Normal-Normal Hierarchies For Continuous Outcomes

Why Use a Normal-Normal Hierarchy?

In a trial with several clinical subgroups, each subgroup may have its own treatment effect on a continuous outcome. An estimate from a small subgroup is often imprecise, while estimates from larger subgroups are more precise. A normal-normal hierarchy represents this variation explicitly: each subgroup has a true effect, and those effects are modeled as arising from a shared distribution. This allows information to be shared without requiring every subgroup effect to be identical.

The model below concerns a continuous treatment effect, such as a difference in mean change from baseline, measured in the outcome’s units. The sign convention must be stated in context. For example, if lower symptom scores are better and the effect is defined as treatment minus control, a negative effect favors treatment.

The first distribution describes uncertainty in the data estimate around its subgroup’s true effect. The second describes how true effects vary across subgroups. These are distinct sources of variation: \(s_j^2\) is sampling variance, whereas \(\tau^2\) is between-subgroup variance. In this tutorial, inference for \(\theta_j\) is conditional on specified values of \(\mu\) and \(\tau^2\). Treating them as fixed means their uncertainty is not propagated in the calculations.

The sampling model is exact if the subgroup estimate is normally distributed with known standard error. In practice, \(y_j\) may be a treatment-effect estimate and \(s_j\) its estimated standard error; the normal sampling distribution is then an approximation, and the calculation treats the reported \(s_j\) as fixed. The model also assumes that the estimates measure comparable treatment effects, use consistent outcome units and direction, and are independent across subgroups conditional on the effects. Shared controls or other design features can violate that last assumption.

Conditional Posterior for One Subgroup

With \(\mu\) and \(\tau^2\) fixed, the prior distribution for \(\theta_j\) is normal and its likelihood contribution from \(y_j\) is normal. Their product gives a normal conditional posterior. To derive it, retain the terms involving \(\theta_j\) in the prior density multiplied by the likelihood:

Expand the two squared terms and collect the quadratic and linear terms in \(\theta_j\). The coefficient of \(\theta_j^2\) is \(1/s_j^2+1/\tau^2\), and the coefficient of \(-2\theta_j\) is \(y_j/s_j^2+\mu/\tau^2\). Completing the square identifies the posterior variance and mean:

The formula uses variances, not standard errors, in the precision terms: the data precision is \(1/s_j^2\), and the prior precision is \(1/\tau^2\). The posterior variance is positive when both variances are positive. The conditional posterior distribution describes uncertainty about the subgroup’s true effect after observing its estimate, under the specified hierarchy.

Interpreting the Hierarchy in Trial Context

A hierarchy represents partial pooling: subgroup effects may differ, but their distribution is linked through \(\mu\) and \(\tau^2\). Conditional on the hyperparameters, the posterior for a subgroup uses its own data estimate and the common distribution as a reference. The amount of influence from each depends on its precision. A precise subgroup estimate has greater data precision; a small \(\tau^2\) gives the distribution of subgroup effects greater precision. The separate derivation and interpretation of the resulting shrinkage weight are not needed to use the conditional posterior formula.

Exchangeability is the substantive justification for sharing information. It means that, before seeing subgroup-specific outcomes, the labels of the subgroups do not determine different prior distributions for their effects: the effects are treated as draws from the same distribution. It does not mean the effects must be equal. Clinical knowledge may make exchangeability implausible, for example if subgroups differ in disease mechanism or treatment exposure. In that case, a single common distribution may not represent the relevant variation.

Connecting Arm-Level Summaries to the Model

For a two-arm trial, the continuous subgroup estimate often comes from a difference in sample means. If treatment and control observations are independent within subgroup \(j\), and the outcome variances in the two arms are \(\sigma_{Tj}^2\) and \(\sigma_{Cj}^2\), then the estimated treatment-minus-control effect is \(\bar{x}_{Tj}-\bar{x}_{Cj}\). Its sampling variance, when those variances are known, is \(\sigma_{Tj}^2/n_{Tj}+\sigma_{Cj}^2/n_{Cj}\). Estimated variances commonly replace the unknown variances, giving an estimated standard error. The resulting estimate and standard error can be used as \(y_j\) and \(s_j\) under the normal sampling approximation.

Diagnostics, Alternatives, and Limitations

A normal-normal model can be mathematically convenient yet clinically inadequate. Check whether treatment effects are defined consistently, whether the endpoint is measured on a common scale, and whether subgroups are plausibly exchangeable. Compare subgroup estimates and their standard errors visually. Large disagreements relative to the reported uncertainty can signal that a common normal distribution is questionable, although disagreement alone does not establish which model is correct. Also check trial design: shared controls can induce dependence among effect estimates, so treating them as conditionally independent may understate or misrepresent uncertainty.

The normal model allows effects on the entire real line. That can be reasonable for a mean difference, but may be unsuitable when the effect is constrained or strongly skewed. The plug-in treatment of \(s_j\), \(\mu\), and \(\tau^2\) also understates the uncertainty that would arise if their estimation were uncertain. If \(\tau^2\) is set to zero, the prior assigns all subgroup effects the same value \(\mu\), corresponding to complete pooling conditional on \(\mu\). Separate subgroup analyses instead omit the shared distribution and do not borrow information. The hierarchical model lies between those approaches, but its usefulness depends on defensible assumptions and sensitivity to the chosen hyperparameters.

A common mistake is to say that the hierarchy makes all subgroup effects equal because they share a common mean. That conclusion confuses the mean of the effect distribution with each subgroup’s effect: the model allows \(\theta_j\) to vary around \(\mu\), with variation governed by \(\tau^2\). Another mistake is to use \(s_j\) where the formula requires \(s_j^2\). The data precision is \(1/s_j^2\), not \(1/s_j\); the units provide a check, since both \(s_j^2\) and \(\tau^2\) must be variances in squared outcome units.