Combining Multiple Historical Control Studies

Why Keep Historical Studies Separate?

When several historical control studies are available, their observations can inform a common control parameter. A power prior makes that borrowing explicit by raising each study’s likelihood contribution to a discount power. Keeping the likelihoods separate preserves a useful record of which data contribute what, and how strongly each study contributes.

The separate contributions matter even when all studies measure the same outcome. Studies may differ in design, follow-up, population, or relevance to the current trial. A discount records how much of a study’s likelihood is used in the prior for the current parameter; it does not alter the observed data or establish that the study is comparable.

Definition: Let $D_i$ denote the data from historical control study $i$, and let $L_i(\theta)=p(D_i\mid\theta)$ be that study’s likelihood for a shared control parameter $\theta$. With baseline prior density $\pi_0(\theta)$ and fixed study discount $a_i\in[0,1]$, the combined power-prior kernel is $$ \pi(\theta\mid D_1,\ldots,D_m,\mathbf a)\ \propto\ \pi_0(\theta)\prod_{i=1}^{m}L_i(\theta)^{a_i}. $$ Each $a_i$ controls the strength of study $i$’s likelihood contribution: $a_i=0$ removes that contribution, while $a_i=1$ uses its full likelihood. Values between zero and one use a fractional likelihood contribution.

For a conditional power prior with fixed discounts, proportionality means that the expression is normalized over $\theta$. Its normalizing constant is needed to define a probability density, but it does not change the relative contributions in the kernel. The displayed product also presumes the studies contribute distinct, conditionally independent data given $\theta$; it is not appropriate to multiply likelihoods that count the same participants twice.

How the Separate Contributions Combine

The product form follows from the likelihood factorization. If the historical data sets are conditionally independent given the shared parameter, their joint likelihood is $\prod_i L_i(\theta)$. Applying a separate power to each factor gives $\prod_i L_i(\theta)^{a_i}$. Taking logs turns the product into a sum:

\log\left\{\prod_{i=1}^{m}L_i(\theta)^{a_i}\right\} =\sum_{i=1}^{m}a_i\log L_i(\theta).

Thus a study contributes its own log likelihood, multiplied by its own discount. This is the key bookkeeping rule: apply a discount to the corresponding study’s contribution before adding the contributions. If every study has the same discount $a$, this reduces to $\{\prod_i L_i(\theta)\}^{a}$. With different discounts, the product generally cannot be represented by raising the unweighted pooled likelihood to a single common power.

For binary control outcomes, suppose study $i$ observes $y_i$ events among $n_i$ participants, with event probability $\theta$. Under a binomial model, $L_i(\theta)$ is proportional to $\theta^{y_i}(1-\theta)^{n_i-y_i}$. The binomial coefficient does not depend on $\theta$, so it can be omitted from the kernel when the discounts are fixed. The combined likelihood kernel is then:

\prod_{i=1}^{m}L_i(\theta)^{a_i} \ \propto\ \theta^{\sum_i a_i y_i}(1-\theta)^{\sum_i a_i(n_i-y_i)}.

Worked Example: Writing Three Study Contributions Separately

Suppose three hypothetical historical control studies report event counts of 3/20, 7/30, and 5/25. For illustration, assume all three estimate the same event probability $\theta$, and use fixed discounts $a_1=1$, $a_2=0.5$, and $a_3=0$. Write the combined discounted likelihood kernel.

The study likelihoods, omitting binomial coefficients that are constant in $\theta$, are $L_1(\theta)\propto\theta^3(1-\theta)^{17}$, $L_2(\theta)\propto\theta^7(1-\theta)^{23}$, and $L_3(\theta)\propto\theta^5(1-\theta)^{20}$. Apply the discount to each likelihood separately:

L_1(\theta)^1L_2(\theta)^{0.5}L_3(\theta)^0 \propto \theta^{3+3.5}(1-\theta)^{17+11.5} = \theta^{6.5}(1-\theta)^{28.5}.

Study 1 contributes 3 event and 17 non-event likelihood units; study 2 contributes 3.5 event and 11.5 non-event units; study 3 contributes nothing because its discount is zero. These are fractional likelihood exponents, not claims that the studies observed fractional participants. The assumed shared parameter and independence are modeling conditions here, not conclusions proved by the counts.

A Two-Study Binary Example

In the next example, the data are hypothetical. The binomial sampling model assumes independent outcomes within each study, a common event probability for participants in that study, and a shared $\theta$ across the two historical studies for purposes of borrowing. That shared-parameter assumption is substantive: it says the studies’ event probabilities are represented by the same parameter in this model. We also assume the studies do not overlap in participants.

Worked Example: Combining Two Discounted Control Likelihoods

Study 1 observes 8 events among 40 controls, and study 2 observes 18 among 60 controls. Use discounts $a_1=0.50$ and $a_2=0.25$. Find the combined likelihood kernel, then show its posterior kernel under a Beta(2,2) baseline prior.

The event and non-event counts are 8 and 32 in study 1, and 18 and 42 in study 2. Omitting constants that do not depend on $\theta$, the discounted likelihood contributions are:

L_1(\theta)^{0.50}\propto\theta^{0.50(8)}(1-\theta)^{0.50(32)} =\theta^4(1-\theta)^{16},

L_2(\theta)^{0.25}\propto\theta^{0.25(18)}(1-\theta)^{0.25(42)} =\theta^{4.5}(1-\theta)^{10.5}.

Multiplying the contributions adds their exponents: $4+4.5=8.5$ for events and $16+10.5=26.5$ for non-events. Therefore the combined discounted likelihood kernel is $\theta^{8.5}(1-\theta)^{26.5}$.

A Beta(2,2) baseline prior has kernel $\theta^{2-1}(1-\theta)^{2-1}=\theta(1-\theta)$. Multiplying by the historical kernel gives $\theta^{9.5}(1-\theta)^{27.5}$, which is the kernel of a Beta(10.5,28.5) distribution. This is the conditional posterior for $\theta$ if these historical likelihood contributions are used with that baseline prior and there are no additional current-trial observations in the update. The posterior mean is $10.5/(10.5+28.5)=10.5/39=0.2692$. The calculation combines fractional likelihood exponents; it does not turn the historical sample sizes into literal fractional enrolment counts.

Interpreting Contributions and Checking the Model

The discount $a_i$ is a power on study $i$’s likelihood, not a probability that the study is valid and not an estimate of its event rate. A discount near one lets that study’s likelihood have nearly its full influence in the prior kernel; a discount near zero makes its influence small. The influence on a posterior summary depends on the baseline prior and on all the other contributions as well, so the discount alone does not specify a posterior effect size.

Before multiplying contributions, check that the studies have been represented in a way that makes a shared parameter scientifically defensible. In a binary example, this includes confirming that the event definition and assessment window match, that denominators and missing outcomes are handled consistently, and that the target control population is relevant. Conditional independence also requires avoiding duplicated participants or data derived from overlapping cohorts. If event risks genuinely differ across studies, a common $\theta$ may be a poor representation, even if the likelihood arithmetic is correct.

A practical diagnostic is to inspect study-level event rates and likelihood contributions before forming the product. Marked differences in observed rates, outcome ascertainment, or study design can signal that the shared-parameter assumption deserves scrutiny. A large discrepancy is not, by itself, proof that the model is false: sampling variation can produce different observed rates. But a combined analysis should not hide such discrepancies behind a single pooled exponent.

Worked Example: Comparing Discounted Likelihood Support

Use the two studies from the preceding example and their discounts. Compare their combined discounted likelihood at $\theta=0.25$ with its value at $\theta=0.30$. State what the ratio means. This calculation uses the binomial likelihood contributions including the discounts; constants cancel in the ratio.

At $\theta=0.25$, study 1 contributes $0.50[8\log(0.25)+32\log(0.75)]=-10.1481$, and study 2 contributes $0.25[18\log(0.25)+42\log(0.75)]=-9.2590$. Their total discounted log likelihood, excluding constants, is $-19.4071$.

At $\theta=0.30$, study 1 contributes $0.50[8\log(0.30)+32\log(0.70)]=-10.5227$, and study 2 contributes $0.25[18\log(0.30)+42\log(0.70)]=-9.1630$. Their total is $-19.6857$.

Using unrounded contributions, the log-likelihood difference at 0.25 versus 0.30 is approximately $0.2785780$. Therefore the ratio of the combined discounted likelihood at 0.25 to that at 0.30 is $\exp(0.2785780)\approx1.3213$. The historical likelihood kernel is about 1.321 times as large at $\theta=0.25$ as at $\theta=0.30$; this is a relative likelihood comparison, not a posterior probability that either value is true.

Alternatives, Limitations, and a Common Pitfall

A pooled analysis with one common discount applies the same power to every historical likelihood. It can be useful when the studies are judged equally relevant, and the product can then be written as the pooled likelihood raised to that common power. Separate powers retain the study-by-study contribution and allow the specified amount of borrowing to differ across studies. Neither form automatically accounts for differences in study populations or outcome processes; that requires a model that represents those differences.

Other alternatives include using no historical likelihood, or modeling variation among study-specific control rates rather than imposing one shared rate. These approaches answer different modeling questions. A power prior with fixed discounts is transparent about how likelihood contributions are weighted, but its conclusions can be sensitive to the specified discounts and to the shared-parameter assumption. It does not, on its own, repair bias in a historical study or guarantee compatibility with a current trial.

Common pitfall: It is tempting to pool the event counts first and then apply one discount, even when the studies have different discounts. That replaces the separate weighted contributions with a different model. In the two-study example, the correct event exponent is $0.50(8)+0.25(18)=8.5$, not a single discount times the unweighted total of 26 events. Always multiply each study’s log likelihood by its own discount before summing.

Key takeaway: A multi-study power prior is built from a product of study-level likelihood contributions, each raised to its specified discount. The product is only as meaningful as the assumptions that the studies provide distinct data and that their likelihoods describe a shared parameter in a way relevant to the current analysis.

Check Your Understanding

Answer each question, then compare with the solutions below.

Two studies have likelihoods $L_1(\theta)$ and $L_2(\theta)$, with discounts 1 and 0. What is their combined discounted likelihood contribution, up to a factor that does not depend on $\theta$?
Study A observes 4 events among 20 participants, and study B observes 6 among 30. With discounts 0.5 and 0.25, respectively, what are the combined event and non-event exponents in the binary likelihood kernel?
Why is multiplying two historical likelihoods inappropriate if the studies contain overlapping participants?
Does a likelihood ratio of 1.3213 at $\theta=0.25$ versus $\theta=0.30$ mean there is a 1.3213 probability that $\theta=0.25$? Explain.

Solutions. (1) The contribution is $L_1(\theta)^1L_2(\theta)^0=L_1(\theta)$; study 2 is omitted. (2) The event exponent is $0.5(4)+0.25(6)=3.5$. The non-event exponent is $0.5(16)+0.25(24)=14$. Thus the kernel is proportional to $\theta^{3.5}(1-\theta)^{14}$. (3) The product would count some outcomes more than once and would not represent a likelihood from conditionally independent study data. (4) No. It is a ratio of likelihood values at two parameter values, not a probability; posterior probabilities require a prior and integration over parameter values.