Stratified Power Priors for Control Outcomes

Why Stratify Historical Control Information?

Historical controls may differ from a current trial in characteristics related to the control outcome. If those characteristics define clinically meaningful strata, a single pooled historical event rate can obscure important differences. Stratification instead models the control outcome separately within each stratum and lets historical observations contribute to the corresponding stratum-specific model.

The key modeling decision is not merely to divide the data into groups. It is to specify which historical observations are considered informative about each current stratum, how strongly they contribute, and whether a common outcome probability within that stratum is credible across studies. A power prior discounts a historical likelihood; stratifying the model allows the likelihood contributions and, where justified, their discounts to be defined by stratum.

Model and Assumptions

For the binary outcome, assume that, conditional on \(p_s\), outcomes are independent Bernoulli observations within stratum \(s\). Thus the historical count has likelihood \(y_{hs}\mid p_s\sim\operatorname{Binomial}(n_{hs},p_s)\), with likelihood proportional to \(p_s^{y_{hs}}(1-p_s)^{n_{hs}-y_{hs}}\). Suppose the current control data contain \(y_{Cs}\) events among \(n_{Cs}\) participants in stratum \(s\), with the same binomial model conditional on \(p_s\).

The main transportability assumption is that, within a given stratum, the historical and current control outcomes share the same \(p_s\). This requires more than matching stratum labels: eligibility criteria, endpoint definitions, follow-up, outcome ascertainment, and relevant care should make the outcomes sufficiently comparable. Conditional independence and a correctly specified stratum-specific outcome model are also required. The model below treats the \(p_s\) values as separate parameters: it does not borrow across strata unless a further model explicitly links them.

Let the baseline prior be independent across strata, \(p_s\sim\operatorname{Beta}(\alpha_{0s},\beta_{0s})\), with \(\alpha_{0s}>0\) and \(\beta_{0s}>0\). With fixed discounts, the joint posterior density is proportional to the baseline prior multiplied by the current likelihood and the powered historical likelihoods:

The prior parameters ensure a proper posterior in this beta-binomial model, including when a stratum has no current observations or receives no historical borrowing. For fixed discounts, conjugacy gives the update directly. The discount \(a_{hs}\) scales both the historical event count and the non-event count; it does not change the observed historical rate \(y_{hs}/n_{hs}\).

Why the Update Has This Form

For one historical study and one stratum, raising the binomial likelihood to \(a_{hs}\) gives a factor proportional to \(p_s^{a_{hs}y_{hs}}(1-p_s)^{a_{hs}(n_{hs}-y_{hs})}\). Multiplying these factors over historical studies adds their powered event and non-event counts to the exponents. Multiplication by the current likelihood adds its ordinary event and non-event counts, while the beta prior contributes exponents \(\alpha_{0s}-1\) and \(\beta_{0s}-1\). Collecting the powers of \(p_s\) and \(1-p_s\) gives the beta posterior parameters shown above.

This derivation also explains what “borrowing within strata” means in this model: a historical observation contributes only to the parameter for its own stratum. The effective historical sample size contributed to stratum \(s\) is \(\sum_h a_{hs}n_{hs}\). It may be fractional, because a power discount is a likelihood weight, not a literal count of participants.

Stratum-Specific Borrowing in Practice

The discount can vary across strata, across historical studies, or both. A smaller discount gives that historical study less influence on that stratum’s posterior. This flexibility should reflect a defensible assessment of relevance, rather than an attempt to force the posterior toward a desired result. The stratum definitions, baseline priors, and discount choices should be specified and justified before interpreting the results.

Multiple Studies and Unequal Relevance

When several historical studies report outcomes in the same strata, each contributes a powered likelihood. Because all studies contribute to the same \(p_s\), the model assumes their outcomes are exchangeable with current outcomes within that stratum after discounting. Different discounts can reflect different levels of relevance, but discounting alone does not repair an incompatible endpoint, a missing stratum, or a systematic change in care.

Diagnostics, Alternatives, and Limitations

Before using a stratified power prior, check that the required data are available and that the model’s assumptions are plausible. A stratum with few or no historical participants has little or no direct historical information, even if the overall historical sample is large. A stratum with no current participants may still have a posterior based on its prior and historical data, but the analysis then relies especially strongly on transportability. If a study does not report outcomes in a stratum, its likelihood cannot contribute to that stratum without additional modeling assumptions.

Check outcome definitions, follow-up windows, eligibility criteria, and stratum coding across studies. Also inspect event rates study by study within each stratum. Large differences can signal residual population differences, calendar-time changes, or incompatible measurement. A fixed discount reduces influence but does not establish that the remaining information is unbiased. Sparse strata can produce unstable estimates, and extensive subdivision can leave the analysis dependent on the baseline prior. Stratification also addresses only the recorded factors used to define strata; it does not eliminate confounding from unmeasured differences.

The main alternatives impose different assumptions. A pooled power prior uses one control probability and allows all historical observations to contribute to it; this is simpler but assumes no relevant outcome differences across the pooled groups. A separate-parameter stratified prior permits stratum risks to differ but does not share information across strata. A hierarchical model can partially pool stratum risks, but it introduces a distributional model and additional assumptions. The appropriate choice depends on the scientific structure and available data; a more complex model is not automatically more robust.

For non-binary outcomes, the same organizing principle applies when the likelihood factorizes by stratum: specify the stratum-specific outcome model and power the relevant historical likelihood contributions. The conjugate beta-binomial formulas above are specific to binary outcomes. For other likelihoods, the posterior update and effective information measure depend on the model and should not be inferred by mechanically treating non-binary observations as counts.