Linear regression with left‐censored covariates and outcome using a pseudolikelihood approach

Environmental toxicology studies often involve sample values that fall below a laboratory procedure's limit of quantification. Such left‐censored data give rise to several problems for regression analyses. First, both covariates and outcome may be left censored. Second, the transformed toxicant levels may not be normal but mixtures of normals because of differences in personal characteristics, for example, exposure history and demographic factors. Third, the outcome and covariates may be linear functions of left‐censored variates, such as averages and differences. Fourth, some toxicant levels may be functions of other toxicant levels resulting in a recursive system. In this paper, marginal and pseudolikelihood‐based methods are proposed for estimation of the means and covariance matrix of variates found in these four settings. Next, linear regression methods are developed allowing outcomes and covariates to be linear combinations of left‐censored measures. This is extended to a recursive system of modeling equations. Bootstrap standard errors and confidence intervals are used. Simulation studies demonstrate that the proposed methods are accurate for a wide range of study designs and left‐censoring probabilities. The proposed methods are illustrated through the analysis of an ongoing community‐based study of polychlorinated biphenyls, which motivated the proposed methodology.