Generalized linear mixed model for binary outcomes when covariates are subject to measurement errors and detection limits

Longitudinal measurement of biomarkers is important in determining risk factors for binary endpoints such as infection or disease. However, biomarkers are subject to measurement error, and some are also subject to left‐censoring due to a lower limit of detection. Statistical methods to address these issues are few. We herein propose a generalized linear mixed model and estimate the model parameters using the Monte Carlo Newton‐Raphson (MCNR) method. Inferences regarding the parameters are made by applying Louis's method and the delta method. Simulation studies were conducted to compare the proposed MCNR method with existing methods including the maximum likelihood (ML) method and the ad hoc approach of replacing the left‐censored values with half of the detection limit (HDL). The results showed that the performance of the MCNR method is superior to ML and HDL with respect to the empirical standard error, as well as the coverage probability for the 95% confidence interval. The HDL method uses an incorrect imputation method, and the computation is constrained by the number of quadrature points; while the ML method also suffers from the constrain for the number of quadrature points, the MCNR method does not have this limitation and approximates the likelihood function better than the other methods. The improvement of the MCNR method is further illustrated with real‐world data from a longitudinal study of local cervicovaginal HIV viral load and its effects on oncogenic HPV detection in HIV‐positive women.