A Stochastic Model for Analysis of Longitudinal AIDS DataJ. M. G. Taylor; W. G. Cumberland; J. P. SyJournal of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994), pp. 727-736.Stable URL:http://links.jstor.org/sici?sici=0162-1459%28199409%2989%3A427%3C727%3AASMFAO%3E2.0.CO%3B2-1Journal of the American Statistical Association is currently published by American Statistical Association.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/astata.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgFri Feb 1 17:20:45 2008A Stochastic Model for Analysis of Longitudinal AIDS Data J. M. G. TAYLOR, W. G. CUMBERLAND,and J. P. SY* 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,(ttj) is the measurement of subject i at time t,, X(t,)cu represents fixed effect terms, Z(t,])b, represents random effect terms, W,(t,) is a stochastic process allowing correlation between measurements, and c, is measurement error. In the simplest case, X(tii) and Z(t,.) contain the times of measurements. For W,(t,), we use a two-parameter integrated Omstein-Uhlenbeck (OU) process. The OU process is the continuous ihean 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. I.INTRODUCTION The concept of tracking for longitudinal data is popular where X (to) a represents fixed effects, Z(tu) bi represents with epidemiologists and has been discussed in the statistical random effects, and E,, is an independent, identically distrib- literature. Foulkes and Davis (1981) and McMahan (1981) uted (iid) measurement error. developed indices of tracking based on the concept of the A second approach is to specify a structure for the within- maintenance over time of the relative or rank position of subject covariance based possibly on a stochastic process. the response variable among the group of subjects. Ware and For example, it might be assumed that Yi(t,) = X(t,])a Wu (198 1) regarded tracking as the ability to predict future + Wi (ti,), where W, (to) is a realization of a stochastic process. observations for each individual, leading to a definition that Other authors have suggested a combination of these two a population tracks if for each individual, the expected values approaches (Chi and Reinsel 1989; Cullis and McGilchrist of serial measurements are given by a polynomial function 1990; Diggle 1988; Jones and Bodi-Boateng 1991). The of time. These two concepts are closely related, but not iden- computational aspects of fitting these models to data are tical. In this article we adopt the first concept of tracking nontrivial and have been discussed by Jennrich and and apply it to the first derivative of the response variable. Schluchter (1986) and Lindstrom and Bates (1988). Thus we regard a population as having derivative trachng In this article we focus on a model consisting of a com- if the relative ranks of the rate of change of the outcome bination of fixed effects, random effects, a specific stochastic variable are maintained over time. process, and measurement error. In particular, we assume The typical structure of longitudinal data is numerous that measurements of a possibly multivariate response variable Yi(tu)= X(tu)a+Z(ti,)bi+ Wi(tg)+cu.on each subject. The measurements will usually be at un- equally spaced time intervals, and the number of measure- Our emphasis will be on interpreting the stochastic process ments per subject may differ between subjects. There could with particular regard to the concept of derivative tracking. also be covariates, possibly time varying, that influence the We propose using an integrated Ornstein-Uhlenbeck (IOU) response variable. The aim in the analysis of such data is to process for the stochastic process. The motivation for this understand the changes in the mean structure of the response choice of stochastic process arose from an AIDS study of the variable with time, to understand the effect of the covariates natural history of CD4 T-cell numbers. CD4 numbers are a on the response variable, and to understand the within- critical aspect of the immune system, with low values indi- subject correlation structure. cating more severe immune deficiency.