Shared Commensurability Across Endpoints

Why Share a Commensurability Parameter?

A clinical trial may measure several outcomes, such as symptom improvement and functional status. Each outcome can have its own treatment effect and its own historical reference. A model can preserve those endpoint-specific effects while allowing evidence from all endpoints to inform a common question: how tightly should current effects be centered on their historical references? This is shared commensurability. It does not assert that the treatment effects are equal, but it can make borrowing at one endpoint depend on data from another.

That cross-endpoint influence can be useful when there is a credible reason to expect similar degrees of current-versus-historical agreement across outcomes. It can be misleading when that reason is absent, or when dependence among outcomes is ignored. The key is to distinguish a shared borrowing-strength parameter from a shared treatment effect.

Model and Assumptions

Let $j=1,\ldots,J$ index endpoints. Let $Y_j$ be the current-trial estimate of the treatment effect for endpoint $j$, with known standard error $s_j$. Let $\theta_j$ be the current treatment effect, and let $\eta_j$ be its historical reference. The effects and references must use the same contrast direction and units within each endpoint. Treat the references as fixed in the model below; their uncertainty is not propagated.

Definition: A shared commensurate-precision model has endpoint-specific effects $\theta_j$, each centered on its own reference $\eta_j$, and a common positive precision $\lambda$: $$ Y_j\mid\theta_j\sim N(\theta_j,s_j^2),\qquad \theta_j\mid\lambda\sim N\left(\eta_j,\frac{1}{\lambda q_j}\right),\qquad \lambda\sim p(\lambda),\quad \lambda>0. $$ Here $q_j>0$ is a prespecified endpoint-specific scale factor. The model assumes that the sampling distributions are conditionally independent across endpoints given all $\theta_j$, and, separately, that the effects $\theta_j$ are mutually independent conditional on $\lambda$. The prior $p(\lambda)$ must be proper for the resulting posterior to be a probability distribution.

The shared $\lambda$ controls prior precision around each endpoint’s reference: a larger $\lambda$ means tighter commensurability, while a smaller $\lambda$ allows greater departure. The $q_j$ values accommodate endpoint-specific scales; they should be chosen so that the intended meaning of a common $\lambda$ is scientifically coherent. Because $\theta_j$ remains distinct for every endpoint, the model does not pool the treatment effects themselves.

The two independence assumptions have different jobs. Conditional independence of the sampling distributions concerns how the current estimates are generated given their true effects. Conditional independence of the effects given $\lambda$ is an additional prior assumption. Together they imply that the marginal current estimates are independent given $\lambda$, which is what permits a product of endpoint marginal likelihoods. Sampling independence alone is not enough.

What the Shared Parameter Learns

Integrating out $\theta_j$ gives a normal distribution for each current estimate. The result follows because $Y_j$ can be written as $\eta_j+(\theta_j-\eta_j)+(Y_j-\theta_j)$, a sum of independent normal deviations with variances $1/(\lambda q_j)$ and $s_j^2$. Thus, under the two conditional independence assumptions,

Y_j\mid\lambda\sim N\left(\eta_j,\ s_j^2+\frac{1}{\lambda q_j}\right), \qquad p(\lambda\mid Y_1,\ldots,Y_J) \propto p(\lambda)\prod_{j=1}^J \frac{1}{\sqrt{s_j^2+1/(\lambda q_j)}} \exp\left\{-\frac{(Y_j-\eta_j)^2}{2\left(s_j^2+1/(\lambda q_j)\right)}\right\}.

The normalizing constants common across values of $\lambda$ have been omitted. Each endpoint contributes a marginal likelihood that weighs both its deviation from the historical reference and its uncertainty. Consequently, an endpoint close to its reference can support larger $\lambda$, while an endpoint far from its reference can favor smaller $\lambda$. The posterior for $\lambda$ reflects their combined evidence, not a separate commensurability assessment for each endpoint.

Conditional on $\lambda$ and the observed estimate, the posterior for an endpoint effect is also normal. Completing the square in the product of the normal sampling density and normal prior gives

\theta_j\mid Y_j,\lambda\sim N\left(\eta_j+w_j(\lambda)(Y_j-\eta_j),\ \frac{s_j^2}{1+\lambda q_j s_j^2}\right), \qquad w_j(\lambda)=\frac{1}{1+\lambda q_j s_j^2}.

Thus, conditional on $\lambda$, the posterior mean moves from the historical reference toward the current estimate by fraction $w_j(\lambda)$. Greater $\lambda$ decreases that fraction. The final posterior for $\theta_j$ averages over uncertainty in $\lambda$, and so can be a mixture rather than a single normal distribution.

Worked Example: Endpoint-Specific Effects at a Fixed Shared Precision

Suppose two continuous outcomes have historical references $\eta_1=1.0$ mmHg and $\eta_2=0.0$ points, with current estimates $Y_1=1.5$ mmHg and $Y_2=-0.3$ points. Their standard errors are $s_1=0.4$ mmHg and $s_2=0.6$ points. For illustration, set $q_1=q_2=1$ and condition on $\lambda=2$. This calculation assumes the model’s normal sampling distributions and uses the fixed references as specified.

For endpoint 1, $w_1=1/(1+2(0.4^2))=1/1.32=0.7576$. The posterior mean is $1.0+0.7576(1.5-1.0)=1.379$ mmHg, and the posterior variance is $0.4^2/1.32=0.1212\;(\text{mmHg})^2$, giving a posterior standard deviation of $0.3482$ mmHg.

For endpoint 2, $w_2=1/(1+2(0.6^2))=1/1.72=0.5814$. Its posterior mean is $0+0.5814(-0.3-0)=-0.1744$ points. Its posterior variance is $0.6^2/1.72=0.2093\;\text{points}^2$, giving a posterior standard deviation of $0.4575$ points.

The effects remain endpoint-specific and the data have different units. The same $\lambda$ does not produce the same shrinkage weight: each weight also depends on $q_j$ and $s_j$. These are conditional calculations at a fixed $\lambda$, not posterior summaries after learning $\lambda$.

Cross-Endpoint Influence in a Shared Update

The following finite-prior example makes the influence visible without requiring numerical integration. Suppose $q_1=q_2=1$, both standard errors are $0.5$, and both historical references are zero. The observed estimates are $Y_1=0$ and $Y_2=0.8$. Assign prior probability $0.5$ to each of $\lambda=4$ and $\lambda=0.25$. The model assumes independent endpoint effects conditional on $\lambda$, as well as independent sampling errors.

Worked Example: An Agreeing Endpoint Changes Borrowing Elsewhere

After integrating out each endpoint effect, the variance of $Y_j$ is $V(\lambda)=0.5^2+1/\lambda$. Hence $V(4)=0.50$ and $V(0.25)=4.25$. For an observed difference $d=Y_j-\eta_j$, the ratio of its marginal density at $\lambda=4$ to that at $\lambda=0.25$ is

R(d)=\sqrt{\frac{4.25}{0.50}}\exp\left\{-\frac{d^2}{2}\left(\frac{1}{0.50}-\frac{1}{4.25}\right)\right\}.

For endpoint 1, $d=0$, so $R(0)=\sqrt{8.5}=2.915$. For endpoint 2, $d=0.8$, so $R(0.8)=2.915\exp\{-0.32(2-0.2353)\}=2.915\exp(-0.5647)=1.658$. The joint likelihood ratio is approximately $4.832$ (more precisely, $4.83247$). With equal prior odds, the posterior probability of $\lambda=4$ is $0.8285$.

Endpoint 1 alone would give posterior probability $2.915/(1+2.915)=0.7446$ for $\lambda=4$; endpoint 2 alone would give $1.658/(1+1.658)=0.6238$. Their combined data favor tighter commensurability more than either endpoint alone. In particular, the estimate at endpoint 1, which agrees with its reference, increases the posterior probability of a tight prior for endpoint 2, even though endpoint 2 has a substantial discrepancy.

For endpoint 2, the conditional posterior mean is $0+0.5(0.8)=0.400$ when $\lambda=4$, and $0+0.9412(0.8)=0.7529$ when $\lambda=0.25$. Averaging these conditional means using the posterior probabilities gives $0.8285(0.400)+0.1715(0.7529)=0.4605$ points. This posterior mean shows the consequence of sharing: endpoint 1’s agreement contributes to the degree of shrinkage at endpoint 2. Whether that consequence is appropriate depends on the scientific justification for a common commensurability parameter.

Dependence Changes the Marginal Likelihood

If endpoint estimates are correlated, the product likelihood above generally does not apply. Even when the endpoint effects are independent conditional on $\lambda$, correlated sampling errors make the marginal estimates dependent given $\lambda$. The correct likelihood must retain their joint covariance. Assuming independence in that setting can change the posterior for the shared parameter and, through it, every endpoint’s borrowing.

Worked Example: Correcting for Correlated Endpoint Estimates

Suppose two endpoint estimates have standard errors of 1 and sampling-error correlation $0.8$. Both historical references are zero, and both observed estimates are $0.5$. The effects are assumed independent conditional on $\lambda$, with $q_1=q_2=1$, and the prior again assigns equal probability to $\lambda=4$ and $\lambda=0.25$. The sampling covariance is $\begin{pmatrix}1&0.8\\0.8&1\end{pmatrix}$. After integrating out the effects, the covariance of the estimate vector is $\Sigma_\lambda=\begin{pmatrix}1+1/\lambda&0.8\\0.8&1+1/\lambda\end{pmatrix}$.

At $\lambda=0.25$, the diagonal entries are 5 and $\det(\Sigma_{0.25})=25-0.64=24.36$. At $\lambda=4$, they are 1.25 and $\det(\Sigma_4)=1.5625-0.64=0.9225$. For the observed vector $d=(0.5,0.5)$, the quadratic form is $d^\mathsf{T}\Sigma_\lambda^{-1}d=0.5/(a+0.8)$, where $a$ is the diagonal entry. It is $0.5/5.8=0.08621$ at $\lambda=0.25$ and $0.5/2.05=0.2439$ at $\lambda=4$.

The correct joint density ratio is therefore $\sqrt{24.36/0.9225}\exp\{-0.5(0.2439-0.08621)\}=4.75$, approximately. Equal prior odds give posterior probability $4.75/(1+4.75)=0.826$ for $\lambda=4$. If one incorrectly set the covariance to zero and multiplied independent marginal densities, the ratio would instead be $4\exp\{-0.5(0.4-0.1)\}=3.443$, giving posterior probability $3.443/(1+3.443)=0.775$. The difference is not rounding: the two calculations use different likelihood assumptions. A joint model, not a product of marginal contributions, is required here.

When Is Shared Borrowing Defensible?

A shared parameter is most credible when the endpoints have a substantive reason to exhibit a similar degree of alignment between historical and current treatment effects. That rationale should be considered alongside scale, measurement, and trial-design differences. A common $\lambda$ is not automatically meaningful just because outcomes were measured in the same trial. Endpoint-specific $q_j$ values can address scale differences, but they cannot repair a mistaken scientific assumption that the endpoints should share their degree of historical agreement.

Useful diagnostics and sensitivity checks include:

Inspect endpoint discrepancies and uncertainty. Compare $Y_j-\eta_j$ with its sampling error for each endpoint. A large discrepancy at one endpoint alongside close agreement at others signals a potential cross-endpoint conflict, not proof that any endpoint is invalid.
Check dependence. Review the design and endpoint construction for shared measurements, repeated observations, or other sources of correlated estimates. If dependence is material, use a joint sampling model with its covariance rather than a product likelihood.
Vary the prior and scale choices. Examine whether posterior borrowing changes substantially under plausible choices of $p(\lambda)$ and $q_j$. The interpretation of $\lambda$ depends on these specifications.
Compare with less shared alternatives. Separate endpoint-specific commensurability parameters avoid direct sharing through a single $\lambda$, but may be weakly informed with sparse data. No borrowing avoids that cross-endpoint pathway but discards potentially relevant information. A shared parameter is a modeling choice between these approaches, not a universal default.

The model also treats each historical reference as fixed. If its uncertainty matters, it should be represented in a model that propagates that uncertainty; otherwise the analysis can overstate how precisely the historical target is known. Likewise, a shared parameter does not by itself account for all forms of endpoint dependence, measurement incompatibility, or changes between historical and current populations.

Common Mistake

Common mistake: “The current endpoint estimates are conditionally independent given their treatment effects, so their marginal likelihoods multiply given $\lambda$.” This conclusion does not follow: the endpoint effects could still be dependent given $\lambda$, inducing dependence among the marginal estimates. The product is justified only if the sampling distributions are conditionally independent and the endpoint effects are also mutually independent conditional on $\lambda$, as specified above. Otherwise, specify a joint distribution and use its joint marginal likelihood.

A second interpretive pitfall is to describe shared commensurability as forcing equal treatment effects. It does not: each endpoint has a distinct $\theta_j$ and a distinct reference $\eta_j$. The shared assumption is that one parameter governs the precision of all endpoint-specific deviations. That assumption can still create substantial cross-endpoint influence, so posterior results should be read in light of it.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

What two conditional independence assumptions justify multiplying the endpoint marginal likelihood contributions for $\lambda$?
With $q_j=1$, $s_j=0.5$, and $\lambda=2$, calculate the conditional shrinkage weight $w_j(\lambda)$.
Why can an endpoint that agrees with its historical reference change the posterior treatment effect at a different endpoint?
If two endpoint estimates have correlated sampling errors, what should replace the product of separate marginal likelihoods?

Solutions. (1) The sampling distributions must be conditionally independent given all endpoint effects, and the endpoint effects must be mutually independent conditional on $\lambda$. (2) $w_j=1/(1+2(1)(0.5^2))=1/1.5=0.6667$. (3) Its agreement can alter the posterior distribution of the shared $\lambda$; that distribution determines the amount of shrinkage at every endpoint. (4) Use the joint marginal likelihood derived from the joint sampling distribution and the specified joint prior for the endpoint effects.