Calendar-Time Decay Power Priors

Why Calendar Time Can Determine a Borrowing Weight

Historical control data may be less informative for a current trial when clinical practice, eligibility, outcome assessment, or supportive care has changed over time. Calendar-time decay provides a prespecified rule for assigning smaller power-prior weights to older studies. It turns study age into a numerical discount; it does not establish that an older study is biased or that a newer study is comparable.

The rule needs a clear time origin. For each historical study, define its age as the elapsed calendar time from a protocol-specified reference date for that study to a protocol-specified anchor date, such as the start of the current trial. A study-level date might be the midpoint of recruitment or the end of recruitment. The choice matters: it should be stated in advance and used consistently. If the study spans a long period, a single study-level age is an approximation.

Definition: Let $d_j\geq 0$ be the age of historical study $j$, measured in a specified time unit, and let $\lambda\geq 0$ be a prespecified decay rate measured in the reciprocal unit. Its calendar-time decay weight is $$ a_j=\exp(-\lambda d_j). $$ Thus $0<a_j\leq 1$ for finite $d_j$. A weight of 1 applies no time discount; a weight near 0 gives that study little influence through its likelihood. If $\lambda=0$, all studies receive weight 1, regardless of age.

The exponential rule is often easier to communicate through its half-life $h>0$, the time required for a study's weight to fall to one half. The two parameterizations are equivalent:

\lambda=\frac{\log(2)}{h}, \qquad a_j=\exp\left(-\frac{\log(2)}{h}d_j\right)=2^{-d_j/h}.

The time unit for $d_j$ and $h$ must match. If age is measured in years, $h$ is in years and $\lambda$ is per year. A half-life is not an estimate of how quickly clinical practice truly changes; it is a convenient way to express a chosen decay rule.

From Calendar Age to a Power Prior

Suppose historical study $j$ contributes likelihood $L_j(\theta)$ for a parameter $\theta$, and let $\pi_0(\theta)$ be the prior before borrowing those historical data. With fixed, prespecified study weights, the historical contribution to a power prior is

\pi(\theta\mid D_0,\mathbf{a}) \propto \pi_0(\theta)\prod_{j=1}^{J}L_j(\theta)^{a_j}, \qquad a_j=\exp(-\lambda d_j).

Here $D_0$ denotes the historical data and $\mathbf{a}$ the collection of weights. Raising a likelihood to a power discounts its contribution to the parameter distribution. A weight is therefore not the probability that a study is relevant, nor the fraction of patients who would have been eligible for the current trial. In a simple binomial model, a weight can act like a fractional contribution to the likelihood, but the original study still had its actual observed sample size.

This construction assumes that the study likelihoods can be expressed in terms of a parameter or model that is meaningful for the current borrowing problem. The alignment of endpoint definitions and populations, the handling of missing outcomes, and the assessment of data reliability remain important. Calendar decay does not repair incompatibility, missing-data problems, or unreliable measurements. It only applies the chosen age-based rule to likelihood contributions.

Worked Example: Weights From a Four-Year Half-Life

Suppose three hypothetical historical control studies enrolled 50, 40, and 30 participants, respectively. Their prespecified study ages are 2, 5, and 8 years. Use a four-year half-life to calculate their decay weights and weighted participant-count contributions.

The rate is $\lambda=\log(2)/4=0.1733$ per year. The weight for age $d_j$ is $2^{-d_j/4}$:

a_1=2^{-2/4}=0.7071,\qquad a_2=2^{-5/4}=0.4204,\qquad a_3=2^{-8/4}=0.2500.

The weights decrease with age, as intended. Multiplying each weight by its study's observed participant count gives:

50(0.7071068)=35.3553,\qquad 40(0.4204482)=16.8179,\qquad 30(0.2500)=7.5000.

Using the unrounded weights, the total weighted count is $50(2^{-2/4})+40(2^{-5/4})+30(2^{-8/4})=59.673267\ldots$, or approximately 59.67 weighted participants. The displayed contributions rounded separately sum to 59.68; that difference is only rounding. The weighted total is a useful summary in this simple setting, not a literal count of participants.

Calculation Checks and a Short Justification

The half-life formula follows directly from the definition. At age $h$, the weight must equal one half, so $\exp(-\lambda h)=1/2$. Taking logarithms gives $-\lambda h=-\log(2)$, hence $\lambda=\log(2)/h$. Substitution into the exponential weight gives $a_j=2^{-d_j/h}$. This also verifies the boundary and monotonicity properties: age zero gives weight 1, and increasing age reduces weight when $\lambda>0$.

\frac{d}{d d_j}\exp(-\lambda d_j) =-\lambda\exp(-\lambda d_j)\leq 0.

For a positive rate, the derivative is strictly negative, so older studies have strictly smaller weights. If the rate is zero, the derivative is zero and there is no decay. These are basic checks for a calculation or implementation. Also check that all ages use the same origin, all durations use the same unit, and the rate has the reciprocal unit. Entering an age in months with a rate specified per year can produce a numerically valid but substantively wrong weight.

Worked Example: Convert a Rate to Weights

A protocol specifies $\lambda=0.18$ per year, with study ages of 0.5, 2, and 6 years. Calculate each weight and check its interpretation.

Using $a_j=\exp(-\lambda d_j)$:

a_1=\exp[-(0.18)(0.5)]=\exp(-0.09)=0.9139,

a_2=\exp[-(0.18)(2)]=\exp(-0.36)=0.6977, \qquad a_3=\exp[-(0.18)(6)]=\exp(-1.08)=0.3396.

The oldest study receives the smallest weight. The rate corresponds to a half-life of $\log(2)/0.18=3.851$ years. In particular, a study aged 3.851 years has weight $\exp[-0.18(3.851)]$, approximately 0.500. These weights describe likelihood discounting; they do not mean that the respective studies have a 91.39%, 69.77%, or 33.96% probability of being relevant.

Worked Use in a Binary Control Model

A simple example shows how age-based weights enter an analysis. Assume each historical study has a binomial likelihood for a common control-event probability $p$: study $j$ observes $y_j$ events among $n_j$ participants, with $Y_j\mid p\sim\mathrm{Binomial}(n_j,p)$. For this illustration, assume conditional independence across studies, a common $p$, and a $\mathrm{Beta}(1,1)$ prior. The common-probability assumption is substantial: the calculation is not evidence that control risks are truly identical across study eras.

Worked Example: Weighted Historical Events and Current Controls

Consider two hypothetical historical studies. The first observed 18 events among 60 participants at an age of 1 year; the second observed 12 among 50 participants at an age of 3 years. A current trial has 9 events among 20 control participants. Use a three-year half-life, the $\mathrm{Beta}(1,1)$ prior, and fixed weights for the historical likelihoods.

The historical weights are $a_1=2^{-1/3}=0.7937$ and $a_2=2^{-3/3}=0.5000$. For a binomial likelihood, $L_j(p)\propto p^{y_j}(1-p)^{n_j-y_j}$. Raising it to $a_j$ multiplies both exponents by $a_j$. Combining the current control likelihood with the two powered historical likelihoods therefore gives posterior parameters

\alpha=1+9+(0.7937005)(18)+(0.5000)(12)=30.2866,

\beta=1+(20-9)+(0.7937005)(60-18)+(0.5000)(50-12)=64.3354.

Thus the posterior distribution is $\mathrm{Beta}(30.2866,64.3354)$. Its mean is $30.2866/(30.2866+64.3354)=0.3201$, or about 32.01% for the control-event probability under this model. The weighted historical count is $0.7937(60)+0.5000(50)=72.622$ participants. It is not the observed historical sample size, which is 110, and it is not a model-free effective sample size. The posterior calculation depends on the common-$p$ binomial assumptions and the stated prior.

Choosing, Comparing, and Checking the Decay Rule

Calendar-time decay is one way to set fixed study-specific power-prior weights. It is useful when recency is a scientifically motivated concern and a simple, transparent rule is desired. It should be compared with plausible alternatives before being adopted.

Approach	How it assigns influence	Key distinction
Calendar-time decay	Uses a prespecified function of study age, such as $\exp(-\lambda d_j)$.	Older means less weight by construction; the rule does not inspect outcomes or measure reliability.
Common fixed discount	Applies the same chosen discount to each historical study.	Does not distinguish studies by age.
Reliability-based weighting	Uses an assessment of data quality or reliability to set weights.	Targets a different concern; a recent study need not be reliable.
Explicit time-varying control model	Represents calendar-time change in the model for the control outcome.	Models change rather than merely reducing older likelihood contributions.

These approaches answer different questions and are not automatically interchangeable. A prespecified decay rate avoids choosing weights after seeing which value gives a preferred result, but it does not make the rate uniquely correct. Sensitivity analysis across defensible fixed rates can show how dependent conclusions are on that choice; the earlier material on sensitivity analysis across fixed discounts is relevant to that comparison.

Useful diagnostics include tabulating study date, calculated age, weight, and weighted contribution; plotting weight against age; and checking that the weights follow the prespecified curve. Investigate a weight that rises with age, exceeds 1, or changes unexpectedly when the date unit is converted. Confirm that “age” was calculated from the stated anchor rather than from inconsistent dates across studies. In a likelihood calculation, compare the study-specific weighted exponents with the original event and non-event counts to catch accidental use of $1-a_j$ or repeated application of a weight.

A central limitation is that calendar age is only a proxy for possible change. It cannot detect an abrupt change that occurred between two otherwise close dates, nor can it guarantee comparability for two studies conducted at the same time. A smooth exponential curve may be inappropriate if the scientific concern suggests a threshold or a different temporal pattern. The chosen half-life should therefore be justified scientifically and specified before examining the current outcomes where possible. Even a small weight can leave a historical study influential if its likelihood is highly informative, so the weight alone is not a complete measure of impact.

Common Mistakes

Confusing a weight with a relevance probability. A value of 0.5 means that the likelihood is raised to the power 0.5 under this construction; it does not mean there is a 50% probability the study applies.
Mixing time units. A rate of 0.2 per year must be multiplied by age in years, not age in months without conversion.
Calling weighted counts observed counts. Weighted counts are fractional likelihood contributions in a model, not a revision of the historical data.
Assuming recency resolves incompatibility. A recent study can still differ in endpoint, population, or data quality. Decay is not a substitute for evaluating those issues.
Adding rounded contributions and reporting that as the exact total. Sum the unrounded contributions first, then round the total; separately rounded terms can add to a slightly different displayed value.

Key takeaway: Calendar-time decay translates a prespecified rate or half-life into a power-prior weight for each historical study. Its meaning depends on the chosen time origin, the likelihood model, and the scientific assumptions that make historical borrowing defensible.

Check Your Understanding

Answer each question, then compare with the solutions below.

A study is 6 years old and the prespecified half-life is 3 years. What is its weight?
For $\lambda=0.10$ per year, calculate the weight for a study aged 4 years. What half-life does this rate imply?
Explain why a power-prior weight of 0.4 is not a 40% probability that a study is relevant.
Name two assumptions or limitations that should be checked before interpreting weighted historical data in a common-probability binomial model.

Solutions. (1) $a=2^{-6/3}=2^{-2}=0.2500$. (2) $a=\exp[-(0.10)(4)]=\exp(-0.4)=0.6703$. The half-life is $\log(2)/0.10=6.931$ years. (3) The weight is an exponent applied to the study likelihood, not a probability model for whether the study is relevant. (4) Examples include the common control-event probability and conditional independence assumptions; also assess endpoint and population compatibility, missing outcomes, and reliability. Calendar decay itself does not resolve these issues.