Why analyze conflict on the binary scale?
For a binary clinical endpoint, the data are usually reported as the number of responses among the number treated. A historical trial may suggest a high response rate, while the current trial observes fewer responses. The practical question is not simply whether the counts differ: it is how much information from the historical data should influence the current response-rate estimate, given the model and its assumptions.
This tutorial works through two approaches: a robust mixture prior, which allows the current response rate to be informed either by a historical component or by a protective component, and a commensurate prior, which relates current and historical effects through a precision parameter. These methods make different assumptions and summarize borrowing differently. Neither turns a conflict diagnostic into a decision that one model is automatically correct.
Worked robust-mixture analysis
Suppose a hypothetical historical trial observed 35 responses among 50 participants, and a current trial observed 6 among 20. For illustration, use a historical-informed prior component \(\operatorname{Beta}(7,3)\) for \(p_C\), calibrated to have mean 0.70 and effective prior sample size 10, and a protective component \(\operatorname{Beta}(1,1)\). The protective component is uniform over response probabilities; it is not a point mass at no borrowing. Set the prior probability of the historical component to \(w=0.80\).
These choices are modeling assumptions, not a claim that all 50 historical participants are being added as current observations. They make the calculation transparent while allowing the current data to shift component membership. The beta-binomial update assumes conditional binomial sampling with a common response probability within each component and independent current outcomes given that probability. Any clinical factors that make the populations non-comparable would require additional modeling or a narrower borrowing target.
Worked Example: Updating a Historical Mixture Component
Under the historical-informed component, the current likelihood is proportional to \(p_C^6(1-p_C)^{14}\). Beta conjugacy gives
Its posterior mean is \(13/(13+17)=0.4333\). Under the protective component, the posterior is \(\operatorname{Beta}(1+6,1+14)=\operatorname{Beta}(7,15)\), with mean \(7/22=0.3182\). The current-only estimate is \(6/20=0.3000\); both posterior means exceed it because the component priors put weight on response rates above 0.30.
To update the component probabilities, compare their marginal likelihoods. The common binomial coefficient cancels, so their ratio is
Here \(B(a,b)\) is the beta function. The ratio is below one: these current data are less predictive under the historical-informed component than under the protective component. Bayes' rule gives the posterior historical-component probability
The posterior for \(p_C\) is therefore a mixture of \(\operatorname{Beta}(13,17)\) with weight 0.4819 and \(\operatorname{Beta}(7,15)\) with weight 0.5181. Its mean is \(0.4819(13/30)+0.5181(7/22)\approx0.3737\). The historical component's posterior weight has fallen from 0.80 to about 0.48, indicating appreciable discounting in this model. It does not mean that 48.19% of individual historical participants were borrowed.
What the mixture calculation does and does not say
The key justification is Bayes' rule applied to the latent component indicator. If \(Z=1\) denotes the historical-informed component and \(Z=0\) the protective component, then \(P(Z=1)=w\), and the current data have component-specific marginal likelihoods \(m_1\) and \(m_0\). Thus
This equation explains why conflict can reduce borrowing: when the observed count is relatively unlikely under the historical-informed prior, its marginal likelihood is smaller and its posterior component odds decline. The posterior response-rate distribution is still a mixture, rather than a single beta distribution with a newly estimated effective sample size. Reporting only its mean can hide that structure; a credible interval or posterior probabilities of clinically relevant response rates should also be obtained from the full mixture when making an actual clinical decision.
Worked Example: Sensitivity to the Mixture Weight
Keep both beta components and the observed current count fixed, but change the prior historical-component weight. The likelihood ratio remains \(R\approx0.2325\). If \(w=0.50\), the posterior historical weight is \(0.50R/[0.50R+0.50]\approx0.1886\). The posterior mean is then \(0.1886(0.4333)+0.8114(0.3182)\approx0.3399\).
If \(w=0.90\), the posterior component weight is \(0.90R/[0.90R+0.10]\approx0.6766\), and the posterior mean is \(0.6766(0.4333)+0.3234(0.3182)\approx0.3961\). Thus the posterior mean ranges from about 0.340 to 0.396 across these prior-weight choices, compared with 0.300 from the current sample alone. This is a focused sensitivity analysis: it changes the prior odds while keeping the likelihood and component distributions fixed.
Commensurate prior for a binary response
A commensurate model uses a continuous measure of similarity rather than a discrete choice between mixture components. One convenient working analysis is on the logit scale. Write \(\theta_C=\log\{p_C/(1-p_C)\}\) and \(\theta_H=\log\{p_H/(1-p_H)\}\). A commensurate prior can be written \(\theta_C\mid\theta_H,\tau\sim N(\theta_H,1/\tau)\), where larger \(\tau\) concentrates the current effect more tightly around the historical effect. If uncertainty in the historical estimate is represented by \(\theta_H\mid D_H\) approximately \(N(\hat\theta_H,V_H)\), then integrating over that uncertainty gives the working prior \(\theta_C\mid D_H,\tau\sim N(\hat\theta_H,V_H+1/\tau)\).
For a current binomial count, a normal approximation to the logit likelihood has estimate \(\hat\theta_C=\log\{(x_C/n_C)/(1-x_C/n_C)\}\) and estimated variance \(V_C\approx1/x_C+1/(n_C-x_C)\), when both response and nonresponse counts are sufficiently large for the approximation. The conjugate normal update below relies on this approximation, not on an exact normal likelihood for the binomial count. The historical summary variance may include nuisance-parameter uncertainty; it must be defined from the fitted historical model rather than assumed to equal a raw-count variance.
Worked Example: Commensurate Logit Update
Consider hypothetical historical data of 70 responses among 100 participants. Suppose a fitted historical logistic model, including its specified uncertainty adjustments, gives \(\hat\theta_H=\log(0.70/0.30)=0.8473\) and posterior variance \(V_H=0.04960\). In a current trial, 14 of 35 participants respond. The current logit estimate is \(\hat\theta_C=\log(0.40/0.60)=-0.4055\), and the binomial information approximation gives
Take \(\tau=20\). The historical-informed prior variance for the current logit is \(V_H+1/\tau=0.04960+0.05000=0.09960\), so its precision is \(1/0.09960=10.0402\). The posterior precision is \(10.0402+8.400=18.4402\), and the posterior mean is
The posterior variance is \(1/18.4402\approx0.05423\), with posterior standard deviation about 0.2329. Transforming the posterior mean gives \(\operatorname{logit}^{-1}(0.2767)\approx0.5687\). This is the response probability corresponding to the posterior mean logit, not the posterior mean response probability; the latter requires averaging the inverse-logit over the logit posterior. The contribution of historical-prior precision to the total posterior precision is \(10.0402/18.4402\approx0.5445\). This is a precision-based borrowing summary for this normal update, not a proportion of historical patients.
The arithmetic follows directly from multiplying two normal densities. For prior mean \(m_0\), prior precision \(q_0\), and likelihood estimate \(y\) with precision \(q_C\), the posterior density is proportional to \(\exp\{-q_0(\theta-m_0)^2/2-q_C(\theta-y)^2/2\}\). Completing the square gives posterior precision \(q_0+q_C\) and mean \((q_0m_0+q_Cy)/(q_0+q_C)\). Here \(q_0=10.0402\), \(q_C=8.400\), \(m_0=0.8473\), and \(y=-0.4055\). The two estimates conflict, so the posterior mean lies between them, with its location determined by their precisions.
Worked Example: Sensitivity to Commensurate Precision
Using the same historical and current summaries, set \(\tau=5\). Then the prior variance is \(0.04960+0.20000=0.24960\), its precision is about 4.0064, and the posterior mean logit is \(\{4.0064(0.8473)+8.400(-0.4055)\}/(4.0064+8.400)\approx-0.0009\). Its inverse-logit is about 0.4998.
At \(\tau=20\), the posterior mean logit was about 0.2767. Increasing \(\tau\) from 5 to 20 makes the prior more concentrated around the historical effect, so it increases the historical contribution and moves the posterior toward the historical logit. The data have not changed; the difference is entirely due to a modeling choice. A sensitivity analysis should therefore report the range of conclusions across clinically defensible \(\tau\) values rather than select a value solely because it produces a preferred estimate.
Comparison, diagnostics, and limitations
| Feature | Robust mixture prior | Commensurate prior |
|---|---|---|
| Borrowing structure | Posterior weight shifts between specified historical and protective components. | Current and historical effects are linked through a precision parameter. |
| Useful summary | Posterior component probability, alongside the full mixture posterior. | Posterior effect and sensitivity to \(\tau\); precision contribution can summarize the normal update. |
| Main sensitivity | Component distributions and prior mixture weight. | Commensurate precision and historical-effect uncertainty. |
| Important limitation | A protective component is still a prior distribution and may itself influence the result. | A single normal commensurability structure may not represent abrupt population differences. |
For either model, check that the endpoint definition, follow-up window, eligibility criteria, and response ascertainment are sufficiently comparable. Examine whether each fitted model can reproduce the observed current count and whether conclusions are stable to defensible prior choices. With sparse counts, the logit normal approximation can be poor or undefined when all participants respond or none respond; use an appropriate binomial likelihood and a suitable computational method instead. In a mixture fit, inspect component-specific predictive fit and the posterior component probability. In a commensurate fit, inspect posterior sensitivity to \(\tau\), the uncertainty assigned to the historical effect, and the adequacy of the logit likelihood approximation.
Neither approach adjusts automatically for covariate imbalance, changes in standard of care, or differences in outcome measurement. A mixture can downweight historical information when the observed count conflicts with it, but it cannot identify why the conflict occurred. A commensurate prior can represent graded similarity, but a large \(\tau\) does not establish that the populations are clinically exchangeable. These models quantify conclusions conditional on assumptions; external scientific judgment remains essential.
Common mistake
A common mistake is to interpret the posterior historical-component probability as a formal test of whether historical and current response rates are equal, or as the probability that the historical data are valid. It is neither. It is the posterior probability of a component label under the particular mixture prior, component distributions, and observed likelihood. Likewise, a commensurate precision is not evidence that the populations truly share an effect: it is a parameter controlling the prior relationship. Report the model, assumptions, and sensitivity along with the borrowing summary.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In the mixture example, what are the two posterior component distributions and their means?
- Why did the posterior historical-component probability decrease from 0.80?
- What does the commensurate precision \(\tau\) control?
- For 14 responses among 35 participants, verify the logit-likelihood variance approximation.
- Why is a posterior component probability not a test of equality of historical and current response rates?
Solutions. (1) The historical component gives \(\operatorname{Beta}(13,17)\), mean \(13/30=0.4333\); the protective component gives \(\operatorname{Beta}(7,15)\), mean \(7/22=0.3182\). (2) The current data have lower marginal likelihood under the historical-informed component: \(R\approx0.2325<1\), which reduces its prior odds under Bayes' rule. (3) It controls the concentration of the current-effect prior around the historical effect; larger \(\tau\) means stronger concentration. (4) \(1/14+1/(35-14)=1/14+1/21=0.11905\), whose reciprocal is 8.400. (5) It is the posterior probability of a specified mixture label, conditional on the model and its prior choices, not a probability that two parameters are equal or that a null hypothesis is true.