Estimation of ROC curves based on stably distributed biomarkers subject to measurement error and pooling mixtures

Additive measurement errors and pooling design are objectively two different issues, which have been separately and extensively dealt with in the biostatistics literature. However, these topics usually correspond to problems of reconstructing a summand's distribution of the biomarker by the distribution of the convoluted observations. Thus, we associate the two issues into one stated problem. The integrated approach creates an opportunity to investigate new fields, e.g. a subject of pooling errors, issues regarding pooled data affected by measurement errors. To be specific, we consider the stated problem in the context of the receiver operating characteristic (ROC) curves analysis, which is the well-accepted tool for evaluating the ability of a biomarker to discriminate between two populations. The present paper considers a wide family of biospecimen distributions. In addition, applied assumptions, which are related to distribution functions of biomarkers, are mainly conditioned by the reconstructing problem. We propose and examine maximum likelihood techniques based on the following data: a biomarker with measurement error; pooled samples; and pooled samples with measurement error. The obtained methods are illustrated by applications to real data studies.