Handling covariates subject to limits of detection in regression

In the environmental health sciences, measurements of toxic exposures are often constrained by a lower limit called the limit of detection (LOD), with observations below this limit called non-detects. Although valid inference may be obtained by excluding non-detects in the estimation of exposure effects, this practice can lead to substantial reduction in power to detect a significant effect, depending on the proportion of censoring and the closeness of the effect size to the null value. Therefore, a variety of methods have been commonly used in the environmental science literature to substitute values for the non-detects for the purpose of estimating exposure effects, including ad hoc values such as and LOD. Another method substitutes the expected value of the non-detects, i.e., E[X|X a parts per thousand currency sign LOD] but this requires that the inference be robust to mild miss-specifications in the distribution of the exposure variable. In this paper, we demonstrate that the estimate of the exposure effect is extremely sensitive to ad-hoc substitutions and moderate distribution miss-specifications under the conditions of large sample sizes and moderate effect size, potentially leading to biased estimates. We propose instead the use of the generalized gamma distribution to estimate imputed values for the non-detects, and show that this method avoids the risk of distribution miss-specification among the class of distributions represented by the generalized gamma distribution. A multiple imputation-based procedure is employed to estimate the regression parameters. Compared to the method of excluding non-detects, the proposed method can substantially increase the power to detect a significant effect when the effect size is close to the null value in small samples with moderate levels of censoring ( a parts per thousand currency sign 50%), without compromising the coverage and relative bias of the estimates.