Assessing the effect of interventions in the context of mixture distributions with detection limits

Many quantitative assay measurements of metabolites of environmental toxicants in clinical investigations are subject to left censoring due to values falling below assay detection limits. Moreover, when observations occur in both unexposed individuals and exposed individuals who reflect a mixture of two distributions due to differences in exposure, metabolism, response to intervention and other factors, the measurements of these biomarkers can be bimodally distributed with an extra spike below the limit of detection. Therefore, estimating the effect of interventions on these biomarkers becomes an important and challenging problem. In this article, we present maximum likelihood methods to estimate the effect of intervention in the context of mixture distributions when a large proportion of observations are below the limit of detection. The selection of the number of components of mixture distributions was carried out using both bootstrap-based and cross-validation-based information criterion. We illustrate our methods using data from a randomized clinical trial conducted in Qidong, People's Republic of China.