Fixed Power Priors for Binary Control Rates

Why Discount Historical Control Data?

Historical control outcomes can inform a current trial’s control event rate, but they may come from a different population, setting, or period. A power prior provides a direct way to temper their contribution: it raises the historical likelihood to a prespecified power between zero and one. The resulting prior is then combined with the current-trial control likelihood. The discount is a modeling choice, not a quantity estimated from the current outcomes in the fixed-discount approach.

The earlier material on the power prior definition and fractional likelihoods establishes the general construction. Here the historical outcome is binary, so the likelihood has a simple form and the prior can be expressed with a beta distribution. Assume throughout that the outcome is a specified event, such as a response or an adverse event; \(p\) is the probability of that event among current-trial control participants.

Definition: Suppose the historical control data consist of \(y_h\) events among \(n_h\) participants, modeled as \(Y_h\mid p\sim\operatorname{Binomial}(n_h,p)\). Let \(\pi_0(p)\) be a baseline prior density and let \(a\in[0,1]\) be a fixed discount chosen before examining current-trial outcomes. The fixed power prior density is proportional to \(\pi_0(p)L_h(p)^a\), where \(L_h(p)\) is the historical likelihood. At \(a=0\), the historical data contribute nothing; at \(a=1\), the historical likelihood is used at full power.

Constructing the Binary Power Prior

For a binomial model, the historical likelihood is

L_h(p)=\binom{n_h}{y_h}p^{y_h}(1-p)^{n_h-y_h},\qquad 0<p<1.

Taking this likelihood to the power \(a\) gives a kernel proportional to \(p^{a y_h}(1-p)^{a(n_h-y_h)}\). The binomial coefficient does not depend on \(p\), so it affects only the proportionality constant. If the baseline prior is \(\operatorname{Beta}(\alpha_0,\beta_0)\), with positive shape parameters, its density is proportional to \(p^{\alpha_0-1}(1-p)^{\beta_0-1}\). Multiplying the baseline density by the powered likelihood gives

\pi(p\mid D_h,a)\ \propto\ p^{\alpha_0+a y_h-1}(1-p)^{\beta_0+a(n_h-y_h)-1}.

This kernel is a beta density. Thus the normalized fixed power prior is \(\operatorname{Beta}(\alpha_0+a y_h,\ \beta_0+a(n_h-y_h))\). This construction requires \(\alpha_0>0\), \(\beta_0>0\), and \(a\in[0,1]\); these conditions give a proper beta distribution even when the historical data contain no events or all events.

The power \(a\) discounts both the historical event count and the historical non-event count. It does not change the observed historical rate \(y_h/n_h\); rather, it changes how strongly those data shape the prior relative to the baseline prior. For \(a=0\), the result is just the baseline prior. For \(a=1\), it is the ordinary beta-binomial update of that baseline prior using the historical data.

Worked Example: Building a Prior From Historical Controls

Suppose a hypothetical historical control group has 18 events among 60 participants. Use a baseline \(\operatorname{Beta}(1,1)\) prior and a fixed discount \(a=0.25\). Calculate the power prior and its mean.

The historical event count is \(y_h=18\); the non-event count is \(n_h-y_h=60-18=42\). Discounting these contributions gives \(0.25(18)=4.5\) and \(0.25(42)=10.5\). Therefore,

p\mid D_h,a=0.25\sim \operatorname{Beta}(1+4.5,\ 1+10.5) =\operatorname{Beta}(5.5,11.5).

A \(\operatorname{Beta}(\alpha,\beta)\) distribution has mean \(\alpha/(\alpha+\beta)\), so the prior mean is \(5.5/(5.5+11.5)=5.5/17=0.3235\). This is a prior mean event probability of about 32.4%, shaped by the baseline prior and the discounted historical data. It is not the historical event rate itself, which is \(18/60=0.300\).

Updating With Current Control Outcomes

Let \(y_c\) be the number of events among \(n_c\) current-trial control participants. Conditional on \(p\), assume \(Y_c\sim\operatorname{Binomial}(n_c,p)\). Combining this likelihood with the fixed power prior gives the posterior

p\mid D_h,D_c,a\sim \operatorname{Beta}\bigl(\alpha_0+a y_h+y_c,\ \beta_0+a(n_h-y_h)+(n_c-y_c)\bigr).

This update uses the discounted historical contributions and the current-trial counts. Current outcomes enter at full likelihood power in this model. The formula assumes that historical and current control outcomes are conditionally independent given the same event probability \(p\), and that each group’s outcomes follow the stated binomial model. The next tutorial derives this beta-binomial update in detail; here it is used to show how the fixed prior operates in practice.

Worked Example: Combining Historical and Current Controls

Continue with the hypothetical historical data of 18 events among 60 participants, baseline \(\operatorname{Beta}(1,1)\), and fixed discount \(a=0.25\). Suppose the current control group has 12 events among 40 participants. Find the posterior distribution and mean.

The historical contribution gives \(\operatorname{Beta}(5.5,11.5)\), as calculated above. The current group contributes 12 events and \(40-12=28\) non-events. Therefore,

p\mid D_h,D_c,a\sim \operatorname{Beta}(5.5+12,\ 11.5+28) =\operatorname{Beta}(17.5,39.5).

The posterior mean is \(17.5/(17.5+39.5)=17.5/57=0.3070\). Under the model, the posterior distribution for the control event probability has mean 30.7%. The current observed rate is \(12/40=0.300\); the posterior mean is slightly higher because the historical information and baseline prior also contribute.

For this interpretation to be useful, the historical and current outcomes must represent a sufficiently comparable control event probability for the shared-\(p\) model to be reasonable. The calculation does not establish that comparability; it assumes it.

Checking the Assumptions and the Discount

A fixed power prior is transparent, but its simplicity relies on substantive and statistical assumptions. Before using it, check that the binary event is defined consistently across data sources, the denominators count eligible participants, and the observation periods and ascertainment rules are sufficiently comparable. The binomial model also treats outcomes within each group as independent with a common event probability. Clustering, repeated observations, or important variation in risk across participants may make that model inadequate.

A shared \(p\) for historical and current controls is especially consequential: it treats the event probability as stable across the two settings. Discounting weakens the historical contribution but does not model why the rates might differ. If the populations or care practices differ materially, a small discount may reduce influence without resolving the model mismatch. A fixed discount should therefore be justified in advance and examined over plausible values.

Worked Example: Assessing Sensitivity to the Discount

Consider hypothetical data with 8 events among 20 historical controls and 14 events among 20 current controls. Use a \(\operatorname{Beta}(1,1)\) baseline prior. Compare fixed discounts \(a=0.10\) and \(a=0.80\). The historical event rate is \(8/20=0.400\), whereas the current rate is \(14/20=0.700\).

For \(a=0.10\), the discounted historical counts are \(0.10(8)=0.8\) events and \(0.10(12)=1.2\) non-events. After adding the current counts, the posterior is

\operatorname{Beta}(1+0.8+14,\ 1+1.2+6) =\operatorname{Beta}(15.8,8.2).

Its mean is \(15.8/(15.8+8.2)=15.8/24=0.6583\). For \(a=0.80\), the historical contributions are \(6.4\) events and \(9.6\) non-events, giving

\operatorname{Beta}(1+6.4+14,\ 1+9.6+6) =\operatorname{Beta}(21.4,16.6).

Its mean is \(21.4/(21.4+16.6)=21.4/38=0.5632\). The more heavily discounted historical data pull the posterior mean less toward the lower historical event rate. These posterior means are not tests of whether the two rates differ; they illustrate how a prespecified modeling choice affects the combined analysis.

Alternatives, Diagnostics, and Limitations

The main simple alternatives clarify what the fixed power prior is doing. With \(a=0\), the analysis uses the baseline prior and current controls only. With \(a=1\), historical data enter at full likelihood power, under the same shared-rate assumption. A value strictly between zero and one interpolates by discounting historical likelihood contributions. These options are not interchangeable evidence-based conclusions: they encode different assumptions about how informative the historical controls should be.

Useful diagnostics begin with the data and the model, not merely the final posterior. Compare event definitions, follow-up duration, eligibility criteria, and observed rates. Investigate whether the current control rate is plausible given the historical data and the selected discount. Recompute key posterior summaries across a clinically defensible range of fixed discounts. Large changes indicate that conclusions depend materially on the borrowing choice and should be reported as such. If incompatibility is serious, reducing \(a\) alone may not be adequate; an analysis that represents differences between studies may be more appropriate.

The fixed-discount approach has limitations. It does not learn the discount from the data, distinguish sources of between-study differences, or guarantee that historical evidence is transportable to the target population. Its beta-binomial form is convenient, but the result depends on the baseline prior, the event definition, and the discount. Sensitivity analysis and a clear account of the assumptions are therefore part of the analysis, not optional decoration.

Wrong reasoning: “The historical rate is close to the current rate, so borrowing is automatically safe.”
Correction: Similar observed rates do not prove that the control populations or measurement processes are comparable. Assess design and clinical context as well as rates.
Wrong reasoning: “A discount of 0.25 means the historical event rate itself is multiplied by 0.25.”
Correction: The discount multiplies the historical likelihood contributions, including both event and non-event counts. It does not replace the historical rate with a quarter of its value.
Wrong reasoning: “A fixed discount protects against any historical-current conflict.”
Correction: Discounting reduces historical influence but does not model incompatibility. Examine sensitivity and whether a shared event probability is credible.

Key takeaway: For binary controls, a fixed power prior combines a baseline beta prior with discounted historical events and non-events. Current outcomes then update that prior at full likelihood power, provided the binomial and shared-rate assumptions are defensible.

Check Your Understanding

Try each problem, then check the solutions below.

With baseline \(\operatorname{Beta}(2,3)\), 10 events among 25 historical controls, and \(a=0.40\), what is the fixed power prior?
For the prior in Question 1, suppose the current controls have 4 events among 15 participants. What is the posterior mean event probability?
What does \(a=0\) imply about the contribution of historical data?
Why does a fixed discount not, by itself, resolve a substantial difference in event rates or trial settings?

Solutions. 1. There are \(25-10=15\) historical non-events. The discounted counts are \(0.40(10)=4\) and \(0.40(15)=6\), so the prior is \(\operatorname{Beta}(2+4,3+6)=\operatorname{Beta}(6,9)\). 2. The current group has \(15-4=11\) non-events, giving posterior \(\operatorname{Beta}(6+4,9+11)=\operatorname{Beta}(10,20)\). Its mean is \(10/30=0.3333\), or 33.3%. 3. At \(a=0\), the powered historical likelihood is constant in \(p\), so the prior remains the baseline prior and historical outcomes do not affect it. 4. A discount only reduces the historical likelihood’s influence. It does not represent how populations, care, or event measurement differ, so the shared-rate assumption still needs justification.