Treatment-By-Biomarker Interaction Models

Why Model Treatment-by-Biomarker Interaction?

In a randomized trial, an overall treatment effect can conceal different effects in biomarker-defined populations. A model with a treatment-by-biomarker interaction represents those differences directly. Its coefficients do not automatically answer every clinical question, however: interpretation depends on the outcome scale and on how treatment and biomarker status are coded. In particular, a logistic interaction describes effect modification on the odds-ratio scale, not necessarily on the risk-difference scale.

Here the biomarker defines two baseline populations: biomarker-negative and biomarker-positive. Let \(T=1\) denote assignment to the experimental treatment and \(T=0\) assignment to control; let \(B=1\) denote biomarker-positive status and \(B=0\) biomarker-negative status. The treatment effect in each group is a population-specific estimand, as in the earlier material on population-specific estimands. Since treatment is randomized, comparisons by assigned treatment within a baseline biomarker group can estimate causal effects of assignment in that group, subject to the trial’s outcome and analysis assumptions. The biomarker itself is not randomized, so its coefficient does not establish a causal effect of biomarker status.

Continuous Outcomes: The Linear Interaction Model

For a continuous endpoint \(Y\), a basic model is

The conditional mean is specified as a linear function of treatment, biomarker status, and their product. For standard linear-model inference, observations must be independent and the conditional error variance must be modeled appropriately. In the classical model, errors have mean zero and common variance \(\sigma^2\) across the four groups. Normally distributed errors, along with these appropriate variance assumptions, are required for exact small-sample \(t\)-inference. Approximate normality supports only approximate inference; with sufficiently large samples, inference may instead rely on large-sample approximations, and heteroskedasticity-robust standard errors can address unequal variances under their conditions.

The fitted means for the four treatment-biomarker combinations follow by substituting their indicator values:

Subtracting control from treatment within each biomarker group gives the treatment effects:

Thus \(\beta_T\) is the effect in the reference group, \(B=0\), not an overall treatment effect. The coefficient \(\beta_B\) is the biomarker-positive versus biomarker-negative mean difference under control. The interaction coefficient is the difference between the two treatment effects, in the units of \(Y\). If lower endpoint values are better, a negative treatment effect indicates a reduction in the endpoint.

Uncertainty for a Subgroup Treatment Effect

A subgroup effect that is a sum of coefficients needs the variance of that sum, not just the standard error of either coefficient. For the positive group, the estimated effect is \(\hat{\beta}_T+\hat{\beta}_{TB}\), so

The covariance term can increase or decrease the variance. A confidence interval for the subgroup effect is then formed using its standard error and an appropriate reference distribution. In small samples under the classical linear model, exact \(t\)-inference requires normally distributed errors and the common-variance assumptions described above. In a large sample, a normal approximation may be used when justified. An interval for \(\beta_{TB}\) answers a different question: it quantifies uncertainty about the difference between subgroup effects.

Binary Outcomes: Logistic Interaction Model

For a binary outcome \(Y\), where \(Y=1\) denotes the event of interest, a common model is logistic regression:

This model assumes independent outcome observations and a correctly specified conditional log-odds structure for the modeled groups. In a randomized trial, treatment assignment is independent by design, but the outcome model still needs adequate information in each treatment-by-biomarker cell. Sparse cells can produce imprecise estimates or separation, in which the maximum-likelihood coefficient may be infinite or fail to converge. Logistic-regression Wald intervals and tests are generally large-sample approximations, not exact small-sample \(t\)-procedures.

As with the linear model, \(\beta_T\) is the treatment effect for \(B=0\), but now it is a log odds ratio. The positive-group log odds ratio is \(\beta_T+\beta_{TB}\). Exponentiating gives the odds ratios:

Consequently, \(\beta_{TB}=0\) means equal treatment odds ratios across the biomarker groups. It does not imply equal risk differences, and a nonzero interaction on the log-odds scale need not correspond to the same magnitude of interaction on another scale. To report probabilities or risk differences, calculate fitted probabilities for each treatment-biomarker combination.

Fitting, Diagnostics, and Interpretation

With binary \(T\) and \(B\), the interaction is represented by the product \(TB\). A model without that term constrains the treatment contrast to be the same across biomarker groups on its modeled scale. Fitting the interaction model allows the four group means or log odds to vary freely, subject to the chosen model. The following sequence helps keep the analysis interpretable.

The interaction estimate is not a substitute for each subgroup treatment estimate. A confidence interval for the interaction may include zero even when one subgroup estimate is individually precise, or exclude zero while one subgroup estimate remains uncertain. Likewise, comparing whether one subgroup has a “significant” treatment result and another does not is not a valid test that their treatment effects differ; that comparison requires the interaction contrast itself.

If treatment or biomarker coding changes, the coefficients’ reference interpretations change. For example, reversing the biomarker coding makes the treatment main effect refer to the other group and reverses the sign of the interaction contrast, although fitted group-specific effects remain the same. Reporting the coding and the calculated subgroup effects prevents a coefficient label from being mistaken for a clinical conclusion.

Common Mistakes and Limits

• Reading the treatment main effect as an overall effect. In an interaction model it is the treatment effect only for the reference biomarker group. Calculate \(\beta_T+\beta_{TB}\) for the other group.
• Interpreting a logistic interaction as a risk-difference change. It is a log odds-ratio contrast. Obtain fitted probabilities in each cell and subtract them if risk differences are the target.
• Comparing subgroup p-values instead of testing effect difference. A difference between “significant” and “not significant” is not itself evidence of interaction. Test or estimate \(\beta_{TB}\) on the specified model scale.
• Ignoring model assumptions or sparse data. Residual patterns, unequal variance, separation, or very small treatment-by-biomarker cells can undermine standard errors and fitted estimates. Report diagnostics and use a model or inferential method suitable for the data.
• Giving the biomarker a causal interpretation. Randomization supports causal treatment comparisons within baseline biomarker groups; it does not randomize biomarker status or show that changing the biomarker would change response.

Interaction models are useful because they make treatment-effect heterogeneity explicit, but a binary subgroup analysis has limits. It depends on a defensible biomarker definition, can lose precision when the sample is divided into cells, and may invite overinterpretation if subgroup contrasts were selected after examining outcomes. A prespecified analysis and clear scale-specific estimates are important, especially when subgroup claims are part of the trial’s confirmatory objectives.