A Latent Variable Model for Measurement Error in a Skewed Regressor

When modeling covariate measurement error in regression, it is typically assumed that the measurement error acts in an additive or multiplicative manner on the true covariate value. However, such assumptions do not hold for the measurement error of sleep-disordered breathing (SDB). The true covariate is the severity of SDB, and the observed surrogate covariate is the number of breathing pauses during unit time of sleep, which has a non-negative skewed distribution with a point mass at zero. We propose a latent variable measurement error model to characterize the error structure in this situation and implement it in longitudinal and cross-sectional data analysis settings. A regression calibration approach with a structural assumption is used to estimate parameters. Our model demonstrates that in some situations, supplemental data are not needed for the measurement error model to be identifiable.