Multivariate Autoregressive Linear Mixed Effects Models

  • Ikuko FunatogawaEmail author
  • Takashi Funatogawa
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)


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.


Autoregressive linear mixed effects model Equilibrium Linear mixed effects model Longitudinal Multivariate 


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Copyright information

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Statistical Data ScienceThe Institute of Statistical MathematicsTachikawaJapan
  2. 2.Clinical Science and Strategy DepartmentChugai Pharmaceutical Co. Ltd.ChūōJapan

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