Inference for biomedical data by using diffusion models with covariates and mixed effects

Neurobiological data such as electroencephalography measurements pose a statistical challenge due to low spatial resolution and poor signal‐to‐noise ratio, as well as large variability from subject to subject. We propose a new modelling framework for this type of data based on stochastic processes. Stochastic differential equations with mixed effects are a popular framework for modelling biomedical data, e.g. in pharmacological studies. Whereas the inherent stochasticity of diffusion models accounts for prevalent model uncertainty or misspecification, random‐effects model intersubject variability. The two‐layer stochasticity, however, renders parameter inference challenging. Estimates are based on the discretized continuous time likelihood and we investigate finite sample and discretization bias. In applications, the comparison of, for example, treatment effects is often of interest. We discuss hypothesis testing and evaluate by simulations. Finally, we apply the framework to a statistical investigation of electroencephalography recordings from epileptic patients. We close the paper by examining asymptotics (the number of subjects going to ∞) of maximum likelihood estimators in multi‐dimensional, non‐linear and non‐homogeneous stochastic differential equations with random effects and included covariates.