Posterior Decision Rules for Either-Or Endpoint Criteria

When Success Means Meeting Any One Criterion

A clinical trial may assess more than one clinically relevant endpoint, while its decision rule may require evidence for at least one of several prespecified benefits. For example, a treatment might qualify by exceeding a meaningful benefit threshold on either symptom improvement or physical function. This is an either-or rule: the trial’s posterior success event is the union of the endpoint-specific events.

The distinction between a union and separate marginal probabilities is central. Knowing the posterior probability that each criterion is met does not, by itself, tell us the probability that at least one is met. The criteria can overlap in the posterior distribution, and the amount of overlap changes the union probability. The joint posterior distribution, not just a list of marginal probabilities, is therefore needed to evaluate the rule.

Define the Events Before Calculating

Each \(A_j\) must express a clinically interpretable criterion for a specified estimand. For example, let \(\theta_1\) be the treatment benefit on a symptom scale, with positive values favoring treatment, and let \(\theta_2\) be the treatment benefit on a function scale, also oriented so positive values favor treatment. If the minimum clinically meaningful benefits are \(m_1\) and \(m_2\), then the endpoint events could be \(A_1=\{\theta_1>m_1\}\) and \(A_2=\{\theta_2>m_2\}\). The either-or event is \(A_1\cup A_2\). The criteria should also specify the relevant population, analysis time, and endpoint definitions.

The threshold \(\gamma\) and the endpoint criteria should be prespecified. Choosing criteria after examining the posterior can make the rule look more favorable than the rule actually evaluated. The posterior probability threshold is a feature of this decision rule; it is not automatically a frequentist significance level or a universal standard.

The posterior model must provide a joint distribution for the endpoint parameters. It should represent dependence induced by shared participants, common controls, correlated outcomes, and any other relevant features of the design and analysis. The marginal posterior for each \(\theta_j\) may be useful, but it is not enough to calculate a union probability unless the overlap between the events is also known.

Why Marginal Probabilities Are Not Enough

For two criteria, the inclusion-exclusion identity follows by partitioning the union into disjoint events: \(A_1\cup A_2\) consists of \(A_1\) and the part of \(A_2\) not already in \(A_1\). Therefore,

The intersection term is what prevents double-counting posterior draws in which both criteria are satisfied. If the intersection is unknown, the marginals alone do not give an exact union probability. For \(K\) criteria, inclusion-exclusion continues with pairwise intersections, triple intersections, and higher-order intersections. In practice, calculating the union directly from joint posterior draws is often simpler and less error-prone.

Estimating the Union From Posterior Draws

Suppose the fitted model provides \(S\) draws \(\theta^{(1)},\ldots,\theta^{(S)}\) from the joint posterior distribution of all parameters needed to assess the criteria. For each draw, evaluate every criterion, then record whether at least one is met. The posterior probability is estimated by the fraction of draws for which the union event occurs.

where \(I\{\cdot\}\) equals 1 when the stated event is true and 0 otherwise. This calculation automatically accounts for overlap, provided each draw represents the joint posterior. It also avoids needing to calculate all intersection probabilities separately.

Assumptions, Diagnostics, and Alternatives

A meaningful union rule requires more than correct arithmetic. The endpoint criteria must be clinically coherent, their directions and units must be correct, and the posterior model must be suitable for the trial data and estimands. The joint posterior must preserve relevant dependence. If posterior draws were generated separately for each endpoint and then paired arbitrarily, their apparent overlap would not generally represent the true joint posterior.

For simulation-based evaluation, inspect convergence and mixing for the parameters and for the union-event indicator itself. A chain may explore endpoint parameters adequately overall yet move slowly between regions where different criteria are met. Check chain agreement and effective sample size, and assess Monte Carlo error when the estimate is near \(\gamma\). Also verify that the event coding matches the prespecified rule: a reversed benefit direction, an incorrect margin, or an accidental requirement that all criteria be met can change the result.

An either-or rule differs from a co-primary rule, where success requires all specified criteria to be met. It also differs from reporting several marginal posterior probabilities without combining them into a single rule. A composite endpoint is another alternative: it defines one outcome or estimand rather than the union of separate endpoint-specific events. These approaches answer different clinical questions and should not be interchanged after seeing the results.

For many criteria, the union can be broad: adding another criterion can only leave the union probability unchanged or increase it. That is a mathematical property, not evidence that each endpoint is independently persuasive. Prespecification, transparent endpoint reporting, and evaluation of the full joint event help prevent an expanding list of opportunities from being mistaken for a stronger single-endpoint result.