Linking Historical and Current Treatment Effects
Historical clinical-trial data can inform a current trial, but only to the extent that the historical and current treatment effects are sufficiently similar for the question at hand. A commensurate prior expresses that relationship directly: it treats the current effect as centered on a historical effect, while allowing their difference to vary. The strength of that link is controlled by a precision parameter.
This construction is useful when complete pooling seems too strong, but treating historical evidence as wholly unrelated to the current trial would discard potentially relevant information. The central modeling choice is not simply whether to borrow. It is how tightly the current effect should be linked to the historical one, on the effect scale being analyzed.
The precision is measured in the reciprocal squared units of the treatment effect. If \(\theta_C\) is a log odds ratio, \(\tau\) is in inverse squared log-odds-ratio units. If the effect is a risk difference in percentage points, the units are inverse squared percentage points. The numerical value of \(\tau\) cannot be interpreted without its scale.
A full borrowing model also needs a distribution for the historical effect given the historical data. Write this as \(p(\theta_H\mid D_H)\), where \(D_H\) denotes historical data. The current data \(D_C\) contribute a likelihood \(p(D_C\mid\theta_C)\). The resulting structure is
Here \(p(\theta_H\mid D_H)\) carries historical uncertainty forward, while \(p(\theta_C\mid\theta_H,\tau)\) describes how far the current effect is allowed to differ from the historical one. In an analysis using this structure, the historical data inform \(\theta_H\), and the current data inform \(\theta_C\); the data sources are modeled as conditionally independent given their respective effects. A coherent historical model and a clear account of how historical evidence enters the analysis are essential to avoid inadvertently counting the same data twice.
What the Precision Controls
Conditional on a fixed historical effect, the prior standard deviation of the difference is \(1/\sqrt{\tau}\). Thus a high precision concentrates prior mass near \(\theta_C=\theta_H\), indicating strong borrowing, while a low precision spreads mass over larger discrepancies, indicating weak borrowing. For any tolerance \(d>0\), the model implies
where \(\Phi\) is the standard normal cumulative distribution function. This probability is a prior statement about the difference, not a probability that the historical and current effects are identical. With finite \(\tau\), exact equality has probability zero under the continuous normal distribution.
The limiting cases help clarify the interpretation. As \(\tau\) becomes very large, the conditional distribution concentrates increasingly tightly around the historical effect, approaching the shared-effect assumption as a limit. As \(\tau\) becomes very small, the conditional distribution becomes increasingly diffuse. Setting \(\tau=0\) does not give a proper normal distribution: its formal variance would be infinite. In practice, a weak-link model needs a finite positive precision or another explicitly specified proper distribution.
Worked Example: Interpreting Precision for a Log Odds Ratio
Suppose the historical treatment effect is a log odds ratio of \(\theta_H=0.30\), and the current effect is given a commensurate prior with fixed precision \(\tau=25\) per squared log-odds-ratio unit. Interpret the implied prior difference.
By definition, $$ \theta_C-\theta_H\mid\theta_H,\tau \sim N\!\left(0,\frac{1}{25}\right) = N(0,0.04). $$ The prior standard deviation of the difference is \(\sqrt{0.04}=0.20\) log-odds-ratio units. A central 95% prior interval for the difference is $$ 0\pm 1.96(0.20)=(-0.392,\,0.392). $$
Conditional on the stated historical effect, the corresponding 95% prior interval for the current log odds ratio is \(0.30+(-0.392,0.392)=(-0.092,\,0.692)\). On the odds-ratio scale, the current-to-historical odds-ratio ratio is \(\exp(\theta_C-\theta_H)\); exponentiating the difference limits gives approximately \((0.676,\,1.480)\). This is a prior range for the ratio of effects, not an interval estimated from current-trial outcomes. The example illustrates a moderately tight link on the log scale; whether that is clinically acceptable depends on the application.
Worked Example: Comparing Two Borrowing Strengths
Suppose effects are measured as risk differences in percentage points and the historical effect is 8 percentage points. Compare \(\tau=0.04\) and \(\tau=1\), with precision units of inverse squared percentage points.
For \(\tau=0.04\), the discrepancy has standard deviation $$ \frac{1}{\sqrt{0.04}}=5 $$ percentage points. Its central 95% prior interval is \(0\pm1.96(5)=(-9.80,9.80)\) percentage points. Adding the historical effect gives a conditional prior interval for the current effect of \((-1.80,17.80)\) percentage points.
For \(\tau=1\), the discrepancy standard deviation is \(1\) percentage point, giving a central 95% prior interval of \((-1.96,1.96)\) percentage points and a current-effect interval of \((6.04,9.96)\) percentage points. The second prior concentrates much more strongly near the historical effect, so it implies stronger borrowing.
For another comparison, the prior probability that the absolute discrepancy exceeds 5 percentage points under \(\tau=0.04\) is $$ P(|\theta_C-\theta_H|>5\mid\tau=0.04) = 2\Phi(-5\sqrt{0.04}) = 2\Phi(-1) \approx 0.317. $$ Under \(\tau=1\), it is \(2\Phi(-5)\approx 5.73\times10^{-7}\). The precision value therefore has a direct interpretation only in combination with a discrepancy threshold and the effect scale.
Historical Uncertainty Is Not the Same as Commensurability
The conditional prior describes uncertainty about \(\theta_C-\theta_H\) once \(\theta_H\) is specified. But the historical effect is rarely known exactly. Its uncertainty must also be represented, rather than replacing \(\theta_H\) with a single historical estimate as though that estimate were error-free.
A simple normal case makes the distinction explicit. If the historical posterior is normal and the conditional commensurate prior is normal, then the marginal prior for the current effect is also normal: its mean is the historical posterior mean, while its variance includes both historical uncertainty and uncertainty about the difference. This result follows by writing the current effect as the sum of the historical effect and a mean-zero discrepancy.
Worked Example: Carrying Historical Uncertainty Forward
Suppose a historical analysis gives \(\theta_H\mid D_H\sim N(0.20,0.01)\), where the effect is a risk difference on the proportion scale. Set \(\tau=100\) per squared proportion unit. Find the resulting marginal prior distribution for \(\theta_C\) before observing current-trial data.
The commensurate model gives \(\theta_C=\theta_H+\delta\), where, conditional on \(\theta_H\), \(\delta\sim N(0,1/100)\). Under this construction, the discrepancy is independent of the historical effect. Therefore the mean is \(0.20+0=0.20\), and the variance is the sum of the two independent variances: $$ \operatorname{Var}(\theta_C\mid D_H,\tau) = 0.01+\frac{1}{100} = 0.02. $$ Thus \(\theta_C\mid D_H,\tau\sim N(0.20,0.02)\), with standard deviation \(\sqrt{0.02}\approx0.1414\). Its central 95% prior interval is $$ 0.20\pm1.96(0.1414)\approx(-0.0772,\,0.477). $$
The conditional standard deviation of the difference is \(0.10\), but the marginal standard deviation of the current effect is larger because it also includes uncertainty in the historical effect. Treating the historical posterior mean as known would omit the historical variance \(0.01\) and understate the uncertainty in the current-effect prior.
Assumptions, Diagnostics, and Alternatives
The model is most defensible when the historical and current parameters represent the same treatment-effect estimand on the same scale. That requires attention to the population, comparator, endpoint definition, follow-up, analysis conventions, and relevant features of trial conduct. If these differ in ways that change the treatment effect, a narrow prior centered on \(\theta_H\) may be misleading even if the numerical effect estimates look similar.
The normal discrepancy assumption is also substantive: it assigns a symmetric distribution to deviations in either direction and gives nonzero prior probability to arbitrarily large deviations. It may be a useful approximation on an approximately unbounded scale, but it can be unsuitable when the effect is tightly bounded or when asymmetry is clinically important. The conditional prior does not itself correct bias in the historical estimate or adjust for changes in care.
Before fitting a model, examine the implied discrepancy distribution on the clinical scale. Useful checks include calculating prior probabilities for prespecified meaningful differences, plotting the current-effect prior predictive distribution, and checking whether it assigns substantial mass to implausible effects. A prior predictive check here asks what values the model considers plausible before seeing current data; it is not evidence that historical and current effects are compatible. If the implied distribution is too concentrated, it can force strong borrowing. If it is too diffuse, the historical data may contribute little useful information.
A commensurate prior differs from a shared-effect model, which imposes equality rather than allowing a variable discrepancy. It also differs from a robust mixture prior: a mixture represents borrowing through alternative prior components and their weights, whereas the commensurate prior represents it through a distribution for the difference between effects. Neither construction automatically guarantees protection from conflict; assumptions and prior implications still require scrutiny.
| Approach | How historical information enters | Key implication |
|---|---|---|
| Shared-effect model | Uses one common treatment effect | Assumes the historical and current effects are equal |
| Commensurate prior | Centers the current effect on the historical effect with variance \(1/\tau\) | Allows differences, with their scale controlled by precision |
| Robust mixture prior | Combines borrowing and protective prior components | Represents alternative borrowing regimes through mixture weights |
Common Mistakes and Practical Limits
- Reading high precision as high uncertainty. Precision is the reciprocal of variance. Larger \(\tau\) means smaller discrepancy variance and a tighter link.
- Ignoring units. The same numerical precision cannot be carried unchanged from proportions to percentage points. A change of scale requires the precision to change accordingly.
- Plugging in a historical point estimate without its uncertainty. This treats the historical effect as known and can make the current prior artificially narrow.
- Calling a tight prior proof of exchangeability. The prior encodes an assumption about the effect difference; it does not establish that trials are exchangeable or that their clinical settings are comparable.
The commensurate construction is a clear way to express graded borrowing, but a fixed precision is a modeling choice, not a data-driven diagnosis of comparability. Its usefulness depends on whether the historical and current effects are meaningfully aligned and whether the implied range of discrepancies is acceptable for the clinical decision.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(\tau=16\), what is the conditional standard deviation of \(\theta_C-\theta_H\)?
- A prior uses a risk difference in proportions. What variance and precision correspond to a discrepancy standard deviation of 0.02?
- Suppose the historical posterior variance is 0.04 and the commensurate discrepancy variance is 0.09. What is the marginal prior variance for the current effect under the normal construction?
- Why does a large precision not establish that the historical and current trials are clinically comparable?
Solutions. 1. The standard deviation is \(1/\sqrt{16}=0.25\) effect units. 2. The variance is \(0.02^2=0.0004\) proportion-squared units, so the precision is \(1/0.0004=2{,}500\) inverse proportion-squared units. 3. The variances add, giving \(0.04+0.09=0.13\). 4. A large precision makes the model concentrate the current effect near the historical effect; it is an assumption encoded in the prior. It does not demonstrate that differences in populations, comparators, endpoints, or trial conduct are negligible.