Multiple Historical Sources with Commensurate Priors

Why Give Historical Sources Separate Links?

Clinical trials may have several historical estimates of a treatment effect: for example, estimates from distinct trials, populations, or periods. Treating all these sources as if they were equally relevant can conceal important differences. A commensurate model instead gives each source its own link to the current treatment effect. A source can then contribute substantially when its estimate is precise and its link permits close agreement, while another source contributes less.

The key is that these contributions are determined by both the uncertainty in a source’s estimate and the precision of its commensurability link. The resulting posterior is straightforward when these quantities are fixed. When link precisions are uncertain, the conditional normal posteriors are averaged over their posterior distribution; the result need not itself be normal.

Model and Assumptions

Suppose the current trial provides an approximately normal estimate \(y\) of a treatment effect \(\theta\), with known sampling variance \(s^2\). For each of \(J\) historical sources, let \(h_j\) be its estimated effect and \(r_j^2\) its known sampling variance. Effects are measured on the same scale and in the same direction across sources. For example, a log hazard ratio is dimensionless, and negative values indicate a lower hazard in the treatment group under the stated coding.

Definition: A normal multiple-source commensurate model has current likelihood \(y\mid\theta\sim N(\theta,s^2)\), historical likelihoods \(h_j\mid\theta_j\sim N(\theta_j,r_j^2)\), and separate links \(\theta\mid\theta_j,\omega_j\sim N(\theta_j,1/\omega_j)\), for \(j=1,\ldots,J\). Here \(\theta_j\) is the effect in historical source \(j\), and \(\omega_j>0\) is that source’s commensurability precision. Larger \(\omega_j\) means a tighter link; smaller \(\omega_j\) allows greater differences.

For the formulas below, assume the current estimate and historical estimates are conditionally independent given their respective parameters, and that the source-specific parameters and link deviations are independent across sources. Use a diffuse (flat) baseline prior for \(\theta\) and diffuse priors for the \(\theta_j\). The current likelihood makes the posterior for \(\theta\) proper. This setup treats the reported standard errors as known and uses normal approximations for the estimates; it does not account for uncertainty in those standard errors.

A source’s historical estimate is not treated as the true effect without uncertainty. Integrating over its unknown \(\theta_j\) gives an effective variance that includes both sampling uncertainty and the link variance:

q_j=r_j^2+\frac{1}{\omega_j}, \qquad \lambda_j=\frac{1}{q_j}.

Thus, \(\lambda_j\) is the effective precision contributed by source \(j\). It is not generally equal to either the precision of the historical estimate alone or the link precision alone.

Deriving the Combined Posterior

The integration that yields \(q_j\) can be seen directly from the model. Conditional on \(\theta_j\), the historical estimate has variance \(r_j^2\), while the current effect differs from \(\theta_j\) with variance \(1/\omega_j\). With a diffuse prior on \(\theta_j\), integrating that parameter produces a normal contribution centered at \(h_j\), with the two variances added.

\begin{aligned} \int N(h_j\mid\theta_j,r_j^2)\, N(\theta\mid\theta_j,1/\omega_j)\,d\theta_j &\ \propto\ N\!\left(h_j\mid\theta,\ r_j^2+1/\omega_j\right). \end{aligned}

The proportionality omits factors that do not depend on \(\theta\). The variance sum follows because the difference between \(h_j\) and \(\theta\) contains the historical sampling error and the link deviation. Since sources are independent under the model, their contributions multiply. Combining those factors with the current normal likelihood gives a normal posterior for \(\theta\), conditional on fixed link precisions.

\begin{aligned} V_{\mathrm{post}} &=\left(\frac{1}{s^2}+\sum_{j=1}^{J}\lambda_j\right)^{-1},\\ m_{\mathrm{post}} &=V_{\mathrm{post}}\left(\frac{y}{s^2} +\sum_{j=1}^{J}\lambda_j h_j\right),\\ \theta\mid y,h_1,\ldots,h_J,\omega_1,\ldots,\omega_J &\sim N(m_{\mathrm{post}},V_{\mathrm{post}}). \end{aligned}

The expression for \(m_{\mathrm{post}}\) is a precision-weighted average. The current estimate has precision \(1/s^2\), and source \(j\) has effective precision \(\lambda_j\). Dividing each precision by the total precision gives its weight in the posterior mean. Each weight is nonnegative, and all the weights sum to one. The posterior precision is the sum of the current and effective historical precisions.

For example, this result follows by collecting the quadratic terms in the log posterior. The current likelihood contributes \(-(y-\theta)^2/(2s^2)\), and historical source \(j\) contributes \(-(h_j-\theta)^2/(2q_j)\). The coefficient of \(-\theta^2/2\) is \(1/s^2+\sum_j1/q_j\); completing the square therefore gives the stated variance and mean.

Worked Example: Two Historical Log-Hazard Estimates

Consider a hypothetical trial with current log hazard ratio estimate \(y=0.40\) and sampling variance \(s^2=0.2525\), giving current precision \(1/s^2\approx3.9604\). Two hypothetical historical sources have estimates \(h_1=-0.20\) and \(h_2=-0.50\), each with sampling variance \(r_j^2=0.01\). Suppose the fixed link variances are \(1/\omega_1=0.03\) and \(1/\omega_2=0.01\). These values assume independent normal estimates on the same log-hazard-ratio scale and fixed positive link precisions.

The effective variances and precisions are \(q_1=0.01+0.03=0.04\), \(\lambda_1=1/0.04=25\), and \(q_2=0.01+0.01=0.02\), \(\lambda_2=1/0.02=50\). The posterior precision is \(3.9604+25+50=78.9604\). The posterior mean is

m_{\mathrm{post}} =\frac{3.9604(0.40)+25(-0.20)+50(-0.50)} {3.9604+25+50} =\frac{1.58416-5.0000-25.0000}{78.9604} \approx -0.3599.

The posterior variance is \(1/78.9604\approx0.0126646\), and the posterior standard deviation is approximately \(0.1125\) log-hazard-ratio units. A conditional 95% normal credible interval is approximately \(-0.3599\pm1.96(0.1125)\), or \((-0.5804,-0.1393)\). Under this model and the fixed precisions, the posterior probability that the current log hazard ratio lies in this interval is 0.95. The interval is on the log scale; it does not assert a 95% frequentist coverage probability for this particular realized interval.

The source weights are \(25/78.9604\approx0.3166\) and \(50/78.9604\approx0.6332\), while the current-data weight is \(3.9604/78.9604\approx0.05016\). The second source contributes more because its effective precision is larger. These calculations describe the specified model; they do not establish that either historical source is clinically appropriate.

Different Sources Can Have Different Contributions

The source weights make the role of each link explicit. A source with a precise historical estimate can still have limited influence if its link is loose. Conversely, a tight link can yield substantial influence when the historical estimate is sufficiently precise. The following example separates those effects by assigning different effective precisions to two hypothetical sources.

Worked Example: Unequal Borrowing From Two Sources

Suppose a hypothetical current treatment-effect estimate is \(y=0.10\) with standard error \(s=0.25\), so its variance is \(0.0625\) and precision is \(16\). Historical source 1 has \(h_1=0.20\), sampling variance \(r_1^2=0.01\), and link variance \(1/\omega_1=0.03\). Source 2 has \(h_2=-0.40\), sampling variance \(r_2^2=0.16\), and link variance \(1/\omega_2=0.09\). Assume independent normal estimates and fixed positive link precisions, as above.

For source 1, \(q_1=0.01+0.03=0.04\), so \(\lambda_1=25\). For source 2, \(q_2=0.16+0.09=0.25\), so \(\lambda_2=4\). The total posterior precision is \(16+25+4=45\), and

m_{\mathrm{post}} =\frac{16(0.10)+25(0.20)+4(-0.40)}{45} =\frac{1.6+5.0-1.6}{45} =\frac{5.0}{45} \approx0.1111.

The posterior variance is \(1/45\approx0.02222\), giving standard deviation \(0.1491\). A conditional 95% normal credible interval is approximately \(0.1111\pm1.96(0.1491)=(-0.1811,0.4033)\), in treatment-effect units. The normalized weights are \(16/45\approx0.3556\) for current data, \(25/45\approx0.5556\) for source 1, and \(4/45\approx0.08889\) for source 2. Source 2 pulls toward its negative estimate, but its larger effective variance limits that pull.

In this normal model with fixed variances, the posterior mean must lie between the smallest and largest of the estimates receiving positive weight. If it does not, check the signs, precision calculations, and denominator. The posterior variance must also be smaller than the current variance when at least one historical precision is positive. These are useful arithmetic checks, not tests of whether the modeling assumptions are scientifically reasonable.

Precision Can Dominate a Source’s Direction

A historical estimate’s direction alone does not determine its influence. To see this, compare two sources whose estimates are relatively close to zero but whose effective variances differ. The source with the smaller effective variance can dominate the posterior even when its estimate is not the closest to the current estimate.

Worked Example: A Highly Precise Link

Suppose a hypothetical current estimate is \(y=0.00\) with variance \(s^2=0.16\), so its precision is \(6.25\). Source 1 has \(h_1=0.30\), sampling variance \(0.04\), and link variance \(0.05\). Source 2 has \(h_2=-0.10\), sampling variance \(0.005\), and link variance \(0.005\). Treat all variances as known and positive, and assume the same independent normal model.

The effective variances are \(q_1=0.04+0.05=0.09\) and \(q_2=0.005+0.005=0.010\), so the effective precisions are approximately \(11.1111\) and \(100\). Thus

m_{\mathrm{post}} =\frac{6.25(0)+11.1111(0.30)+100(-0.10)} {6.25+11.1111+100} =\frac{3.3333-10.0000}{117.3611} \approx-0.0568.

The posterior variance is approximately \(1/117.3611=0.008521\), and the posterior standard deviation is approximately \(0.0923\). The weights are approximately \(0.05325\) for current data, \(0.09467\) for source 1, and \(0.8521\) for source 2. Although source 2 is only one input among three estimates, its high effective precision makes it the principal contributor to the posterior mean. This is a consequence of the stated variances and links, not a general endorsement of the source.

Fixed and Uncertain Link Precisions

The closed-form expressions above are conditional on fixed \(\omega_j\). In a fully Bayesian analysis, one may instead assign prior distributions to those precisions and update them using the data. For each value of the precision vector, the conditional posterior for \(\theta\) has the normal form already derived. The overall posterior averages those conditional distributions over the posterior distribution of the precisions:

p(\theta\mid y,h_1,\ldots,h_J) =\int p(\theta\mid y,h_1,\ldots,h_J,\omega_1,\ldots,\omega_J)\, p(\omega_1,\ldots,\omega_J\mid y,h_1,\ldots,h_J)\, d\omega_1\cdots d\omega_J.

This averaging generally produces a mixture of normal distributions, not one normal distribution with a single precision-weighted mean. Reporting the conditional formula as if the precisions were known would ignore uncertainty in the links. Conversely, a calculation with fixed precisions is valid as a conditional analysis when those values are deliberately held fixed.

A common simplification is to pool historical estimates first and then apply one commensurability link to the pooled result. That can be reasonable under additional assumptions, but it removes the ability to let sources have different link variances. It can also obscure differences in their sampling precision. Separate links preserve the source-specific contributions, at the cost of more parameters and stronger requirements for specifying or estimating them.

Assumptions, Pitfalls, and Limitations

Common scale and estimand. Historical and current estimates must refer to comparable treatment effects, use consistent coding, and be expressed on the same scale. A normal calculation cannot repair a mismatch in population or estimand.
Approximate normality and known variances. The formulas assume normal summary-data likelihoods with known sampling variances. Strong skewness, sparse data, or poorly estimated standard errors may make this approximation inadequate.
Conditional independence. The simple sum of source precisions assumes independent source information conditional on the parameters. Overlapping patients, reused controls, or correlated estimates require a model that accounts for their covariance.
Link specification. A small link variance means strong commensurability and can produce substantial borrowing. A large one weakens that source’s contribution. The model does not make a scientifically unjustified link appropriate merely by expressing it probabilistically.
Common pitfall: using link precision as the source weight. It is wrong to say that a source with \(\omega_j=50\) contributes precision 50. Its effective precision is \(1/(r_j^2+1/\omega_j)\), which incorporates historical sampling uncertainty as well as link uncertainty.
Common pitfall: interpreting a narrow interval as evidence of agreement. A high total precision can produce a narrow posterior interval even when sources point in different directions. The interval describes uncertainty under the specified model; it does not by itself validate the assumptions or the sources.

Compared with a single pooled historical estimate, separate commensurability links retain source-specific uncertainty and allow unequal borrowing. Compared with analyzing only the current data, they can increase precision, but that gain depends on the historical likelihoods and link assumptions. These normal formulas are also not a substitute for endpoint-specific likelihoods when the available data are binary counts or event-time records rather than approximately normal effect estimates.

Key takeaway: For each historical source, add its sampling variance and link variance to obtain its effective variance. In the fixed-precision normal model, the reciprocal of that sum is the source’s contribution to posterior precision; contributions from distinct sources are calculated separately and then combined with the current likelihood.

Check Your Understanding

Answer each question using the fixed-precision normal model. Solutions are included below for checking your work.

A source has historical variance \(0.04\) and link variance \(0.06\). What is its effective precision?
Why can a source with a tight commensurability link still have a modest effective precision?
Current data have estimate \(0\) and precision \(10\). One historical source has estimate \(0.5\) and effective precision \(10\). Find the posterior mean and variance under a flat baseline prior.
What assumption is violated if two historical estimates share the same control patients but are treated as independent?
If the commensurability precisions are uncertain, is the marginal posterior for the current effect necessarily normal? Explain.

Solutions. (1) The effective variance is \(0.04+0.06=0.10\), so the effective precision is \(1/0.10=10\). (2) Because the effective variance also includes historical sampling variance; even a small link variance cannot remove uncertainty in the historical estimate. (3) The posterior precision is \(10+10=20\), so the variance is \(1/20=0.05\). The mean is \(\{10(0)+10(0.5)\}/20=0.25\). (4) Conditional independence is violated: the estimates are correlated through their shared controls, so their precisions cannot simply be added as if independent. (5) No. It is generally a mixture of the conditional normal posteriors over the posterior distribution of the commensurability precisions, and a mixture of normals need not be normal.