Testing SNP-By-Environment Interaction

Why Test SNP-By-Environment Interaction?

A genetic association may differ across environmental conditions. For example, the change in a quantitative trait associated with one additional effect allele could be larger among people with high exposure than among people with low exposure. A regression model can represent this pattern with a genotype-by-environment product term. The term tests whether the SNP association changes with environment on the scale of the model; it does not, by itself, show that changing the environment would cause a change in the genetic effect.

Let \(G\) be an individual's effect-allele dosage, usually coded \(0\), \(1\), or \(2\), and let \(E\) be an environmental variable. Let \(C\) denote covariates included to adjust for relevant features of the sample. For a continuous environment, it is often convenient to center \(E\), so that \(E=0\) represents a meaningful value such as the sample mean. Centering helps interpretation but does not remove confounding or change the fitted interaction pattern.

Regression Models and the Interaction Null

For a quantitative trait \(Y\), the linear model with a continuous environment is

Here \(\beta_{GE}\) is the interaction coefficient. The conditional mean is modeled as linear in the included predictors. For conventional standard errors and tests, observations are assumed independent and errors are assumed to have mean zero conditional on the predictors; constant error variance is also required for the usual homoskedastic standard errors. Exact small-sample \(t\)-based inference additionally relies on normally distributed errors. If the variance assumption is questionable, a suitable robust standard error may be used, but it does not repair a misspecified mean model or dependence between observations.

For a binary trait \(Y_i\), a logistic model is

This model assumes independent Bernoulli outcomes conditional on the predictors and a correctly specified conditional log-odds model. The fit must also be estimable: sparse genotype-environment combinations or separation can produce unstable or non-finite estimates. For case-control samples, interpreting the fitted odds-ratio slopes requires the usual condition that selection into the sample depends on outcome, not additionally on genotype or environment within outcome and covariate strata.

The interaction coefficient is not the SNP effect at every environment value. In either model, the SNP slope at \(E=e\) is \(\beta_G+\beta_{GE}e\). In a logistic model this is a log odds ratio per additional effect allele, so its exponentiation gives the conditional odds ratio. The main-effect coefficient \(\beta_G\) is the SNP effect at \(E=0\); the environment main effect \(\beta_E\) is the environment slope at \(G=0\). Centering \(E\) makes these reference values more useful, but does not alter \(\beta_{GE}\) or its test.

Deriving the Meaning of the Product Term

The interpretation follows directly by comparing the model's SNP slopes at two environmental values. Holding covariates fixed, the linear-model conditional mean at \(G=g\) and \(E=e\) contains the terms \(\beta_Gg+\beta_Ee+\beta_{GE}ge\). Its change for a one-unit increase in dosage is therefore

Consequently, the difference between the SNP slopes at \(E=e_2\) and \(E=e_1\) is

Thus \(\beta_{GE}=0\) exactly when the linear SNP slope does not vary with \(E\). In logistic regression, the same algebra applies to the log odds. Exponentiating the difference gives the ratio of the conditional SNP odds ratios at the two environment values: \(\exp\{\beta_{GE}(e_2-e_1)\}\).

Categorical Environments and Joint Tests

If environment has categories rather than a meaningful numeric scale, represent it with indicator variables and interact each indicator with genotype. For a three-category environment with “low” as the reference, let \(I_M\) and \(I_H\) indicate middle and high categories. The linear predictor includes \(\beta_GG+\beta_MI_M+\beta_HI_H+\beta_{GM}GI_M+\beta_{GH}GI_H\), along with the intercept and covariates. The SNP slopes are \(\beta_G\) in the low group, \(\beta_G+\beta_{GM}\) in the middle group, and \(\beta_G+\beta_{GH}\) in the high group. The interaction null is the joint hypothesis \(H_0:\beta_{GM}=\beta_{GH}=0\), with two restrictions. A single coefficient test would not test this full null.

Model Checks, Alternatives, and Limitations

Before interpreting an interaction, check that its coding matches the scientific question. A linear dosage-by-environment term assumes an additive dosage effect and a change in that effect that is linear in a continuous \(E\). If the exposure is categorical, use the appropriate indicator interactions. If a linear dosage assumption is unsuitable, genotype indicators can be interacted with environment, but this adds parameters and requires an appropriately defined joint test. The interaction is scale-dependent: a null interaction on the log-odds scale does not imply a null interaction on the probability-difference scale.

Also examine the data support for the comparisons being made. If one genotype is rare at high exposure, the fitted interaction may rely heavily on a small number of observations. For a categorical environment, inspect genotype counts within every group; for a continuous environment, check whether genotypes occur across the exposure range. Sparse cells, near-collinearity between product and main-effect terms, an extreme exposure range, or separation in logistic regression can lead to large standard errors, unstable estimates, or convergence warnings. A small \(p\)-value does not resolve these problems.

The product-term model is generally preferable to running separate association tests in exposure subgroups and informally comparing their \(p\)-values. A significant result in one subgroup and a nonsignificant result in another does not establish that the effects differ: the subgroup estimates may differ little, with different precision. A formal interaction test directly assesses the difference in effects. Conversely, an additive-only model is simpler and can be more precise when interaction is absent, but it constrains the SNP slope to be the same across environments.

As with other single-variant models, correct adjustment and sound sampling matter. Population structure, relatedness, or omitted covariates associated with genotype, environment, and outcome can distort the interaction estimate. Including main effects for both genotype and environment is essential when including their product; otherwise, the product coefficient can absorb main-effect patterns. Statistical interaction is not automatically biological interaction or causal effect modification. Causal interpretation requires additional design and assumptions beyond the regression specification.