Multivariate Autoregressive Linear Mixed Effects Models
Previous chapters discussed linear mixed effects models and autoregressive linear mixed effects models for analysis of longitudinal data. This chapter discusses multivariate extensions of these models. In longitudinal clinical studies, multivariate responses are often collected at each measurement time point from each subject. When two response variables, such as an efficacy measurement and a safety measurement are obviously correlated, there are advantages in analyzing the bivariate responses jointly. Parathyroid hormone (PTH) and serum calcium (Ca) measurements in the treatment of secondary hyperparathyroidism in chronic hemodialysis patients provide an example in which joint bivariate responses are of interest. We introduce multivariate longitudinal data and explain bivariate autoregressive linear mixed effects models in which the current responses are regressed on the previous responses of both variables, fixed effects, and random effects. The dependent bivariate responses approach equilibria, and the equilibria are modeled using fixed and random effects. These type of profiles are observed in long-term clinical studies. We also explain bivariate linear mixed effects models.
KeywordsAutoregressive linear mixed effects model Equilibrium Linear mixed effects model Longitudinal Multivariate
- Funatogawa I, Funatogawa T (2008) State space representation of an autoregressive linear mixed effects model for the analysis of longitudinal data. In: JSM Proceedings, biometrics section. American Statistical Association, pp 3057–3062Google Scholar
- Harvey AC (1993) Time series models, 2nd edn. The MIT PressGoogle Scholar
- Kurokawa K, Akizawa T, Suzuki M, Akiba T, Nishizawa Y, Ohashi Y, Ogata E, Slatopolsky E (2000) Effect of long-term administration of 22-oxacalcitriol (OCT) on secondary hyperparathyroidism in hemodialysis patients. Kidney Dial 48:875–897 (in Japanese)Google Scholar
- Searl SR (1982) Matrix algebra useful for statistics. WileyGoogle Scholar