Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter

Abstract

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

Keywords: continuous-discrete extended Kalman filter; continuous-time model; mixed-effects model; nonlinear model.

Figure 1.

Phase portraits (A1-D1) and over-time trajectories (A2-D2) of the ODE equations $d\mathrm{P}\mathrm{E}(t)=-\mathrm{e}\mathrm{x}\mathrm{p}\left({\rho }_1+{\rho }_{12}(\mathrm{N}\mathrm{E}(t)-1.5)\right)(\mathrm{P}\mathrm{E}(t)-2.5)dt$, and $d\mathrm{N}\mathrm{E}(t)=-\mathrm{e}\mathrm{x}\mathrm{p}\left({\rho }_2+{\rho }_{21}(\mathrm{P}\mathrm{E}(t)-2.5)\right)(\mathrm{N}\mathrm{E}(t)-1.5)dt$ with different parameter values of ${\rho }_1={\rho }_2$ and ${\rho }_{12}={\rho }_{21}$.

Figure 2.

Smoothed emotion state estimates for 3 random individuals from the ADID study

Figure 3.

Observed and predicted emotion trajectory for a random individual from the ADID study

Figure 4.

Simulated irregularly spaced observed and latent scores of positive and negative emotions for a single subject when $Q=0.2$, $nP=100$, $nT=100$

Figure 5.

The effects of spacing on the performance measures under typical simulation conditions when $N_p=200$, $N_t=150$, and $Q=(\mathrm{a})$ 0.2 and (b) 0.5. The letters “F”, “I”, “N”, “E”, and “R” are respectively short for fixed-effects, initial condition, process noise, measurement error, and random-effects parameters. The value of the performance measures were averages of the related measures within each sub-category of parameters.

Figure 6.

The effects of $N_p$ and $N_t$ on the performance measures for irregularly spaced simulated data when (a) $Q=0.2$ and (b) $Q=0.5$. The letters “F”, “I”, “N”, “E”, and “R” are respectively short for fixed-effects, initial condition, process noise, measurement error, and random-effects parameters. The value of the performance measures were averages of the related measures within each sub-category of parameters.

Figure 7.

The effects of process noise variance (Q) on the performance measures under the simulation condition $N_p=200$ and $N_t=150$ for (a) regularly and (b) irregularly spaced data. The letters “F”, “I”, “N”, “E”, and “R” are respectively short for fixed-effects, initial condition, process noise, measurement error, and random-effects parameters. The value of the performance measures were averages of the related measures within each sub-category of parameters.