Coding Genotypes and Exposures for Interaction

Why Coding Choices Matter

A genotype and an exposure do not enter a regression model as abstract labels: each must be represented by numerical variables. The representation determines which groups are compared directly, what a coefficient means, and which patterns the model allows. A model can therefore be technically fit but answer a different question from the one intended if its coding is not made explicit.

For example, coding genotypes as 0, 1, and 2 treats each additional copy of an allele as the same-sized step. Coding the same three genotypes with indicator variables instead allows separate comparisons of each genotype with a chosen reference group. Neither encoding is automatically correct for every analysis. The choice should match the scientific contrast of interest and the assumptions the analyst is prepared to make.

In the equations below, let \(G\) denote genotype, with values 0, 1, and 2 for the number of copies of a specified allele, and let \(E\) denote an exposure. Let \(Y\) be a quantitative outcome. We focus on how to encode \(G\) and \(E\), and on the contrasts represented by the resulting coefficients.

Four Common Genotype Codings

For a biallelic variant, suppose \(G=0\), \(G=1\), and \(G=2\) correspond to zero, one, or two copies of the allele being counted. The labels must be stated: counting the other allele reverses the ordering. Three commonly used one-degree-of-freedom codings and a categorical coding are shown below.

Here \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. In the allele-count model, a single coefficient imposes a linear trend across the three genotype values: the modeled difference from \(G=0\) to \(G=1\) is the same as the modeled difference from \(G=1\) to \(G=2\). The label “additive” in this coding describes the allele-count scale. By itself, it does not mean that an outcome effect is additive on every other scale.

Dominant and recessive coding each combine two genotype groups. They therefore impose equality of the modeled means within the combined groups, apart from other terms in the model. Categorical coding does not impose an ordered trend: with \(G=0\) as reference, it estimates separate \(G=1\) versus \(G=0\) and \(G=2\) versus \(G=0\) contrasts. For three genotype categories, this uses two indicator variables rather than one.

Binary Exposure Coding and Reference Groups

For a binary exposure, one common coding sets \(E=0\) for unexposed and \(E=1\) for exposed. In a model containing an intercept, the unexposed group is the exposure reference. Reversing the codes makes the exposed group the reference. The observed data have not changed, but coefficient values and their direct comparisons have.

For illustration, consider a categorical genotype model with \(G=0\) as the genotype reference. Define \(D_1=I(G=1)\) and \(D_2=I(G=2)\). A model with a binary exposure and genotype-by-exposure terms can be written as follows. The error term represents variation around the conditional mean.

Under this coding, \(\beta_0\) is the modeled mean for \(G=0,E=0\). The terms \(\beta_1\) and \(\beta_2\) compare \(G=1\) and \(G=2\), respectively, with \(G=0\) when \(E=0\). The coefficient \(\beta_E\) compares \(E=1\) with \(E=0\) within the reference genotype \(G=0\). The remaining coefficients describe how those genotype comparisons change between the two exposure values. Their exact contrasts follow from the coding and model, not just from the names of the coefficients.

Continuous Exposure Coding

A continuous exposure is usually retained as a numerical variable when the intended model assumes a linear relationship over the range being analyzed. If \(E\) is measured in a stated unit, its coefficient is a change in the modeled outcome per one-unit increase in \(E\), conditional on the other model terms. This coding does not divide participants into exposed and unexposed groups.

Centering replaces \(E\) with \(E_c=E-c\), where \(c\) is a chosen value, often a scientifically meaningful exposure level or the sample mean. In a model with an intercept, the intercept then corresponds to \(E=c\). Centering changes the point at which genotype reference contrasts are stated, but it does not change the unit of exposure or the fitted linear trends. Rescaling, such as replacing years with decades, changes the numerical size and units of the exposure coefficient.

For categorical genotype coding and a continuous exposure, a useful model is

Here \(\beta_0\) is the modeled mean for \(G=0\) at \(E=c\). The coefficients \(\beta_1\) and \(\beta_2\) compare the other genotypes with \(G=0\) at that exposure value. The coefficient \(\beta_E\) is the modeled exposure slope for \(G=0\), in outcome units per exposure unit. The interaction terms encode differences between genotype-specific exposure slopes and the reference-genotype slope. With an allele-count variable instead of \(D_1,D_2\), the model imposes a more restricted pattern across genotype values.

Choosing and Reporting a Coding

The appropriate coding depends on the scientific comparison and the structure one is willing to impose. Allele-count coding is compact and can be useful when a linear per-allele pattern is a reasonable working assumption. Dominant and recessive codings answer specific carrier-status or two-copy comparisons, respectively, but combine genotypes that categorical coding keeps distinct. Categorical coding is more flexible for three genotype groups, at the cost of estimating more parameters.

For a continuous exposure, retain its quantitative scale when a linear exposure pattern is a reasonable model over the relevant range. If exposure categories are scientifically motivated, state the cut points and the reference category. Dichotomizing a continuous measure solely for convenience changes the question and discards within-category distinctions. Centering can make reference contrasts more meaningful; it does not itself make a nonlinear relationship linear.

Before fitting or interpreting a model, state the allele being counted, the genotype coding, the exposure coding and units, and the reference values. For example: “Genotype was coded as the number of copies of allele A; exposure was measured in dose units per day and centered at 4; genotype 0 was the reference.” This information makes the model’s contrasts reproducible and interpretable.

Model Conditions and a Common Pitfall

The worked regression examples use conditional mean models. For ordinary least squares coefficient estimates to represent the stated modeled contrasts, the mean function must be specified as written and the relevant design columns must be estimable from the data. Exact conventional small-sample inference additionally relies on assumptions such as independent errors with mean zero and constant variance; normally distributed errors give the usual exact \(t\)- and \(F\)-based results under the linear model. The examples calculate coding contrasts from means and do not claim statistical significance.

Sparse genotype-by-exposure cells can make some contrasts imprecise or inestimable. For instance, if no participants have \(G=2,E=1\), a model that requires that cell mean cannot estimate it from the data without additional assumptions. Coding choices cannot create information that is absent. Analysts should check that the groups and exposure ranges needed for their intended contrasts are represented.

• Mistake: “The allele-count coefficient is simply the difference between \(G=2\) and \(G=0\).” Correction: It is a fitted per-allele slope under the linear trend constraint; twice that slope equals the model-implied \(G=2\) versus \(G=0\) difference, not necessarily the observed difference when the trend does not fit.
• Mistake: “The genotype main-effect coefficient is a genotype comparison for everyone.” Correction: In a model with genotype-by-exposure terms, it is a comparison at the exposure reference value—zero for uncentered exposure, or the centering value for centered exposure.
• Mistake: “Changing the 0 and 1 labels changes the data or underlying group means.” Correction: It changes the parameterization and direct reference contrasts. State the coding before interpreting coefficients.
• Mistake: “Dominant and recessive coding are just alternative labels for categorical coding.” Correction: They combine genotype categories and impose equality within the combined groups; categorical coding can estimate separate means for all three genotypes.