Borrowing Across Subgroups Without Forced Pooling

Why Borrowing May Differ Across Subgroups

A clinical trial may contain subgroups whose historical evidence is more or less relevant to the current trial. For example, the treatment effect in one age group may agree closely with its historical estimate, while another group’s current estimate may differ substantially. A model that uses a single commensurability parameter across subgroups can blur this distinction: it may borrow too little for the compatible subgroup, too much for the conflicting one, or settle on a compromise that fits neither well.

The aim here is not to make the subgroup effects equal. Instead, each current subgroup effect is linked to its own historical subgroup estimate, with a separate parameter controlling the strength of that link. This allows the data to support different degrees of borrowing across subgroups. The approach is meaningful only when subgroup definitions, outcome measurement, treatment contrast, and relevant aspects of the clinical setting are sufficiently aligned to make each historical comparison interpretable.

The precision \(\tau_j\) is not the precision of the current data. It describes how tightly the current effect is centered on the historical effect. In the examples below, historical uncertainty is represented by an approximate normal distribution \(\eta_j\mid D_{H,j}\sim N(h_j,v_j)\), where \(D_{H,j}\) denotes the historical data, \(h_j\) is the historical estimate, and \(v_j\) is its variance. The current estimate \(y_j\) has sampling model \(y_j\mid\theta_j\sim N(\theta_j,s_j^2)\). We assume the historical and current data are conditionally independent given their respective effects, and that the normal approximations are adequate for the estimates being modeled.

What the Separate Parameters Do

Conditional on \(\eta_j\) and \(\tau_j\), the current effect has prior distribution \(N(\eta_j,\tau_j^{-1})\). Integrating over uncertainty in the historical effect gives a marginal prior for the current effect, conditional on the historical data and \(\tau_j\):

Consequently, the current estimate’s marginal distribution given \(\tau_j\) is \(N(h_j,s_j^2+q_j)\). This distribution supplies evidence about whether the current estimate is compatible with the historical estimate at the specified value of \(\tau_j\). Conditional on \(\tau_j\), normal-normal updating gives the posterior variance and mean:

The posterior for \(\theta_j\) averages these conditional normal distributions over uncertainty about \(\tau_j\). If \(\tau_j\) has a discrete prior, this is a mixture of normal posteriors, not a single normal distribution in general. A separate \(\tau_j\) is assigned to each subgroup; that does not mean the subgroup effects are pooled or assumed equal.

Learning Compatibility Separately

A common specification gives each subgroup its own prior over commensurability states. For a transparent illustration, suppose each subgroup independently has prior probability 0.5 of a strong-commensurability state \(\tau=1\), and probability 0.5 of a weak-commensurability state \(\tau=0.05\). These numerical values are illustrative choices, not universal definitions of strong and weak borrowing. In both states below, take \(s^2=1\) and \(v=0.25\), in percentage-points squared, and set the historical estimates to \(h=0\). Then \(q\) is 1.25 in the strong state and 20.25 in the weak state; the corresponding marginal variances of \(y\) are 2.25 and 21.25.

Bayes’ rule updates each subgroup’s state probabilities using that subgroup’s own current estimate. For equal prior state probabilities, the posterior odds of the strong state against the weak state equal the ratio of the two normal densities at the observed estimate. A current estimate close to \(h_j\) tends to favor strong commensurability; a more distant estimate tends to favor weak commensurability. The evidence is relative to the specified variances and prior states, not a test that determines whether the historical data are “valid.”

Why a Shared Parameter Can Be Misleading

A shared commensurability parameter makes the degree of historical borrowing common across subgroups. That can be a useful simplification when a common degree of compatibility is substantively plausible, but it cannot represent the contrasting evidence in the example above. With independent subgroup-specific parameters, A’s data update A’s commensurability state and B’s data update B’s state. With a shared two-state parameter, the data from both subgroups update one common state.

Assumptions, Diagnostics, and Limitations

The normal calculations rely on adequate normal approximations for the historical and current estimates, appropriate variance estimates, and a meaningful alignment of the subgroup-specific estimands. The historical estimate’s uncertainty must be carried into \(v_j\); treating \(h_j\) as known when it is not can make borrowing appear more secure than it is. The analysis also assumes that differences between historical and current effects are represented adequately by the commensurability model. Differences in patient mix, outcome definitions, follow-up, or treatment implementation may make a raw subgroup comparison misleading.

Useful checks include plotting current and historical subgroup estimates with uncertainty intervals, examining the likelihood of the observed difference under each commensurability state, and checking that fitted state probabilities and posterior estimates are not dominated by arbitrary prior choices. In computational fits, subgroup-specific state probabilities, posterior means, and effective sample sizes can be examined together. A very high probability for one state from a small amount of current information may indicate a highly influential prior or an overly optimistic variance model, rather than strong empirical evidence.

The main alternatives answer different questions. No borrowing leaves the current subgroup estimates independent of historical information. A common commensurability parameter constrains subgroups to share the strength of historical linkage. Separate commensurability parameters allow that strength to differ, but do not on their own create a model in which subgroup effects are exchangeable or shrink toward one another. Each approach depends on assumptions about which information is relevant. The separate model can also be unstable with many small subgroups, because each subgroup supplies limited evidence about its own compatibility.

Common Mistake

• Wrong reasoning: “The subgroups have the same treatment effect because they use the same historical control or share a commensurability model.”
• Correction: Separate commensurability parameters link each current effect to its own historical counterpart. They permit different borrowing strengths; they do not impose equality of effects or establish that historical and current populations are exchangeable.
• Another pitfall: Treating a posterior probability for a strong-borrowing state as a formal conflict-test result or as proof that the historical data are compatible. It is conditional on the model, state values, and prior probabilities.