There is no impact of exposure measurement error on latency estimation in linear models

Identification of the latency period for the effect of a time-varying exposure is key when assessing many environmental, nutritional, and behavioral risk factors. A pre-specified exposure metric involving an unknown latency parameter is often used in the statistical model for the exposure-disease relationship. Likelihood-based methods have been developed to estimate this latency parameter for generalized linear models but do not exist for scenarios where the exposure is measured with error, as is usually the case. Here, we explore the performance of naive estimators for both the latency parameter and the regression coefficients, which ignore exposure measurement error, assuming a linear measurement error model. We prove that, in many scenarios under this general measurement error setting, the least squares estimator for the latency parameter remains consistent, while the regression coefficient estimates are inconsistent as has previously been found in standard measurement error models where the primary disease model does not involve a latency parameter. Conditions under which this result holds are generalized to a wide class of covariance structures and mean functions. The findings are illustrated in a study of body mass index in relation to physical activity in the Health Professionals Follow-Up Study.