Workshop: Compare Robust Mixture and Commensurate Fits

Why compare two borrowing models?

Historical trial data can improve the precision of a current treatment-effect estimate when the trials are sufficiently comparable. But if their underlying effects differ, borrowing too much can pull the current estimate away from what the current trial supports. Robust mixture and commensurate models address this risk in different ways: one represents borrowing through a choice among prior components, while the other represents it through a precision linking historical and current effects.

This workshop compares both approaches using the same hypothetical continuous-outcome summaries. The comparison is about the estimates and diagnostics produced under stated models, not about selecting a model through a conflict test. The introductory material on robust borrowing provides context; here the emphasis is on fitting, checking and interpreting the two classes side by side.

Data and model assumptions

Suppose the treatment effect is a difference in mean systolic blood pressure, measured in mmHg, with positive values favoring the treatment. Let \(\theta_H\) denote the historical effect and \(\theta\) the current-trial effect. To make the calculations transparent, represent the historical information by the distribution \(\theta_H\sim N(10,1)\). This means a historical posterior mean of 10 mmHg and variance 1 \((\mathrm{mmHg})^2\). The current trial reports an estimate \(x\), with likelihood \(x\mid\theta\sim N(\theta,4)\), so its standard error is 2 mmHg.

These summaries are assumed to be approximately normal, to refer to the same treatment contrast and estimand, and to be measured on the same scale. The current likelihood is assumed independent of the historical information conditional on the effects. For the calculations below, the historical distribution and current likelihood variances are treated as known. In practice, the summaries and their uncertainty must be justified, and a normal approximation may be poor for small samples or strongly skewed outcomes.

Definition: A robust mixture prior places prior probability on a set of borrowing components. Here, conditional on the historical summary, \(\theta\) has a strong component \(N(10,1)\) with prior probability \(0.8\) and a weak component \(N(10,25)\) with prior probability \(0.2\). A commensurate model links the effects through \(\theta\mid\theta_H,\tau\sim N(\theta_H,1/\tau)\), where \(\tau\) is the commensurate precision: larger \(\tau\) implies tighter linking and, generally, more borrowing.

The weak mixture component is deliberately broader than the strong component, but it is not a no-borrowing model: it remains centered at the historical estimate. The commensurate model uses a different borrowing mechanism. If \(\tau\) is fixed, the current effect has prior variance from both historical uncertainty and the conditional link. Specifically, integrating over \(\theta_H\) gives \(\theta\sim N(10,1+1/\tau)\). The two model classes therefore need not imply identical prior distributions, even when their centers agree.

Normal updating and the mixture calculation

For a normal prior \(N(m,v)\) and the current likelihood \(x\mid\theta\sim N(\theta,4)\), completing the square gives a normal posterior with variance \(v^*=4v/(4+v)\) and mean \(m^*=(4m+vx)/(4+v)\). The prior variance \(v\) and likelihood variance 4 are variances, not standard errors. In the mixture model, apply this update separately within each component, then update the component probabilities using the current-data predictive densities.

\theta\mid x,k\sim N\left(\frac{4m_k+v_kx}{4+v_k},\frac{4v_k}{4+v_k}\right), \qquad P(k\mid x)=\frac{P(k)f_k(x)}{\sum_j P(j)f_j(x)}, \qquad f_k(x)=\phi\left(x;10,4+v_k\right).

Here \(k\) indexes a mixture component, and \(\phi(x;m,v)\) is the normal density evaluated at \(x\), with mean \(m\) and variance \(v\). The predictive variance is \(4+v_k\), because it combines the prior component variance with the current likelihood variance. The posterior mean for \(\theta\) is the component-posterior-weighted average of the component-specific posterior means.

Justification: Within component \(k\), the prior density contributes a quadratic term \((\theta-m_k)^2/v_k\), and the likelihood contributes \((x-\theta)^2/4\). Adding their coefficients of \(\theta^2\) gives posterior precision \(1/v_k+1/4\), so the posterior variance is \((1/v_k+1/4)^{-1}=4v_k/(4+v_k)\). The corresponding precision-weighted mean is \((m_k/v_k+x/4)/(1/v_k+1/4)=(4m_k+v_kx)/(4+v_k)\). Integrating the likelihood over the prior component yields the predictive distribution \(x\sim N(m_k,4+v_k)\), which supplies the weight update.

Worked Example: Moderate Displacement at 13 mmHg

Let the current estimate be \(x=13\) mmHg, three mmHg above the historical mean. For the strong component, the posterior mean is \((4(10)+1(13))/(4+1)=53/5=10.60\) mmHg, with variance \(4(1)/5=0.8000\). For the weak component, the posterior mean is \((4(10)+25(13))/(4+25)=365/29=12.59\) mmHg, with variance \(4(25)/29=100/29=3.448\).

The predictive distributions for \(x\) are \(N(10,5)\) and \(N(10,29)\), respectively. Their density ratio at 13 is \(f_{\mathrm{strong}}(13)/f_{\mathrm{weak}}(13)=\sqrt{29/5}\exp\{-\frac{9}{2}(1/5-1/29)\}\approx1.144\). Thus the data favor the strong component relative to the weak component at this value, although they are displaced from the historical mean. Updating prior odds gives strong-to-weak posterior odds \(4(1.144)=4.574\), so the strong-component posterior probability is \(4.574/(1+4.574)=0.8206\). It has increased from its prior value of 0.8.

The mixture posterior mean is \(0.8206(10.60)+0.1794(12.59)\approx10.96\) mmHg. For comparison, under a commensurate model with fixed \(\tau=1\), the marginal prior for \(\theta\) is \(N(10,2)\). The posterior mean is \((4(10)+2(13))/(4+2)=11.00\) mmHg and the posterior variance is \(4(2)/6=1.333\). Both fits give similar point estimates here, but their borrowing diagnostics and posterior distributions arise from different assumptions.

Commensurate updating and its diagnostic

With fixed \(\tau\), integrating over \(\theta_H\sim N(10,1)\) gives a marginal prior for \(\theta\) with mean 10 and variance \(v_\tau=1+1/\tau\). The same normal update then gives posterior mean \((4(10)+v_\tau x)/(4+v_\tau)\) and variance \(4v_\tau/(4+v_\tau)\). The weight on the current estimate in the posterior mean is \(v_\tau/(4+v_\tau)\); the remaining weight is on the historical mean. As \(\tau\) increases, \(v_\tau\) decreases and the historical mean receives more weight.

A useful conditional diagnostic is the ratio of commensurate precision to current-likelihood precision. The latter is \(1/4\), so the ratio is \(\tau/(1/4)=4\tau\). This describes the relative precisions in the link and current likelihood, not a probability of agreement or a direct measure of an effective sample size. If \(\tau\) is estimated rather than fixed, examine its posterior relative to its prior and inspect whether the result is sensitive to the prior on \(\tau\). An MCMC fit also requires the chain-mixing and convergence checks appropriate to that computation.

Worked Example: Severe Displacement at 18 mmHg

Now let \(x=18\) mmHg. Under the mixture model, the component predictive density ratio is \(f_{\mathrm{strong}}(18)/f_{\mathrm{weak}}(18)=\sqrt{29/5}\exp\{-\frac{64}{2}(1/5-1/29)\}\approx0.01207\). The current data therefore favor the weak component. The posterior strong-component probability is \(0.8(0.01207)/(0.8(0.01207)+0.2)\approx0.0461\), sharply below its prior value of 0.8.

The within-component posterior means are \((40+18)/5=11.60\) mmHg for the strong component and \((40+25(18))/29=490/29=16.90\) mmHg for the weak component. Averaging them with the updated weights gives \(0.0461(11.60)+0.9539(16.90)\approx16.65\) mmHg. The mixture has moved most of its posterior weight to the weak component, while retaining some strong-component weight.

For the commensurate model with \(\tau=1\), the posterior mean is \((4(10)+2(18))/6=12.67\) mmHg: this fixed, relatively strong link pulls substantially toward the historical estimate. If instead \(\tau=0.05\), then \(v_\tau=1+1/0.05=21\), and the posterior mean is \((4(10)+21(18))/(4+21)=16.72\) mmHg, with variance \(4(21)/25=3.360\). This illustrates how a weaker link can accommodate the displacement. It is a conditional fit at a specified \(\tau\), not evidence that a model with an estimated \(\tau\) would necessarily choose that value.

Worked Example: Concordant Data at 10 mmHg

Suppose instead that \(x=10\) mmHg. In the mixture model, the predictive density ratio is \(\sqrt{29/5}\approx2.408\), so the strong-component posterior probability is \(0.8(2.408)/(0.8(2.408)+0.2)\approx0.9060\). Both component posterior means equal 10 mmHg, so the mixture posterior mean is also 10 mmHg. The higher strong-component probability shows that these data are more predictive under the narrower component; it does not establish that the historical and current effects are identical.

In the commensurate model, both \(\tau=1\) and \(\tau=0.05\) give posterior mean 10 mmHg because the current estimate and historical mean coincide. Their posterior variances differ: \(1.333\) for \(\tau=1\), and \(4(21)/25=3.360\) for \(\tau=0.05\). Thus agreement in point estimates does not mean that the models imply the same uncertainty or borrowing strength.

Comparing fits, diagnosing problems and choosing a model

The mixture posterior component probabilities answer a model-specific question: given this mixture prior and these data, how much posterior probability is assigned to each component? They are useful borrowing diagnostics, but they are not probabilities that the historical and current populations are truly exchangeable. The commensurate precision describes the conditional tightness of the link; when \(\tau\) is uncertain, its posterior and sensitivity to its prior reveal how much the data support a tight or loose connection. These diagnostics are not interchangeable.

Feature	Robust mixture	Commensurate model
Borrowing mechanism	Posterior allocation across specified prior components	Precision of the link between historical and current effects
Primary diagnostic	Posterior component probabilities	Fixed \(\tau\), or posterior for \(\tau\) if it is estimated
Key sensitivity	Component variances and prior mixture weights	Prior on \(\tau\), and how it is estimated
Potential failure	Components may not cover plausible conflict patterns	A fixed tight link may overborrow under conflict

A practical comparison should use the same estimand, current likelihood, historical summary and scale in both fits. Check the prior predictive distributions: could each model plausibly generate the observed current estimate, and are the weak component or low-precision link broad enough for the scientific context? Then compare posterior means and intervals, component weights or \(\tau\), and sensitivity to defensible prior choices. For the examples here, moderate displacement at 13 mmHg does not cause the mixture to downweight its strong component; the predictive density calculation actually raises its posterior weight. At 18 mmHg, the mixture shifts sharply toward its weak component, while the \(\tau=1\) commensurate fit remains pulled toward the historical estimate. A commensurate model with a suitably supported low-\(\tau\) region can instead give a much less borrowed estimate.

These calculations support a conditional, scientific choice rather than a universal winner. If distinct borrowing regimes are meaningful and the component weights and scales can be defended, a mixture makes those regimes explicit. If gradual changes in the strength of a historical-current link are meaningful, a commensurate model can represent that through \(\tau\). Under severe conflict, a fixed commensurate precision should not be mistaken for an adaptive fit; an uncertain precision needs careful prior specification and computational checking. Neither approach repairs differences in estimand, outcome definition, follow-up or patient mix. Those should be investigated before interpreting a borrowing diagnostic as evidence about population change.

Diagnostics can reveal that a fit is behaving implausibly: posterior mixture weights that barely change despite extreme current data may point to an unsuitable weak component or likelihood; a commensurate posterior for \(\tau\) concentrated at a boundary may expose prior or computational sensitivity; and poor MCMC mixing can make summaries unreliable. These are prompts to inspect assumptions and sensitivity, not automatic rules for model selection. In particular, comparing the two posterior estimates alone does not determine which model is correct.

Common pitfall: Seeing a current estimate farther from the historical mean and concluding that the mixture must favor its weak component is incorrect. Component updating depends on the predictive densities, including their spread. At \(x=13\) in the worked example, the strong-component predictive density is about 1.144 times the weak-component density, so the strong-component posterior probability rises from 0.8 to about 0.8206. Calculate the density ratio and update the odds before interpreting the diagnostic.

Check Your Understanding

Use the model assumptions and worked calculations above. The solutions are included for checking your reasoning.

For \(x=13\), does the current data favor the strong or weak mixture component? Why?
Under the commensurate model, what happens to \(v_\tau\) when \(\tau\) decreases?
At \(x=18\), why does the commensurate fit with \(\tau=1\) differ from the fit with \(\tau=0.05\)?
Does a high posterior probability for the strong mixture component prove that the historical and current effects are identical?
Name one assumption that could make this comparison misleading even if the arithmetic is correct.

Solutions. (1) The strong component: its predictive density at 13 is about 1.144 times the weak-component density, so the strong posterior probability increases to about 0.8206. (2) Since \(v_\tau=1+1/\tau\), decreasing \(\tau\) increases \(v_\tau\), making the marginal prior on the current effect broader. (3) A smaller \(\tau\) weakens the link, increasing the current estimate's posterior weight; the posterior means are 12.67 and 16.72 mmHg, respectively. (4) No. It is a conditional component probability under the specified model and prior, not proof of equality. (5) For example, the treatment effect may not have the same estimand or measurement scale across trials, or the normal likelihood approximation may be inadequate.