A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation Models with Mixed Effects

Abstract

Several approaches currently exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA), generalized local linear approximation (GLLA), and generalized orthogonal local derivative approximation (GOLD). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children’s self-regulation.

Dynamical systems modeling of early childhood self-regulation.

“…Quantitative methods for fitting nonlinear ODEs are an exciting possibility for studying self-regulation in early childhood (Chow, Bendezú, Cole, & Ram, 2016). Here, we use a specific class of ODE model, a second order coupled-oscillator, to investigate how PR and EP contribute to the dynamics of young children’s self-regulation.…”

Dynamical Systems Modeling of Couple Interaction: a New Method for Assessing Intervention Impact Across the Transition to Parenthood

“…For instance, uncertainty in the derivative estimation in Stage 1 is not fully reflected in the standard error estimates obtained in Stage 2, and greater inaccuracy in parameter estimates may be expected compared to single-stage approaches. Nevertheless, simulation results reported elsewhere (Chow et al, 2016)) suggested that despite these known limitations, the extent of parameter and standard error inaccuracies remained satisfactory even in models of greater complexity than those considered in the present study, and the novel insights from the current exploratory study provide a foundation –or rather, some much needed incentives—to pursue further validation in future studies.…”

“…The estimates of the smoothed level and derivatives were used as manifest variables in a bivariate mixed effects model to fit ordinary differential equation (ODE) models in a two-stage process (Chow, Bendezú, Cole, & Ram, 2016) using the nlme routine in R (Pinheiro, Bates, DebRoy, Sarkar, & R Core Team, 2014). Based on results from our graphical explorations (see supplementary materials) and our hypotheses of interest, we developed a bivariate model with the couples’ second derivatives in negative behaviors as dependent variables.…”

Validity of dynamical analysis to characterize heart rate and oxygen consumption during effort tests

“…The estimation of the three parameters characterizing the dynamics according to equation 1 is done in a two-step procedure, consisting in first estimating the first derivative of the variable studied over a given number of points with Functional Data Analysis (FDA) regression spline method 10,19 . It consists on generating a B-spline function that fits the outcome to be studied and then estimating the derivative of that function.…”

“…The second approach, based on dynamical system modeling, could allow to more accurately characterize the HR or VO2 response during effort. Dynamical analysis based on differential equations is an active subject of research in the behavioral field since the seminal work of Boker 9 and has led to numerous studies in the field of psychology and to several methodological advances 10 . As approach based on first order differential equation approach show potential ability to adjust HR measurement 11 and VO2 dynamics during variable effort loads 12 , we propose to use a simple first order differential equation coupled with a mixed effect regression to quantify the link between the exercise load during effort test and the resulting HR or VO2 dynamics.…”