Modeling bivariate longitudinal hormone profiles by hierarchical state space models

Ziyue Liu, Anne R. Cappola, Leslie J. Crofford, Wensheng Guo

Research output: Contribution to journalArticle

4 Scopus citations


The hypothalamic-pituitary-adrenal (HPA) axis is crucial in coping with stress and maintaining homeostasis. Hormones produced by the HPA axis exhibit both complex univariate longitudinal profiles and complex relationships among different hormones. Consequently, modeling these multivariate longitudinal hormone profiles is a challenging task. In this article, we propose a bivariate hierarchical state space model, in which each hormone profile is modeled by a hierarchical state space model, with both population-average and subject-specific components. The bivariate model is constructed by concatenating the univariate models based on the hypothesized relationship. Because of the flexible framework of state space form, the resultant models not only can handle complex individual profiles, but also can incorporate complex relationships between two hormones, including both concurrent and feedback relationship. Estimation and inference are based on marginal likelihood and posterior means and variances. Computationally efficient Kalman filtering and smoothing algorithms are used for implementation. Application of the proposed method to a study of chronic fatigue syndrome and fibromyalgia reveals that the relationships between adrenocorticotropic hormone and cortisol in the patient group are weaker than in healthy controls. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)108-118
Number of pages11
JournalJournal of the American Statistical Association
Issue number505
StatePublished - 2014


  • Circadian rhythm
  • Feedback relationship
  • HPA axis
  • Kalman filter
  • Periodic splines

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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