A Worked Binary-Control Borrowing Analysis

Why a Worked Borrowing Analysis Needs Care

A historical control group can add information to a current trial’s control-rate estimate, but it should not automatically count as if every historical participant had been enrolled in the current trial. A fixed power-prior discount gives historical outcomes a specified, reduced contribution. The calculation is simple once the model is explicit; the interpretation depends on the assumptions that justify treating historical and current control outcomes as information about a common rate.

This tutorial works through one hypothetical binary-control analysis from assumptions to posterior interpretation. The discount is fixed in advance for the calculation. It is not learned from the outcomes here.

Model and Assumptions

For this model, each participant has a binary outcome, such as whether a prespecified event occurs within a fixed follow-up period. Conditional on the control rate \(p\), outcomes within each group are modeled as independent Bernoulli trials. Thus the observed counts have binomial likelihoods. The model also assumes that the historical and current outcomes are comparable and can be represented by the same underlying rate \(p\); the discount reduces the historical likelihood’s contribution but does not itself model a difference in rates.

The prior is proper, with \(\alpha_0>0\) and \(\beta_0>0\). The value of \(a\) is fixed for the analysis, and the endpoint definition, observation window, and relevant control conditions are sufficiently aligned that the common-rate model is a meaningful approximation. These are substantive assumptions, not consequences of beta-binomial conjugacy. If follow-up differs, outcome ascertainment changes, or the current population is materially different, discounting alone may not resolve the mismatch.

The beta prior has density proportional to \(p^{\alpha_0-1}(1-p)^{\beta_0-1}\). The historical binomial likelihood, ignoring its factor that does not depend on \(p\), is proportional to \(p^{x_h}(1-p)^{n_h-x_h}\). Raising it to \(a\) gives a factor proportional to \(p^{a x_h}(1-p)^{a(n_h-x_h)}\). The current likelihood contributes \(p^{x_c}(1-p)^{n_c-x_c}\). Multiplying these factors yields the posterior kernel below.

This is the kernel of the beta distribution with the parameters in the definition. The calculation also shows why a fixed discount can produce non-integer beta parameters: the historical counts are weighted, rather than reinterpreted as literal fractional participants.

Worked Binary-Control Analysis

What the Discount Changes

The result can be understood by comparing it with the boundary cases. At \(a=0\), the historical likelihood contributes nothing, so the posterior uses only the prior and current outcomes. At \(a=1\), the historical likelihood is used without discount. These comparisons are useful checks on the calculation and show how the chosen discount changes the balance of information.

Sensitivity and Diagnostics

A fixed discount is an assumption whose consequences should be visible in the results. For the same data, changing the chosen discount changes both posterior location and precision. The historical effective sample size in this simple beta-binomial calculation is \(a n_h\): with \(a=0.25\), it is \(0.25(158)=39.5\). This is a useful description of the weighted historical contribution, not a claim that the analysis observed 39.5 actual participants.

Before interpreting a borrowed posterior, examine whether the data and model make borrowing credible. Useful warning signs include different outcome definitions or follow-up windows, changes in standard care, a different eligibility mix, and data-quality or ascertainment changes. A large discrepancy between historical and current observed rates is also a reason to question whether the common-rate assumption is adequate. With small counts, observed rates can be noisy, so the discrepancy should be considered alongside the design and clinical context rather than treated as a mechanical pass-or-fail test.

A practical calculation check is to verify that historical successes and non-successes sum to \(n_h\), and current successes and non-successes sum to \(n_c\). Then check that \(a=0\) removes historical counts and \(a=1\) restores them at full weight. Finally, check that both beta shape parameters are positive and that the posterior mean lies between 0 and 1. These checks catch arithmetic and implementation errors, but they do not validate the assumptions.

Interpretation and Limitations

The posterior mean of about 31.0% in the main example is the posterior expectation of the current control event rate under the specified model. It is not the current group’s raw event proportion, which is \(3/15=20.0\%\), and it is not a probability that the rate equals 31.0%. The posterior variance describes uncertainty in \(p\) conditional on the data, prior, common-rate model, and fixed discount. A Bayesian credible interval, if reported, would be computed from this posterior distribution and interpreted as a posterior probability statement about \(p\) under those assumptions.

The main alternative in this calculation is to use current data alone, corresponding to \(a=0\). This avoids borrowing but can leave a less precise estimate when the current control sample is small. Full borrowing, \(a=1\), gives historical outcomes their full likelihood contribution, but can be overly influential when historical and current conditions differ. A partial fixed discount lies between these choices in its likelihood contribution; it does not guarantee protection from bias or make non-comparable data comparable.

The analysis is also limited by its simple common-rate structure. It does not adjust for participant-level covariates or explicitly represent different control rates across studies. Its posterior precision can look reassuring even when the common-rate assumption is implausible. Therefore, a numerical posterior should be reported with the data sources, outcome definition, prior, and chosen discount, and interpreted as conditional on those choices.