Commensurate Priors for Binary Endpoints

Why Commensorate Binary Effects?

A historical trial can inform a current trial when their treatment effects are sufficiently similar. For a binary endpoint, a natural effect scale is the log odds ratio. A commensurate prior expresses similarity probabilistically: the current treatment effect is centered on the historical effect, while a precision parameter controls how strongly that link is enforced. High precision means little expected difference; low precision allows substantial difference.

This is not the same as assuming that historical and current event rates are identical. Each trial can have its own baseline event probability. The link concerns the treatment effects, not the control-group risks. This distinction matters when the populations, follow-up periods, or background care produce different baseline risks.

Definition: For trial \(j\), let \(p_{j0}\) and \(p_{j1}\) be the event probabilities in the control and treatment groups. Its log odds treatment effect is \(\delta_j=\operatorname{logit}(p_{j1})-\operatorname{logit}(p_{j0})\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). A commensurate prior for a current effect \(\delta_C\) and historical effect \(\delta_H\) is \(\delta_C\mid\delta_H,\tau\sim N(\delta_H,1/\tau)\), with \(\tau>0\). The precision \(\tau\) governs the prior variance of their difference.

From Binary Counts to Log-Odds Effects

Suppose a trial has \(a\) events among \(n_1\) treated participants and \(c\) events among \(n_0\) control participants. The estimated treatment and control odds are \(a/(n_1-a)\) and \(c/(n_0-c)\), respectively. Their log odds ratio is \(\hat{\delta}=\log\{a(n_0-c)/[(n_1-a)c]\}\). When all four cells are positive, a common large-sample variance estimate is \(1/a+1/(n_1-a)+1/c+1/(n_0-c)\). These are likelihood-based approximations; the binomial model itself specifies the event counts, not a normally distributed effect.

Worked Example: Calculating a Historical Log Odds Ratio

Imagine a hypothetical historical randomized trial with 68 events among 110 treated participants and 55 events among 110 control participants. The estimated log odds ratio is \(\log\{68(110-55)/[(110-68)55]\}=\log(68/42)=0.482\). Thus the estimated treatment odds of the event are about \(\exp(0.482)=1.620\) times the control odds.

The estimated sampling variance is \(1/68+1/42+1/55+1/55=0.07488\), so the standard error is \(\sqrt{0.07488}=0.2736\). This calculation assumes independent participants and a sufficiently regular likelihood for the large-sample approximation. All four cells are positive, and the expected event and non-event counts are not especially small, so the approximation is plausible here, though not guaranteed to be exact.

Commensurate Model and Normal Approximation

For binary counts in trial \(j\), a basic logistic model is \(X_{j1}\sim\operatorname{Bin}(n_{j1},p_{j1})\) and \(X_{j0}\sim\operatorname{Bin}(n_{j0},p_{j0})\), with \(\operatorname{logit}(p_{j0})=\alpha_j\) and \(\operatorname{logit}(p_{j1})=\alpha_j+\delta_j\). The intercept \(\alpha_j\) allows the baseline odds to differ across trials. Conditional on their probabilities, the treatment and control counts are independent binomials; participants are assumed independent within each arm.

The commensurate prior adds \(\delta_C\mid\delta_H,\tau\sim N(\delta_H,1/\tau)\). In an exact analysis, this prior can be combined with the binomial likelihoods. For transparent calculations, the examples below use a normal approximation to the likelihoods: \(\hat{\delta}_H\mid\delta_H\) is approximately \(N(\delta_H,V_H)\), and \(\hat{\delta}_C\mid\delta_C\) is approximately \(N(\delta_C,V_C)\), where \(V_H\) and \(V_C\) are estimated sampling variances.

Assume that the historical effect has a diffuse prior, so its approximate posterior after the historical data is \(N(\hat{\delta}_H,V_H)\). Integrating over that uncertainty gives the current effect, before observing current data, the history-conditioned prior \(N(\hat{\delta}_H,V_H+1/\tau)\). The historical estimate is therefore not treated as known without error.

Posterior result under the normal approximation: Write \(y_H=\hat{\delta}_H\) and \(y_C=\hat{\delta}_C\). Conditional on \(\tau\), the posterior for \(\delta_C\) is normal with variance \(W_\tau=\{1/V_C+1/(V_H+1/\tau)\}^{-1}\) and mean \(m_\tau=W_\tau\{y_C/V_C+y_H/(V_H+1/\tau)\}\). The predictive distribution of \(y_C\), conditional on the historical data and \(\tau\), is \(N(y_H,T_\tau)\), where \(T_\tau=V_C+V_H+1/\tau\).

The posterior expression follows by multiplying the normal likelihood for \(y_C\) by the history-conditioned normal prior for \(\delta_C\). The terms in the exponent that depend on \(\delta_C\) are \[ -\frac{1}{2}\left\{\frac{(y_C-\delta_C)^2}{V_C} +\frac{(\delta_C-y_H)^2}{V_H+1/\tau}\right\}. \] Collecting the coefficient of \(\delta_C^2\) gives posterior precision \(1/V_C+1/(V_H+1/\tau)\); collecting the linear terms gives posterior mean \(W_\tau\{y_C/V_C+y_H/(V_H+1/\tau)\}\). The predictive variance adds the independent variances of the current estimate, historical estimate, and commensurability difference, giving \(T_\tau\).

Worked Analysis with Uncertain Precision

Uncertainty about \(\tau\) can be represented by a prior distribution and updated using the predictive density of the current estimate. The following discrete example makes each step visible. It is a hypothetical normal-likelihood approximation to binary-trial log odds estimates, not a report of an actual clinical study.

Worked Example: Updating Borrowing Strength from Binary-Trial Summaries

Suppose the historical and current estimates are \(y_H=0.4953\) and \(y_C=0.08268\), with approximate variances \(V_H=0.0832\) and \(V_C=0.0827\). Consider two possible commensurability precisions, \(\tau=16\) and \(\tau=0.25\), each with prior probability \(0.5\). The estimates are on the log odds ratio scale; a positive value favors treatment for an endpoint coded as an event that is beneficial.

Conditional posterior when \(\tau=16\). The history-conditioned prior variance is \(0.0832+1/16=0.1457\). Thus \(W_{16}=\{1/0.0827+1/0.1457\}^{-1}=0.0528\). The posterior mean is \(m_{16}=\{1/0.0827+1/0.1457\}^{-1}\{0.08268/0.0827+0.4953/0.1457\}=0.2321\). This stronger-commensurability component pulls the current estimate appreciably toward the historical estimate.

Conditional posterior when \(\tau=0.25\). The history-conditioned prior variance is \(0.0832+1/0.25=4.0832\). Therefore \(W_{0.25}=\{1/0.0827+1/4.0832\}^{-1}=0.08106\), and \(m_{0.25}=\{1/0.0827+1/4.0832\}^{-1}\{0.08268/0.0827+0.4953/4.0832\}=0.0909\). The weakly commensurate component leaves the posterior mean closer to the current estimate.

Update the precision probabilities. Let \(d=y_C-y_H=-0.41262\). The predictive variances are \(T_{16}=0.0827+0.0832+0.0625=0.2284\) and \(T_{0.25}=0.0827+0.0832+4=4.1659\). The likelihood ratio favoring \(\tau=16\) over \(\tau=0.25\) is \(\sqrt{T_{0.25}/T_{16}}\exp\{-d^2(1/T_{16}-1/T_{0.25})/2\}\). Substitution gives \(\sqrt{4.1659/0.2284}\exp\{-(0.41262)^2(1/0.2284-1/4.1659)/2\}\approx4.2708(0.7031)=3.003\). With equal prior probabilities, the posterior probability of \(\tau=16\) is \(3/(1+3)=0.750\), and that of \(\tau=0.25\) is \(0.250\).

Combine the conditional posteriors. The posterior mean is \(0.750(0.2321)+0.250(0.0909)=0.1968\). For a mixture, total variance is the weighted average of within-component variances plus the variance of component means around the mixture mean: \[ \operatorname{Var}(\delta_C\mid\text{data})= 0.750\{0.05280+(0.2321-0.1968)^2\} +0.250\{0.08106+(0.0909-0.1968)^2\} =0.0636. \] Using the unrounded component quantities, the posterior standard deviation is approximately \(0.2521\). Thus the precision uncertainty produces a posterior that averages across both borrowing strengths; it is not equivalent to selecting one component and reporting its conditional variance alone.

The likelihood ratio is based on predictive variances \(T_\tau\), whereas the conditional posterior variances are \(W_\tau\). They answer different questions and must not be interchanged. Here the current estimate is below the historical estimate, yet the high-precision model receives more posterior support than the low-precision model because its predictive density is more concentrated around the historical value and still assigns adequate density to the observed difference.

A Second Conditional Calculation

Worked Example: Interpreting a Fixed Precision

Suppose a hypothetical analysis has \(y_H=0.8\), \(V_H=0.1\), \(y_C=0.2\), and \(V_C=0.1\), and temporarily conditions on \(\tau=4\). The current effect prior variance after accounting for historical uncertainty is \(0.1+1/4=0.35\), with precision \(1/0.35=2.857\). The current likelihood precision is \(1/0.1=10\).

The posterior variance is \(W_4=1/(10+2.857)=0.07778\). Its mean is \(W_4\{0.2/0.1+0.8/0.35\}=0.07778(2+2.286)=0.3333\). The posterior mean lies between the current estimate \(0.2\) and historical estimate \(0.8\), as it must for a normal likelihood and normal prior with positive precisions. This is a conditional calculation: if \(\tau\) is uncertain, the final posterior must also average over its posterior distribution.

Assumptions, Diagnostics, and Limitations

The normal approximation is most credible when both arms have enough events and non-events for the log odds ratio likelihood to be reasonably close to quadratic. A zero event cell makes the usual log odds ratio estimate and reciprocal-count variance undefined. In that setting, do not silently apply the formula above: fit the binomial-logistic model directly with an appropriate proper prior, or use a justified likelihood-based approximation. Arbitrary continuity corrections can materially affect sparse-data estimates.

Before borrowing, check that the endpoint definition, assessment time, treatment contrast, and relevant population features support interpreting the two log odds effects as comparable. The model allows different baseline odds, but it does not repair differences in treatment versions, outcome ascertainment, or trial conduct. Also inspect whether the current likelihood is approximately symmetric on the log odds scale and whether conclusions change under plausible prior distributions for \(\tau\).

A commensurate prior provides a continuous link controlled by precision uncertainty. A robust mixture approach instead places explicit alternative prior components on the current effect, often including a protective component intended to limit borrowing under conflict. Neither approach automatically solves incompatibility: each depends on prespecified distributions and modeling choices. The commensurate formulation is especially interpretable when similarity can be described as a distribution for the difference \(\delta_C-\delta_H\); its limitation is that a symmetric normal link may not suit strongly directional differences or severe conflict.

Common mistake: “The historical estimate is precise, so I can plug it in as the true current prior mean and ignore its uncertainty.” The correction is to propagate historical sampling variance: under the stated model the prior variance for \(\delta_C\) after observing historical data is \(V_H+1/\tau\), not merely \(1/\tau\). Another error is to use the conditional posterior variance \(W_\tau\) to update \(\tau\); precision probabilities are updated through the predictive density with variance \(T_\tau=V_C+V_H+1/\tau\).

The posterior mean and standard deviation are summaries on the log odds ratio scale. Exponentiating a log odds effect gives an odds ratio, but it does not give a risk ratio or a difference in event probabilities. Clinical interpretation should therefore include the endpoint direction and, when useful, predicted event probabilities at clinically relevant baseline risks.

Check Your Understanding

Answer each question, then compare with the solutions below.

If \(\tau\) increases, what happens to the prior variance of \(\delta_C-\delta_H\), and what does that mean for borrowing?
Why does the predictive variance used to update \(\tau\) include \(V_H\) as well as \(V_C\) and \(1/\tau\)?
In the uncertain-precision example, what posterior probability is assigned to \(\tau=16\), and why is it not the prior probability?
What is a key limitation of the usual reciprocal-cell variance formula when a binary trial has a zero event cell?

Solutions. (1) The difference variance is \(1/\tau\), so increasing \(\tau\) decreases it and makes the prior concentrate more tightly around the historical effect. (2) The historical effect is estimated rather than known; its posterior uncertainty \(V_H\) contributes to the predictive variability of the current estimate. (3) It is approximately \(0.750\), because the current estimate's predictive density under \(\tau=16\) is about three times that under \(\tau=0.25\), and the prior probabilities were equal. (4) A zero cell makes the log odds ratio or its estimated variance undefined, so a direct binomial model or another justified method is needed instead of applying the formula mechanically.