Why Compare Fixed Discounts?
When a clinical trial borrows information from historical controls, the amount borrowed can affect both the estimate of the control response rate and its uncertainty. A fixed discount \(a\) scales the historical likelihood: \(a=0\) gives no historical contribution, while \(a=1\) gives the historical data their full likelihood contribution. Intermediate values temper that contribution. Because the choice of \(a\) is consequential, a single analysis at one selected value can conceal how dependent the conclusions are on that choice.
A fixed-discount sensitivity analysis evaluates a prespecified set of plausible values of \(a\), then compares the same posterior summaries across those values. It is not a procedure for finding the discount that produces the most favorable result. The purpose is to show how conclusions change under clearly stated borrowing assumptions, including the no-borrowing and, where appropriate, full-borrowing boundaries.
Model and Assumptions
For illustration, let \(y_c\) of \(n_c\) concurrent control participants have the binary outcome, and let \(y_h\) of \(n_h\) historical control participants have it. Write \(p\) for the common control response probability. Suppose the current and historical counts are conditionally binomial, given \(p\), and take a Beta prior with positive parameters \(\alpha_0\) and \(\beta_0\). The power prior raises the historical likelihood to the fixed power \(a\), where \(0\leq a\leq1\).
This model requires more than correct arithmetic. It assumes that the binary endpoint and its interpretation are sufficiently aligned across data sources; that the counts represent the stated denominators and outcomes; and that a shared \(p\) is a defensible starting model. The discount moderates the historical likelihood but does not itself correct a different endpoint definition, a calendar-time shift, outcome missingness, or other bias. The current and historical observations are treated as conditionally independent given \(p\). The prior parameters and the grid of \(a\) values should be specified and justified rather than chosen after inspecting which result is most attractive.
The current likelihood contributes \(p^{y_c}(1-p)^{n_c-y_c}\). Raising the historical likelihood to \(a\) contributes \(p^{a y_h}(1-p)^{a(n_h-y_h)}\), up to factors not involving \(p\). Multiplying these terms by the Beta prior gives the posterior kernel below. This short derivation also makes clear why the historical success and failure counts are both discounted.
For a Beta posterior with parameters \(A\) and \(B\), its mean is \(A/(A+B)\) and its standard deviation is \(\sqrt{AB/[(A+B)^2(A+B+1)]}\). These summaries describe the posterior for \(p\) conditional on the specified discount. A useful grid compares them at each value, along with any other summaries tied to the clinical decision, such as a posterior quantile or the posterior probability that \(p\) exceeds a prespecified benchmark.
Planning the Grid and the Comparisons
The grid should represent substantively different borrowing assumptions, not merely a convenient sequence of decimals. Include \(a=0\) as the concurrent-data-only reference. If full borrowing is scientifically interpretable, include \(a=1\) as a boundary scenario. Choose intermediate values to span plausible degrees of borrowing and to reveal changes that matter to the clinical decision. Equal spacing can be useful, but is not mandatory; a denser set in a region where summaries change quickly may be more informative.
Before calculating results, state the parameter and summaries to be compared, the units, the clinical benchmark if one is relevant, and how differences will be judged. Use the same prior, outcome definition, and summary definitions at every grid point. It is often informative to report the posterior mean and standard deviation, and to add an interval or decision-relevant posterior probability when that is part of the analysis plan. A table makes the comparison auditable. Do not interpret a change in the posterior mean alone as a change in the probability of clinical benefit.
Worked Example: Comparing Posterior Mean and Precision Across a Grid
Suppose, hypothetically, 18 of 40 concurrent control participants and 64 of 100 historical control participants respond. Use a \(\operatorname{Beta}(1,1)\) prior and the prespecified grid \(a=0,0.25,0.50,0.75,1\). The posterior parameters are \(A=19+64a\) and \(B=23+36a\). For example, at \(a=0.50\), \(A=51\), \(B=41\), so the posterior mean is \(51/92=0.5543\), and the standard deviation is \(\sqrt{51(41)/(92^2\cdot93)}=0.05154\).
| Discount \(a\) | Posterior distribution | Posterior mean | Posterior SD |
|---|---|---|---|
| 0 | Beta(19, 23) | 0.4524 | 0.07590 |
| 0.25 | Beta(35, 32) | 0.5224 | 0.06057 |
| 0.50 | Beta(51, 41) | 0.5543 | 0.05154 |
| 0.75 | Beta(67, 50) | 0.5726 | 0.04554 |
| 1 | Beta(83, 59) | 0.5845 | 0.04121 |
The historical response rate is higher than the concurrent response rate, so increasing \(a\) shifts the posterior mean upward in this example. The posterior standard deviation also declines across this particular grid. The table does not establish that a higher response rate is clinically adequate: that judgment requires a prespecified benchmark and, if relevant, a posterior probability or interval interpreted against it. All rates here are proportions, not percentages; multiplying by 100 would express them as percentages.
Worked Example: Checking the Arithmetic at an Intermediate Discount
In a second hypothetical study, 10 of 20 concurrent controls and 64 of 80 historical controls respond. With a \(\operatorname{Beta}(1,1)\) prior, evaluate \(a=0.50\). The historical contributions are \(0.50(64)=32\) responses and \(0.50(16)=8\) nonresponses. Thus \(A=1+10+32=43\), \(B=1+10+8=19\), and \(A+B=62\).
The posterior is \(\operatorname{Beta}(43,19)\); its mean is a response probability of about 0.6935 and its standard deviation is about 0.05808. The computation checks both historical components separately. Discounting only the historical successes, or discounting the entire posterior after combining the data, would not implement this power prior.
Worked Example: Choosing a Grid Around a Clinical Benchmark
Suppose, hypothetically, 12 of 30 concurrent controls and 45 of 60 historical controls respond. Use a \(\operatorname{Beta}(1,1)\) prior. Before analysis, the team identifies \(a=0\), \(0.20\), \(0.50\), and \(1\) as useful scenarios and a response probability of 0.55 as a clinical benchmark. The posterior parameters are \(A=13+45a\) and \(B=19+15a\).
At \(a=0\), the posterior is \(\operatorname{Beta}(13,19)\), with mean \(13/32=0.4063\). At \(a=0.20\), it is \(\operatorname{Beta}(22,22)\), with mean \(22/44=0.5000\). At \(a=0.50\), it is \(\operatorname{Beta}(35.5,26.5)\), with mean \(35.5/62=0.5726\). At \(a=1\), it is \(\operatorname{Beta}(58,34)\), with mean \(58/92=0.6304\).
The posterior mean passes the benchmark between \(a=0.20\) and \(a=0.50\). That observation signals that this particular summary is sensitive to the discount; it does not mean that the probability \(P(p>0.55\mid y_c,y_h,a)\) has been calculated, nor does it establish a decision. If that posterior probability is the decision-relevant summary, calculate and report it at every grid value using the same benchmark. The grid is clinically interpretable because its values and benchmark were specified as scenarios, rather than chosen to make the posterior mean cross 0.55.
Interpreting Variation and Finding Problems
Read the results as conditional statements: “At \(a=0.50\), the posterior mean response probability is …” This makes the role of the assumption explicit. If estimates or decision-relevant summaries change little over a clinically plausible range, the finding is relatively insensitive to that range. If they change materially, the report should identify which summaries move, where the change occurs, and which assumptions about historical relevance drive it. A narrow posterior at a large discount is not evidence by itself that the historical data are exchangeable with the current controls.
Use the grid to check calculations and model behavior, not to assume a universal direction of change. When historical and current response rates differ, increasing the discount will generally pull the posterior mean toward the historical rate in this conjugate model. If the rates are similar, the mean may change little. Posterior uncertainty can also change with \(a\); do not assume every posterior summary must vary monotonically. Verify calculated parameters, denominators, and summaries at the boundaries and at intermediate values.
Warning signs include a grid that omits the no-borrowing reference, an undocumented prior, different endpoint definitions across scenarios, or summaries that silently change from one row to another. A large discrepancy between historical and concurrent outcomes is a substantive warning about the common-\(p\) assumption, even if the posterior calculations are correct. Conversely, a stable result across a coarse grid does not prove robustness to historical-data bias or to values between grid points. If there is sharp variation, add prespecified or transparently justified intermediate values and investigate whether the model assumptions are defensible.
How This Analysis Differs From Alternatives
The \(a=0\) analysis uses only concurrent controls, while \(a=1\) gives the historical likelihood full weight under the shared-parameter model. A fixed-discount sensitivity analysis places these reference cases beside intermediate, explicitly chosen values. It differs from a single-discount analysis by making dependence on the discount visible; it differs from a random-discount or model-averaging approach because it does not assign probabilities to discounts or combine their posteriors. Those alternatives answer different questions. A grid of fixed analyses describes conditional results across assumptions; it does not identify which discount is true or replace clinical and data-quality judgments about historical relevance.
The approach is limited by the model and the grid. A common response probability may be too simple when studies differ in important ways, and discounting alone cannot represent every source of noncomparability. A finite grid only evaluates its chosen points. Posterior summaries also depend on the prior and on the selected estimand. Report enough detail for another analyst to reproduce each row: counts, prior parameters, discount values, posterior parameters, and the definition of every summary.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- With \(a=0\), what contribution do historical successes and failures make to the posterior parameters?
- For a \(\operatorname{Beta}(1,1)\) prior, 8 responses among 20 concurrent controls, and 30 responses among 50 historical controls, calculate the posterior parameters at \(a=0.50\).
- Why is a posterior mean of 0.60 not the same statement as \(P(p>0.55\mid\text{data})=0.60\)?
- If increasing \(a\) moves the posterior mean toward the historical rate, what should be checked before interpreting that movement as clinically informative?
Solutions. (1) At \(a=0\), both historical contributions are zero, so the posterior uses only the prior and concurrent likelihood. (2) The historical data have 30 responses and 20 nonresponses, contributing 15 and 10 after discounting. Thus \(A=1+8+15=24\) and \(B=1+12+10=23\), giving \(\operatorname{Beta}(24,23)\). (3) The posterior mean summarizes the average value of \(p\); the posterior probability is the area of the posterior distribution above 0.55. They are different summaries and require different calculations. (4) Check that outcome definitions and denominators align, that a shared response probability is a defensible model, and that the prior and discount grid were specified consistently. Discounting does not by itself remove bias or explain historical-current differences.