A Stochastic Model for Analysis of Longitudinal AIDS Data

• Published in: Journal of the American Statistical Association Vol. 89; no. 427; pp. 727 - 736
• Main Authors: Taylor, J. M. G., Cumberland, W. G., Sy, J. P.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis Group 01.09.1994; American Statistical Association; Taylor & Francis Ltd
• Subjects: Acquired immune deficiency syndrome; AIDS; Applications; Applications and Case Studies; Biology, psychology, social sciences; Brownian motion; Correlations; Covariance; Datasets; Exact sciences and technology; Infections; Insurance, economics, finance; Mathematical models; Mathematics; Medical sciences; Modeling; Ornstein-Uhlenbeck process; Parametric models; Probability and statistics; Reliability, life testing, quality control; Repeated measures; Sciences and techniques of general use; Statistics; Stochastic models; Stochastic processes; Tracking; Infection; Stochastic model; Autoregression; Viral disease; Repeated measurement; AIDS; Ornstein Uhlenbeck process; Slope
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1994.10476806

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.