Counts Recorded Over Unequal Observation Windows

Counts Depend on Both Events and Observation Time

A count records how many events occurred. But the same count can mean different things if one observation covers a week and another covers a year. Conversely, a longer observation window can produce a larger count even when the underlying event rate is unchanged. In a generalized linear mixed model (GLMM), the response remains the count; the length of time at risk is additional information used to specify its expected value.

For example, counting hospital admissions over follow-up requires distinguishing the number of admissions from the time each person was followed and eligible to be admitted. A count of four admissions over two years and a count of four over eight years are not, by themselves, evidence of the same event rate.

Exposure must have a meaningful, consistent unit, such as person-years, machine-hours, or days at risk. It is not necessarily the calendar duration of a study: if a person enters late, leaves early, or is not at risk during part of follow-up, their relevant exposure can be less than the study's full duration. Exposure is not the count, and it is not a binomial denominator unless the response process is actually a set of fixed opportunities with success or failure outcomes.

The Exposure Offset in a Count GLMM

Suppose the conditional event rate for unit \(j\) in cluster \(i\) is \(\lambda_{ij}\), measured as expected events per unit of exposure. Under a constant-rate Poisson process, the expected count over exposure \(t_{ij}\) is \(t_{ij}\lambda_{ij}\). A common Poisson GLMM specifies the rate through fixed and random effects:

Here \(x_{ij}\) and \(z_{ij}\) are the fixed-effect and random-effect design vectors, \(\beta\) is the fixed-effect coefficient vector, and \(b_i\) is the cluster-specific random-effect vector. Taking the logarithm of the mean gives

The term \(\log(t_{ij})\) is an offset: a known model term whose coefficient is fixed at 1. It makes the expected count proportional to exposure while allowing the covariates and random effects to describe the rate. Thus, holding the covariates and random effects fixed, doubling exposure doubles the expected count. A fixed effect \(\beta_k\) instead compares rates: increasing its predictor by one unit multiplies the conditional rate, and hence the conditional expected count at the same exposure, by \(\exp(\beta_k)\).

These assumptions matter. Unequal exposure does not automatically make a Poisson model appropriate, nor does adding an offset correct omitted predictors, dependence, or an unsuitable variance distribution. A negative binomial GLMM can also include \(\log(t_{ij})\) as an offset when its conditional mean is proportional to exposure. Its additional variance structure addresses a different feature of the data; it does not remove the need to define exposure or justify the mean scaling.

Why the Offset Has Its Particular Form

The log offset follows from the rate definition, rather than being a discretionary adjustment. If \(\lambda_{ij}\) is the expected number of events per unit exposure, then the expected number in an observation window of length \(t_{ij}\) is \(t_{ij}\lambda_{ij}\). If the log rate is the linear predictor \(\eta_{ij}=x_{ij}^{\mathsf T}\beta+z_{ij}^{\mathsf T}b_i\), then \(\lambda_{ij}=\exp(\eta_{ij})\), so

This also explains why the offset coefficient is 1. Exposure multiplies the rate before the logarithm is taken; its log therefore enters additively without an estimated multiplier. If the coefficient on \(\log(t_{ij})\) were estimated freely, the model would no longer impose the proportional scaling implied by a rate per unit exposure.

Worked Rate Comparisons

A model without exposure adjustment can confuse a longer window with a higher event rate. The following hypothetical example shows the difference using a simple Poisson model with a group-specific rate. It illustrates the offset calculation; it does not include a cluster random effect.

In a GLMM, the rate comparison is usually conditional on the random effects: the fixed-effect rate ratio compares observations with the same random-effect values and other modeled predictors, at equal exposure. The offset accounts for exposure, while the random effects account for the modeled cluster-level variation. They have distinct roles.

Choosing and Checking an Exposure Model

Including the offset is appropriate when exposure is known and the scientific model says that the expected count scales in proportion to it. A simple alternative is to use the rate \(Y_{ij}/t_{ij}\) as a continuous response. That changes the response distribution: the observed rate is not generally Poisson, can take noninteger values, and has a variability that depends on exposure. A count likelihood with an offset uses the count's support and lets the exposure inform its mean directly.

Another alternative is to enter \(\log(t_{ij})\) as an ordinary predictor and estimate its coefficient. This is useful only when the aim and assumptions justify estimating an exposure elasticity rather than fixing proportional scaling. For example, a coefficient of 0.8 would imply that multiplying exposure by two multiplies the expected count by \(2^{0.8}\approx1.741\), not by 2. It is not the standard offset model for a rate per unit exposure.

Check how the model behaves against the way observation windows were formed. Useful diagnostics and questions include:

• Plot or summarize observed counts and fitted counts against exposure. Large systematic discrepancies can indicate a poor mean specification or inadequate covariates.
• Check whether follow-up time is actually time at risk. Calendar time that includes periods when an event could not occur is not the appropriate denominator for that rate.
• Investigate whether the rate changes over the window. If it does, a single constant-rate term may be inadequate; time-varying predictors or a model that represents changing rates may be needed.
• Check whether exposure is related to unmodeled risk, event occurrence, or dropout. An offset corrects the arithmetic of unequal exposure; it does not by itself correct informative observation or selection.
• Check the conditional variance and dependence assumptions separately. Extra-Poisson variation or within-cluster dependence may remain even with the correct offset, motivating a different count distribution or dependence structure.

Exposure values must be positive in the ordinary log-offset formulation, because \(\log(0)\) is undefined and a zero-time window contains no opportunity for the modeled events. Such a row usually contributes no count information for that rate and should not be treated as an ordinary positive-exposure observation. Also, a zero count with positive exposure is entirely valid: it means no events were observed during that window, not that the exposure was zero.

The main limitation is that the offset imposes proportional scaling rather than discovering it from the data. If longer observation windows have systematically different event rates, or if exposure is measured inaccurately, the rate interpretation may be misleading. Choosing a count distribution and random-effects structure also requires attention to the response process; the offset alone cannot repair a misspecified GLMM.