Joint Posterior Success Rules for Co-Primary Endpoints

Why Joint Success Is a Separate Question

A trial with co-primary endpoints is successful only if every prespecified endpoint criterion is met. For example, a treatment might need to produce both a sufficiently large reduction in symptom severity and a sufficiently large improvement in physical function. Knowing the posterior probability of meeting each criterion separately does not, by itself, give the probability of meeting both.

The missing ingredient is the joint posterior distribution of the endpoint effects. If favorable effects on the endpoints tend to occur together in posterior draws, the chance of meeting both criteria may be relatively high. If one endpoint tends to be favorable when the other is not, the joint chance may be lower. The calculation must therefore preserve the dependence represented by the model.

The direction and scale of each effect must be stated carefully. If larger values are favorable, a criterion might be \(\theta_j\geq \delta_j\), where \(\delta_j\) is a prespecified clinically meaningful threshold. If smaller values are favorable, the effect can be defined with its sign reversed or the success set can be written in the appropriate direction. Criteria may also be interval-based. The joint event is the intersection of the endpoint-specific success events, not a new average of the endpoint effects.

The probability \(P(A\mid D)\) describes uncertainty about the joint event conditional on the observed data, model, and prior. It is not the probability that the model is true, nor does it establish that treatment caused an effect beyond the assumptions of the trial and analysis. Its clinical interpretation depends on the endpoints, scales, thresholds, and model being appropriate.

Calculating the Probability From Joint Posterior Draws

Suppose posterior computation supplies \(M\) draws \((\theta_1^{(m)},\ldots,\theta_J^{(m)})\), each retaining the endpoint effects from the same draw \(m\). For each draw, check whether every criterion is satisfied. The fraction of draws satisfying all criteria estimates the joint posterior probability.

Here \(\mathbf{1}\{\cdot\}\) equals 1 when the condition inside the braces is true and 0 otherwise. This estimator follows directly by writing the posterior probability as the posterior expectation of an indicator:

The draws must be joint draws. Pairing \(\theta_1^{(m)}\) with \(\theta_2^{(m)}\) preserves their posterior association. Independently shuffling one endpoint’s draws, or sampling each endpoint separately and pairing arbitrary values, can destroy that association and produce the wrong joint probability. For correlated MCMC output, the estimate is still the indicator average; its Monte Carlo precision depends on the effective information in the indicator sequence, not just the raw number of saved iterations.

What Dependence Changes

For two endpoints, let \(A_1\) and \(A_2\) denote their respective success events. The probability rule \(P(A_1\cap A_2\mid D)=P(A_1\mid D)P(A_2\mid A_1,D)\) shows exactly when multiplying marginal probabilities works: it works if the success events are posterior independent, so that \(P(A_2\mid A_1,D)=P(A_2\mid D)\). Independence of the endpoint effects is sufficient for this condition, but independence should not be assumed just because the endpoints are analyzed separately.

With marginal success probabilities \(p_1=P(A_1\mid D)\) and \(p_2=P(A_2\mid D)\), the joint probability must satisfy \(\max(0,p_1+p_2-1)\leq P(A_1\cap A_2\mid D)\leq\min(p_1,p_2)\). The upper bound holds because meeting both criteria implies meeting each one. For the lower bound, the probability of the union cannot exceed 1, while \(P(A_1\cup A_2\mid D)=p_1+p_2-P(A_1\cap A_2\mid D)\). Rearranging gives the stated lower bound, along with the fact that probabilities cannot be negative. These bounds are useful checks, but they do not determine the joint probability: the joint posterior distribution supplies the additional information.

A Normal-Posterior Illustration

Dependence is often summarized by posterior correlation between continuous endpoint effects, but correlation is not itself the probability of joint success. The effect of correlation also depends on where the success thresholds lie relative to each marginal posterior distribution. A simple special case makes the relationship visible.

Assumptions, Diagnostics, and Common Pitfalls

The calculation is only as meaningful as the event and posterior distribution used to compute it. Before interpreting a joint probability, confirm that each endpoint, effect direction, scale, threshold, and combination rule was specified appropriately for the clinical question. Confirm also that the posterior model represents the endpoint data jointly, including relevant within-patient association when patients contribute measurements to multiple endpoints. A model that imposes independence may be unsuitable if the endpoint data support meaningful dependence.

Practical diagnostics follow from the calculation. Check that each paired draw uses effects from the same posterior iteration and that every endpoint criterion is applied with the correct inequality and units. Compare the joint estimate with the marginal probabilities and the bounds above; a result outside those bounds signals a calculation or coding error. If MCMC is used, inspect sampling and mixing for the joint quantities and for the indicator of joint success. Few effective draws near the success boundary can make the estimated probability imprecise even when each marginal probability appears stable.

The estimated fraction is subject to Monte Carlo error. For \(M\) independent posterior draws and estimated probability \(\hat{q}\), its approximate Monte Carlo standard error is \(\sqrt{\hat{q}(1-\hat{q})/M}\). MCMC draws are generally dependent, so substituting the raw number of iterations for \(M\) can understate this error; precision should reflect the effective sample size of the indicator sequence.

• Wrong: “The marginal probabilities are 0.80 and 0.70, so the probability of both is 0.56.” Correction: That product requires posterior independence of the success events. Otherwise, estimate the joint probability from paired draws or the specified joint posterior.
• Wrong: “The endpoints are positively correlated, so the joint success probability must be high.” Correction: Correlation alone is not enough; the marginal locations and thresholds matter too. Calculate the probability of the actual joint event.
• Wrong: “I can pair any draw for endpoint 1 with any draw for endpoint 2.” Correction: Arbitrary pairing removes the dependence in the joint posterior and can change the answer.

A marginal-probability calculation answers how likely each criterion is to be met separately. A product calculation adds an independence assumption. A joint posterior calculation answers the all-criteria question while retaining the dependence in the model. This rule concerns the conjunction of co-primary criteria; it is distinct from a rule in which success requires only one of several endpoint criteria.

The result remains conditional on the chosen model, prior, data, and thresholds. It does not remove the need to justify clinical thresholds, assess model adequacy, or consider whether the chosen endpoints capture the intended benefit. When those inputs are uncertain or poorly specified, a precise-looking joint probability can still be misleading.