Inference for biomedical data using diffusion models with covariates and mixed effects
Résumé
Neurobiological data such as EEG 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 modeling framework for this type of data based on stochastic processes. Stochastic differential equations with mixed effects are a popular framework for modeling biomedical data, e.g., in pharmacological studies. While the inherent stochasticity of diffusion models accounts for prevalent model uncertainty or misspec-ification, random effects take care of the inter-subject variability. The 2-layer stochasticity, however, renders parameter inference challenging. This is especially true for more complex model dynamics, and only few theoretical investigations on the asymptotic behavior of estimates exist. This article adds to filling this gap by examining asymptotics (number of subjects going to infinity) of Maximum Likelihood estimators in multidimensional, nonlinear and non-homogeneous stochastic differential equations with random effects and included covariates. Estimates are based on the discretized continuous-time likelihood and we investigate finite-sample and discretization bias. In applications, the comparison of, e.g., different experimental conditions such as placebo vs. treatment, is often of interest. We suggest a hypothesis testing approach and evaluate the test's performance by simulations. Finally, we apply the framework to a statistical investigation of EEG recordings from epileptic patients.
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