Intra and Interindividual Variation Modeling: Bayesian Mixed-Effects Nonstationary Latent Differential Equation Model

Abstract

Longitudinal analysis are powerful methods to estimate change over time. The combination of nomothetic and idiographic approaches within longitudinal analysis would allow to answer questions related to intra and interindividual variability in one integrated method. In order to have lag independent results, longitudinal analysis should be made with a continuous-time method. Continuous-time methods take into account that the phenomena of interest does not stop existing between measurement time points. Differential equations modeling is a method that studies intraindividual variability as a form of continuous-time modeling, which can be implemented as fixed-effects or mixed-effects. Mixed-effects models allows to integrate interindividual variability, and properly estimate non-dependent data. Latent Differential Equation (LDE) model is a method to estimate differential equation models from a familiar framework in psychology (Structural Equation Modeling). This dissertation tend to extend the LDE by adding the mixed-effects, estimating subject and sample parameters, including interindividual variability on parameters of interest. The analyses were implemented from the Bayesian framework, this framework provides several advantages, one of them being that allow us to make direct inference of the estimated parameters. The proposed model (Bayesian Mixed-Effects Nonstationary Latent Differential Equation Model, BMNLDE) was tested in a simulation, and exemplify with a real data describing the sedentary behavior in older adults. The simulation shows that the BMNLDE model estimate parameters with low bias, the 95\% Credible Interval coverage is unreliable when the model presents low frequency and high damping. The frequency of the oscillating processes was the main factor that affected bias and CI coverage. The BMNLDE model showed to be an appropriate method to include intra and interindividual variability. The simulation was capable to demonstrate the conditions in which the model performs as desire, and under which conditions the model does not perform as desire. The real data example shows an application of the BMNLDE model, were the BMNLDE model describes the oscillating behavior of the sedentary behavior. It also shows how it can used to compare parameters of interest between groups.