Unequal Subgroup Sizes In Hierarchical Models

Why Unequal Basket Sizes Matter

In a basket trial, subgroups can differ substantially in how many patients they contribute. When a hierarchical model partially pools subgroup estimates, a smaller basket generally receives more influence from the shared distribution than a larger basket does. This is not an arbitrary adjustment: with fewer observations, the subgroup-specific data are less informative relative to the prior distribution.

The key qualification is that a larger shrinkage weight does not guarantee a larger observed movement toward the shared center. The movement also depends on how far the subgroup’s observed estimate is from that center. We will separate these two ideas using the beta-binomial model, then check the same principle in a normal-normal model.

Binary Outcomes: The Size-Dependent Weight

Suppose there are \(J\) baskets. For basket \(j\), let \(n_j\) be the number of patients and \(y_j\) the number who respond. Assume, conditionally on the basket response probability \(p_j\), that responses are independent Bernoulli outcomes. Thus \(y_j\mid p_j\sim\operatorname{Binomial}(n_j,p_j)\). For this calculation, take a common beta prior \(p_j\sim\operatorname{Beta}(a,b)\), with \(a>0\) and \(b>0\). The prior mean is \(m=a/(a+b)\), and \(a+b\) is its concentration.

This is a conditional calculation for specified \(a\) and \(b\). Its simple form also describes a beta-binomial hierarchy conditional on those shared hyperparameters. It assumes a common prior distribution across baskets and the same definition of response and observation window in each basket.

The posterior distribution and mean follow from multiplying the binomial likelihood by the beta prior:

Writing \(y_j=n_jr_j\) and \(k=a+b\), the posterior mean can be rearranged as

Why the Smaller Basket Shrinks More

The derivation shows that the data weight depends on the basket size and the prior concentration. With \(k=a+b\) held fixed, \(w_j=n_j/(n_j+k)\) increases as \(n_j\) increases. Equivalently, the weight on the prior center decreases. The actual change in the estimate is

This equation gives both the direction and the magnitude of the movement. If \(r_j\) is above \(m\), its posterior mean lies between \(m\) and \(r_j\); if it is below \(m\), the posterior mean also lies between them. For two baskets with the same observed proportion and the same distance from \(m\), the smaller basket moves farther toward \(m\). But if their observed proportions differ, the smaller basket need not move farther in absolute terms: a large basket very far from \(m\) can move more than a small basket close to \(m\).

Unequal Sizes and Unequal Observed Rates

When basket rates differ, it is useful to compute both the weight and the distance from the center. The weight describes how the model combines the data and prior; the realized movement is the product of the prior-center weight and the observed deviation. Comparing weights alone cannot establish which basket’s estimate moves more in percentage-point terms.

A Short Derivation of the Size Effect

The beta-binomial expression makes the comparison especially direct. For fixed concentration \(k>0\), the data weight is \(w(n)=n/(n+k)\). If \(n_2>n_1>0\), then

Therefore the larger basket has the greater data weight and the smaller prior-center weight. The amount of shrinkage in the posterior mean, in absolute terms, is

For the same \(|r-m|\), this quantity decreases with \(n\). The result is conditional on a fixed shared prior concentration. In a full Bayesian analysis where the hyperparameters are uncertain, posterior inference averages over their uncertainty; simply substituting posterior averages of \(a\) and \(b\) into this formula need not give the exact marginal posterior mean.

The Same Principle in a Normal-Normal Model

For a continuous basket outcome, suppose patient-level observations in basket \(j\) are independent normal observations with known within-basket variance \(\sigma^2\), so \(\bar{x}_j\mid\theta_j\sim N(\theta_j,\sigma^2/n_j)\). Suppose also that \(\theta_j\sim N(\mu,\tau^2)\), with \(\mu\) and \(\tau^2\) treated as fixed for this calculation. The posterior mean combines the basket mean and the shared center:

The data weight increases with \(n_j\), because the sampling variance \(\sigma^2/n_j\) falls as the basket gets larger. Thus, under these assumptions, a small basket’s mean is pulled more toward \(\mu\). The formula also shows why unequal sizes matter even when the observed means are identical.

Assumptions, Diagnostics, and Alternatives

Unequal-size shrinkage is a consequence of the assumed information structure, not a guarantee that borrowing is appropriate. In the binary example, the calculation treats responses as conditionally independent within each basket and uses a common beta distribution for basket probabilities. It also presumes comparable response definitions and follow-up. If these conditions are implausible, the resulting common center may be a poor reference for some baskets.

A useful diagnostic is to compare each basket’s observed rate, posterior mean, sample size, and distance from the fitted shared center. Check that the direction of shrinkage is sensible and that differences between baskets are not being hidden by a strong shared prior. In a fully Bayesian model, inspect whether posterior uncertainty in the shared parameters materially changes the basket estimates; conditional weights alone do not show that uncertainty. Unusually strong pulls or a center poorly aligned with clinically comparable baskets are reasons to question the modeling assumptions and examine sensitivity to the prior and hierarchy.

The main alternatives differ in how much information is shared. No pooling estimates each basket separately and does not use a common center. Complete pooling treats all baskets as having one response probability, removing basket-specific variation. Partial pooling retains basket-specific probabilities while connecting them through a shared distribution. Unequal-size shrinkage is characteristic of partial pooling: each basket contributes information according to its precision relative to the shared distribution.

A common mistake is to say, “The smaller basket must have the larger posterior change because it has fewer patients.” Fewer patients imply a larger prior-center weight under the stated model, but the absolute change also depends on the observed rate’s distance from the center. Another mistake is to treat a displayed weight as a universal property of the basket: it depends on the prior concentration (or, in the normal model, the between-basket and within-basket variances). Always state the model and the center against which shrinkage is being measured.