Commensurate Borrowing Across Successive Trial Cohorts

Why Borrow Across Successive Cohorts?

A trial may enroll a control cohort in one period and a later control cohort under an updated protocol, at a new site, or alongside a new treatment cohort. If the control response rates are sufficiently similar, the earlier cohort can add information about the later cohort’s control rate. But similarity cannot simply be assumed: changes in eligibility, supportive care, outcome assessment, or the population may make the earlier outcomes less relevant. A commensurate prior gives a structured way to express how strongly the current cohort’s control parameter is linked to the earlier cohort’s parameter.

This tutorial develops a normal approximation for binary control outcomes on the logit scale. It makes the borrowing calculation transparent and allows closed-form posterior calculations. The approximation is most defensible when both cohorts have enough events and non-events for their observed logits to be reasonably well approximated by normal distributions. It is not a substitute for checking whether the cohorts are clinically comparable.

Model and Borrowing Precision

Let \(x_j\) of \(n_j\) participants in cohort \(j\) have the binary outcome, where \(j=0\) denotes the earlier cohort and \(j=1\) the current cohort. Write \(p_j\) for that cohort’s outcome probability and \(\eta_j=\operatorname{logit}(p_j)=\log\{p_j/(1-p_j)\}\). Within each cohort, assume independent Bernoulli outcomes with a constant probability \(p_j\). Conditional on its parameter, the cohort’s count therefore has a binomial distribution.

For calculation, approximate each cohort’s logit estimate \(y_j=\operatorname{logit}(x_j/n_j)\) by a normal observation with variance \(v_j=1/x_j+1/(n_j-x_j)\). This is a large-sample approximation obtained from the binomial likelihood and the delta method. It requires \(0<x_j<n_j\); when a cohort has zero events or zero non-events, the estimate and this variance formula fail, and a binomial-logit analysis with an appropriate prior or another carefully justified method is needed.

Suppose the earlier logit has a normal prior, \(\eta_0\sim N(m_0,V_0)\). Combining that prior with the earlier cohort’s approximate normal likelihood gives an earlier-cohort posterior \(\eta_0\mid\text{earlier data}\sim N(m_H,V_H)\), where

After integrating over uncertainty in \(\eta_0\), the commensurate prior for \(\eta_1\) is \(N(m_H,V_H+1/\tau)\). The earlier cohort therefore contributes prior precision \(B=1/(V_H+1/\tau)\) to the current logit. The current likelihood contributes precision \(L=1/v_1\). The resulting posterior mean and variance are

The fraction \(B/(L+B)\) is a useful measure of the borrowing contribution to total posterior precision in this approximation. It is not a fraction of participants literally transferred between cohorts, nor a universal measure of borrowing independent of the current sample size. As the current cohort grows, its likelihood precision increases and the earlier cohort generally supplies a smaller fraction of the total precision.

Worked Calculations

How to Choose and Assess the Degree of Borrowing

The examples fixed \(\tau\) to make the effect of commensurability explicit. In an analysis, \(\tau\) may be assigned a prior and estimated jointly with the cohort parameters. Then uncertainty about commensurability must be propagated into the current posterior; one should not simply substitute a single estimated value of \(\tau\) and ignore its uncertainty. Another practical approach is to prespecify a small set of plausible borrowing strengths and report sensitivity analyses. A fixed strong value is not evidence that the cohorts are exchangeable.

Before borrowing, check whether the cohorts share a clinically meaningful outcome definition, follow-up window, eligibility criteria, and relevant care context. Compare observed rates and uncertainty, but do not treat a nonsignificant difference as proof of agreement: small cohorts can fail to reveal important differences. Inspect the posterior under little or no borrowing as well as under the intended commensurability assumptions. If the clinical or operational context suggests a shift, a sensitivity analysis that substantially reduces borrowing is especially important.

The normal approximation has additional failure modes. Sparse counts, estimates near zero or one, or a small sample can make the logit-normal approximation inaccurate. A logit estimate is undefined at a boundary count. In those settings, use a model that directly represents the binomial data, and verify the prior’s behavior and posterior computation. If a hierarchical or mixture model is used to let the data adapt borrowing, inspect computation diagnostics and the posterior for the borrowing parameter; apparent computational stability does not establish clinical comparability.

Commensurate borrowing is related to, but distinct from, the power-prior approach used for historical controls. A power prior discounts a historical likelihood; the commensurate formulation instead specifies a distribution for the current parameter conditional on an earlier parameter. Both can be useful, and neither automatically protects against a poorly chosen model or an unrecognized cohort shift. The robust mixture borrowing approach from the earlier material on robust mixture borrowing offers a different way to retain a low-borrowing possibility when a shift is suspected. These approaches encode different assumptions, so results should not be treated as interchangeable.

Derivation and Interpretation

The posterior formulas follow directly from normal-normal updating. Given the earlier posterior, write \(\eta_0\mid\text{earlier data}\sim N(m_H,V_H)\), and independently introduce \(\epsilon\sim N(0,1/\tau)\). The commensurate model says \(\eta_1=\eta_0+\epsilon\). The sum of independent normal variables is normal, so marginally \(\eta_1\sim N(m_H,V_H+1/\tau)\). Combining that prior with \(y_1\mid\eta_1\sim N(\eta_1,v_1)\) gives posterior precision \(1/(V_H+1/\tau)+1/v_1=B+L\), and the posterior mean is the precision-weighted mean of \(m_H\) and \(y_1\). Thus

This equation shows why \(B/(L+B)\) is a useful borrowing measure: it is exactly the weight assigned to the earlier posterior mean in the current posterior mean under this approximation. It also shows why the weight depends on both commensurability and current-data precision. A large \(\tau\) alone does not guarantee that earlier data dominate; the current cohort’s size and observed information matter as well.

The model is a conditional relationship between cohort parameters, not a claim that the earlier outcomes caused the current outcomes. It also borrows only for the modeled control response parameter; it does not resolve bias from differences in patient mix or care. Report the cohort data, model, prior or sensitivity range for \(\tau\), degree-of-borrowing summary, and the no- or low-borrowing comparison so readers can see how conclusions depend on the link.