Bayesian Analysis of Mixed-effect Regression Models Driven by Ordinary Differential Equations

• Published in: Sankhyā. Series B (2008) Vol. 83; no. 1; pp. 3 - 29
• Main Authors: Tan, Qianwen, Ghosal, Subhashis
• Format: Journal Article
• Language: English
• Published: New Delhi Springer Science + Business Media 01.05.2021; Springer India
• Subjects: Mathematics and Statistics; Statistics; Primary 62J02; Secondary 62G20; ODE models; Two-step method; 62G08; Random effects; Longitudinal data; Bayesian inference; 62F15; B-splines
• DOI: 10.1007/s13571-019-00199-6
• ISSN: 0976-8386

Non-linear regression models with regression functions specified by ordinary differential equations (ODEs) involving some unknown parameters are used to model dynamical systems appearing in pharmacokinetics and pharmacodynamics, viral dynamics, engineering, and many other fields. We consider the situation where multiple subjects are involved, each of which follow the same ODE model, with different parameters related by a linear regression model in certain observable covariates in the presence of a random effect. We follow a Bayesian two-step method, where first a nonparametric spline model is used and then a posterior on the parameter of interest is induced by a suitable projection map depending on the system of ODEs. Our main contribution is accommodating mixed effects within the Bayesian two-step method by using a further projection on the space of linear combinations of covariates. We describe efficient posterior computational techniques based on direct sampling and optimization. We show that the parameters of interest are estimable at the parametric rate and Bayesian credible sets have the correct frequentist coverage in large samples. By an extensive simulation study, we show the effectiveness of the proposed method. We apply the proposed method to an intravenous glucose tolerance test study.