A Worked Survival-Control Borrowing Analysis

Why Survival-Control Borrowing Needs Care

Historical controls can add information when the current trial has relatively few control participants or events. For survival outcomes, however, a participant who has not had the event by the end of observation is not an event-free participant with unlimited follow-up: that person contributes a known amount of time without an event and is then right-censored. A borrowing analysis must preserve both pieces of information. It must also make explicit how strongly the historical event and follow-up information is allowed to influence the current analysis.

This tutorial develops a fixed-discount power-prior analysis for a constant-hazard survival model. It is a worked model-based analysis of a control-group event-time distribution, not a guarantee that historical and current participants are exchangeable. The discount controls the historical likelihood's statistical influence; it does not repair differences in populations, outcome definitions, or care.

Model, Censoring, and the Discounted Likelihood

Definition: In an exponential survival model, the event time \(T\) has hazard \(\lambda>0\), constant over time, and survival function \(S(t\mid\lambda)=\exp(-\lambda t)\). For participant \(i\), let \(y_i\) be the observed time to event or censoring, whichever comes first, and let \(\delta_i=1\) if the event is observed and \(0\) if the observation is right-censored. Define \(D=\sum_i\delta_i\) as the event count and \(Y=\sum_i y_i\) as total observed person-time. The likelihood, omitting factors that do not depend on \(\lambda\), is \(L(\lambda)=\lambda^D\exp(-\lambda Y)\).

The likelihood follows because an observed event at time \(y_i\) contributes the exponential density \(\lambda\exp(-\lambda y_i)\), whereas right-censoring at \(y_i\) contributes the probability of surviving to that time, \(\exp(-\lambda y_i)\). Multiplying these contributions gives one factor of \(\lambda\) for each observed event and an exponential term involving all observed time. Thus censored participants contribute follow-up time but no event factor.

Assume the current control data have event count \(D_C\) and total person-time \(Y_C\). Let historical control data have corresponding values \(D_H\) and \(Y_H\). A fixed power-prior weight \(a\), where \(0\leq a\leq1\), raises the historical likelihood to the power \(a\). The current data are not discounted. Use a proper Gamma prior with shape \(\alpha_0>0\) and rate \(\beta_0>0\), written \(\lambda\sim\operatorname{Gamma}(\alpha_0,\beta_0)\).

Model assumptions: Event times follow the stated constant-hazard model within each cohort; censoring is independent of event time under the analysis (or independent conditional on modeled information); and the event definition and time scale are sufficiently comparable for the historical likelihood to inform the current control hazard. The cohorts' observations are independent, and \(a\) is fixed for this calculation. The Gamma rate \(\beta_0\), and all follow-up totals, are measured in time units; \(\lambda\) is measured in inverse time units.

Posterior Distribution and Its Justification

The historical likelihood raised to \(a\) is proportional to \(\lambda^{aD_H}\exp(-a\lambda Y_H)\). Multiplying it by the current likelihood and the Gamma prior kernel gives the posterior kernel:

\begin{aligned} \pi(\lambda\mid\text{data},a) &\propto \lambda^{\alpha_0-1}\exp(-\beta_0\lambda) \lambda^{D_C}\exp(-Y_C\lambda) \left\{\lambda^{D_H}\exp(-Y_H\lambda)\right\}^{a}\\ &= \lambda^{\alpha_0+D_C+aD_H-1} \exp\left[-(\beta_0+Y_C+aY_H)\lambda\right]. \end{aligned}

This is the kernel of a Gamma density. To see the normalization, a Gamma distribution with shape \(\alpha\) and rate \(\beta\) has density proportional to \(\lambda^{\alpha-1}\exp(-\beta\lambda)\) for \(\lambda>0\). Matching powers and rates yields the posterior directly:

Result: With a fixed power-prior weight \(a\), the posterior is \(\lambda\mid\text{data},a\sim\operatorname{Gamma}(\alpha_0+D_C+aD_H,\ \beta_0+Y_C+aY_H)\). Its posterior mean hazard is \((\alpha_0+D_C+aD_H)/(\beta_0+Y_C+aY_H)\). The quantities \(aD_H\) and \(aY_H\) are discounted likelihood contributions, not literal fractional people or a claim that the historical observations were partially observed.

For this posterior, the model-based survival function at time \(t\), averaged over uncertainty in \(\lambda\), is \(E[\exp(-t\lambda)\mid\text{data},a]\). Integrating the Gamma density gives \((\beta/(\beta+t))^\alpha\), where \(\alpha=\alpha_0+D_C+aD_H\) and \(\beta=\beta_0+Y_C+aY_H\). This posterior average is distinct from the survival curve evaluated at the posterior mean hazard, \(\exp[-tE(\lambda\mid\text{data},a)]\).

Worked Example: Combining Current and Historical Right-Censored Data

Consider hypothetical control cohorts with follow-up measured in months. In the historical cohort, 8 participants have events at 4 months, 4 have events at 8 months, 20 are censored at 6 months, and 8 are censored at 12 months. Thus \(D_H=12\), and \(Y_H=8(4)+4(8)+20(6)+8(12)=280\) person-months. In the current cohort, 3 participants have events at 5 months, 2 have events at 10 months, and 15 are censored at 10 months. Thus \(D_C=5\), and \(Y_C=3(5)+2(10)+15(10)=185\) person-months.

Use the hypothetical prior \(\lambda\sim\operatorname{Gamma}(1,20)\), with a fixed historical discount \(a=0.5\). The prior mean hazard is \(1/20=0.0500\) events per person-month. The posterior parameters are

\alpha=1+5+0.5(12)=12,\qquad \beta=20+185+0.5(280)=345\text{ person-months}.

Therefore \(\lambda\mid\text{data},a=0.5\sim\operatorname{Gamma}(12,345)\), with posterior mean hazard \(12/345=0.03478\) events per person-month. The historical crude event rate is \(12/280=0.04286\) per person-month, greater than the current crude rate \(5/185=0.02703\). Accordingly, in this example the posterior mean hazard increases as more historical information is borrowed.

At 12 months, the posterior average survival probability under the model is \((345/(345+12))^{12}=(345/357)^{12}\approx0.6635\). This is the posterior expectation of the model's survival probability at 12 months for a control participant represented by this hazard model. It is not a 12-month survival estimate for every patient subgroup, nor does it account for uncertainty about whether the historical cohort is comparable.

What the Discount Does in This Example

A fixed weight changes the historical likelihood's contribution to both the event-count term and the exposure-time term. In the example, increasing \(a\) adds historical events to the posterior shape and historical person-time to its rate. The posterior mean is a ratio of these updated quantities, so it is not generally correct to say that every increase in borrowing must move the posterior mean in one direction. The direction depends on the historical event rate, current data, and prior.

Worked Example: Comparing Several Fixed Discounts

Use the same data and prior, now comparing \(a=0\), \(0.25\), \(0.5\), and \(1\). For each weight, calculate the posterior shape, rate, and mean hazard from the result above. For example, at \(a=0.25\), the shape is \(1+5+0.25(12)=9\), and the rate is \(20+185+0.25(280)=275\), so the mean hazard is \(9/275=0.03273\) per person-month.

Weight \(a\)	Posterior distribution	Posterior mean hazard (per person-month)
0	\(\operatorname{Gamma}(6,205)\)	0.02927
0.25	\(\operatorname{Gamma}(9,275)\)	0.03273
0.5	\(\operatorname{Gamma}(12,345)\)	0.03478
1	\(\operatorname{Gamma}(18,485)\)	0.03711

Here the means rise with the discount because the historical event rate, \(0.04286\), is greater than the current rate, \(0.02703\). For instance, the full-borrowing calculation is \((1+5+12)/(20+185+280)=18/485=0.03711\) per person-month. This pattern is a property of these data and prior, not a universal rule. The \(a=0\) analysis is the current-data-only posterior; \(a=1\) uses the historical likelihood without discounting.

Diagnostics, Alternatives, and Limitations

Before interpreting the posterior, check whether the model and borrowing setup are credible. A constant hazard implies an exponential event-time distribution, so inspect survival patterns for substantial departures from that shape, such as a changing hazard over time. Check event counts, follow-up totals, time units, event definitions, and censoring indicators against the underlying records. A discrepancy in the coding of censoring can change the likelihood substantially. Also compare the historical and current cohorts' observed survival patterns and follow-up support. Agreement in crude rates alone does not establish exchangeability, while a large discrepancy is a warning that borrowing may be sensitive to assumptions.

The exponential model is parsimonious and makes the conjugate calculation transparent, but a constant hazard may be implausible. A Weibull survival model allows a time-varying hazard under its parametric form, while Kaplan–Meier estimation provides a nonparametric survival estimate rather than this Gamma-conjugate hazard calculation. Those approaches answer related questions but do not automatically perform the same fixed power-prior borrowing. The appropriate alternative depends on the survival shape, data support, and inferential target.

A fixed discount is a sensitivity choice, not evidence that historical data are exchangeable. A practical analysis should show how conclusions change over substantively plausible values of \(a\), rather than presenting one chosen value as definitive. This tutorial conditions on a fixed \(a\); it does not estimate the discount from the data. The constant-hazard model also does not represent competing events as a separate event process. If censoring depends on event risk in a way not handled by the model, or if the cohorts differ materially in baseline risk or care, the simple likelihood combination can be misleading.

Worked Example: The Cost of Treating Censoring as Events

Return to the current cohort in the first example: 5 observed events among 20 participants, with 185 person-months of total observed follow-up. The other 15 participants are censored at 10 months. With \(a=0.5\), the correct posterior is \(\operatorname{Gamma}(12,345)\), so its mean is \(12/345=0.03478\) events per person-month.

Suppose an analyst incorrectly counts all 20 participants as having events but leaves the follow-up total unchanged. The resulting, incorrect shape would be \(1+20+0.5(12)=27\), and the rate would remain \(345\) person-months. The erroneous mean is \(27/345=0.07826\) events per person-month, more than twice the correct value. The error arises because censoring at 10 months establishes survival through 10 months; it does not establish an event at 10 months. The proper likelihood includes that participant's 10 person-months in \(Y_C\), but contributes nothing to \(D_C\).

Common Mistakes

Counting censored participants as events. A right-censored observation contributes observed follow-up time, not an event factor.
Discounting only the historical event count. Raising the historical likelihood to \(a\) discounts both \(D_H\) and \(Y_H\), as shown by the terms \(aD_H\) and \(aY_H\).
Calling \(aD_H\) a fractional number of patients. It is a powered-likelihood contribution; it does not describe a literal fractional cohort.
Assuming more borrowing always raises or lowers the hazard estimate. The direction depends on the relative historical and current event rates and on the prior.
Interpreting the posterior as proof of cohort comparability. The calculation uses a comparability assumption; it does not validate that assumption.
Confusing average survival with survival at the average hazard. In general, \(E[\exp(-t\lambda)\mid\text{data}]\) is not \(\exp[-tE(\lambda\mid\text{data})]\).

Key takeaway: For an exponential survival-control analysis, each observed event contributes a hazard factor and every participant's observed time contributes exposure. A fixed power prior discounts both historical contributions. The resulting posterior is easy to calculate, but its clinical interpretation depends on the survival model, censoring assumptions, and defensibility of borrowing.

Check Your Understanding

Try each question first, then check the brief solutions below.

In an exponential model, a cohort has 4 events and 90 person-months of total follow-up. Write its likelihood up to proportionality.
If the historical cohort has \(D_H=10\), \(Y_H=200\), and \(a=0.3\), what are its discounted event and exposure contributions?
With prior \(\operatorname{Gamma}(2,30)\), current data \(D_C=3,\ Y_C=100\), and historical data \(D_H=6,\ Y_H=150\), calculate the posterior at \(a=0.5\).
Why does a participant censored after 8 months contribute to total person-time but not to the event count?

Solutions. (1) \(L(\lambda)\propto\lambda^4\exp(-90\lambda)\). (2) The contributions are \(aD_H=3\) and \(aY_H=60\) person-months. (3) The posterior is \(\operatorname{Gamma}(2+3+0.5(6),\,30+100+0.5(150))=\operatorname{Gamma}(8,205)\), with mean hazard \(8/205=0.03902\) per person-month. (4) Censoring at 8 months says that the participant remained event-free through the observed 8 months; it provides exposure information but no observed event time.