Why Work Through the Model Numerically?
A commensurate prior links a current-trial treatment effect to a historical-trial treatment effect, while allowing the strength of that link to vary. The precision of the link determines how much historical information can influence the current estimate. A numerical analysis makes this influence visible: it shows where the posterior estimate lies, how its uncertainty changes, and what happens when the link itself is uncertain.
The model here concerns two approximately normal trial-level estimates of the same treatment-effect scale. Its calculations are appropriate only when that common scale and the normal approximation are defensible. They do not make different populations, outcome definitions, or treatment contrasts automatically comparable.
Model and Assumptions
The working assumptions are that the two estimates are conditionally independent given their respective effects, their reported standard errors are treated as known, and both effects use the same scale and direction. A diffuse flat prior for \(\theta_H\) is used here so that its posterior given the historical estimate is \(N(\hat\theta_H,V_H)\). The precision \(\tau\) is either fixed for a sensitivity calculation or assigned a prior distribution and updated from the data.
Integrating over uncertainty in \(\theta_H\) gives the current effect a prior distribution conditional on \(\tau\):
The historical estimate's sampling uncertainty remains in this prior variance. Treating \(\hat\theta_H\) as if it were the true historical effect would omit \(V_H\) and generally overstate the precision of the borrowed information.
Posterior for a Fixed Link Precision
For a fixed \(\tau\), the current likelihood has precision \(1/V_C\), and the commensurate prior has precision \(q_\tau=1/(V_H+1/\tau)\). Multiplying the two normal densities gives a normal posterior. Its variance is the reciprocal of the sum of precisions, and its mean is the precision-weighted average:
Equivalently, \(m_\tau=(1-B_\tau)\hat\theta_C+B_\tau\hat\theta_H\), where \(B_\tau=q_\tau/(1/V_C+q_\tau)\). This weight is the posterior mean's historical contribution for the stated model and fixed \(\tau\). It is not a general measure of effective sample size, nor does it mean the historical study is literally a fraction of the current trial.
Worked Example: Posterior With a Specified Precision
Suppose a hypothetical historical trial estimates a treatment effect of \(\hat\theta_H=1.20\) units with standard error \(0.30\) units. A hypothetical current trial estimates \(\hat\theta_C=0.80\) units with standard error \(0.25\) units. Assume independent normal estimates, known standard errors, a common effect scale, and \(\tau=4\) per squared effect unit.
The variances are \(V_H=0.30^2=0.0900\) and \(V_C=0.25^2=0.0625\). The historical-informed prior variance for the current effect is \(V_H+1/\tau=0.0900+0.2500=0.3400\), so \(q_\tau=1/0.3400=2.9412\) per squared unit. The current likelihood precision is \(1/0.0625=16.0000\) per squared unit.
Substitution gives \(W_\tau=1/(16.0000+2.9412)=0.05280\) squared units, with posterior standard deviation \(\sqrt{0.05280}=0.2298\) units. The posterior mean is \(m_\tau=(16.0000(0.80)+2.9412(1.20))/(18.9412)=0.8622\) units.
The historical weight is \(B_\tau=2.9412/18.9412=0.1553\). Thus the posterior mean is pulled 15.53% of the way from the current estimate toward the historical estimate. A normal 95% credible interval conditional on this fixed \(\tau\) is \(0.8622\pm1.96(0.2298)=(0.412,\ 1.312)\) units. Under the specified model and fixed precision, 95% of the posterior probability for \(\theta_C\) lies in that interval.
How the Precision Changes the Result
The next table holds the two trial estimates and their standard errors fixed and varies \(\tau\). The limiting case \(\tau\to0\) makes the link variance \(1/\tau\) diverge, so the historical information's contribution tends to zero. As \(\tau\to\infty\), the link becomes exact, but the historical estimate still has variance \(V_H\); it does not become known without error.
Worked Example: Sensitivity Across Precision Values
Use the hypothetical data from the preceding example. For each precision, calculate \(q_\tau=1/(0.0900+1/\tau)\), then use the posterior formulas. For \(\tau=0.5\), \(q_\tau=1/(0.0900+2.0000)=0.4785\); hence \(m_\tau=(12.8000+0.4785(1.20))/(16.0000+0.4785)=0.8116\), and \(W_\tau=1/16.4785=0.06069\). For \(\tau=40\), \(q_\tau=1/(0.0900+0.0250)=8.6957\); hence \(m_\tau=(12.8000+8.6957(1.20))/(24.6957)=0.9409\), and \(W_\tau=1/24.6957=0.04049\). The \(\tau=4\) results were calculated above.
| \(\tau\) (per squared unit) | \(B_\tau\) | Posterior mean (units) | Posterior SD (units) | Conditional 95% credible interval (units) |
|---|---|---|---|---|
| 0.5 | 0.02904 | 0.8116 | 0.2463 | (0.329, 1.294) |
| 4 | 0.1553 | 0.8622 | 0.2298 | (0.412, 1.312) |
| 40 | 0.3521 | 0.9409 | 0.2012 | (0.546, 1.335) |
The larger precision draws the posterior mean toward the higher historical estimate and reduces posterior standard deviation. The intervals do not simply move inward around the current estimate: their centers also change. All intervals in the table condition on the selected value of \(\tau\), so they do not include uncertainty about which precision is appropriate.
Allowing Uncertainty in the Borrowing Link
If \(\tau\) is uncertain, the posterior for \(\theta_C\) averages the fixed-\(\tau\) posteriors over the posterior distribution of \(\tau\). In general this is a mixture, not a single normal distribution. The historical and current estimates inform \(\tau\) through their difference: after integrating out \(\theta_H\), the difference \(\hat\theta_C-\hat\theta_H\), conditional on \(\tau\), has variance \(V_C+V_H+1/\tau\). A large discrepancy is less compatible with a very precise link, all else equal.
The following finite mixture makes that update explicit. A discrete prior is used only to keep the arithmetic transparent; a continuous prior can be handled by numerical integration or simulation.
Worked Example: Updating a Two-Value Precision Prior
For the same hypothetical trial estimates, assign prior probability \(0.5\) to each of \(\tau=0.5\) and \(\tau=4\). Conditional on \(\tau\), the difference \(\hat\theta_C-\hat\theta_H=-0.40\) units has variance \(S_\tau=V_C+V_H+1/\tau\). Thus \(S_{0.5}=0.0625+0.0900+2.0000=2.1525\) and \(S_4=0.0625+0.0900+0.2500=0.4025\), in squared units.
The ratio of the marginal density of the observed difference under \(\tau=4\) to that under \(\tau=0.5\) is \(\sqrt{2.1525/0.4025}\exp\{-0.40^2(1/0.4025-1/2.1525)/2\}=1.969\) (approximately). With equal prior probabilities, the posterior probability of \(\tau=4\) is \(1.969/(1+1.969)=0.663\), and that of \(\tau=0.5\) is \(0.337\). The estimates are more compatible with the tighter link than the looser link in this comparison, but substantial posterior probability remains on the looser link.
The conditional current-effect posteriors are \(N(0.8116,0.06069)\) for \(\tau=0.5\) and \(N(0.8622,0.05280)\) for \(\tau=4\), where the second argument is the variance. The mixture posterior mean is \(0.337(0.8116)+0.663(0.8622)=0.8451\) units. Its variance includes both within-component variance and between-component variation:
The posterior standard deviation is therefore about \(0.2367\) units. This exceeds either component's standard deviation because it also reflects uncertainty about which link precision applies. A normal interval formed from the mixture mean and standard deviation is not automatically an exact 95% credible interval; a mixture interval should instead be obtained from the mixture's cumulative distribution.
Derivation, Diagnostics, and Alternatives
The fixed-\(\tau\) posterior follows by completing the square in the product of the current likelihood and the normal prior for \(\theta_C\). Writing \(a=\hat\theta_C\), \(b=\hat\theta_H\), and \(q=q_\tau\), the terms involving \(\theta_C\) in the log density are proportional to \(-\frac{1}{2}\{(\theta_C-a)^2/V_C+q(\theta_C-b)^2\}\). The coefficient of \(\theta_C^2\) is \(1/V_C+q\), giving posterior variance \(W_\tau=(1/V_C+q)^{-1}\). The coefficient of \(\theta_C\) is \(a/V_C+qb\), giving mean \(W_\tau(a/V_C+qb)\). This verifies the precision-weighted formulas rather than assuming that the historical effect is known exactly.
Useful checks include confirming the units of every variance and precision, verifying that \(B_\tau\) lies between 0 and 1, and checking limiting behavior: as \(q_\tau\) approaches zero, \(m_\tau\) approaches \(\hat\theta_C\) and \(W_\tau\) approaches \(V_C\). Under a tighter link, the estimate should move toward \(\hat\theta_H\), not away from it, and the conditional variance should not increase. If the historical and current estimates disagree sharply, inspect how the posterior for \(\tau\) changes and whether the normal likelihoods and common-effect interpretation remain plausible. These checks do not establish clinical comparability; they flag calculations or assumptions that need scrutiny.
A fixed precision provides a transparent sensitivity analysis but treats the borrowing strength as known. A prior on \(\tau\) propagates uncertainty about the link, though results can depend on that prior. Robust mixture approaches instead represent borrowing through distinct components, such as a borrowing component and a protective component; they can offer a different form of adaptation but require their own component and weight choices. None of these approaches repairs biased historical estimates or unmeasured differences between trials. The normal model can also be unsuitable when the effect estimate is strongly skewed, standard errors are poorly estimated, or the outcome scale is not comparable.
Common Mistakes and Interpretation
- Ignoring historical uncertainty: using \(1/\tau\) alone as the prior variance for \(\theta_C\) treats \(\theta_H\) as known. With the stated diffuse-prior model, the variance is \(V_H+1/\tau\).
- Reading precision backward: a larger \(\tau\) means smaller link variance, hence a tighter link. It does not mean less borrowing.
- Calling a conditional interval unconditional: an interval calculated at one fixed \(\tau\) excludes uncertainty in \(\tau\). A mixture posterior must be summarized as a mixture.
- Equating borrowing with validity: a posterior pulled toward the historical estimate is a consequence of the model, not evidence that the historical and current trials are exchangeable or clinically equivalent.
The posterior mean and interval describe uncertainty about the current treatment effect under the assumed likelihoods, prior for the historical effect, and commensurability model. In a clinical interpretation, state the effect's units and direction—for example, which sign favors treatment. Avoid interpreting the interval as a repeated-sampling guarantee or as evidence that the model assumptions are true.
Check Your Understanding
Answer each question using the definitions and calculations in this tutorial. Solutions follow the questions.
- If \(V_H=0.16\) squared units and \(\tau=4\) per squared unit, what is the variance of the commensurate prior for \(\theta_C\) after observing the historical estimate?
- With a fixed link and \(q_\tau=2\), \(V_C=0.25\), \(\hat\theta_C=0.5\), and \(\hat\theta_H=1.0\), calculate the posterior mean and variance.
- Why is the posterior under uncertain \(\tau\) generally not a normal distribution?
- In the two-value example, why did the posterior favor \(\tau=4\) over \(\tau=0.5\), and what does that result not establish?
Solutions. (1) \(V_H+1/\tau=0.16+0.25=0.41\) squared units. (2) \(W=1/(1/0.25+2)=1/6=0.1667\) squared units and \(m=(4(0.5)+2(1.0))/6=0.6667\) units. (3) It averages normal conditional posteriors with different means and variances; such a mixture is not generally normal. (4) The observed estimates differed by only \(0.40\) units, which had higher marginal density under the tighter link in this calculation. This is evidence only within the specified model and prior; it does not prove that the trials are clinically comparable or that borrowing is valid.