Correlating Two Viral Load Assays with Known Detection Limits

Abstract

A timely objective common to many HIV studies involves assessing the correlation between two different measures of viral load obtained from each of a sample of patients. This correlation has scientific utility in a number of contexts, including those aimed at a comparison of competing assays for quantifying virus and those aimed at determining the level of association between viral loads in two different reservoirs using the same assay. A complication for the analyst seeking valid point and interval estimates of such a correlation is the fact that both variables may be subject to left censoring due to values below assay detection limits. We address this problem using a bivariate normal likelihood that accounts for left censoring of two variables that may have different detection limits. We provide simulation results to evaluate sampling properties of the resulting correlation estimator and compare it with ad hoc estimators in the presence of nondetects. In an effort to obtain improved confidence interval properties relative to the Wald approach, we evaluate and compare profile likelihood-based intervals. We apply the methods to HIV viral load data on women and infants from a trial in Bangkok, Thailand, and we discuss an extension of the original model to accommodate interval censoring arising due to the study design.

Regression models for censored serological data

“…Also let i~'z1;:::; c be left-censored data and i5c+1,..., n be right-censored data. Then, a generalization of the log-likelihood shown above becomes (Breen, 1996;Lyles et al, 2001;Zhang & Sun, 2010): MI is an alternative method for fitting regression equations when censoring occurs. A simple version of the MI method was used that consisted of two steps: (i) estimation of the model parameters using the censored regression method, and (ii) using these estimated parameters to draw random samples (from a normal distribution) to replace the censored observations.…”

Assay validation for left-censored data

“…However, because the X L 's and Y L 's are left censored versions of (X; Y ), these sample estimates are biased for the mean and covariance matrix of (X; Y ). An ML approach under normality assumptions [4] for estimating in the presence of left censored data can be easily extended to estimate the concordance correlation coe cient if we insert the ML estimates for  and into equation (1). Let {x Li ; y Li }; i = 1; : : : ; N , be a random sample from random variables (X L ; Y L ).…”

“…where V i is the working covariance matrix for Z i , and the expressions for U(Â) = (U [1]; U [2]; U [3]; U [4]) and the derivative matrix D i = @U=@Â are presented in the Appendix. The GEE approach uses empirical covariance estimates to adjust for a mis-speciÿed covariance structure without the loss of much e ciency [5,6].…”

“…This ad hoc approach is obviously biased and tends to produce larger mean estimates and smaller variance estimates than would have been obtained without lower limits of detection. Lyles et al [4] proposed a maximum likelihood (ML) method based on normality assumptions to estimate the correlation coe cient between two left censored variables. Their method can be extended to estimate the concordance correlation coe cient.…”

Maximum Pairwise Pseudo-likelihood Estimation of the Covariance Matrix from Left-Censored Data

“…For the bivariate problem, Lyles et al (2001) used maximum likelihood based on a bivariate normal distribution to simultaneously estimate the means, variances, and correlation between two left-censored log-transformed variables. The goal in this paper is to develop likelihood-based methods for estimating the mean vector and covariance and correlation matrices from left-censored data for the p-variate problem ( p ≥ 2).…”