June 2024 Semiparametric bivariate hierarchical state space model with application to hormone circadian relationship
Mengying You, Wensheng Guo
Author Affiliations +
Ann. Appl. Stat. 18(2): 1275-1293 (June 2024). DOI: 10.1214/23-AOAS1834


The adrenocorticotropic hormone and cortisol play critical roles in stress regulation and the sleep-wake cycle. Most research has been focused on how the two hormones regulate each other in terms of short-term pulses. Few studies have been conducted on the circadian relationship between the two hormones and how it differs between normal and abnormal groups. The circadian patterns are difficult to model as parametric functions. Directly extending univariate functional mixed effects models would result in a large dimensional problem and a challenging nonparametric inference. In this article we propose a semiparametric bivariate hierarchical state space model in which each hormone profile is modeled by a hierarchical state space model with nonparametric population-average and subject-specific components. The bivariate relationship is constructed by concatenating two latent independent subject-specific random functions specified by a design matrix, leading to a parametric inference on the correlation. We propose a computationally efficient state-space EM algorithm for estimation and inference. We apply the proposed method to a study of chronic fatigue syndrome and fibromyalgia and discover an erratic regulation pattern in the patient group in contrast to a circadian regulation pattern conforming to the day–night cycle in the control group.

Funding Statement

This research was supported by NIH R01 Grants DK117208 and HL161303.


We would like to thank the Editor and reviewers for their careful reviews that substantially improved the presentation of this paper.


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Mengying You. Wensheng Guo. "Semiparametric bivariate hierarchical state space model with application to hormone circadian relationship." Ann. Appl. Stat. 18 (2) 1275 - 1293, June 2024. https://doi.org/10.1214/23-AOAS1834


Received: 1 March 2023; Revised: 1 August 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1834

Keywords: circadian rhythm , functional mixed effects model , state space EM algorithm , time-varying correlation

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 2 • June 2024
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