Power Priors in Logistic Control Models

Why Discount Historical Binary Outcomes?

A historical control group may help estimate the event probability expected in a new trial’s control group. The challenge is that historical and current outcomes need not be equally informative about that probability: the populations, care standards, or conduct of the study may differ. A power prior offers a direct way to temper the historical likelihood. In a logistic model, the discount applies to the likelihood as a whole; it does not guarantee a closed-form posterior update.

Here the model is deliberately limited to a single control event probability. This makes the likelihood and the role of the discount transparent. It does not adjust for patient-level covariates.

Logistic Likelihood for Historical Controls

Let \(Y_{Hi}\) indicate whether historical control participant \(i\) experiences the event: \(Y_{Hi}=1\) for an event and \(Y_{Hi}=0\) otherwise. Assume these outcomes are independent and have a common event probability \(p_H\), with

Y_{Hi}\mid\beta \sim \operatorname{Bernoulli}(p_H), \qquad \operatorname{logit}(p_H)=\beta, \qquad \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

The parameter \(\beta\) is the historical control log odds of the event. The model assumes that the event probability is the same for all participants represented by this control group. If \(y_H=\sum_i Y_{Hi}\) events are observed among \(n_H\) participants, the likelihood is

L_H(\beta) = \prod_{i=1}^{n_H} \left(\frac{e^\beta}{1+e^\beta}\right)^{Y_{Hi}} \left(\frac{1}{1+e^\beta}\right)^{1-Y_{Hi}} = \frac{e^{y_H\beta}}{(1+e^\beta)^{n_H}}.

Equivalently, \(y_H\) has a binomial distribution with size \(n_H\) and probability \(p_H\); the binomial coefficient in that likelihood does not depend on \(\beta\). For inference about \(\beta\) with a fixed discount, it cancels from posterior ratios, but it is part of the full data likelihood.

Definition: For a fixed discount \(a_0\) between 0 and 1 and a baseline prior density \(\pi_0(\beta)\), the power prior based on historical control data \(D_H\) is proportional to \(L_H(\beta)^{a_0}\pi_0(\beta)\). The discount raises the historical likelihood to a power; it does not change the observed outcomes or assert that a fractional number of participants was enrolled.

For a proper baseline prior, a normalized version for fixed \(a_0\) is

\pi(\beta\mid D_H,a_0) = \frac{L_H(\beta)^{a_0}\pi_0(\beta)} {\displaystyle \int L_H(u)^{a_0}\pi_0(u)\,du}.

At \(a_0=0\), the historical likelihood contributes no information; at \(a_0=1\), it contributes at full likelihood strength. For intermediate values, its log-likelihood contribution is multiplied by \(a_0\). When the discount itself is random, the normalizing factor can depend on that discount and must be treated accordingly; the fixed-discount examples below do not require that extension.

Worked Example: Discounting a Historical Likelihood Comparison

Suppose a hypothetical historical control group has 18 events among 60 participants. Compare the historical likelihood at \(p_H=0.30\) with its value at \(p_H=0.40\), then repeat the comparison after using \(a_0=0.40\).

At a specified probability \(p\), the binomial likelihood, omitting the common binomial coefficient, is \(p^{18}(1-p)^{42}\). Thus the log-likelihood difference, first probability minus second, is

\log L_H(0.30)-\log L_H(0.40) = 18\log\left(\frac{0.30}{0.40}\right) + 42\log\left(\frac{0.70}{0.60}\right) \approx 1.296.

The unpowered likelihood ratio is therefore \(\exp(1.296)\approx 3.655\): the observed historical outcomes give about 3.655 times as much likelihood to \(p_H=0.30\) as to \(p_H=0.40\). With \(a_0=0.40\), the log-likelihood difference becomes \(0.40(1.296)=0.5184\), and the powered likelihood ratio is \(\exp(0.5184)\approx 1.679\). The historical data still favor 0.30, but less strongly.

This comparison concerns the historical likelihood, not the posterior probability that either event probability is true. A posterior comparison also depends on the baseline prior and, if present, the current data.

What the Discount Does to the Model

Taking logs makes the weighting explicit:

\log L_H(\beta)^{a_0} = a_0\left[y_H\beta-n_H\log(1+e^\beta)\right].

Both the event contribution and the non-event contribution are weighted by the same \(a_0\). For this simple intercept-only model, the powered likelihood has the same parameter-dependent form as a binomial likelihood with exponents \(a_0y_H\) and \(a_0(n_H-y_H)\). Those fractional exponents are a mathematical consequence of weighting. They are not new observed counts or a claim that the data were actually collected from a smaller group.

The discount also scales the historical likelihood’s contribution to the score and curvature. Differentiating its log gives

\frac{\partial}{\partial\beta}\log L_H(\beta)^{a_0} = a_0\{y_H-n_Hp_H\}, \qquad \frac{\partial^2}{\partial\beta^2}\log L_H(\beta)^{a_0} = -a_0n_Hp_H(1-p_H).

Thus reducing \(a_0\) weakens how strongly the historical outcomes pull the estimate and reduces their contribution to local information. It does not itself assess whether the historical control group is relevant; the choice of studies and discount remains a substantive modeling decision.

Worked Example: Interpreting Weighted Historical Contributions

Suppose a hypothetical historical control group has 12 events among 40 participants, and the fixed discount is \(a_0=0.25\). Write the powered likelihood and calculate the weighted event and non-event exponents.

The binomial likelihood, apart from its parameter-independent coefficient, is \(p_H^{12}(1-p_H)^{28}\). Raising it to \(0.25\) gives

L_H(\beta)^{0.25} \ \propto\ p_H^{0.25(12)}(1-p_H)^{0.25(28)} = p_H^3(1-p_H)^7.

The weighted exponents are 3 for events and 7 for non-events, for a total of 10 weighted observations in this particular intercept-only likelihood. This is an algebraic description of the likelihood’s weighting, not a literal replacement of 40 people by 10 people. The model still depends on the original historical outcomes, and other quantities—such as a normalized prior when the discount varies—cannot be inferred simply by relabeling the data as 10 participants.

Combining Historical and Current Control Data

To update with current control outcomes, let \(y_C\) of \(n_C\) current control participants experience the event. If the historical and current controls are assumed to share the same log odds \(\beta\), their likelihood contributions combine with the baseline prior as

\pi(\beta\mid D_H,D_C,a_0) \ \propto\ \pi_0(\beta)L_H(\beta)^{a_0}L_C(\beta), \qquad L_C(\beta)=\frac{e^{y_C\beta}}{(1+e^\beta)^{n_C}}.

This formulation requires a proper posterior distribution. A proper baseline prior is a straightforward way to ensure that condition. In particular, logistic likelihoods can exhibit separation, in which the likelihood keeps increasing as a coefficient moves without bound. In the intercept-only setting, all observed outcomes being events or all being non-events produces this boundary behavior. An improper baseline prior can then lead to an improper posterior; a proper prior avoids relying on the data likelihood alone to control the tails.

Even with a proper prior, the posterior generally has no standard closed-form distribution. The logistic likelihood contains the term \((1+e^\beta)^{-n}\), which does not combine with common normal priors into a familiar conjugate family. Posterior summaries are therefore typically obtained using numerical integration, quadrature, or suitable simulation methods. The power-prior construction specifies the target density; it does not provide a closed-form posterior update.

Worked Example: A Numerical Posterior Mode

Consider hypothetical historical control data with 18 events among 60 participants and current control data with 10 events among 40 participants. Use a fixed historical discount \(a_0=0.50\) and the proper baseline prior \(\beta\sim N(0,2.5^2)\). Find the equation for the posterior mode and take a numerical step toward its solution.

Ignoring constants that do not depend on \(\beta\), the log posterior is

\ell(\beta) = (0.50\cdot18+10)\beta - (0.50\cdot60+40)\log(1+e^\beta) - \frac{\beta^2}{2(2.5^2)} = 19\beta-70\log(1+e^\beta)-\frac{\beta^2}{12.5}.

Differentiating, a mode solves

\ell'(\beta)=19-70\operatorname{logit}^{-1}(\beta)-\frac{\beta}{6.25}=0, \qquad \ell''(\beta)=-70p(1-p)-\frac{1}{6.25},

where \(p=\operatorname{logit}^{-1}(\beta)=e^\beta/(1+e^\beta)\). Starting at \(\beta=-1\), \(p\approx0.26894\), so \(\ell'(-1)\approx19-70(0.26894)+0.16=0.3341\), and \(\ell''(-1)\approx-70(0.26894)(0.73106)-0.16=-13.923\). One Newton update is

\beta_{\mathrm{new}} = -1-\frac{0.3341}{-13.923} \approx -0.9760.

Further numerical iteration gives a mode of approximately \(-0.9761\), corresponding to an event probability of approximately 0.2737. This is the posterior mode under the stated model and prior, not the posterior mean. Computing other posterior summaries requires evaluating or sampling from the posterior distribution.

Assumptions, Diagnostics, and Alternatives

The borrowing interpretation depends on assumptions that are stronger than the algebra. In this basic model, historical outcomes are conditionally independent Bernoulli observations with a common event probability; current and historical controls share the same \(\beta\); and the endpoint and population definitions support treating that parameter as meaningfully shared. The fixed discount reduces the historical likelihood’s influence but does not correct a mismatch in endpoint definitions, study conduct, or underlying event risk.

Useful checks include examining event and non-event counts, confirming that the likelihood and discount have been coded consistently, inspecting the posterior over \(\beta\), and checking whether posterior summaries change substantially across plausible discount choices. If all outcomes fall in one category, note the resulting boundary tendency of the likelihood and verify that the prior makes the posterior proper. In computational work, check numerical convergence and compare results across reasonable numerical settings or independent runs. A stable computation cannot rescue a scientifically implausible shared-parameter assumption.

The main alternatives answer different questions. With no historical borrowing, the current control likelihood is combined only with the baseline prior. Full borrowing uses the historical likelihood at \(a_0=1\); partial borrowing uses a value between 0 and 1. A conjugate beta-binomial model is available for a binomial probability with a beta prior, but it does not provide the same logistic-regression parameterization. In this tutorial’s intercept-only model the probability can be written directly as \(p_H\); a logistic formulation becomes especially useful when modeling coefficients, but ordinary logistic likelihoods generally do not yield a conjugate posterior. Covariate-adjusted borrowing requires additional modeling choices beyond this single shared event probability.

Common pitfall: It is incorrect to say that \(a_0=0.25\) means the historical data become exactly one quarter of a real study, or that it produces a closed-form posterior. The correction is that the log likelihood is multiplied by 0.25, so its influence is reduced; the posterior still depends on the baseline prior and any current data, and usually requires numerical inference.

Key takeaway: A logistic power prior contributes \(a_0\log L_H(\beta)\) to the log posterior. It discounts the historical likelihood under a shared-parameter model; it does not, by itself, establish historical compatibility or create a conjugate posterior.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A historical control group has 9 events among 30 participants. Write its likelihood in terms of \(\beta\), where \(\operatorname{logit}(p_H)=\beta\).
For that group, what are the weighted event and non-event exponents when \(a_0=0.40\)? What do they mean?
In the combined posterior, what changes when the historical discount is reduced while current data and the baseline prior remain fixed?
Why does a proper baseline prior matter when all observed outcomes are events?
Does a power-prior discount establish that historical and current control populations are comparable? Explain.

Solutions. (1) \(L_H(\beta)=e^{9\beta}/(1+e^\beta)^{30}\). (2) The exponents are \(0.40(9)=3.6\) and \(0.40(21)=8.4\); they describe weighted likelihood contributions, not actual fractional participant counts. (3) The historical log-likelihood contribution is multiplied by a smaller value, so it exerts less influence; the current likelihood and baseline prior are unchanged. (4) With all outcomes events, the likelihood increases toward its boundary as \(\beta\) grows. A proper prior can control the posterior tails and ensure a proper posterior. (5) No. The discount tempers the historical likelihood, but the shared-parameter assumption and clinical comparability must still be justified.