Correlating two continuous variables subject to detection limits in the context of mixture distributions

In individuals who are infected with human immunodeficiency virus (HIV), distributions of quantitative HIV ribonucleic acid measurements may be highly left censored with an extra spike below the limit of detection LD of the assay. A two-component mixture model with the lower component entirely supported on [0, LD] is recommended to model the extra spike in univariate analysis better. Let LD1 and LD2 be the limits of detection for the two HIV viral load measurements. When estimating the correlation coefficient between two different measures of viral load obtained from each of a sample of patients, a bivariate Gaussian mixture model is recommended to model the extra spike on [0, LD1] and [0, LD2] better when the proportion below LD is incompatible with the left-hand tail of a bivariate Gaussian distribution. When the proportion of both variables falling below LD is very large, the parameters of the lower component may not be estimable since almost all observations from the lower component are falling below LD. A partial solution is to assume that the lower component's entire support is on [0, LD1]x[0, LD2]. Maximum likelihood is used to estimate the parameters of the lower and higher components. To evaluate whether there is a lower component, we apply a Monte Carlo approach to assess the p-value of the likelihood ratio test and two information criteria: a bootstrap-based information criterion and a cross-validation-based information criterion. We provide simulation results to evaluate the performance and compare it with two ad hoc estimators and a single-component bivariate Gaussian likelihood estimator. These methods are applied to the data from a cohort study of HIV-infected men in Rio de Janeiro, Brazil, and the data from the Women's Interagency HIV oral study. These results emphasize the need for caution when estimating correlation coefficients from data with a large proportion of non-detectable values when the proportion below LD is incompatible with the left-hand tail of a bivariate Gaussian distribution.