Study-Specific Power Prior Discounts

Why Give Historical Studies Different Discounts?

A combined historical likelihood can treat several studies as sources of evidence about one shared control parameter. But treating them as one undifferentiated data set may be inappropriate: studies can differ in design, population, conduct, or relevance to the current trial. A power prior allows the analyst to discount historical likelihood contributions. When each study receives its own discount, more relevant studies can retain more influence while less relevant studies contribute less.

The key point is that a study-specific discount changes the relative weighting of the historical likelihood contributions. It is not simply a change in the total amount of historical information. The weights must therefore be specified and interpreted study by study.

Definition: Suppose there are $J$ historical studies with data $D_1,\ldots,D_J$, and study $j$ has likelihood $L_j(\theta\mid D_j)$ for a common parameter $\theta$. A study-specific power-prior discount is a vector $\mathbf{a}=(a_1,\ldots,a_J)$, where each $a_j$ is usually fixed in the interval $[0,1]$. The combined discounted historical likelihood is $$ L_{\mathrm{hist}}(\theta\mid \mathbf{D},\mathbf{a})=\prod_{j=1}^{J}L_j(\theta\mid D_j)^{a_j}. $$ A value $a_j=1$ retains study $j$'s likelihood at full power, $a_j=0$ removes its likelihood contribution, and an intermediate value discounts it.

For a fixed vector $\mathbf{a}$, the corresponding power-prior distribution is proportional to a baseline prior density $\pi_0(\theta)$ times this combined likelihood. If the power prior is normalized, its normalizing constant must also be included; it can depend on the specified discounts. The focus here is the likelihood contribution and how distinct fixed discounts alter it.

The construction assumes that the studies provide likelihoods for the same parameter $\theta$, conditional on the model, and that their data are independent across studies. The model must also be suitable for each study's outcome and design. A discount changes the strength of a likelihood contribution; by itself, it does not make incompatible study populations or outcome definitions comparable.

How the Weights Enter the Likelihood

For independent studies, the ordinary combined likelihood is the product of their likelihoods. Raising each study's likelihood to its own power gives a direct way to change each study's influence. Taking logarithms makes this especially clear: the log of the combined discounted likelihood is the sum of the study-specific log likelihoods, each multiplied by its own discount.

\log L_{\mathrm{hist}}(\theta\mid \mathbf{D},\mathbf{a}) =\sum_{j=1}^{J}a_j\log L_j(\theta\mid D_j).

Thus a study with $a_j=0.8$ contributes four times as much log likelihood as it would with $a_j=0.2$, for the same observed data and parameter value. This comparison concerns that study's own contribution; the overall impact on inference also depends on its sample size and observed outcomes.

For binary outcomes, let study $j$ observe $y_j$ events among $n_j$ participants, and let $\theta$ be a common event probability. Under a binomial model, the likelihood is $L_j(\theta)=\binom{n_j}{y_j}\theta^{y_j}(1-\theta)^{n_j-y_j}$. The discounted product, apart from factors not depending on $\theta$, is

L_{\mathrm{hist}}(\theta\mid \mathbf{D},\mathbf{a}) \ \propto\ \theta^{\sum_j a_jy_j} (1-\theta)^{\sum_j a_j(n_j-y_j)}.

The binomial coefficients are part of each exact likelihood, and are raised to the corresponding powers in the literal product. They do not depend on $\theta$, so they can be omitted when focusing only on the likelihood's shape as a function of $\theta$. The weighted success and failure totals in the displayed kernel need not be integers. They are likelihood contributions, not counts of actual participants.

Interpretation: A study-specific discount scales the study's log-likelihood contribution. It does not change the study's observed event rate, turn fractional contributions into observed patients, or guarantee that the study is compatible with the other studies.

Worked Calculations

Worked Example: A Weighted Historical Estimate

Suppose two hypothetical historical control studies measure the same binary outcome. Study 1 observes 8 events among 20 participants; study 2 observes 18 events among 30 participants. Assign discounts $a_1=0.25$ and $a_2=0.80$. Find the maximum-likelihood estimate from the combined discounted historical likelihood.

For binomial likelihoods, the estimate that maximizes the weighted likelihood is the weighted event total divided by the weighted sample-size total. The weighted event contribution is $0.25(8)+0.80(18)=2+14.4=16.4$. The weighted total is $0.25(20)+0.80(30)=5+24=29$.

\hat{\theta}_{\mathrm{hist}} =\frac{\sum_j a_jy_j}{\sum_j a_jn_j} =\frac{16.4}{29} =0.5655.

The estimate from the discounted historical likelihood is therefore about 0.5655, or 56.55%. The weighted totals are fractional likelihood contributions, not a claim that the studies enrolled 29 actual participants with 16.4 events. For comparison, a common discount of 0.5 would yield $(0.5(8)+0.5(18))/(0.5(20)+0.5(30))=13/25=0.5200$, a different estimate because it gives the two studies equal discounts rather than the specified unequal ones.

Worked Example: Comparing Study-Specific and Common Weighting

Consider two hypothetical studies, each with 10 participants. Study 1 has 2 events and study 2 has 8. Compare the historical likelihood ratio for $\theta=0.6$ against $\theta=0.4$ under discounts $a_1=0.2$ and $a_2=0.9$. For one study with $y$ events among $n$, the binomial coefficient cancels in this ratio, leaving $(0.6/0.4)^y(0.4/0.6)^{n-y}$.

For study 1, the log likelihood ratio is $2\log(1.5)+8\log(2/3)=-2.43279$. After applying its discount, its contribution is $0.2(-2.43279)=-0.486558$. For study 2, the log likelihood ratio is $8\log(1.5)+2\log(2/3)=2.43279$. Its discounted contribution is $0.9(2.43279)=2.189511$.

\log\frac{L_{\mathrm{hist}}(0.6)}{L_{\mathrm{hist}}(0.4)} =-0.486558+2.189511 =1.702953, \qquad \frac{L_{\mathrm{hist}}(0.6)}{L_{\mathrm{hist}}(0.4)} =\exp(1.702953)\approx 5.49.

Under these specified weights, the combined discounted historical likelihood is about 5.49 times as large at 0.6 as at 0.4. With a common discount of 0.5, the two studies' unweighted log likelihood ratios cancel, so the combined ratio is 1. This illustrates that a common discount cannot reproduce the relative weighting in the specified study-specific vector.

Worked Example: Sample Size and Discount Both Matter

Three hypothetical historical studies report, respectively, 20 events among 50 participants, 7 among 10, and 8 among 20. Assign discounts $0.10$, $0.80$, and $0.25$. Calculate the weighted estimate and compare each study's weighted total with the others.

The weighted event total is $0.10(20)+0.80(7)+0.25(8)=2+5.6+2=9.6$. The weighted sample-size total is $0.10(50)+0.80(10)+0.25(20)=5+8+5=18$.

\hat{\theta}_{\mathrm{hist}}=\frac{9.6}{18}=0.5333.

The resulting estimate is 53.33%. The three studies contribute weighted totals of 5, 8, and 5, respectively. Although study 2 is the smallest, its high discount gives it the largest weighted total. If all studies instead received the common discount 0.5, the estimate would be $(10+3.5+4)/(25+5+10)=17.5/40=0.4375$. The contrast shows why both the sample size and the assigned discount matter: a discount is applied to the entire study likelihood, while the likelihood's information also depends on the study's data and size.

Justification for the Weighted Estimate

The weighted estimate used in the examples follows directly from maximizing the binomial likelihood kernel. Define $Y_a=\sum_j a_jy_j$ and $N_a=\sum_j a_jn_j$. The log likelihood kernel is $Y_a\log(\theta)+(N_a-Y_a)\log(1-\theta)$. Differentiating with respect to $\theta$ gives the score equation below.

\frac{d}{d\theta}\log L_{\mathrm{hist}}(\theta) =\frac{Y_a}{\theta}-\frac{N_a-Y_a}{1-\theta}=0 \quad\Longrightarrow\quad \hat{\theta}_{\mathrm{hist}}=\frac{Y_a}{N_a}.

When $0<Y_a<N_a$, the second derivative is $-Y_a/\theta^2-(N_a-Y_a)/(1-\theta)^2$, which is negative for $0<\theta<1$. Therefore the score's solution is the unique maximum. If $Y_a=0$, the maximum is at the boundary $\theta=0$; if $Y_a=N_a$, it is at $\theta=1$. If all discounts are zero, there is no historical likelihood contribution and this historical-only estimate is undefined.

Choosing and Checking Study-Specific Discounts

Discounts should reflect a defensible judgment about how informative each historical study is for the target trial, not merely a preference for a desired result. Relevant considerations can include differences in eligibility criteria, treatment versions, outcome measurement, follow-up, or clinical practice. Those differences may motivate smaller weights, but a power prior does not repair a misspecified common-parameter model. The clinical and statistical justification for borrowing must be made separately from the arithmetic of applying the discounts.

A practical check is to inspect each study's contribution to the log likelihood or weighted total and compare the resulting inference across plausible discount vectors. In the binary common-parameter case, $a_jn_j$ is a useful summary of the study's weighted sample-size contribution, but it is not by itself a complete measure of influence: the observed event pattern also matters. A large study with a very small discount may contribute less than a smaller study with a large discount.

Approach	How it weights studies	Useful distinction
Study-specific fixed discounts	Each study has its own $a_j$.	Can represent different prespecified levels of relevance.
One common fixed discount	All studies use the same $a$.	Changes the overall strength of historical evidence but not its relative weighting across studies.
No borrowing from historical studies	All historical contributions are excluded.	A reference point for assessing how much conclusions depend on historical evidence.

The comparison is about weighting choices, not a claim that one approach is universally preferable. A common discount is simpler and may be suitable when studies are comparably relevant. Separate discounts allow more differentiation but require more study-specific justification. In either case, examining sensitivity to reasonable alternatives helps reveal whether a conclusion depends heavily on a particular weighting decision.

Diagnostics, Limitations, and a Common Pitfall

Warning signs include a combined estimate that changes sharply under small, clinically plausible changes to the discounts; a single study dominating despite weak relevance; or substantial disagreement between studies that a shared-$\theta$ model cannot represent. These are reasons to review the assumptions, data harmonization, and discount choices. They are not proof that a particular alternative weight is correct. In particular, choosing discounts after inspecting which values produce a favorable treatment conclusion risks making the analysis outcome-driven.

A common mistake is to say, “Study 1 and study 2 are both discounted, so using a common discount gives the same historical weighting as using separate discounts.” That reasoning confuses discounting the overall likelihood with changing the relative weights. For a common positive discount $a$, the log likelihood is multiplied by $a$ as a whole. For distinct values, each study's log likelihood is scaled differently; the relative contribution can therefore change, as the likelihood-ratio example demonstrates.

Another limitation is that a likelihood power is a modeling device, not a literal reduction in the number of participants. Weighted totals can be fractional, and standard uncertainty calculations based on an ordinary sample of that fractional size should not be inferred without a model that justifies them. The power-prior formulation also does not automatically account for between-study heterogeneity. If a shared parameter is implausible, merely assigning study-specific discounts may not be enough.

Key takeaway: With study-specific power-prior discounts, each historical study contributes $a_j\log L_j(\theta)$ to the combined log likelihood. Distinct discounts can change the balance of historical evidence, not just its overall amount; the assumptions and rationale for each weight remain essential.

Check Your Understanding

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

In a two-study analysis, what does $a_1=0$ imply about study 1's likelihood contribution?
Two studies each have 10 participants. One has 2 events and the other has 8. If both receive the same discount, why does that discount not change the combined likelihood ratio between $\theta=0.6$ and $\theta=0.4$ in this example?
A hypothetical study has 5 events among 12 participants and receives discount $0.4$. What are its weighted event and total contributions?
In the three-study example above, which study has the largest weighted sample-size contribution, and what is that contribution?
Why does assigning a small discount not, on its own, resolve serious incompatibility between a historical study and the target trial?

Solutions. (1) Its likelihood is raised to the zero power, so it contributes the factor $L_1(\theta)^0=1$ and has no effect on the combined historical likelihood. (2) The studies' log likelihood ratios are equal in magnitude and opposite in sign; multiplying both by the same discount preserves their cancellation. (3) The weighted event contribution is $0.4(5)=2$, and the weighted total is $0.4(12)=4.8$; these are likelihood contributions, not observed counts. (4) Study 2 contributes $0.80(10)=8$, larger than the 5 contributed by each other study. (5) Discounting changes the strength of that study's likelihood contribution but does not change its outcome definition, population, or the model's assumption that the studies inform a common parameter.

Approach	How it weights studies	Useful distinction
Study-specific fixed discounts	Each study has its own \(a_j\).	Can represent different prespecified levels of relevance.
One common fixed discount	All studies use the same \(a\).	Changes the overall strength of historical evidence but not its relative weighting across studies.
No borrowing from historical studies	All historical contributions are excluded.	A reference point for assessing how much conclusions depend on historical evidence.