Analysis of left-censored longitudinal data with application to viral load in HIV infection

The classical model for the analysis of progression of markers in HIV-infected patients is the mixed effects linear model. However, longitudinal studies of viral load are complicated by left censoring of the measures due to a lower quantification limit. We propose a full likelihood approach to estimate parameters from the linear mixed effects model for left-censored Gaussian data. For each subject, the contribution to the likelihood is the product of the density for the vector of the completely observed outcome and of the conditional distribution function of the vector of the censored outcome, given the observed outcomes. Values of the distribution function were computed by numerical integration. The maximization is performed by a combination of the Simplex algorithm and the Marquardt algorithm. Subject-specific deviations and random effects are estimated by modified empirical Bayes replacing censored measures by their conditional expectations given the data. A simulation study showed that the proposed estimators are less biased than those obtained by imputing the quantification limit to censored data. Moreover, for models with complex covariance structures, they are less biased than Monte Carlo expectation maximization (MCEM) estimators developed by Hughes (1999) Mixed effects models with censored data with application to HIV RNA Levels. Biometrics 55, 625-629. The method was then applied to the data of the ALBI-ANRS 070 clinical trial for which HIV-1 RNA levels were measured with an ultrasensitive assay (quantification limit 50 copies/ml). Using the proposed method, estimates obtained with data artificially censored at 500 copies/ml were close to those obtained with the real data set.