Worked Continuous Endpoint Posterior Analysis

From Trial Summaries to an Endpoint Comparison

A continuous endpoint comparison should answer a clinically interpretable question: how far apart are the treatment and control arm means, in the units actually measured? A posterior distribution for each arm mean provides the pieces for that comparison. Under a specified normal conjugate model, the update can be calculated from each arm’s sample size and sample mean, then combined to obtain a posterior distribution for the raw mean difference.

This analysis concerns a two-arm, independent-groups trial and a continuous endpoint measured on the same scale in both arms. For the examples, the endpoint is systolic blood pressure at 12 weeks, measured in mmHg, and lower values are favorable. The data and prior information below are hypothetical. The model treats the within-arm variances as known; this is an assumption of the worked analysis, not a claim that a trial generally knows its population variance exactly.

The normal sampling assumption describes the individual outcomes within each arm; under it, the sample mean has distribution \(\bar{x}_j\mid\mu_j\sim N(\mu_j,\sigma_j^2/n_j)\). The analysis uses the sample size and sample mean as the arm’s data summary. In a randomized trial, random assignment supports a between-arm comparison, but it does not itself guarantee normal outcomes, independent observations, or correctly specified variances.

The Conjugate Update for Each Arm

The prior mean \(m_{0j}\) and variance \(v_{0j}\) describe uncertainty about the arm mean before observing this trial’s endpoint data. The sample mean contributes information with variance \(\sigma_j^2/n_j\). Multiplying the normal prior density by the normal sampling density gives a normal posterior. Its precision (the reciprocal of variance) is the sum of the prior precision and the data precision.

Here \(m_j\) and \(v_j\) are the posterior mean and variance, respectively. The posterior mean is a precision-weighted average of the prior mean and the observed sample mean. More participants, or a smaller known outcome variance, increases the data precision and gives the sample mean more weight. A more precise prior gives the prior mean more weight.

Expanding the two squared terms, the coefficient of \(\mu_j^2\) inside the brackets is \(1/v_{0j}+n_j/\sigma_j^2\), and the coefficient of the linear term is \(-2(m_{0j}/v_{0j}+n_j\bar{x}_j/\sigma_j^2)\). Completing the square therefore gives a normal kernel with variance equal to the reciprocal of the total precision and mean equal to that variance times the combined precision-weighted sum. This yields the update above.

Posterior Difference and Uncertainty

Define the raw treatment effect as \(\delta=\mu_T-\mu_C\). Under the model, the two arm posteriors are independent normal distributions. A difference of independent normal random variables is normal: its mean is the difference of their means, and its variance is the sum of their variances. Consequently, the posterior distribution of \(\delta\) is

The posterior standard deviation of the difference, \(\sqrt{v_T+v_C}\), describes its posterior uncertainty in the endpoint’s original units. A central 95% equal-tail credible interval for a normal posterior is its mean plus or minus 1.96 posterior standard deviations. This interval contains 95% of the posterior probability for \(\delta\), conditional on the model, prior, and observed data.

Assumptions, Diagnostics, and Alternatives

Before interpreting the results, check whether the model describes the design and endpoint adequately. The independent-arm calculation is inappropriate if the observations are paired or otherwise dependent in a way the model ignores. Repeated measurements from one participant, clustering by site, or shared family membership may require a model that represents the dependence. The normal likelihood and known variance also need to be defensible for the endpoint and sample sizes. With small samples, pronounced skewness or influential outliers can make the normal summary model a poor description.

The prior deserves a check as well. A prior mean of 135 mmHg with standard deviation 20 mmHg is broad relative to the assumed standard deviation of an individual outcome, but it may still affect the result. Compare its implications with plausible clinical ranges and consider whether reasonable changes to its mean or variance would materially change the posterior. If variance is not known, using a known-variance update without justification understates the uncertainty in the variance; a model that estimates variance, such as the shared-unknown-variance approach, is an alternative. A paired endpoint needs a model for within-person differences, not two independent arm updates.

The raw mean difference is often preferable when the endpoint scale is clinically meaningful: it directly states the estimated contrast in mmHg, points, or another recorded unit. A standardized mean difference expresses a contrast relative to a standard deviation and can help compare effects measured on different scales, but it no longer reports the change in the original clinical units. The present analysis deliberately reports the raw contrast and does not replace the endpoint with a standardized effect.

This conjugate analysis is compact and transparent, but its simplicity is also a limitation: it conditions on a chosen normal likelihood, known variances, a particular prior, and complete observed endpoint summaries. It does not by itself address missing outcomes, nonadherence, multiplicity, baseline adjustment, or treatment-policy choices about the estimand. Such features should be handled by a model and analysis plan that explicitly represent them, rather than assumed away by the algebra of the normal update.