On estimation of bivariate biomarkers with known detection limits

To estimate the correlation coefficient between two measurements of viral load obtained from HIV-infected individuals, a Bernoulli/bivariate normal mixture model has been applied to provide a better fit for the high proportions of left censoring due to assay measurements falling below limits of detection (LD). In this paper, we investigate the relationship among several alternative likelihood-based methods; particularly we derive an explicit relationship between the maximum likelihood estimates based on the Bernoulli/bivariate normal mixture model for all data pairs and a left-truncated bivariate normal model for data pairs with one or two detectable values. With a case study and a set of Monte Carlo simulations, we demonstrated the potential impact of using an incorrect likelihood on the estimation of correlation coefficient from two random variables with known LD. Furthermore, we investigated the performance of nonparametric methods using Kendall's tau and Spearman's rank correlation with tie correction for left censoring, which have been used in practice. When data are simulated from a left-censored bivariate normal or a left-censored Bernoulli/bivariate normal mixture distribution, both Kendall's tau and Spearman's rank correlation tend to give biased estimates. The biases increase as the probability in the lower component and the proportions of left censoring increases. Further research on nonparametric methods is needed to deal with left censoring, especially in the presence of mixture distributions.