Why a Submission Must Explain Borrowing
External data can strengthen a clinical-trial analysis when they provide relevant information about the treatment context, disease course, or control outcome. But borrowing makes a substantive assumption: some part of the external evidence is informative for the parameter in the current trial. A submission should therefore explain not only which evidence is used, but also how much influence it receives, under what conditions, and how the resulting estimates differ from reasonable alternatives.
This tutorial focuses on the justification and consequences of borrowing choices. The earlier material on Assessing Relevance and Quality of External Data concerns whether a source is suitable to consider. Here, the question is how a chosen source enters the analysis. Borrowing cannot repair a mismatch in population, endpoint, follow-up, or outcome measurement. Nor does a discount parameter turn an unsuitable source into a suitable one: it only reduces the source's mathematical influence under the specified model.
A Power-Discounted Binomial Model
A simple model makes the implications of a borrowing choice explicit. Let \(Y_T\) be the number of responses among \(n_T\) current-trial participants, and let \(p_T\) be the current-trial response probability. Suppose \(Y_T\mid p_T\sim\operatorname{Binomial}(n_T,p_T)\). Let \(Y_E\) be the number of responses among \(n_E\) participants in an external data set. For a basic power-prior analysis, the external likelihood is raised to a prespecified power \(a\), where \(0\leq a\leq1\).
This expression is based on treating the external outcomes as informative about \(p_T\), after applying the discount. It does not mean that external participants were randomized into the current trial or that the two data sets are automatically exchangeable. The analysis still depends on the appropriateness of the shared parameter and on the quality and comparability of the evidence. With fixed \(a\), the posterior follows by multiplying the current-trial likelihood, the powered external likelihood, and the beta prior. The beta prior is conjugate to the binomial likelihood.
The powers of \(p_T\) and \(1-p_T\) identify the posterior beta parameters. For fixed \(a\), any likelihood factor independent of \(p_T\) cancels when the posterior is normalized. The discount therefore contributes \(aY_E\) response-equivalents and \(a(n_E-Y_E)\) nonresponse-equivalents. These can be fractional; they are a mathematical representation of weighted evidence, not literal fractional people.
In this binomial setting, the external effective sample size is \(a n_E\). That is an accounting description of the likelihood contribution, not a universal measure of how much information a data set contains. Information also depends on the outcome model, the prior, data quality, and whether the external evidence really bears on the current parameter.
Worked Consequences of Alternative Discounts
Worked Example: Comparing No, Partial, and Full Borrowing
Hypothetical data. Suppose the current trial observes 13 responses and 9 nonresponses among 22 participants. The external data contain 30 responses and 20 nonresponses among 50 participants. Use a baseline prior \(p_T\sim\operatorname{Beta}(1,1)\). Calculate the posterior under \(a=0\), \(a=0.2\), and \(a=1\).
At \(a=0\), the external data contribute nothing. The posterior is \(\operatorname{Beta}(1+13,1+9)=\operatorname{Beta}(14,10)\), with mean \(14/24=0.5833\). The beta posterior standard deviation is \(\sqrt{\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)]}\), so here it is \(\sqrt{140/(24^2\cdot25)}=0.09860\).
At \(a=0.2\), the external data contribute \(0.2(30)=6\) response-equivalents and \(0.2(20)=4\) nonresponse-equivalents. The posterior is \(\operatorname{Beta}(1+13+6,1+9+4)=\operatorname{Beta}(20,14)\). Its mean is \(20/34=0.5882\), and its standard deviation is \(\sqrt{20(14)/(34^2\cdot35)}=\sqrt{280/40460}\approx0.08319\).
At \(a=1\), the external data contribute all 30 responses and 20 nonresponses. The posterior is \(\operatorname{Beta}(44,30)\), with mean \(44/74=0.5946\) and standard deviation \(\sqrt{44(30)/(74^2\cdot75)}=\sqrt{1320/410700}\approx0.05669\).
The posterior means rise modestly across these choices because the external response rate, \(30/50=0.6000\), is slightly higher than the current-trial rate, \(13/22=0.5909\). The posterior standard deviation decreases as more evidence is included under this shared-parameter model. These summaries show consequences of the assumed discounts; they do not establish which discount is justified.
How to Justify the Amount of Borrowing
A defensible borrowing choice connects the statistical weight to evidence that was assessed before looking at the desired result. The rationale should identify the external source and explain the relevant similarities and differences between it and the current trial. Useful considerations include eligibility criteria, baseline risk, treatment setting, endpoint definition, assessment schedule, follow-up duration, concomitant care, and data completeness. The explanation should distinguish features that support transportability from features that limit it.
The submission should then make the translation from those considerations to the model explicit. For a fixed power prior, report \(a\), the external sample size, and the resulting effective sample size \(a n_E\). If several sources or cohorts are used, state whether each has its own discount \(a_j\), and show how each enters the likelihood. If borrowing is restricted to a subset, define that subset using reproducible criteria. Avoid vague statements such as “the external data were down-weighted because they were less reliable” without specifying the weight, the rule for applying it, and the evidence that motivates the rule.
Identify the population, endpoint, time point, and parameter to which the external evidence is intended to contribute.
Give the model, discount parameter or source-specific weights, and any prespecified inclusion or exclusion conditions.
Report the external sample size and its effective contribution under the chosen weights, alongside the current-trial information.
Compare the chosen analysis with clearly defined alternatives, including no borrowing, and explain what changes in the estimate and uncertainty.
State what the discount does not address, including residual differences between sources and assumptions about the endpoint and data generation.
Worked Example: Translating a Discount Into Effective Evidence
Hypothetical data. An external source has 52 responses and 28 nonresponses among 80 participants. The current trial has 10 responses and 10 nonresponses among 20 participants. A prespecified analysis uses \(a=0.25\) and a baseline \(\operatorname{Beta}(1,1)\) prior.
The external effective sample size is \(0.25(80)=20\) participants. Its contributions are \(0.25(52)=13\) response-equivalents and \(0.25(28)=7\) nonresponse-equivalents. Thus the posterior is \(\operatorname{Beta}(1+10+13,1+10+7)=\operatorname{Beta}(24,18)\). Its mean is \(24/42=0.5714\), and its standard deviation is \(\sqrt{24(18)/(42^2\cdot43)}=\sqrt{432/75852}\approx0.07546\).
With no external borrowing, the posterior would be \(\operatorname{Beta}(11,11)\), with mean \(0.5000\) and standard deviation \(\sqrt{121/(22^2\cdot23)}\approx0.1043\). The external response rate is \(52/80=0.6500\), above the current rate \(10/20=0.5000\), so partial borrowing pulls the posterior mean upward and reduces its standard deviation. A submission should explain why a quarter-power contribution is appropriate; the arithmetic alone cannot justify that choice.
Conditions and Multiple Sources
Some analyses use a fixed discount for all external observations; others define distinct source-specific contributions. A source-specific power model can be written with external cohorts \(j=1,\ldots,J\), response counts \(Y_{Ej}\), sizes \(n_{Ej}\), and powers \(a_j\). Under the same beta-binomial assumptions, the posterior parameters add each cohort's weighted response and nonresponse counts. This makes it possible to reflect prespecified differences in relevance or quality without pretending all sources are interchangeable.
Worked Example: Different Weights for Two External Sources
Hypothetical data. A current trial has 12 responses and 8 nonresponses. Source A has 18 responses and 12 nonresponses, with prespecified \(a_A=0.5\). Source B has 8 responses and 12 nonresponses, with prespecified \(a_B=0.25\). Use \(p_T\sim\operatorname{Beta}(1,1)\).
Source A contributes \(9\) response-equivalents and \(6\) nonresponse-equivalents. Source B contributes \(2\) response-equivalents and \(3\) nonresponse-equivalents. The posterior is therefore \(\operatorname{Beta}(1+12+9+2,\ 1+8+6+3)=\operatorname{Beta}(24,18)\), with mean \(24/42=0.5714\). The weighted external total is \(0.5(30)+0.25(20)=20\) effective participants.
For comparison, using Source A alone gives \(\operatorname{Beta}(1+12+9,\ 1+8+6)=\operatorname{Beta}(22,15)\), with mean \(22/37=0.5946\). Source B has a lower observed response rate, \(8/20=0.4000\), so its partial contribution pulls the combined posterior mean downward. The example illustrates why source-specific weights and source-specific contributions should be reported rather than summarized only as “external data were used.”
Conditions for borrowing should be operational. A submission might specify that a source is eligible only if it uses the same endpoint definition and a prespecified follow-up window, or that a cohort is excluded if required baseline variables are unavailable. The criteria should be determined without selecting them to produce a preferred treatment conclusion. If the weight itself is estimated from the current data rather than fixed, the model, prior for that weight, and role of the data in determining it must be stated; this is a different analysis from simply choosing a fixed \(a\).
Comparing Approaches and Recognizing Limitations
| Approach | What it assumes or does | What to explain |
|---|---|---|
| No borrowing | Uses the current-trial likelihood and baseline prior, with no external likelihood contribution. | Why external evidence is not used, or why this is a useful reference analysis. |
| Full borrowing | Uses the external likelihood at full power for the target parameter. | Why the shared-parameter assumption is plausible and how differences between sources are addressed. |
| Fixed power discount | Uses a prespecified fraction of the external likelihood. | Why that fraction is defensible and what effective sample size it implies. |
| Source-specific discounts | Assigns different fixed powers to distinct external sources or cohorts. | Why the sources receive different weights and how each contribution is calculated. |
A smaller discount generally makes the external observations less influential in a power-prior model, but it does not guarantee a particular change in every posterior summary. The direction of the change depends on how the external outcomes compare with the current data, and the uncertainty also depends on the model and prior. Report posterior quantities under alternatives rather than relying on a verbal claim that a model is “conservative.”
A power discount is not a correction for unmeasured differences, selection bias, or outcome misclassification. It also does not demonstrate that external and current participants are exchangeable. Relevant diagnostics include checking that counts, follow-up, and endpoint definitions used in the likelihood match the submission's description; verifying the effective contribution numerically; and confirming that the stated eligibility rules were applied as prespecified. If the posterior is unexpectedly dominated by external information, first check the coding and the implied effective sample size, then explain the model's assumptions clearly.
The alternatives serve different purposes. No borrowing provides a direct current-data reference. Full borrowing represents the strongest shared-parameter assumption in this simple model. A fixed discount makes the degree of borrowing transparent, while source-specific discounts can represent deliberate distinctions among sources. None is automatically preferable in all settings. The choice should be justified by the scientific relationship between sources and the intended regulatory question, with the consequences of other plausible choices visible to the reviewer.
Common Mistakes
- Calling a discount “conservative” without quantifying it. A label does not reveal the contribution. State \(a\), the effective sample size, and the weighted event counts.
- Treating fractional contributions as actual participants. A power of \(0.2\) applied to 30 responses contributes six response-equivalents to the likelihood, not six newly observed people.
- Assuming down-weighting removes incompatibility. A lower weight reduces influence but does not establish that different populations or endpoints measure the same parameter.
- Reporting only the chosen model. Without a no-borrowing reference and clearly defined alternative weights, readers cannot see how much the conclusion depends on the borrowing choice.
- Choosing a weight after seeing which one gives the preferred result. Prespecify the rule and justify it from source relevance and quality, not from the desired posterior conclusion.
Check Your Understanding
Try each question, then compare your reasoning with the brief solutions.
- An external binomial data set has 40 responses and 60 nonresponses among 100 participants. With \(a=0.3\), what are its effective sample size and weighted event contributions?
- Why is it incorrect to interpret \(a=0.3\) as saying that 30% of the external participants were randomly selected into the current trial?
- In a power-prior model, what should a submission report in addition to the chosen discount value?
- The external response rate is lower than the current-trial response rate. Under the shared-\(p_T\) model, what direction would partial borrowing generally move the posterior mean, and what qualification is needed?
Solutions. (1) The effective sample size is \(0.3(100)=30\); the contributions are \(0.3(40)=12\) response-equivalents and \(0.3(60)=18\) nonresponse-equivalents. (2) The discount raises a likelihood to a power; it reweights evidence mathematically and does not describe participant selection or randomization. (3) Report the external source and size, model and assumptions, weighted contributions or effective sample size, borrowing conditions, and consequences under alternatives such as no borrowing. (4) The posterior mean will generally be pulled toward the lower external rate, but the exact change depends on the current data, baseline prior, and model; the direction is not a justification for using the source.