(N(1-q)(1-r)\). Substituting these expected sizes gives the planning approximation:

From Standard Error to Power and Sample Size

For a two-sided test at level \(\alpha\), a large-sample normal approximation rejects when the absolute test statistic exceeds \(z_{1-\alpha/2}\), the \(1-\alpha/2\) quantile of the standard normal distribution. Under a true interaction \(\delta\), the statistic is approximately normal with mean \(\lambda=\delta/SE(\hat{\delta})\) and variance 1. Thus its power is:

For a negative interaction, \(\lambda\) is negative, but the two-sided power is unchanged if the sign is reversed. A convenient approximation for a target power \(1-\beta\) is to require \(\lvert\lambda\rvert\approx z_{1-\alpha/2}+z_{1-\beta}\). Solving for the total sample size yields:

The squared effect size makes this expression apply to either direction of interaction. It assumes independent genotype and exposure, the stated binary coding, a common outcome variance, and the large-sample approximation. With genotype–exposure dependence, replace the expected cell frequencies with appropriate joint frequencies; the simple product formula no longer applies.

Exposure Variance and Genotype Coding

An exposure need not be binary. For a quantitative exposure \(E\), consider a linear mean model with \(G\), \(E\), and their product, and let \(G\) and \(E\) be independent. Centering either variable changes the interpretation of main effects but not the product-term interaction coefficient. If \(G\) is binary with frequency \(q\), and the exposure has variance \(v=\operatorname{Var}(E)\), the residual variation in the product after accounting for the intercept and main effects is \(q(1-q)v\). Under the same independent, constant-variance outcome model, this gives:

Here \(\beta_{GE}\) is measured in outcome units per exposure unit per genotype-group difference. For a binary exposure, \(v=r(1-r)\), recovering the earlier formula. A more variable exposure can provide more information about a linear interaction, but only relative to the chosen exposure scale and model. If genotype and exposure are correlated, or the exposure-response relationship is nonlinear, this simple variance expression is not sufficient.

For an additive genotype dosage coded 0, 1, 2, the same logic uses the genotype variance in place of \(q(1-q)\), under independence of genotype and exposure and the stated linear model. Under Hardy–Weinberg equilibrium with allele frequency \(f\), dosage variance is \(2f(1-f)\). The model, not just the label “genotype,” determines which variance belongs in the calculation.

Binary Outcomes and the Logistic Interaction Scale

For a binary outcome, a common target is the product-term coefficient in a logistic model:

The parameter \(\beta_{GE}\) is a log-odds interaction: it describes how the exposure-associated change in log odds differs between genotype groups. It is not an additive risk difference. In a saturated model with four genotype–exposure cells, the large-sample variance of the interaction estimate is approximately the sum of the four inverse cell informations. If the event probability in cell \((g,e)\) is \(p_{ge}\), that is:

This expression reflects that a cell's information for its log odds is approximately \(n_{ge}p_{ge}(1-p_{ge})\), and the interaction contrast combines the four cell logits with coefficients \(+1\) or \(-1\). If all cell event probabilities are close to a common value \(p\), and genotype and exposure are independent, it simplifies to:

A rough sample-size calculation then replaces \(\sigma^2\) by \(1/[p(1-p)]\) in the normal-outcome formula and replaces the mean interaction \(\delta\) by the log-odds interaction \(\beta_{GE}\). This approximation can be poor when event probabilities differ substantially across cells. In that case, calculate information using the anticipated \(p_{ge}\) values and joint cell frequencies, or use a simulation that matches the proposed analysis.

Imbalance, Assumptions, and a Common Pitfall

For the binary mean-scale interaction, the frequency factor \(q(1-q)r(1-r)\) is largest when both frequencies are 0.5. Moving either frequency toward 0 or 1 reduces information at a fixed total \(N\). The same pattern has a direct cell-count interpretation: when a joint group is rare, its reciprocal count in the variance formula is large. Increasing total sample size can compensate, but the required increase can be substantial. For a continuous exposure, low exposure variance similarly weakens information for the product term.

These calculations presume the outcome model is correctly specified and that the interaction effect is the one the study intends to detect. They also presume independent participants and the assumed genotype–exposure distribution. Population structure, relatedness, clustering, covariate adjustment, missing data, unequal sampling, and exposure measurement error can change the effective information. A planning calculation should reflect the analyzed sample rather than merely the number recruited; for example, if only a proportion \(c\) is expected to have complete data, recruitment needs to be roughly \(N/c\) to yield the calculated analyzable sample size.

• Wrong reasoning: “The total sample is large, so the interaction must be well powered.”
• Correction: Interaction precision depends on outcome information across joint genotype–exposure groups. Calculate expected cell sizes and inspect their contributions to the standard error.
• Wrong reasoning: “The interaction effect is negative, so a positive critical-value sum cannot be used for sample size.”
• Correction: For a two-sided test, use \(\lvert\lambda\rvert\approx z_{1-\alpha/2}+z_{1-\beta}\), or equivalently square the signed effect when solving for \(N\). The sample-size approximation depends on \(\delta^2\) or \(\beta_{GE}^2\), so either interaction direction is covered.