From Discounted Historical Data to Beta Parameters
When a trial borrows information from historical controls, the nominal number of historical patients is not necessarily the number of patients’ worth of information that enters the analysis. A power discount reduces the contribution of the historical likelihood. In the beta-binomial model, that reduction has a direct count-scale interpretation: it scales the historical successes and failures, and therefore scales their total contribution to the beta prior.
The quantity of interest here is the effective sample size (ESS) of the discounted historical contribution. We will define it as the number of binomial observations represented by the historical contribution on the beta count scale. This convention makes the calculation transparent, but it does not turn fractional counts into actual patients or guarantee that historical data are exchangeable with current controls.
Here an event is the binary outcome designated as a success in the model; it need not be a clinically beneficial outcome. The outcome definition must be fixed and consistent across historical and current data. We assume historical outcomes follow a binomial model with the same event probability \(\theta\) used in the prior for the current control parameter. The discount \(a\) is treated as fixed for this calculation.
Why the Effective Sample Size Is \(a n_H\)
For historical data with \(y_H\) events among \(n_H\) patients, the binomial likelihood as a function of the event probability is proportional to \(\theta^{y_H}(1-\theta)^{n_H-y_H}\). A power prior raises this likelihood to the power \(a\). Multiplication by a beta prior kernel then gives the updated beta exponents.
The last line is the kernel of a beta distribution. Thus the beta parameters after adding the discounted historical contribution are \(\alpha_0+a y_H\) and \(\beta_0+a(n_H-y_H)\). This derivation also shows why both outcome counts must be discounted: discounting only successes, for example, would not correspond to raising the entire binomial likelihood to the power \(a\).
The beta concentration, the sum of its two parameters, is often used as a count-scale measure of information. In this convention, a historical contribution of \(a n_H\) has the same parameter-count increment as \(a n_H\) binomial observations. The value can be fractional: for example, a contribution of 15.5 is not a claim that half a patient was enrolled. It summarizes the likelihood’s discounted contribution.
Worked Calculations
Worked Example: A Quarter-Weighted Historical Sample
Suppose a hypothetical historical control group has 18 events among 60 patients. Use a fixed discount \(a=0.25\) and a starting prior \(\mathrm{Beta}(1,1)\). Find the discounted beta parameters and the historical ESS.
There are \(60-18=42\) historical non-events. The discounted success increment is \(0.25(18)=4.5\), and the discounted failure increment is \(0.25(42)=10.5\). Therefore the beta distribution after incorporating this historical contribution is $$ \mathrm{Beta}\bigl(1+4.5,\ 1+10.5\bigr)=\mathrm{Beta}(5.5,11.5). $$
The historical ESS is the sum of the two discounted increments: $$ 4.5+10.5=15.0, \qquad a n_H=0.25(60)=15.0. $$ The full beta concentration is \(5.5+11.5=17.0\), of which 2.0 comes from the starting \(\mathrm{Beta}(1,1)\) prior and 15.0 from the discounted historical data.
Worked Example: Separating the Historical ESS From the Prior
In a hypothetical analysis, 12 of 40 historical control patients experience the event. The starting prior is \(\mathrm{Beta}(2,3)\), and the fixed discount is \(a=0.60\). Calculate the updated beta parameters, historical ESS, and posterior mean under this beta distribution.
There are \(40-12=28\) non-events. The discounted increments are \(0.60(12)=7.2\) for events and \(0.60(28)=16.8\) for non-events. Thus the beta parameters are $$ \alpha=2+7.2=9.2, \qquad \beta=3+16.8=19.8. $$
The historical ESS is \(7.2+16.8=24.0\), also \(0.60(40)=24.0\). The starting prior contributes \(2+3=5\) to the concentration, so total concentration is \(9.2+19.8=29.0\). The mean of a \(\mathrm{Beta}(\alpha,\beta)\) distribution is \(\alpha/(\alpha+\beta)\), giving $$ \frac{9.2}{9.2+19.8}=\frac{9.2}{29.0}=0.3172. $$ This is the mean of the resulting beta distribution for the event probability, not an estimate that 31.72% of actual current-trial patients will necessarily experience the event.
Worked Example: Same ESS, Different Historical Event Rates
Consider two hypothetical historical datasets, each with 50 patients and discount \(a=0.40\). In dataset A, 5 patients experience the event; in dataset B, 25 do. Use the starting prior \(\mathrm{Beta}(1,1)\). Compare their ESS values and resulting beta distributions.
Both datasets have historical ESS \(0.40(50)=20.0\). For dataset A, there are 45 non-events, so the discounted increments are \(0.40(5)=2\) and \(0.40(45)=18\). The resulting distribution is $$ \mathrm{Beta}(1+2,1+18)=\mathrm{Beta}(3,19), \qquad \text{mean}=\frac{3}{22}=0.1364. $$
For dataset B, there are 25 non-events. The increments are \(0.40(25)=10\) for events and \(0.40(25)=10\) for non-events. The resulting distribution is $$ \mathrm{Beta}(1+10,1+10)=\mathrm{Beta}(11,11), \qquad \text{mean}=\frac{11}{22}=0.5000. $$
The two historical contributions have equal ESS because each represents 20 discounted observations on the count scale. Their beta distributions differ because ESS describes the total contribution, while the split between discounted events and non-events determines where that contribution places the distribution.
Assumptions, Checks, and Interpretations
The algebraic identity \(a n_H\) follows from the power-weighted binomial likelihood. Its substantive interpretation depends on the model being appropriate. In particular, the model assumes that historical and current controls can be represented by a common event probability \(\theta\), conditional on the features included in the analysis. Issues such as changes in eligibility, endpoint definition, outcome ascertainment, or standard of care can undermine that assumption. A power discount reduces the historical likelihood’s contribution; by itself, it does not correct a mismatch between populations or outcomes.
Before reporting an ESS, check the inputs and the meaning of the discount. The event count must satisfy \(0 \leq y_H \leq n_H\), the discount must satisfy \(0 \leq a \leq 1\), and the non-event count must be \(n_H-y_H\). Verify that the two discounted increments sum to \(a n_H\). If the calculated total does not, a count or arithmetic error is likely.
The ESS also helps explain two limiting cases. With \(a=0\), the historical contribution is zero and the beta parameters remain at their starting values. With \(a=1\), the entire historical likelihood contributes, and the count-scale ESS is \(n_H\). For an intermediate discount, the ESS lies between zero and \(n_H\). These checks are useful diagnostics for a calculation, but they do not establish that the chosen discount is clinically or statistically appropriate.
| Quantity | Expression | Interpretation |
|---|---|---|
| Discounted success increment | \(a y_H\) | Historical event contribution on the beta count scale |
| Discounted failure increment | \(a(n_H-y_H)\) | Historical non-event contribution on the beta count scale |
| Historical count-scale ESS | \(a n_H\) | Total discounted historical contribution |
| Full beta concentration | \(\alpha_0+\beta_0+a n_H\) | Starting-prior concentration plus historical contribution |
A raw sample-size comparison is the most direct alternative: it reports \(n_H\), the number of historical patients, without discounting. This is appropriate when describing enrollment, but it does not represent the borrowed contribution when \(a<1\). At the other extreme, reporting only the full beta concentration can obscure how much came from the starting prior. For transparency, report the starting prior, the discounted increments, and the historical ESS separately.
The count-scale ESS is not a universal measure of every kind of posterior information. It summarizes the additive beta-parameter contribution under this model. It does not describe whether the historical and current populations are clinically comparable, and it does not measure the effective number of independent patients if the binomial model is inappropriate. For example, clustering, overdispersion, or an outcome definition that changes across studies can make a simple binomial contribution misleading.
Common Mistakes and Failure Modes
- Using \(a y_H\) as the ESS. This counts only discounted events. The ESS includes both event and non-event contributions: \(a y_H+a(n_H-y_H)=a n_H\).
- Calling the full beta concentration the historical ESS. The full concentration includes \(\alpha_0+\beta_0\). Subtract that starting-prior concentration to isolate the historical increment.
- Reporting a fractional ESS as actual enrollment. A value such as 15.0 is a count-scale summary of discounted likelihood information, not a literal patient count.
- Assuming equal ESS means equal prior distributions. Equal totals do not imply equal event rates or equal beta parameters. The event/non-event split also matters.
- Treating discounting as a fix for incompatibility. A smaller \(a\) reduces influence but does not make different endpoints, populations, or care settings equivalent. Investigate comparability rather than relying on the ESS alone.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A hypothetical historical group has 14 events among 35 patients, with \(a=0.20\). What are the discounted success and failure increments, and what is the historical ESS?
- A starting prior is \(\mathrm{Beta}(1,2)\). A historical contribution adds 6 to the beta success parameter and 9 to the failure parameter. What are the historical ESS and full beta concentration?
- Two historical datasets have the same sample size and discount, but different event counts. Must their ESS values differ? Must their resulting beta means be equal? Explain.
- For a historical dataset with \(n_H=80\), what are the historical ESS values at \(a=0\) and \(a=1\), and what do they represent?
Solutions. (1) There are \(35-14=21\) non-events. The increments are \(0.20(14)=2.8\) and \(0.20(21)=4.2\); the ESS is \(2.8+4.2=7.0\), also \(0.20(35)=7.0\). (2) The historical ESS is \(6+9=15\). The starting concentration is \(1+2=3\), so the full concentration is \(3+15=18\). (3) Their ESS values must be equal because both equal \(a n_H\), but their beta means need not be equal because the event and non-event increments can be split differently. (4) At \(a=0\), the ESS is \(0(80)=0\), so the historical data add no count-scale contribution. At \(a=1\), it is \(1(80)=80\), the full historical sample-size contribution on this count scale.