Unequal Follow-Up And Person-Time

Why Unequal Follow-Up Changes the Count Model

A count is hard to compare across subjects when their observation periods differ. One subject may have 3 events during 6 months of observation, while another has 3 during 4 years. The counts are equal, but the amount of time in which events could be observed is not. A rate model accounts for that difference by using each subject’s person-time: the duration that subject was actually observed and at risk of the event.

This distinction is central to fitting a Poisson model with unequal follow-up. A subject’s expected count depends both on their event rate and on their exposure duration. The rate describes events per unit of time; the expected count is the rate multiplied by the time available for events to occur.

Person-time is not necessarily the same as time since study entry or total calendar time. If a subject enters late, leaves early, is lost to follow-up, or ceases to be at risk, their contribution ends when their observable at-risk time ends. The precise rule depends on the event and study design. For recurrent events, follow-up may continue after an event; for an event that ends the period at risk, it may not.

Why Person-Time Adds Across Follow-Up Periods

The basic justification for person-time comes from how event counts behave over time. Suppose that, conditional on a subject’s covariates, events follow a homogeneous Poisson process with rate \(\lambda\) during the time the subject is at risk. Counts in disjoint time intervals are then independent Poisson random variables, and the mean count in an interval of length \(e\) is \(e\lambda\).

Proof. Let \(Y_j\) be the count in interval \(j\). Under the stated Poisson-process assumption, the \(Y_j\) are independent and \(Y_j\sim\operatorname{Poisson}(\lambda e_j)\). For \(0\leq s\leq1\), the probability generating function of \(Y_j\) is \(E[s^{Y_j}]=\exp\{\lambda e_j(s-1)\}\). Independence gives the generating function of the total count:

This is the probability generating function of a Poisson random variable with mean \(\lambda\sum_{j=1}^m e_j\). Therefore the total count has that distribution, and the accumulated exposure is the sum of the interval lengths. \(\square\)

The constant-rate condition matters. If the rate changes with calendar time or with a subject’s changing characteristics, adding durations and multiplying by one rate may not describe the mean correctly. The exposure and rate model must reflect the relevant time structure; simply recording total follow-up does not make changing rates constant.

Putting Person-Time Into the Poisson Regression

When covariates explain differences in rates, write the rate as \(\lambda_i=\exp(x_i^\mathsf{T}\beta)\), where \(x_i\) is the covariate vector and \(\beta\) is the coefficient vector. The conditional mean count is then \(\mu_i=E_i\exp(x_i^\mathsf{T}\beta)\). Taking logarithms shows exactly how the exposure enters the linear predictor.

The term \(\log(E_i)\) is an offset: it enters with coefficient fixed at one, rather than being estimated as another predictor. The linear predictor \(x_i^\mathsf{T}\beta\) describes the log rate, while the full expression describes the log expected count. In particular, the fitted rate and fitted count are distinct quantities:

This form follows directly from the Poisson likelihood. Assume the counts are conditionally independent given their covariates and exposures, and \(Y_i\mid x_i,E_i\sim\operatorname{Poisson}\{E_i\exp(x_i^\mathsf{T}\beta)\}\). For positive \(E_i\), the log-likelihood is

This is the Poisson log-likelihood with log exposure added to each linear predictor. Differentiating with respect to \(\beta\) gives the score \(\sum_i x_i\{Y_i-E_i\exp(x_i^\mathsf{T}\beta)\}\): the exposure affects the fitted mean through the offset, not through an estimated exposure coefficient. This derivation assumes the \(E_i\) are observed, positive exposures and that the specified conditional Poisson model is appropriate.

Interpreting Rates and Counts

With unequal follow-up, a fitted count cannot be interpreted as if every subject had the same time at risk. At fixed covariates, doubling exposure doubles the fitted expected count, but it does not change the fitted rate. Thus a longer-followed subject can have a larger fitted count even when their modeled event rate is identical to that of a shorter-followed subject.

The exposure unit must be stated when reporting a rate. A rate of \(1.4\) events per person-year is not a count of \(1.4\) events for every subject, and it is not automatically a one-year risk. It is a rate relative to person-time under the model. The expected count over a specified duration is obtained by multiplying that rate by the duration in the matching units.

Person-time modeling is useful when individuals enter or leave observation at different times, provided that each exposure is defined consistently with the event process and the model’s assumptions. It does not by itself correct informative loss to follow-up, unmeasured differences in risk, or a rate that changes over time in an unmodeled way. Those features require an appropriate model or study design; an offset alone cannot resolve them.