Interpreting Interactions in Logistic and Poisson Models

Why the Link Scale Matters

A treatment-by-subgroup coefficient in a regression model describes how the treatment coefficient changes on the model’s link scale. It does not, by itself, state how much treatment changes the probability of an outcome or its rate within either subgroup. This distinction is especially important for logistic and log-link Poisson models: their links transform the outcome mean, so a constant contrast on the link scale generally translates into different contrasts on the outcome scale.

The practical task is therefore twofold. First, translate the fitted interaction into the subgroup-specific relative effect represented by the model: an odds ratio for logistic regression or a rate ratio for a log-link Poisson model. Then, when the scientific question concerns absolute outcomes, calculate subgroup-specific predicted probabilities or rates and contrast those predictions directly.

Logistic Regression: Interaction on the Log-Odds Scale

Let \(A\) be a binary treatment indicator, with \(A=1\) for treatment and \(A=0\) for control, and let \(S\) be a binary, pre-treatment subgroup indicator. Suppose \(Y\) is a binary outcome. A logistic model with a treatment-by-subgroup interaction is

This model assumes that the conditional log odds are represented by the stated linear predictor. In particular, with binary \(S\), it allows a distinct treatment log-odds contrast in each subgroup. The coefficients use \(A=0,S=0\) as the reference combination. They describe conditional probabilities for the specified model; a causal interpretation requires additional design and identification conditions discussed below.

To see the interpretation, set \(S=0\). The treatment coefficient in the linear predictor is \(\beta_A\), so the treatment odds ratio is \(e^{\beta_A}\). Setting \(S=1\) makes the treatment coefficient \(\beta_A+\beta_{AS}\), giving a treatment odds ratio of \(e^{\beta_A+\beta_{AS}}\). Their ratio is

Thus \(\beta_{AS}=0\) means equal treatment odds ratios across the two subgroups under this model. It does not mean equal treatment risk differences, nor does it mean that treatment has no effect in either subgroup.

Why Equal Odds Ratios Need Not Mean Equal Risk Differences

The logistic transformation is nonlinear. Consequently, even a model with no treatment-by-subgroup term can imply different probability changes in subgroups with different baseline risks. The following example isolates this point: it has no interaction on the log-odds scale, yet its predicted risk differences are unequal.

Log-Link Poisson Models: Interaction on the Log-Rate Scale

For count outcomes, a Poisson model with a log link can represent rates when observations have differing amounts of exposure time. Let \(Y\) be a count and \(T\) the exposure time. A common model is

Here \(\log(T)\) is an offset: its coefficient is fixed at 1. The model states that the expected count is proportional to exposure time and that the log rate follows the stated linear predictor. Equivalently, the conditional expected rate is \(E(Y\mid A,S,T)/T\). For interpreting the mean and rate ratios, the essential requirement is that this conditional mean model is appropriate. If conventional Poisson likelihood standard errors are used, the additional Poisson distribution and variance assumptions matter for those standard errors; they are not needed for the algebraic interpretation of the fitted mean coefficients.

At \(S=0\), the treatment rate ratio is \(e^{\gamma_A}\). At \(S=1\), it is \(e^{\gamma_A+\gamma_{AS}}\). Thus \(e^{\gamma_{AS}}\) is the ratio of the subgroup-specific treatment rate ratios. A positive \(\gamma_{AS}\) means the treatment rate ratio is larger in \(S=1\) than in \(S=0\); it does not, by itself, mean that the treatment rate is higher in absolute terms in that subgroup.

From Model Predictions to Causal Subgroup Effects

The equations above first describe conditional associations under specified regression models. To interpret a predicted treatment contrast as causal, the subgroup must be defined before treatment, treatment versions and outcomes must be well defined, and the usual identification conditions must be credible. In a randomized study, random assignment within the relevant population supports exchangeability between treatment groups. In an observational study, conditional exchangeability requires adequate control of treatment-outcome confounding within the subgroup. In either setting, consistency and positivity are also required: observed outcomes must correspond to the assigned treatment, and each treatment must be possible for the covariate profiles being compared.

Under these conditions, model predictions for \(A=1\) and \(A=0\) within a subgroup can be used to form subgroup-specific causal contrasts, provided the model correctly represents the conditional outcome mean for the target profiles. The logistic model produces probabilities by applying the inverse-logit function to its linear predictor. The Poisson model produces rates by exponentiating the rate linear predictor. Contrasting those predictions on the probability or rate scale answers an absolute-effect question. The interaction coefficient instead answers a relative-effect question on the model’s link scale.

Common Mistake: Treating the Interaction Coefficient as an Absolute Effect

A frequent error is to read a negative logistic interaction coefficient as “treatment reduces risk by this many percentage points more in the subgroup.” That statement is wrong: the coefficient is a difference in log odds ratios, and its exponent is a ratio of odds ratios. The corresponding probability-scale subgroup contrasts require inverse-logit predictions under both treatment conditions and then subtraction. Likewise, a Poisson interaction coefficient describes a log rate-ratio contrast, not a difference in event rates.

A second error is to conclude that a zero interaction coefficient means no subgroup heterogeneity of any kind. It means no interaction on the model’s link scale under the specified model. As the logistic example showed, probability-scale treatment differences can vary even when the log-odds interaction is zero. Always name the scale of the effect and the scale on which the interaction is being described.