Prior Predictive Implications of a Treatment Contrast

Why Examine the Contrast Before the Trial?

For a binary endpoint, a prior can imply treatment and control response rates that look reasonable when considered one at a time, yet imply implausibly large differences between them. A prior predictive check of treatment contrast makes that implication visible before trial outcomes are observed. For example, a prior may place substantial probability on treatment improving the response rate by 40 percentage points, even if the expected rates in each arm seem modest.

It is important to distinguish two quantities. The latent treatment contrast is a function of unknown response probabilities; its distribution under the parameter prior is an induced prior distribution. Future event counts, or a difference calculated from those counts, are observable random quantities; their distribution after averaging over the parameter prior is the prior predictive distribution. The latter includes both parameter uncertainty and sampling variation.

The sign has a clinical interpretation only after the endpoint is defined. If the event is a desirable response, positive \(\Delta\) favors treatment; if the event is an adverse outcome, negative \(\Delta\) may favor treatment. The numerical contrast is always treatment minus control, but the clinical direction of benefit depends on what counts as an event.

Model and Assumptions

Suppose the trial will enroll fixed numbers \(n_T\) and \(n_C\) in the two arms, and each participant has a binary endpoint recorded once. Conditional on the arm-specific response probabilities, assume outcomes are independent within and between arms, with a common response probability within each arm. The sampling model is

For the calculations below, the priors are independent beta distributions:

These assumptions mean that each arm’s response probability is constant across its participants, that each participant contributes one binary outcome, and that the two arm probabilities are independent before data are observed. Independence of the priors is a modeling choice, not a consequence of randomization. If the design or prior specification instead links the arm probabilities, the induced contrast and prior predictive calculations must reflect that dependence. The calculations also concern the specified endpoint and follow-up definition; a change in either can change what response rates are clinically plausible.

Summarizing the Induced Prior for the Contrast

For a beta random variable with parameters \(a,b\), the mean is \(a/(a+b)\), and the variance is \(ab/[(a+b)^2(a+b+1)]\). Under independent arm priors, linearity of expectation and independence give

The contrast mean alone is not a measure of how much prior probability lies on clinically large effects. Its variance, quantiles, and probabilities of crossing prespecified clinical thresholds are also informative. Because beta-distributed probabilities are bounded between zero and one, \(\Delta\) is bounded between negative one and one, but that mathematical bound does not guarantee clinical plausibility.

From Latent Contrasts to Prior Predictive Outcomes

Let \(D=X_T/n_T-X_C/n_C\) denote the observed difference in response proportions in a future trial. The prior predictive distribution is obtained by averaging the conditional sampling distribution over the prior for the arm probabilities. In integral form,

Here \(\pi_T\) and \(\pi_C\) are the prior densities. Under the independent beta priors and binomial sampling model, integrating each arm separately gives beta-binomial probabilities:

where \(B(\cdot,\cdot)\) is the beta function. The joint prior predictive probability is the product of the two arm probabilities under the stated independence assumptions. The distribution of \(D\) follows by summing joint probabilities for all count pairs that produce the same difference. It is generally not the same distribution as the induced prior of \(\Delta\).

Practical Checks and Common Pitfalls

A useful check starts with clinical thresholds stated in the units of the response-rate difference. If an improvement of 10 percentage points is considered important, for example, inspect the induced prior probability of \(\Delta>0.10\) for a desirable event, as well as the probability of contrasts in the opposite direction. Also examine predictive probabilities for future observable summaries when the intended question concerns what trial results may look like. These are different checks and should be labeled accordingly.

When an exact distribution or tail probability is inconvenient, simulation can approximate it. For each simulated replicate, draw \(p_T\) and \(p_C\) from their joint prior, calculate \(\Delta\), then independently draw \(X_T\) and \(X_C\) from their conditional binomial models. Summaries of the \(\Delta\) draws describe its induced prior; summaries of \(X_T,X_C\), or \(D\), describe prior predictive quantities. Keeping these outputs separate prevents a common terminology and interpretation error.

• Check the event definition. Confirm that a positive treatment-minus-control contrast corresponds to the intended clinical direction, rather than assuming that all events are beneficial.
• Check the contrast, not only the arm means. Plausible prior means do not ensure plausible differences, as the first worked example illustrates.
• Check dependence assumptions. If arm probabilities are dependent under the prior, the contrast variance includes a covariance term: \(\operatorname{Var}(\Delta)=\operatorname{Var}(p_T)+\operatorname{Var}(p_C)-2\operatorname{Cov}(p_T,p_C)\). Independent-arm calculations would then be inappropriate.
• Check the predictive design. Event-count predictions depend on sample sizes and the binomial sampling model. Different allocation, follow-up, or outcome definitions require corresponding changes to the predictive calculation.

A frequent mistake is to call the distribution of \(p_T-p_C\) the prior predictive distribution. That is incorrect: it is the induced prior distribution of a latent parameter contrast. The correction is to reserve “prior predictive” for future observable quantities, such as the counts or their observed difference, which average sampling variation over the parameter prior. Another mistake is to infer that a prior is clinically sensible because its expected contrast is reasonable. The expectation gives only a center; tail probabilities, spread, and the predictive implications for the planned design also matter.

The main alternatives are complementary rather than interchangeable. Analytic beta-binomial calculations give exact event-count probabilities under independent beta priors and fixed sample sizes. Simulation is flexible for more complicated joint priors or derived summaries, but its Monte Carlo error should be considered when probabilities are small. Directly specifying a prior on a treatment contrast can focus elicitation on the clinical effect of interest, while priors on arm response probabilities make the baseline rates explicit; either approach must imply valid response probabilities and a coherent joint model. This tutorial’s independent beta-binomial calculations do not address dependence, covariate adjustment, missing outcomes, or changing event risk over time.