A Stochastic Model for Analysis of Longitudinal AIDS Data

Abstract

In this paper we analyze serial CD4 T-cell measurements from the Los Angeles portion of the Multicenter AIDS Cohort Study. Our emphasis is on developing a plausible and parsimonious model to describe the stochastic process underlying the patterns of CD4 measurements. The stochastic process that we use enables us to investigate the concept of derivative tracking, for which it is assumed that the rank order of the individual's slopes is maintained over time. A general model for the analysis of longitudinal repeated measures data is

where Y_i (t_ij ) is the measurement of subject i at time t_ij, X(t_ij )α represents fixed effect terms, Z(t_ij )b_i represents random effect terms, W_i (t_ij ) is a stochastic process allowing correlation between measurements, and ε _ij is measurement error. In the simplest case, X(t_ij ) and Z(t_ij ) contain the times of measurements. For W_i (t_ij ), we use a two-parameter integrated Ornstein-Uhlenbeck (OU) process. The OU process is the continuous mean zero Gaussian Markov process, which includes Brownian motion and white noise as special limiting cases. This model is a continuous-time version of an AR(1) process for the deviations of the derivative of y from the expected derivative of y with respect to t. This approach is flexible and tractable as the covariance structure has a closed-form expression. The model allows unequally spaced observations and can be generalized to multivariate responses. This model enables one to assess whether individuals maintain their trajectories; that is, whether their slope of Y tracks. We find no evidence in the data that the slopes of the CD4 values track.

Key Words:

• AIDS
• Ornstein-Uhlenbeck process
• Repeated measures
• Tracking