Borrowing Historical Information For Joint Models

Why Borrow Historical Joint Outcomes?

Dose-finding trials often have limited current-trial data at each dose. Earlier studies may contain useful evidence about both efficacy and toxicity, but simply pooling all observations can be misleading: treatment versions, eligibility, endpoint definitions, follow-up, or patient mix may differ. Borrowing is a modeling decision that lets historical evidence inform current estimates while making its influence explicit.

For joint dose models, the target is not just an efficacy rate and a toxicity rate considered separately. Their dependence matters: the frequency of patients who experience both outcomes can affect a dose’s benefit-risk assessment. The likelihood construction for joint dose models provides the foundation; here the focus is on incorporating external information and assessing whether it is compatible with current evidence.

Representing the Joint Outcomes

For a simple illustration, suppose efficacy and toxicity are each recorded as binary outcomes over prespecified assessment windows. At dose \(d\), define \(p_{d,et}=P(E=e,T=t\mid d)\), where \(e,t\in\{0,1\}\). The four probabilities sum to one. This representation retains the association between outcomes: in general, \(p_{d,11}\) is not determined by the marginal efficacy and toxicity probabilities alone.

For each dose, let \(n_{d,et}\) be the number of current-trial patients in category \((e,t)\), and let \(n^{(H)}_{d,et}\) be the corresponding historical count. If patients within each source are independent and have the same category probabilities at a given dose, the likelihood is multinomial. When dose levels or patient characteristics differ, the model must account for those differences rather than treating unlike observations as exchangeable.

Before borrowing, check that the historical and current data use compatible outcome definitions, assessment periods, dose mappings, and estimands. Also consider whether supportive care, eligibility, or covariate distributions have changed. A matching dose label alone does not establish comparability. If patient mix differs, a model may need to standardize or condition on relevant covariates; otherwise apparent conflict may reflect composition rather than a changed dose-outcome relationship.

Power Priors: Explicitly Discounting Historical Data

A power prior raises the historical likelihood to a power \(a\), with \(0\leq a\leq1\). A value of one gives the historical likelihood its full nominal contribution; zero gives it none. Intermediate values discount its information. With a fixed \(a\), a baseline prior \(\pi_0(\mathbf p_d)\), and historical data \(D_H\), the prior for the current dose probabilities is proportional to \(\pi_0(\mathbf p_d)L_H(\mathbf p_d)^{a}\).

For the multinomial model, a convenient baseline prior is Dirichlet with parameters \(\boldsymbol{\alpha}_0=(\alpha_{00},\alpha_{10},\alpha_{01},\alpha_{11})\), all positive. Its density is proportional to the product of each category probability raised to its parameter minus one. The historical likelihood contributes one factor \(p_{d,et}\) for every historical observation in category \((e,t)\). Raising that likelihood to \(a\) contributes \(a n^{(H)}_{d,et}\) to the exponent of \(p_{d,et}\). The current likelihood then contributes \(n_{d,et}\). Therefore the posterior is Dirichlet with parameters \(\alpha_{0,et}+a n^{(H)}_{d,et}+n_{d,et}\).

This result assumes the same four-category multinomial structure applies to both sources at that dose, that observations are independent within source conditional on the probabilities, and that \(a\) is fixed in advance for this calculation. The fractional historical counts are a likelihood discount, not literal fractional patients. If \(a\) is estimated or assigned a prior, the normalization of the power prior and the dependence between \(a\) and the model parameters need careful treatment; the simple conjugate update above is conditional on fixed \(a\).

For a Dirichlet posterior with parameters \(\beta_{et}\), the posterior mean of a category probability is \(\beta_{et}/\beta_+\), where \(\beta_+=\sum_{et}\beta_{et}\). Thus the posterior mean efficacy probability is \((\beta_{10}+\beta_{11})/\beta_+\), and the posterior mean toxicity probability is \((\beta_{01}+\beta_{11})/\beta_+\). These are summaries of the joint model, not separate analyses that discard outcome dependence.

Diagnosing Conflict and Limiting Borrowing

A high borrowing weight can dominate a small current sample. Conflict diagnostics should therefore be planned before examining results, interpreted in context, and followed by sensitivity analyses. Begin with a clinical and data review: check outcome coding, follow-up completeness, endpoint windows, dose levels, and relevant differences in patient characteristics. Then compare historical and current summaries at doses with data, including all four joint categories where sample sizes allow.

For a quick marginal screen, an approximate standardized difference for two independent efficacy proportions is \((\hat p_C-\hat p_H)/\sqrt{\hat p_H(1-\hat p_H)/n_H+\hat p_C(1-\hat p_C)/n_C}\). Here \(C\) denotes current data and \(H\) historical data. This is a descriptive diagnostic, not a formal test that decides whether borrowing is valid. Sparse categories, multiple doses, covariate imbalance, and adaptive dose assignment can make such a simple comparison inadequate.

A power prior is transparent and easy to compute, but a single fixed discount applies the same borrowing strength to every joint category in that model. If historical and current efficacy align but toxicity does not, a common \(a\) may be too crude. Conversely, choosing separate discounts after seeing outcome differences can introduce unplanned flexibility. Prespecify the strategy and report sensitivity over scientifically plausible values.

Hierarchical Models and Partial Pooling

A hierarchical model represents study-specific dose-response parameters as related but not necessarily identical. For example, let \(\theta_H\) and \(\theta_C\) be historical and current log-odds of efficacy at a dose. A commensurate model can express the current parameter as conditionally centered on the historical parameter, \(\theta_C\mid\theta_H\sim N(\theta_H,\tau^2)\). Small \(\tau\) implies strong similarity and greater borrowing; large \(\tau\) allows substantial differences and weakens the link. A full model can assign a prior to \(\tau\), or use a prespecified value and examine sensitivity.

For joint outcomes, the corresponding study-specific parameter can be a vector, such as efficacy and toxicity log-odds plus an association parameter, or a vector of multinomial-logit coefficients. A multivariate hierarchical distribution can allow study deviations to be correlated. This matters because efficacy and toxicity may shift together across settings. The hierarchy must still respect the joint outcome likelihood and the structure of dose response; separate univariate borrowing does not automatically preserve joint dependence.

Hierarchical borrowing is often called partial pooling: current estimates share information with historical estimates, but need not be identical to them. Its strength depends on the between-study variation and the current data. The assumption is exchangeability at the level modeled: after accounting for specified differences, the studies are treated as draws from a common distribution. This is a substantive assumption, not a consequence of fitting a hierarchical model.

Choosing and Reporting a Borrowing Strategy

The principal alternatives differ in how they express similarity. A fixed power prior gives a direct, interpretable discount on historical likelihood contributions. A hierarchical model estimates or specifies variation across studies and produces partial pooling. A robust mixture prior offers another safeguard: it combines a component concentrated near the historical information with a broader component, so posterior weight can shift toward the broader component when current data disagree. Each approach encodes assumptions; none guarantees safe borrowing without checking whether the historical evidence is relevant.

Useful diagnostics include side-by-side joint category summaries, dose-specific comparisons, covariate and endpoint audits, and sensitivity of posterior dose summaries to the discount or heterogeneity assumptions. Track whether conclusions change materially: for example, whether the posterior probability of exceeding a toxicity threshold or the selected dose changes when borrowing is weakened. Such sensitivity is itself important information for decision-makers.

A common mistake is to say, “The historical sample is large, so it should dominate the current estimate.” A large historical sample can be highly influential and still be poorly transportable. The correction is to justify comparability first, make borrowing strength explicit, assess conflict, and show how conclusions behave under weaker borrowing. Another mistake is to compare only marginal efficacy and toxicity rates; two sources can have similar margins but very different rates of experiencing both. Inspect the joint distribution as well.