In this paper, we develop Poisson-type regression methods that require the durations of exposure be measured only on a possibly nonrandom subset of the cohort members. These methods can be used to make inferences about the incidence density during exposure as well as the ratio of incidence densities during exposure versus not during exposure. Numerical studies demonstrate that the proposed methods yield reliable results in practical settings. We describe an application to a population-based case-control study assessing the transient increase in the risk of primary cardiac arrest during leisure-time physical activity.