Comparing Shared-Effect And Random-Effects Basket Models

Two Different Claims About Basket Effects

Basket trials study treatment effects in several subgroups, or baskets. A central modeling choice is whether the baskets are assumed to have exactly the same underlying treatment effect, or whether their effects may differ but are related. These assumptions are not interchangeable: they change what the model says about heterogeneity and how it predicts an individual basket’s effect.

A shared-effect model assigns one treatment-effect parameter to all baskets. A random-effects model assigns a basket-specific effect to each basket and models those effects as draws from a common distribution. The second model permits genuine variation among baskets; the first does not. This contrast concerns treatment effects, not baseline risks. As in the earlier material on separating baseline risk and treatment-effect variation, baseline differences can be modeled separately.

For clarity, the worked calculations below use continuous treatment-effect estimates in points. Suppose the basket-level estimate \(Y_j\), conditional on its true effect, follows \(Y_j\mid\theta_j \sim N(\theta_j,s_j^2)\), with known sampling standard error \(s_j\). The usual assumptions for these calculations are that the estimates are unbiased for their basket effects, their sampling variances are correctly specified, and their sampling errors are independent across baskets. These assumptions concern the sampling model; they do not establish that the basket effects are identical or normally distributed.

What the Models Imply

Under the shared-effect model, all basket data contribute information about the same \(\theta\). Conditional on \(\theta\), each estimate has likelihood \(Y_j\mid\theta\sim N(\theta,s_j^2)\). The model therefore rules out true effect heterogeneity: observed differences among \(Y_j\) must be explained by sampling variation. The assumption may be reasonable when treatment effects are expected to be sufficiently alike for the trial’s purpose, but it is strong.

Under the random-effects model, each basket has its own \(\theta_j\). Conditional on fixed \(\mu\) and \(\tau\), the prior distribution for a basket effect is \(N(\mu,\tau^2)\). The parameter \(\mu\) describes the center of the modeled effects, while \(\tau\) is measured in the same units as the treatment effect. A large \(\tau\) allows substantial variation; a value near zero makes the model behave more like a common-effect model. In a full Bayesian analysis, \(\mu\) and \(\tau\) can themselves have prior distributions. The simple calculations here condition on specified values to make the distinction transparent.

With fixed \(\mu,\tau\) such that \(\tau>0\), and a normal sampling model, the basket-specific posterior is also normal. Its variance and mean are

When \(\tau=0\), the prior is a point mass at \(\mu\), so \(\theta_j=\mu\) with probability one.

The posterior mean is a precision-weighted average of the basket’s estimate and the distributional center. This is the source of the model’s partial pooling: a basket estimate is informed by the common distribution as well as by its own data. It does not mean all basket effects are equal.

Worked Posterior Calculations

Justification and Model Checking

The random-effects posterior formula follows by multiplying the normal likelihood and normal distribution for \(\theta_j\). Ignoring constants that do not depend on \(\theta_j\), the conditional posterior density is proportional to

Writing the exponent as \(-\frac12[A\theta_j^2-2B\theta_j+\text{constant}]\), where \(A=\frac{1}{s_j^2}+\frac{1}{\tau^2}\) and \(B=\frac{Y_j}{s_j^2}+\frac{\mu}{\tau^2}\), completing the square gives a normal density with variance \(v_j=(1/s_j^2+1/\tau^2)^{-1}\) and mean \(v_j(Y_j/s_j^2+\mu/\tau^2)\), as stated above. This derivation requires the normal sampling model and normal distribution for the basket effect, as well as fixed hyperparameters for this conditional calculation.

Model choice should not be made from the posterior means alone. The earlier material on posterior predictive checks across subgroups and residual patterns that reveal model misfit provides relevant diagnostics. In this comparison, inspect whether the shared-effect model can plausibly account for the observed spread of basket estimates through sampling error. Under a random-effects model, examine posterior uncertainty about heterogeneity and whether its predictive distributions can reproduce the observed subgroup patterns. A fitted model can still be inadequate if the basket effects are not reasonably represented by its assumed distribution, or if the sampling standard errors are wrong.

The principal alternatives make different commitments. A shared-effect model is efficient when effects really are sufficiently alike, but its subgroup predictions can be overconfident if meaningful heterogeneity exists. A random-effects model accommodates variation and produces basket-specific estimates, but its conclusions depend on the distributional form and on assumptions about \(\mu\) and \(\tau\). An independent analysis assigns each basket its own effect without a common distribution; it avoids a shared-distribution assumption but does not use that structure to stabilize subgroup estimates. Which assumption is defensible depends on biological rationale, trial design, and diagnostics, not simply on which model produces narrower intervals.

Common Pitfalls and Limitations

• Reading a shared effect as evidence of homogeneity. A common-effect model assumes there is no true basket-to-basket variation. It does not test or demonstrate that biological effects are identical.
• Calling the random-effects center every basket’s effect. \(\mu\) is the distributional center, not the effect in each subgroup. Basket-specific posterior estimates depend on both their own data and the assumed distribution.
• Ignoring uncertainty in heterogeneity. The worked calculations condition on fixed \(\tau\). When \(\tau\) is estimated, its posterior uncertainty should be propagated into subgroup and predictive uncertainty; treating a poorly known value as fixed can make conclusions too certain.
• Confusing effect prediction with outcome prediction. Predicting a future basket’s true treatment effect includes between-basket variation under the random-effects model. Predicting a future estimate or patient outcome also includes the relevant sampling or patient-level variation.
• Assuming the random-effects model solves every transportability problem. A normal distribution of effects may be inappropriate, and baskets may differ in ways not captured by a single center and heterogeneity scale. Neither model can repair poorly defined subgroups or an unsuitable sampling model.

The comparison is most useful when the scientific question is explicit: is the target a single effect regarded as common across baskets, or are basket-specific effects of interest and expected to vary? The model should match that target and the clinical rationale. Sensitivity to plausible heterogeneity assumptions is especially important when the data contain few baskets, since those data may provide limited information about between-basket variation.