Logistic regression and latent class models for estimating positivities in diagnostic assays with poor resolution

In biomedical research and diagnostic practice it is common to classify objects dichotomously based on continuous observations (x) measuring some form of biological activity, where some proportion of the objects have a level of activity above background. In this paper, we consider the problem of estimating the proportion of positive objects for a typical assay where: (i) the distribution of x for positive objects is unknown, although (ii) the risk of positivity is known to be a monotonic function of x; and (iii) x has been measured for a set of negative control objects. Monte Carlo simulations evaluating four alternative estimators of the positivity, including novel non-parametric mixture decompositions, indicate that where the positives and negatives have distributions of x with a moderate degree of overlap, a non-parametric decomposition using a latent class model provides precise and close to unbiased estimates. The methods are illustrated using data from an autoradiography assay used in cell biology.