A measurement error model with a Poisson distributed surrogate

We study a linear model in which one of the covariates is measured with error. The surrogate for this covariate is the event count in unit time. We model the event count by a Poisson distribution, the rate of which is the unobserved true covariate. We show that ignoring the measurement error leads to inconsistent estimators of the regression coefficients and propose a set of unbiased estimating equations to correct the bias. The method is computationally simple and does not require using supplemental data as is often the case in other measurement error analyses. No distributional assumption is made for the unobserved covariate. The proposed method is illustrated with an example from the Wisconsin Sleep Cohort Study.