Pitfalls In Interpreting Interaction Coefficients

Why Interaction Coefficients Need Context

A product-term coefficient is not a context-free measure of how two loci “interact.” Its meaning depends on how each genotype is coded, which other terms are in the model, what outcome scale is used, and which effects are being compared. Two analyses can describe the same fitted data while reporting different coefficient values because they use different genotype reference points. Conversely, a coefficient that looks like an interaction can partly reflect omitted main effects.

The central task is therefore interpretation, not just reading the sign or size of a coefficient. First identify the model and its scale; then state the contrast represented by the coefficient; finally check whether that contrast answers the biological or statistical question of interest. All examples below use hypothetical data.

Definition: In a two-locus model with genotype codes \(G\) and \(H\), a product term is the predictor \(GH\). Its coefficient describes an interaction contrast on the model’s specified scale, conditional on the model’s other terms. It is not, by itself, evidence of a biological mechanism or a causal effect.

What the Coefficient Means Depends on Coding

For a quantitative outcome, consider a linear mean model with additive allele dosages \(G,H\in\{0,1,2\}\):

E(Y\mid G,H)=\beta_0+\beta_GG+\beta_HH+\gamma GH.

Here \(\beta_G\) is the slope for a one-allele increase in \(G\) when \(H=0\), while \(\beta_H\) is the slope for a one-allele increase in \(H\) when \(G=0\). The coefficient \(\gamma\) is the change in the \(G\) slope for each one-allele increase in \(H\), and equivalently the change in the \(H\) slope for each one-allele increase in \(G\). In a linear model, these quantities are expressed in units of \(Y\) per allele or per product of allele increments.

This interpretation is conditional on the other locus’s code. The \(G\) slope at \(H=h\) is \(\beta_G+\gamma h\). Thus, \(\beta_G\) is not generally an average effect across the observed \(H\) genotypes; it is the slope at the reference value \(H=0\).

Worked Example: Reversing an Allele-Dosage Code

Suppose a hypothetical quantitative phenotype is measured in millimetres. The fitted conditional mean is \(10+2G+H+0.5GH\), where \(G\) and \(H\) are allele dosages coded from 0 to 2. What do the coefficients imply, and what happens if the alleles at \(G\) are coded in the opposite direction?

At \(H=0\), the predicted change per additional \(G\) allele is \(2+0.5(0)=2\) mm. At \(H=1\), it is \(2+0.5(1)=2.5\) mm; at \(H=2\), it is \(2+0.5(2)=3\) mm. The interaction coefficient is therefore \(0.5\) mm per product of one-allele increments: each additional \(H\) allele increases the \(G\)-dosage slope by 0.5 mm.

Now define the reversed dosage \(G'=2-G\). Substituting \(G=2-G'\) into the same model gives

$$ 10+2(2-G')+H+0.5(2-G')H =14-2G'+2H-0.5G'H. $$

The fitted means have not changed, but the interaction coefficient is now \(-0.5\), and the coefficient of \(H\) has changed from 1 to 2. The sign change reflects the reversal of the \(G\) coding, not a change in the underlying fitted genotype means. The interaction coefficient must be reported with the allele coding and reference values specified.

More generally, if both genotype codes are changed by nonzero linear transformations, the product-term coefficient rescales according to those transformations. Reversing the direction of one dosage changes its sign; reversing both leaves the sign unchanged. Centering or choosing a different reference value can also change the main-effect coefficients, because those coefficients describe slopes at the newly chosen reference points. When comparing reported coefficients, compare the genotype definitions and the model formula before comparing their numerical values.

Why Main Effects Usually Belong With the Product Term

A model containing a product term but omitting one or both corresponding main effects imposes restrictions on the fitted means. In the two-locus genotype setting, the usual hierarchical specification includes the main effects for \(G\) and \(H\) whenever it includes \(GH\). This principle is not a claim that every locus must have a nonzero main effect. It prevents the product term from being forced to account for ordinary differences among genotype groups.

The restriction is especially easy to see when both genotype indicators are binary. Let \(A,B\in\{0,1\}\) and write the four group means as \(\mu_{00},\mu_{10},\mu_{01},\mu_{11}\). In the full model \(\mu_{ab}=\beta_0+\beta_Aa+\beta_Bb+\gamma ab\), the interaction coefficient is the difference-in-differences contrast:

\gamma=(\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}) =\mu_{11}-\mu_{10}-\mu_{01}+\mu_{00}.

This contrast compares the \(A\) difference when \(B=1\) with the \(A\) difference when \(B=0\). If the main effects are omitted, the model becomes \(\mu_{ab}=\beta_0+\gamma ab\). It then forces \(\mu_{00}=\mu_{10}=\mu_{01}\): both single-locus groups must have the same mean as the double-reference group. The coefficient of \(AB\) in that restricted model is not, in general, the difference-in-differences interaction.

Worked Example: A Product Term Absorbs Omitted Main Effects

Suppose hypothetical binary genotype indicators produce these four outcome means, in phenotype units:

A	B	Mean outcome
0	0	10
1	0	12
0	1	13
1	1	20

In the full model, the interaction contrast is

\gamma=20-12-13+10=5.

The main-effect contrasts are also visible: the difference for \(A\) at \(B=0\) is \(12-10=2\), and the difference for \(B\) at \(A=0\) is \(13-10=3\). With all four terms, the model reproduces the four means exactly.

Instead fit only \(\beta_0+\gamma AB\), giving the four cells equal weight. The three cells with \(AB=0\) must share one fitted mean. Least squares sets that common fitted mean to their average, \((10+12+13)/3=11.667\). The fitted mean for cell \(11\) is 20, so the product-term coefficient is \(20-11.667=8.333\). This is not the full-model interaction contrast of 5. It has absorbed some of the differences in the two single-locus cells because the restricted model cannot represent their separate main effects.

The lesson is not that an interaction coefficient must be nonzero whenever the single-locus means differ. It is that a product term in a model without the relevant main effects answers a different, constrained modelling question. A large coefficient from that model cannot be interpreted as a clean two-locus contrast.

Conditional Effects Are Not Automatically Marginal Effects

A conditional effect compares outcomes at a specified value or distribution of the other genotype. A marginal effect averages over a distribution of that genotype. These can differ when the distribution being averaged over differs between the groups being compared. Even when the same distribution is used for both groups, a conditional coefficient and a marginal contrast need not have the same interpretation.

Worked Example: Genotype Composition Changes a Marginal Comparison

Consider hypothetical risks of an outcome in four genotype groups. Let \(G,H\) be binary indicators, with risks \(p_{00}=0.10\), \(p_{10}=0.20\), \(p_{01}=0.30\), and \(p_{11}=0.60\). On the risk-difference scale, the effect of \(G\) is \(0.20-0.10=0.10\) when \(H=0\), and \(0.60-0.30=0.30\) when \(H=1\). The interaction contrast on this scale is \(0.30-0.10=0.20\).

Suppose, in a hypothetical population, 10% of the \(G=0\) group has \(H=1\), while 90% of the \(G=1\) group has \(H=1\). The observed marginal risks are

P(Y=1\mid G=0)=0.90(0.10)+0.10(0.30)=0.12,

P(Y=1\mid G=1)=0.10(0.20)+0.90(0.60)=0.56.

Their difference is \(0.56-0.12=0.44\), which is not either conditional risk difference, 0.10 or 0.30. It also is not the risk difference standardized to a common \(H\) distribution. For example, using 50% \(H=1\) in each group gives risks \(0.20\) for \(G=0\) and \(0.40\) for \(G=1\), a standardized difference of \(0.20\). The marginal comparison of 0.44 reflects both the differing conditional risks and the different \(H\) compositions. It should not be labelled the conditional effect of \(G\).

The same distinction matters in regression. A coefficient in a model containing both loci describes a conditional relationship according to that model. A marginal summary averages over genotype distributions, which must be stated or specified. Neither is automatically causal: causal interpretation requires assumptions and a causal estimand appropriate to the study design, not merely an interaction term.

Interaction Is Specific to the Outcome Scale

“Interaction” can refer to different contrasts on different scales. For a binary outcome, an interaction on the log-odds scale means that the joint odds ratio differs from the product of the two single-locus odds ratios. An interaction on the risk-difference scale instead compares risk differences across genotype groups. A coefficient can be zero on one scale and the corresponding interaction contrast nonzero on another.

Worked Example: No Log-Odds Interaction but a Risk Interaction

Suppose hypothetical risks for binary indicators \(G,H\) are \(p_{00}=0.10\), \(p_{10}=2/11\approx0.1818\), \(p_{01}=0.25\), and \(p_{11}=0.40\). Their odds in the four cells are \(1/9\), \(2/9\), \(1/3\), and \(2/3\), respectively.

Relative to cell \(00\), the odds ratio for \(G\) alone is \((2/9)/(1/9)=2\), and for \(H\) alone it is \((1/3)/(1/9)=3\). The odds in cell \(11\) equal \((1/9)(2)(3)=2/3\). Thus the joint odds ratio is exactly the product of the two single-locus odds ratios. A logistic regression with the corresponding main effects and no \(GH\) term describes these four odds; its log-odds interaction coefficient is zero.

But the risk-difference interaction is

p_{11}-p_{10}-p_{01}+p_{00} =0.40-0.1818-0.25+0.10 \approx 0.0682.

So there is a nonzero interaction contrast on the risk-difference scale despite the absence of a product term on the log-odds scale. This is not a contradiction: the scales define different contrasts. State the scale whenever describing an interaction in a binary-trait analysis.

A scale-specific statistical interaction also does not identify a biological mechanism. A coefficient may summarize how observed outcome values depart from a specified model’s additive or multiplicative structure. By itself, it does not show that one gene alters the molecular action of another, nor does it establish a causal pathway. Biological claims require evidence beyond the statistical pattern, including appropriate design and supporting functional evidence.

A Practical Interpretation Check

Before describing an interaction coefficient, work through the following checks. They keep the numerical result connected to the contrast actually estimated.

Write the model and outcome scale.
State whether the mean, risk, log odds, or another quantity is modelled, and show which main effects and product terms are included.

Specify genotype coding.
Give the allele dosage or indicator definition, direction, and reference values. Check whether a coding reversal would change the coefficient’s sign or reference-point interpretation.

Translate the coefficient into a contrast.
For example, state a slope change per allele increment or a difference-in-differences on a named scale. Do not substitute a marginal interpretation for a conditional one.

Check model hierarchy and scope of the claim.
Confirm the relevant main effects are included, and distinguish a fitted statistical pattern from a biological or causal explanation.

Common Mistakes

Comparing signs without checking allele coding. A sign reversal can arise from reversing one dosage definition while leaving every fitted genotype mean unchanged. Report the coding and interpret the contrast it defines.
Calling a product-term coefficient “the interaction” without naming the scale. A zero coefficient on the log-odds scale does not imply zero interaction on the risk-difference scale. State the outcome scale and contrast.
Dropping main effects because the focus is the interaction. This constrains single-locus genotype means and can make the product term absorb their differences. Include the relevant main effects in the usual hierarchical model.
Reading a conditional coefficient as a population-average effect. A coefficient evaluated at a reference genotype is not automatically an average over genotype frequencies. Define the averaging distribution for marginal summaries.
Turning statistical interaction into a mechanistic claim. A scale-specific departure from a model’s structure does not by itself demonstrate molecular interaction or causation.

Key takeaway: An interaction coefficient is interpretable only after specifying genotype coding, model terms, conditioning, and outcome scale. Its statistical meaning is a contrast under that specification; biological meaning requires additional evidence.

Check Your Understanding

For each question, identify the contrast or assumption that needs attention before interpreting the result.

In a linear dosage model, what does a product-term coefficient describe when the other dosage is increased by one allele?
If one allele dosage is replaced by its reverse coding \(2-G\), what can happen to the product-term coefficient even when fitted means do not change?
For binary indicators \(A\) and \(B\), write the four-cell difference-in-differences that equals the product-term coefficient in the full linear mean model.
Why can a marginal genotype comparison differ from a conditional effect when the other genotype has different distributions across the groups?
Can a logistic model have no product term while the risk-difference interaction is nonzero? Explain what the answer implies about scale.