Priors for Event Rates With Unequal Follow-Up

Why Follow-Up Time Belongs in the Model

In a clinical trial, participants may be observed for different lengths of time. A participant followed for two years has more opportunity to experience an event than one followed for two months. Comparing raw event counts without accounting for this difference can therefore be misleading. A Poisson rate model handles unequal follow-up by treating each participant’s observed time as exposure: more exposure contributes more opportunity for events.

The model does not make unequal follow-up harmless by itself. It assumes that a common event rate is meaningful over the included participants and observation periods, and that the recorded exposure represents time at risk. A prior should be specified on that rate scale, with its units stated explicitly.

Poisson Likelihood and Gamma Prior

Definition: Let $Y_i$ be the event count for participant $i$, and let $T_i$ be that participant’s known exposure time, measured in a stated unit such as person-years. Under a common-rate Poisson model, $$ Y_i\mid \lambda,T_i \sim \operatorname{Poisson}(\lambda T_i), $$ independently across participants, where $\lambda$ is the event rate per unit of exposure. A Gamma prior in the shape-rate parameterization is $$ \lambda \sim \operatorname{Gamma}(a,b), \qquad \pi(\lambda)=\frac{b^a}{\Gamma(a)}\lambda^{a-1}e^{-b\lambda}, \quad \lambda>0. $$ Here $a>0$ is the shape and $b>0$ is the rate parameter of the Gamma distribution. The prior mean is $a/b$, and its variance is $a/b^2$. If $\lambda$ is measured in events per person-year, $b$ has units of person-years.

The Poisson assumption means that, conditional on the rate and exposure, event counts have the specified Poisson distribution. In a recurrent-event setting, it corresponds to a constant event rate and independent increments over time. Participants are assumed to share the modeled rate, or to be sufficiently comparable for a single common rate to be a useful approximation. Exposure must be measured on the time participants were actually at risk; follow-up after an event may not count as exposure if the outcome is a first event.

With unequal follow-up, the relevant summaries are the total count $Y=\sum_i Y_i$ and total exposure $T=\sum_i T_i$. The model does not require all $T_i$ to be equal. The prior is about $\lambda$, not about an event probability over a fixed time window. The distinction matters: a rate has units, while a probability does not.

Eliciting the Prior on a Useful Scale

Elicit prior beliefs in the units clinicians can interpret. For example, ask what average event rate is plausible and how uncertain that rate is, both per person-year. If the elicited prior mean is $m$ and standard deviation is $s$, matching the Gamma mean and variance gives $$ a=\frac{m^2}{s^2}, \qquad b=\frac{m}{s^2}. $$ These parameters are positive whenever $m>0$ and $s>0$. A Gamma prior places probability only on positive rates, but its tail shape still needs checking: a mathematically valid prior may assign substantial probability to clinically implausible rates.

The exposure scale $b$ helps describe how concentrated the prior is, but it is not automatically a literal number of prior participants. Its interpretation depends on the chosen rate parameterization and prior shape. In particular, a prior specified per person-year must not be applied unchanged to a rate expressed per 100 person-years.

Worked Example: Eliciting a Prior From a Mean and Standard Deviation

Suppose a clinical team considers a mean rate of $0.020$ events per person-year plausible, with a prior standard deviation of $0.010$ events per person-year. Find the shape-rate Gamma prior.

The prior variance is $s^2=(0.010)^2=0.000100$ in squared rate units. Therefore, $$ a=\frac{(0.020)^2}{(0.010)^2} =\frac{0.000400}{0.000100}=4, \qquad b=\frac{0.020}{0.000100}=200 \text{ person-years}. $$

The resulting prior is $\lambda\sim\operatorname{Gamma}(4,200)$, with $\lambda$ measured in events per person-year. Its mean is $4/200=0.020$ events per person-year, and its standard deviation is $\sqrt{4/200^2}=0.010$ events per person-year, as required.

Before using it, the team should inspect prior implications over clinically relevant exposure amounts. A prior can match the stated mean and standard deviation yet still put too much probability on very high rates.

Updating With Unequal Follow-Up

For independent Poisson counts, the likelihood for $\lambda$, omitting factors that do not depend on $\lambda$, is $$ L(\lambda)\propto \prod_i (\lambda T_i)^{Y_i}e^{-\lambda T_i} \propto \lambda^{\sum_iY_i}e^{-\lambda\sum_iT_i}. $$ Combining this likelihood with the Gamma prior gives the conjugate posterior:

Theorem: If $\lambda\sim\operatorname{Gamma}(a,b)$ and the counts follow the independent Poisson model with known exposures $T_i$, then $$ \lambda\mid \{Y_i,T_i\}\sim \operatorname{Gamma}\left(a+\sum_iY_i,\;b+\sum_iT_i\right). $$ The posterior mean is $\left(a+\sum_iY_i\right)/\left(b+\sum_iT_i\right)$.

The reason is that multiplying the prior density by the likelihood gives a kernel proportional to $\lambda^{a+\sum_iY_i-1}e^{-(b+\sum_iT_i)\lambda}$, which is the kernel of a Gamma distribution with the stated shape and rate. Thus, unequal participant times enter through their sum; the total count and total exposure, not an unweighted average of individual rates, determine this update.

Worked Example: Updating With Different Amounts of Follow-Up

Use the prior $\operatorname{Gamma}(4,200)$ from the previous example. In a hypothetical trial, one group contributes 20 person-years and has 2 events; another contributes 50 person-years and has 1 event. Assume both groups are appropriately modeled by the same event rate. Find the posterior mean rate.

The total count is $Y=2+1=3$ events, and the total exposure is $T=20+50=70$ person-years. The posterior is $$ \lambda\mid \text{data}\sim \operatorname{Gamma}(4+3,\;200+70) =\operatorname{Gamma}(7,270). $$

Its posterior mean is $$ E(\lambda\mid\text{data})=\frac{7}{270} =0.02593 \text{ events per person-year}. $$ For comparison, the likelihood estimate $Y/T$ is $3/70=0.04286$ events per person-year. The posterior mean is lower because the prior mean is $0.020$ events per person-year and contributes information alongside the observed counts and exposure. The raw total of 3 events alone would not convey the same information without the 70 person-years of exposure.

Rate Units and Prior Predictive Checks

A rate’s numerical value changes when its time unit changes, even though the underlying clinical belief does not. If $\lambda$ is measured in events per person-year and $r=100\lambda$ is the rate in events per 100 person-years, then $$ \lambda\sim\operatorname{Gamma}(a,b) \quad\Longrightarrow\quad r\sim\operatorname{Gamma}\left(a,\frac{b}{100}\right). $$ This follows from the change of scale for a Gamma random variable: multiplying it by 100 divides its shape-rate parameter $b$ by 100. Exposures must use compatible units in the likelihood. A rate per 100 person-years paired with exposure in person-years would produce an incorrect expected count unless the units are converted.

Prior predictive checks make the implications of a prior concrete. For a future exposure $T_*$, a Poisson count conditional on $\lambda$ has mean $\lambda T_*$. Integrating over the Gamma prior gives $$ P(Y_*=0\mid T_*)= \int_0^\infty e^{-\lambda T_*}\pi(\lambda)\,d\lambda =\left(\frac{b}{b+T_*}\right)^a. $$ The final equality follows by integrating the Gamma density with its exponential term updated from $b$ to $b+T_*$. This probability is a useful check on whether the prior predicts plausible event counts for the planned exposure.

Worked Example: Checking a Prior at a Different Rate Scale

For the prior $\lambda\sim\operatorname{Gamma}(4,200)$, express the rate in events per 100 person-years and calculate the prior predictive probability of no events over 5 person-years of future exposure.

Let $r=100\lambda$. Then $r\sim\operatorname{Gamma}(4,2)$: its mean is $4/2=2$ events per 100 person-years, the same belief as $0.020$ events per person-year. For the predictive calculation, use the original per-person-year rate and $T_*=5$ person-years: $$ P(Y_*=0\mid T_*=5) =\left(\frac{200}{200+5}\right)^4 =\left(\frac{40}{41}\right)^4 \approx 0.9060. $$ Thus the prior assigns about a 90.60% probability to observing no events over 5 person-years. This is plausible for a low rate, but the judgment depends on the clinical context. The scale conversion changes the numbers used to describe the rate; it does not change this predictive conclusion.

Assumptions, Diagnostics, and Alternatives

Unequal follow-up is accommodated when exposure time is correctly included, but the common-rate model can fail for other reasons. Check whether event rates vary over calendar time, follow-up time, or clinically important participant subgroups. Clustering of events or substantially more count variation than the Poisson model predicts can indicate overdispersion. A plot of counts against exposure, or comparisons of observed and model-predicted counts across exposure bands, can reveal patterns that a single rate conceals. Prior predictive checks should likewise compare plausible event counts across the range of exposure expected in the trial.

The model also assumes that exposure is meaningfully observed and that participants contribute time at risk under the modeled process. Informative loss to follow-up, or stopping follow-up in a way related to event risk, can undermine the interpretation of a simple rate analysis. If rates differ importantly across time or groups, separate rates or a model that represents those differences may be more suitable than one common $\lambda$. If counts are overdispersed, a model allowing extra variation may be needed. The Gamma prior does not repair a misspecified likelihood.

A fixed-follow-up binary model describes whether each participant has an event by a specified time, rather than modeling counts per unit exposure. It can be natural when follow-up windows are essentially the same and the endpoint is binary. The Poisson rate model is useful when exposure varies or repeated events are counted, subject to its assumptions. These approaches target different data structures; neither is automatically preferable in every trial.

Common pitfall: Dividing each participant’s count by their follow-up time and averaging those individual rates gives equal weight to a participant with little exposure and one with much exposure. Under the common-rate Poisson model, the likelihood estimate is total events divided by total exposure, $Y/T$, and the Gamma posterior uses $a+Y$ and $b+T$. Also check units: a prior on events per person-year cannot be combined directly with exposure recorded in months without conversion.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A prior on a rate per person-year has $a=3$ and $b=150$. What are its mean and variance?
A hypothetical study observes 4 events over 80 person-years. With a $\operatorname{Gamma}(3,150)$ prior, give the posterior distribution and posterior mean.
Explain why unequal follow-up does not require equal weighting of participants’ individual event rates in this model.
For the prior in question 2, what is the prior predictive probability of zero events over 10 person-years?

Solutions. (1) The mean is $a/b=3/150=0.0200$ events per person-year. The variance is $a/b^2=3/150^2=0.0001333$ in squared rate units. (2) The posterior is $\operatorname{Gamma}(3+4,150+80)=\operatorname{Gamma}(7,230)$, with mean $7/230=0.03043$ events per person-year. (3) Each count’s expected value is proportional to its exposure, so the joint likelihood depends on total events and total exposure; averaging individual rates equally discards that exposure weighting. (4) The probability is $(150/(150+10))^3=(15/16)^3\approx0.8240$. It is the prior probability of zero events for that future exposure, not a posterior probability about whether the rate equals zero.