Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations

• Published in: Biometrics 2010-09, Vol.66 (3), p.733-741
• Main Authors: Donnet, Sophie, Foulley, Jean-Louis, Samson, Adeline
• Format: Article
• Language: English
• Subjects: Algorithms; Animals; Applications; Bayes Theorem; Bayesian analysis; Bayesian estimation; BIOMETRIC METHODOLOGY; Biometrics; Chickens - growth & development; Datasets; Differential equations; Euler-Maruyama scheme; Gompertz model; Growth; Growth curves; Humans; Mathematical independent variables; Methodology; Mixed models; Modeling; Models, Theoretical; Odes; Ordinary differential equations; Parametric models; Predictive modeling; Predictive posterior distribution; Statistics; Stochastic differential equation; Stochastic models
• ISSN: 0006-341X; 1541-0420
• DOI: 10.1111/j.1541-0420.2009.01342.x

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.