In clinical studies, often values of a covariate or biomarker are left-censored due to the limit of detection (LOD). An ordinary regression approach that fits a model by simply replacing the left-censored values of the covariate by the LOD or \((1/2)\,\text {LOD}\) generally produces a biased estimator of the covariate effect. In addition, if a covariate is subject to the measurement error, then a naive approach that does not correct for the measurement error can produce an asymptotically biased estimator. In this paper, we propose and explore an innovative method for fitting a logistic regression model to binary data by correcting for both limits of detection and measurement errors in covariates. The finite-sample properties of the proposed estimators are investigated using Monte Carlo simulations. The empirical results are very encouraging, as the proposed method appears to provide unbiased and efficient estimators in the presence of covariates that are subject to the LOD and measurement error. An application is also provided using some actual cardiovascular fitness data obtained from a health survey with measurements on biomarkers and demographic variables.
