Modeling Unequally Spaced Bivariate Growth Curve Data Using a Kalman Filter Approach

ABSTRACT

In many clinical studies, patients are followed over time with their responses measured longitudinally. Using mixed model theory, one can characterize these data using a wide array of across subject models. A state-space representation of the mixed effects model and use of the Kalman filter allows one to have great flexibility in choosing the within error correlation structure even in the presence of missing or unequally spaced observations. Furthermore, using the state-space approach, one can avoid inverting large matrices resulting in efficient computation. The approach also allows one to make detailed inference about the error correlation structure. We consider a bivariate situation where the longitudinal responses are unequally spaced and assume that the within subject errors follows a continuous first-order autoregressive (CAR(1)) structure. Since a large number of nonlinear parameters need to be estimated, the modeling strategy and numerical techniques are critical in the process. We developed both a Visual Fortran^® and a SAS^® program for modeling such data. A simulation study was conducted to investigate the robustness of the model assumptions. We also use data from a psychiatric study to demonstrate our model fitting procedure.

Key Words:

• Growth curve
• Kalman filter
• Longitudinal data
• Multivariate mixed effects model
• State-space approach

Mathematics Subject Classification:

• 62H99

Acknowledgments

The authors wish to thank Dr. C. F. Reynolds and P. Houck for providing study data and valuable comments on the analysis results. We also thank the referee for some useful suggestions. This work was supported in part by Public Health Service grants U10CA-069651 and U10CA-69974 from the National Cancer Institute.