Predictive Means Under Normal Endpoint Models

Why Predict Trial Means?

A prior for a continuous endpoint mean describes uncertainty about the underlying average response. A trial, however, produces a sample mean that also varies from one set of enrolled participants to another. A prior predictive distribution for the sample mean combines these two sources of uncertainty: uncertainty about the endpoint model parameters and sampling variation among participants.

This distinction matters when judging whether a prior is compatible with plausible trial outcomes. A prior mean may be clinically reasonable while its induced distribution for a trial average is too wide, too narrow, or centered somewhere unexpected. The calculations below use fixed sample sizes and normal endpoint models. As in the earlier material on prior predictive distributions for trial data, the aim is to examine outcomes implied by the model before observing trial data.

Model and Assumptions

For one arm, begin with the model \(Y_i\mid\mu,\sigma^2\sim N(\mu,\sigma^2)\), independently for \(i=1,\ldots,n\). The sample size \(n\) is fixed, \(\mu\) is the arm’s population mean, and \(\sigma^2\) is the within-arm variance. Thus, conditional on the parameters, \(\bar{Y}\mid\mu,\sigma^2\sim N(\mu,\sigma^2/n)\).

These results require independent participants with a common normal outcome distribution within the arm, and a prior on \(\mu\) and, where relevant, \(\sigma^2\). They do not account automatically for clustering, repeated measures, changing variance across participants, or informative missingness. Such features require a model that represents them. All means, standard deviations, and outcomes must be expressed in the endpoint’s units; variances are in squared units.

Known Endpoint Variance: A Normal Predictive Distribution

First suppose \(\sigma^2\) is treated as known and the prior for the arm mean is \(\mu\sim N(m_0,\tau^2)\). Conditional on \(\mu\), the sample mean has mean \(\mu\) and variance \(\sigma^2/n\). Integrating over the normal prior gives:

The predictive center is \(m_0\), the prior mean. The predictive variance is the sum of the prior variance \(\tau^2\) and the sampling variance \(\sigma^2/n\). Increasing \(n\) reduces the sampling contribution, but it does not remove uncertainty about \(\mu\): as \(n\) grows, the predictive variance approaches \(\tau^2\).

Uncertain Variance: A Student t Predictive Distribution

In many applications, the within-arm variance is uncertain as well as the mean. One conjugate specification is the normal-inverse-gamma prior. Here the inverse-gamma distribution is parameterized so that a variance \(v\) with shape \(a_0\) and scale \(b_0\) has density proportional to \(v^{-a_0-1}\exp(-b_0/v)\).

A short derivation shows where this result comes from. Conditional on \(\sigma^2=v\), integrating over \(\mu\) combines the prior uncertainty in \(\mu\) with sampling variation:

Write \(c=1/k_0+1/n\) and \(x\) for a possible value of \(\bar{Y}\). Integrating this conditional normal density over the inverse-gamma density gives an integral proportional to:

The proportionality follows by substituting \(u=[b_0+(x-m_0)^2/(2c)]/v\); the remaining integral is a gamma integral. The resulting density has the Student t form with \(2a_0\) degrees of freedom and scale squared \(b_0c/a_0\), which gives the stated scale. The heavier tails relative to a normal distribution reflect uncertainty about the variance as well as the mean.

Two Arms and a Difference of Sample Means

For two independent trial arms, the same reasoning can be applied to each mean. If the arm-specific priors and outcomes are independent, and each arm’s variance is treated as known, then the difference of sample means is also normal. Define the difference as treatment minus control. Its predictive mean is the difference between the prior means, and its predictive variance is the sum of the two prior variances and the two sampling variances.

The independence assumptions are important: the formula for the variance adds the two variances because the arm means are independent. If the prior means are dependent, a covariance term is needed; if the outcomes are dependent across arms, the sampling model must represent that dependence. This calculation concerns a difference on the original endpoint scale, not a standardized effect.

Diagnostics, Alternatives, and Limitations

A useful check is to compare the analytical calculation with simulated prior predictive trial means. Draw parameters from the stated prior, generate \(n\) outcomes from the conditional endpoint model, and calculate \(\bar{Y}\) for each replicate. The simulated center and spread should agree with the derived distribution within Monte Carlo error. A mismatch can reveal a coding error, an inconsistent variance parameterization, or a simulation that does not follow the assumed model.

Also check the model’s implications in endpoint units. If the predictive distribution assigns substantial probability to sample means outside the scale’s possible or clinically credible range, investigate the prior, the assumed normal endpoint model, or both. A normal model has unbounded support; that may be an acceptable approximation for some continuous measures but not for every constrained endpoint. The predictive distribution also changes with \(n\): smaller trials have more sampling variation, while larger trials retain uncertainty from the prior on the mean.

A plug-in calculation that fixes \(\mu=m_0\) and \(\sigma^2\) at selected values gives only a conditional sampling distribution. It omits uncertainty in the parameters and therefore is not the prior predictive distribution. The normal and Student t formulas above are convenient when their assumptions and priors apply. For more complicated priors or models, simulation can approximate the prior predictive distribution without requiring a closed-form result, but it does not remove the need to check the model and parameterization.

These formulas depend on fixed sample sizes, normal outcomes, and the stated prior structures. They do not by themselves diagnose every feature of a trial design or validate a prior. In particular, a plausible predictive mean does not ensure plausible individual outcomes, and a plausible marginal distribution for each arm does not establish that a joint model is appropriate when arms or endpoints are dependent.