Continuous Endpoint Location And Scale Implications

From Prior Parameters to Patient-Level Values

A prior on a continuous endpoint is often specified through parameters such as a mean and a standard deviation. Those parameters are not themselves patient outcomes: the mean describes the location of the endpoint distribution, while the standard deviation describes variation among patients conditional on the parameters. A prior predictive check asks what values these choices imply before observing trial data. The practical question is whether those values, in the endpoint’s actual units, look clinically credible.

For example, a mean prior centered at a clinically reasonable change does not guarantee that individual values are plausible. A large outcome standard deviation can imply many extreme patient values even when the mean is sensible. Conversely, a very small standard deviation can imply an implausibly homogeneous group. Checking location and scale together is therefore essential.

The conditional model assumes that, given \(\mu\) and \(\sigma\), patients have the specified normal distribution. If the model treats patients as conditionally independent, each patient has that same conditional distribution; this assumption may need to be adjusted for clustering or repeated measurements. The normal model also assigns positive probability to every real value. That support is appropriate only if values outside the clinically possible range are impossible or sufficiently rare under the prior to be negligible for the intended use.

Separating Location, Scale, and Uncertainty

There are two kinds of variation to keep distinct. Conditional on a particular \((\mu,\sigma)\), \(\sigma\) describes patient-to-patient spread. Across prior draws, \(\mu\) and \(\sigma\) can also vary because they are uncertain. Consequently, a patient-level prior predictive distribution can be wider than the outcome distribution for any one fixed parameter draw.

A useful exact result is available when the prior for \(\mu\) is normal and \(\sigma\) is fixed. Write \(\mu\sim N(m_\mu,\tau_\mu^2)\), and suppose \(\epsilon_i\sim N(0,\sigma^2)\) independently of \(\mu\), with \(Y_i=\mu+\epsilon_i\). Then the marginal distribution of one patient’s value is normal:

The mean is \(E(Y_i)=E\{E(Y_i\mid\mu)\}=E(\mu)=m_\mu\). For the variance, the law of total variance gives

This addition of variances is not a rule for adding standard deviations. It combines conditional patient variation with uncertainty in the mean. If \(\sigma\) is uncertain, the marginal distribution is generally a mixture of normal distributions, not necessarily normal. Its variance, when finite, is \(E(\sigma^2)+\operatorname{Var}(\mu)\). A prior with an infinite second moment for \(\sigma\) does not yield a finite patient-level predictive variance, even if its central values appear reasonable.

Check the Scale Against the Endpoint

A standard deviation is expressed in the same units as the endpoint, not in squared units. Its square is the variance. This makes unit checks an efficient way to catch specification errors: a mean of 18 metres and a standard deviation of 25 metres can be compared directly, while a variance of 625 metres squared cannot be read as a 625-metre patient spread.

For fixed \(\mu\) and \(\sigma\), the normal model places approximately 95% of individual values within \(\mu\pm1.96\sigma\). This is a model-based range for individual outcomes conditional on those parameter values. It is not a 95% interval for \(\mu\), and when parameters have priors it is not automatically a 95% interval for the prior predictive distribution. The distinction matters because uncertainty about the mean adds further spread.

When \(\sigma\) has a prior, examine its typical values and tails in endpoint units. A prior can have a reasonable median yet still put substantial mass on scales that produce extreme patient values. Looking at several prior draws, alongside clinical reference ranges, helps reveal whether the issue is an implausible mean, an implausible scale, or their combination. If \(\mu\) and \(\sigma\) are dependent under the prior, retain that dependence when generating draws; breaking it by sampling each parameter independently changes the implied outcomes.

From a Parameter Draw to a Trial Summary

A prior predictive check can also examine whether values generated under a parameter draw would lead to recognisable trial summaries. The goal here is not to calculate a sampling distribution for a trial mean. It is to inspect whether a plausible draw of location and scale produces patient values, and hence an observed group, whose scale and range make clinical sense.

Diagnostics, Alternatives, and Limitations

Useful checks include plotting prior draws of \(\mu\) and \(\sigma\) on their natural scales, generating patient values using the joint prior, and comparing those values with clinically meaningful limits or reference data. Review both central outcomes and tails. A check that reports only an average can miss an excessive spread; one that reports only a typical \(\sigma\) can miss a prior tail that generates implausible outcomes. When values are generated computationally, confirm that the recorded units and any transformations are reversed correctly before clinical review.

The main alternative to a prior predictive simulation is an analytic calculation. For a normal prior on \(\mu\) and fixed \(\sigma\), the marginal distribution derived above gives exact probabilities for ranges. This is transparent and easy to audit. Simulation is more flexible when \(\sigma\) is uncertain, priors are dependent, or the model is not normal. It can also expose combinations of parameters that deserve review. In either approach, results should be interpreted under the stated model and prior, not as evidence about what the trial will certainly observe.

A normal model may be a useful approximation for an unbounded measurement or a change score, but it can be a poor choice for a naturally bounded, discrete, or strongly skewed endpoint. Alternatives include models whose support matches the endpoint, or transformations that are scientifically justified and interpretable. Those alternatives still require their own prior predictive checks. A model with correct support can remain implausible in its location or spread.

A common mistake is to treat a credible-looking prior mean as sufficient evidence that the continuous endpoint prior is plausible. For example, a mean centered at 4 on a 0-to-10 scale may seem reasonable, but a large \(\sigma\) can still imply many impossible scores. The correction is to inspect the joint implications of location and scale at the patient level, keeping the endpoint’s units and support explicit. Another mistake is to compare a standard deviation directly with a variance or to carry a scale across endpoints with different units; state the unit every time.