Deriving Normal Commensurate Shrinkage

Why derive the shrinkage?

A commensurate prior links a current treatment effect to a historical treatment effect while allowing them to differ. With a fixed commensurability precision, the link has a transparent normal-model derivation: the current posterior mean is a weighted average of the current estimate and the historical posterior mean. The weights depend on sampling uncertainty, historical uncertainty, and the precision of the link.

The derivation also shows that borrowing is reciprocal. Current data can update the historical effect as well as the current effect. This is a consequence of treating both effects as uncertain, not of treating the historical estimate as a fixed target.

Model and assumptions

Let \(\theta_H\) be the historical treatment effect and \(\theta_C\) the current treatment effect, measured on the same scale and in the same direction. Suppose the historical data have already been summarized by a proper normal posterior distribution. Write its mean as \(m_H\) and its variance as \(V_H\). This summary may, for example, result from a normal prior for \(\theta_H\) and a normal likelihood for a historical estimate.

For the current data, suppose \(y_C\) is an unbiased normal estimate of \(\theta_C\), with known sampling variance \(V_C\). The commensurate prior states that, conditional on the historical effect, the current effect differs from it by a normal deviation with precision \(\tau\).

Here \(\tau\) has units of inverse effect-squared, so \(\tau^{-1}\) is a variance on the effect scale. Larger \(\tau\) means the model expects the two effects to be closer; smaller \(\tau\) allows greater differences. The model also assumes that the two effects are comparable: the endpoint, treatment contrast, population context, and effect scale must support interpreting their difference as a commensurate deviation. The normal sampling model and known variances are working assumptions, not consequences of choosing a commensurate prior.

The historical posterior \(N(m_H,V_H)\) must represent historical information only. Incorporating current data into it first and then using it as the historical input would count the current data twice. The derivation below conditions on the historical data through that posterior and then adds the current likelihood once.

Posterior distributions and derivation

Set \(A=V_H+\tau^{-1}\). By adding independent normal variables, the prior distribution for the current effect after accounting for historical data is \(\theta_C\mid D_H\sim N(m_H,A)\). Updating this normal distribution with the current estimate gives the usual normal-normal posterior. To see the weighted-average form, multiply the prior density by the current likelihood and collect the terms involving \(\theta_C\).

The final line is a precision-weighted average: the current estimate receives more weight when the historical-based prior variance \(A\) is large relative to the current sampling variance \(V_C\). Equivalently, the posterior mean starts at \(m_H\) and moves toward \(y_C\) by a fraction of their difference.

The historical effect also updates. Marginally, the current estimate obeys \(y_C\mid\theta_H,D_H\sim N(\theta_H,W)\), where \(W=\tau^{-1}+V_C\): the estimate differs from \(\theta_H\) both through the commensurate deviation and through current sampling error. Updating \(\theta_H\mid D_H\sim N(m_H,V_H)\) with this likelihood gives

These expressions can also be checked by completing the square in the normal likelihood for \(y_C\) given \(\theta_H\). They show that the historical effect moves toward the current estimate, but its movement is governed by \(V_H\) and the combined uncertainty \(W\). If desired, the posterior covariance of the two effects is \(V_H V_C/(A+V_C)\); their posterior dependence remains because the commensurate prior links them.

How fixed precision controls borrowing

The limiting cases clarify what the commensurability assumption does. As \(\tau\) approaches zero, \(\tau^{-1}\) becomes very large. Then \(A/(A+V_C)\) approaches one, and the current posterior mean approaches \(y_C\); the historical information has little influence. The historical update also vanishes because \(V_H/(V_H+W)\) approaches zero. This is the no-borrowing limit.

As \(\tau\) becomes very large, the commensurate variance approaches zero. The prior enforces \(\theta_C=\theta_H\) increasingly closely, so the current data can strongly update the historical effect and the current effect. In the limit, the two effects are a common effect, and their information is pooled through the historical posterior and current likelihood. A large but finite \(\tau\) does not make the effects exactly equal.

Assumptions, diagnostics, and alternatives

The algebra is exact under the specified normal model, but the model can still be a poor description of a trial. Before using the result, check that the effects refer to compatible estimands and that their sampling variances are credible. A treatment effect on one scale cannot be directly linked to an effect on another without a justified transformation. Also check whether normal sampling summaries are adequate; skewed, sparse, or otherwise non-normal data may not be well represented by a normal estimate with a known variance.

A useful diagnostic is the discrepancy \(y_C-m_H\) in relation to its prior predictive standard deviation \(\sqrt{V_H+\tau^{-1}+V_C}\). A discrepancy large relative to this scale means the current estimate is surprising under the chosen link and uncertainty assumptions. That is a warning to investigate incompatibility and sensitivity, not a formal proof that the studies differ. A fixed \(\tau\) cannot adapt to conflict: it assigns the same linking strength regardless of the observed discrepancy.

The main alternatives answer different modeling questions. Setting borrowing to zero leaves the current analysis independent of the historical effect. A common-effect model assumes the effects are equal, rather than merely close with a finite commensurate variance. Robust mixture approaches can allow borrowing to change according to conflict through their component structure and posterior weights. The fixed-precision model is simpler and gives closed-form updates, but it does not represent uncertainty about \(\tau\) or automatically protect against prior-data conflict.

Common mistake

A frequent error is to interpret larger \(\tau\) as allowing more difference because it is called a precision. That reverses the parameter's role: the variance of the difference is \(\tau^{-1}\), so increasing \(\tau\) decreases the allowed difference and strengthens borrowing. Another error is to call \(A/(A+V_C)\) the historical weight. It is the weight on the current estimate \(y_C\); the historical mean's weight is \(V_C/(A+V_C)\). Keeping the variance and weight definitions beside the formula prevents both mistakes.