Random Slopes For Treatment Effects Across Subgroups

Why Let Treatment Effects Vary Across Subgroups?

In a basket trial, a treatment may be evaluated in several disease subgroups. Subgroups can differ in their untreated response rates, in how strongly the treatment works, or in both. A model that allows only baseline risks to differ captures the first kind of variation, but assumes a common treatment effect on its chosen model scale. A random treatment slope allows the treatment effect itself to vary across subgroups, while still relating those effects through a shared distribution.

The distinction is especially important for binary outcomes. On the logistic scale, subgroup baseline risk is represented by an intercept, whereas the conditional treatment effect is represented by a treatment coefficient. A random intercept alone allows different baseline log odds; it does not create subgroup-specific conditional odds ratios.

A Logistic Model With Random Intercepts and Slopes

Index subgroups by \(j=1,\ldots,J\) and patients within subgroup \(j\) by \(i\). Let \(Y_{ij}\) equal 1 for a patient with the specified binary outcome and 0 otherwise. Let \(T_{ij}=1\) indicate treatment and \(T_{ij}=0\) indicate control. The logistic model is

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The parameter \(\alpha\) is the population-average intercept on the log-odds scale, and \(\beta\) is the population-average treatment coefficient. The subgroup deviations \(u_j\) and \(v_j\) are, respectively, the random intercept and random treatment slope. Thus, subgroup \(j\)’s intercept is \(\alpha+u_j\), and its treatment coefficient is \(\beta+v_j\).

A common exchangeable specification is

The zero means define \(\alpha\) and \(\beta\) as the centers of the subgroup intercept and treatment-coefficient distributions. The variance parameters describe between-subgroup dispersion, and \(\rho\) allows baseline and treatment-effect deviations to be associated. The distribution expresses an exchangeability assumption: before observing subgroup-specific outcomes, the subgroup effects are treated as draws from a shared distribution. It does not assert that the subgroups are clinically identical.

The model also requires a suitable outcome likelihood and treatment allocation. Conditional on the subgroup effects and model parameters, patient outcomes are assumed independent Bernoulli observations with the stated probabilities. For grouped counts, this gives a binomial likelihood within each subgroup and treatment arm. In a randomized basket trial, treatment and control comparisons within a subgroup can support subgroup-specific effects, subject to the usual requirements of valid randomization, outcome measurement, and adequate data. A subgroup containing only treated or only control patients generally cannot, by itself, separate its baseline level from its treatment effect.

A complete Bayesian model additionally assigns prior distributions to \(\alpha\), \(\beta\), and the between-subgroup parameters. Those priors must be proper or otherwise justified for the intended analysis. The specific choice affects inference, especially when data are sparse; it is not determined by the likelihood alone.

The Treatment Contrast Shows What the Slope Means

For a patient in subgroup \(j\), the control linear predictor is \(\alpha+u_j\). The treated linear predictor is \(\alpha+u_j+\beta+v_j\). Their difference is

Because the difference in linear predictors is the log odds ratio, the subgroup-specific conditional odds ratio is \(\exp(\beta+v_j)\). The random intercept cancels. This is the key distinction: \(u_j\) changes both arms’ baseline log odds in subgroup \(j\), whereas \(v_j\) changes the treatment-control contrast.

This is a conditional odds ratio, conditional on subgroup and on the model’s included terms. It is not a risk difference, and it is not generally a marginal odds ratio averaged over subgroups. Even when \(v_j=0\) for every subgroup, different baseline risks can produce different treatment risk differences. Conversely, different \(v_j\) values imply different conditional odds ratios.

Why the Random Intercept Does Not Replace a Random Slope

The distinction follows directly from the model’s treatment contrast. Consider the random-intercept-only model, which sets \(v_j=0\). The log odds ratio in every subgroup is then \(\beta\), no matter how much the \(u_j\) values differ. Thus, a random intercept can accommodate baseline heterogeneity but cannot represent variation in conditional treatment odds ratios.

The converse also matters. A model with a random slope but no subgroup-specific intercept imposes a common baseline log odds across subgroups. If baseline risks differ and this restriction is false, the model may misrepresent the control outcomes, even if the treatment effects vary. Including both terms allows baseline and treatment contrasts to vary separately, and allowing their random effects to be correlated recognizes that the two kinds of subgroup deviation may be associated.

Worked Subgroup Contrasts and Their Limits

Observed treatment-control contrasts can suggest whether a common slope is plausible, but raw subgroup estimates are noisy, particularly with small basket sizes. A hierarchical model estimates subgroup effects jointly and partially pools them through the shared distribution. The calculations below are descriptive contrasts, not fitted posterior estimates.

Alternatives, Diagnostics, and Limitations

A treatment-by-subgroup interaction with fixed subgroup-specific coefficients is another alternative. It estimates distinct effects without treating them as draws from a common distribution. This may be useful when the subgroup set is small and specific comparisons are the focus, but it does not provide the same hierarchical partial pooling assumption.

Several checks are important before interpreting random slopes. First, verify treatment overlap within each subgroup: if treatment is completely confounded with subgroup, within-subgroup treatment contrasts cannot be distinguished from baseline differences. Second, examine whether the outcome and allocation information are adequate for estimating the additional variation; sparse or extreme binary outcomes can leave subgroup effects weakly identified. Third, inspect posterior summaries and uncertainty for subgroup intercepts, slopes, and predictions, alongside convergence diagnostics and model-based predictive checks. In particular, check whether replicated data reproduce response patterns across both treatment arms and subgroups.

A random-slope model is not automatically preferable because its estimates can differ by subgroup. It adds structure and parameters, and its conclusions depend on the exchangeability and prior assumptions. With few subgroups, the between-subgroup slope variation may be difficult to estimate. With little treatment overlap, the data may not distinguish \(u_j\) from \(v_j\) reliably. Sensitivity to reasonable prior choices and to the random-effects structure should therefore be considered.

Common Mistake

• Wrong reasoning: “The subgroups have different control response rates, so the treatment effect must vary.”
• Correction: Different control rates indicate baseline variation, which a random intercept can represent. To claim variation in conditional treatment effects, compare treatment-control contrasts on the model scale; in this logistic model, those are the subgroup log odds ratios \(\beta+v_j\).
• Another wrong inference: “The treatment risk differences differ, so the conditional odds ratios must differ.”
• Correction: A common odds ratio can yield different risk differences at different baseline risks, as the worked calculation showed. State the effect scale before interpreting heterogeneity.