Latent Differential Equation Research Articles

Multilevel Latent Differential Structural Equation Model with Short Time Series and Time-Varying Covariates: A Comparison of Frequentist and Bayesian Estimators

Continuous-time modeling using differential equations is a promising technique to model change processes with longitudinal data. Among ways to fit this model, the Latent Differential Structural Equation Modeling (LDSEM) approach defines latent derivative variables within a structural equation modeling (SEM) framework, thereby allowing researchers to leverage advantages of the SEM framework for model building, estimation, inference, and comparison purposes. Still, a few issues remain unresolved, including performance of multilevel variations of the LDSEM under short time lengths (e.g., 14 time points), particularly when coupled multivariate processes and time-varying covariates are involved. Additionally, the possibility of using Bayesian estimation to facilitate the estimation of multilevel LDSEM (M-LDSEM) models with complex and higher-dimensional random effect structures has not been investigated. We present a series of Monte Carlo simulations to evaluate three possible approaches to fitting M-LDSEM, including: frequentist single-level and two-level robust estimators and Bayesian two-level estimator. Our findings suggested that the Bayesian approach outperformed other frequentist approaches. The effects of time-varying covariates are well recovered, and coupling parameters are the least biased especially using higher-order derivative information with the Bayesian estimator. Finally, an empirical example is provided to show the applicability of the approach.

Separating Long-Term Equilibrium Adaptation from Short-Term Self-Regulation Dynamics Using Latent Differential Equations

Self-regulating systems change along different timescales. Within a given week, a depressed person’s affect might oscillate around a low equilibrium point. However, when the timeframe is expanded to capture the year during which they onboarded antidepressant medication, their equilibrium and oscillatory patterns might reorganize around a higher affective point. To simultaneously account for the meaningful change processes that happen at different time scales in complex self-regulatory systems, we propose a single model that combines a second-order linear differential equation for short timescale regulation and a first-order linear differential equation for long timescale adaptation of equilibrium. This model allows for individual-level moderation of short-timescale model parameters. The model is tested in a simulation study which shows that, surprisingly, the short and long timescales can fully overlap and the model still converges to the reasonable estimates. Finally, an application of this model to self-regulation of emotional well-being in recent widows is presented and discussed.

A Dynamical Systems Investigation of the Co-regulation between Perceived Daily Parental Warmth and Adolescent Attention-deficit/hyperactivity Disorder Symptoms.

Longitudinal research demonstrates that child ADHD symptoms and behaviors exhibit reciprocal associations with parenting behaviors over time. However, minimal research has investigated these associations and their dynamic links at the daily level. Intensive longitudinal data can disentangle stable between-person differences from within-person fluctuations and reveal nuanced short-term family dynamics on a micro timescale. Using 30-day daily diary data from a community sample of 86 adolescents (Mage = 14.5, 55% female, 56% White, 22% Asian) and latent differential equation modeling, this study examined the links between perceived daily parental warmth and ADHD symptoms as coupled dynamical systems. The results show that the magnitude of fluctuations in perceived daily parental warmth generally remains stable, while elevated ADHD symptoms return to their normal level over time. Perceived parental warmth is sensitive to change in ADHD symptoms such that adolescents feel that their parents will fine-tune their warmth with gradual changes when adolescents demonstrate heightened symptoms. There are substantial between-family differences in these regulating system dynamics. Among families with more baseline parental non-harsh discipline, both perceived parental warmth and ADHD symptoms tend to be more stable and fluctuate less often. Intensive longitudinal data and dynamical systems approaches offer a new lens to uncover short-term family dynamics and adolescent adjustment at a refined micro level. Future research should explore antecedents and consequences of between-family differences in these short-term family dynamics on multiple timescales.

A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T

Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., T=50), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of T=56 observations of daily stress for N=44 individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.

What can be learned from couple research: Examining emotional co-regulation processes in face-to-face interactions.

A crucial component of successful counseling and psychotherapy is the dyadic emotion co-regulation process between patient and therapist that unfolds moment to moment during therapy sessions. The major reason for the disappointing progress in understanding this process is the lack of appropriate methods to assess subjectively experienced emotions continuously during therapy sessions without disturbing the natural flow of the interaction. The resulting inability has forced the field to focus on patients' overall emotion ratings at the end of each session with limited predictive value of the dyadic interplay between patient and therapist's emotional states within each session. The current tutorial demonstrates how couple research-confronted with a comparable problem-has overcome this issue by (i) developing a video-based retrospective self-report assessment method for individuals' continuous state emotions without undermining the dyadic interaction and (ii) using a validated statistical tool to analyze the dynamical process during a dyadic interaction. We show how to assess emotion data continuously, and how to unravel self-regulation and co-regulation processes using a Latent Differential Equation Modeling approach. Finally, we discuss how this approach can be applied in counseling psychology and psychotherapy to test basic theoretical assumptions about the co-creation of emotions despite the conceptual differences between couple dyads and therapist-patient dyads. The present method aims to inspire future research activities examining systematic real-time processes between patients and therapists. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Constrained Fourth Order Latent Differential Equation Reduces Parameter Estimation Bias for Damped Linear Oscillator Models

Second-order linear differential equations can be used as models for regulation since under a range of parameter values they can account for a return to equilibrium as well as potential oscillations in regulated variables. One method that can estimate parameters of these equations from intensive time series data is the method of Latent Differential Equations (LDE). However, the LDE method can exhibit bias in its parameters if the dimension of the time delay embedding and thus the width of the convolution kernel is not chosen wisely. This article presents a simulation study showing that a constrained fourth-order Latent Differential Equation (FOLDE) model for the second-order system almost completely eliminates bias as long as the width of the convolution kernel is less than two-thirds the period of oscillations in the data. The FOLDE model adds two degrees of freedom over the standard LDE model but significantly improves model fit.

Empirical Bayes Derivative Estimates

A dynamic system is a set of interacting elements characterized by changes occurring over time. The estimation of derivatives is a mainstay for exploring dynamics of constructs, particularly when the dynamics are complicated or unknown. The presence of measurement error in many social science constructs frequently results in poor estimates of derivatives, as even modest proportions of measurement error can compound when estimating derivatives. Given the overlap in the specification of latent differential equation models and latent growth curve models, and the equivalence of latent growth curve models and mixed models under some conditions, derivatives could be estimated from estimates of random effects. This article proposes a new method for estimating derivatives based on calculating the Empirical Bayes estimates of derivatives from a mixed model. Two simulations compare four derivative estimation methods: Generalized Local Linear Approximation, Generalized Orthogonal Derivative Estimates, Functional Data Analysis, and the proposed Empirical Bayes Derivative Estimates. The simulations consider two data collection scenarios: short time series (≤10 observations) from many individuals or occasions, and long individual time series (25–500 observations). A substantive example visualizing the dynamics of intraindividual positive affect time series is also presented.

A Method of Correcting Estimation Failure in Latent Differential Equations with Comparisons to Kalman Filtering

Studies have used the latent differential equation (LDE) model to estimate the parameters of damped oscillation in various phenomena, but it has been shown that correct, non-zero parameter estimates are only obtained when the latent series exhibits little or no process noise. Consequently, LDEs are limited to modeling deterministic processes with measurement error rather than those with random behavior in the true latent state. The reasons for these limitations are considered, and a piecewise deterministic approximation (PDA) algorithm is proposed to treat process noise outliers as functional discontinuities and obtain correct estimates of the damping parameter. Comprehensive, random-effects simulations were used to compare results with those obtained using a state-space model (SSM) based on the Kalman filter. The LDE with the PDA algorithm (LDEPDA) successfully recovered the simulated damping parameter under a variety of conditions when process noise was present in the latent state. The LDEPDA had greater precision and accuracy than the SSM when estimating parameters from data with sparse jump discontinuities, but worse performance for diffusion processes overall. All three methods were applied to a sample of postural sway data. The basic LDE estimated zero damping, while the LDEPDA and SSM estimated moderate to high damping. The SSM estimated the smallest standard errors for both frequency and damping parameter estimates.

Adaptive Equilibrium Regulation: Modeling Individual Dynamics on Multiple Timescales

Damped Linear Oscillators estimated by 2nd-order Latent Differential Equation have assumed a constant equilibrium and one oscillatory component. Lower-frequency oscillations may come from seasonal background processes, which non-randomly contribute to deviation from equilibrium at each occasion and confound estimation of dynamics over shorter timescales. Boker (2015) proposed a model of individual change on multiple timescales, but implementation, simulation, and applications to data have not been demonstrated. This study implemented a generalization of the proposed model; examined robustness to varied timescale ratios, measurement error, and occasions-per-person in simulated data; and tested for dynamics at multiple timescales in experience sampling affect data. Results show small standard errors and low bias to dynamic estimates at timescale ratios greater than 3:1. Below 3:1, estimate error was sensitive to noise and total occasions; rates of non-convergence increased. For affect data, model comparisons showed statistically significant dynamics at both timescales for both participants.

Integration of Stochastic Differential Equations Using Structural Equation Modeling: A Method to Facilitate Model Fitting and Pedagogy

Stochastic differential equation (SDE) models are a promising method for modeling intraindividual change and variability. Applications of SDEs in the social sciences are relatively limited, as these models present conceptual and programming challenges. This article presents a novel method for conceptualizing SDEs. This method uses structural equation modeling (SEM) conventions to simplify SDE specification, the flexibility of SEM to expand the range of SDEs that can be fit, and SEM diagram conventions to facilitate the teaching of SDE concepts. This method is a variation of latent difference scores (McArdle, 2009; McArdle & Hamagami, 2001) and the oversampling approach (Singer, 2012), and approximates the advantages of analytic methods such as the exact discrete model (Oud & Jansen, 2000) while retaining the modeling fiexibility of methods such as latent differential equation modeling (Boker, Neale, & Rausch, 2004). A simulation and empirical example are presented to illustrate that this method can be implemented on current computing hardware, produces good approximations of analytic solutions, and can flexibly accommodate novel models.

Latent Differential Equation Models for Binary and Ordinal Data

This study evaluates latent differential equation models on binary and ordinal data. Binary and ordinal data are widely used in psychology research and many statistical models have been developed, such as the probit model and the logit model. We combine the latent differential equation model with the probit model through a threshold approach, and then compare the threshold model with a naive model, which blindly treats binary and ordinal data as continuous. Simulation results suggest that the naive model leads to bias on binary data and on ordinal data with fewer than 5 levels, whereas the threshold model is unbiased and efficient for binary and ordinal data. Two example analyses on empirical binary data and ordinal data show that the threshold model also has better external validity. The R code for the threshold model is provided.

Coupled latent differential equation with moderators: Simulation and application.

Latent differential equations (LDE) use differential equations to analyze time series data. Because of the recent development of this technique, some issues critical to running an LDE model remain. In this article, the authors provide solutions to some of these issues and recommend a step-by-step procedure demonstrated on a set of empirical data, which models the interaction between ovarian hormone cycles and emotional eating. Results indicated that emotional eating is self-regulated. For instance, when people do more emotional eating than normal, they will subsequently tend to decrease their emotional eating behavior. In addition, a sudden increase will produce a stronger tendency to decrease than will a slow increase. We also found that emotional eating is coupled with the cycle of the ovarian hormone estradiol, and the peak of emotional eating occurs after the peak of estradiol. The self-reported average level of negative affect moderates the frequency of eating regulation and the coupling strength between eating and estradiol. Thus, people with a higher average level of negative affect tend to fluctuate faster in emotional eating, and their eating behavior is more strongly coupled with the hormone estradiol. Permutation tests on these empirical data supported the reliability of using LDE models to detect self-regulation and a coupling effect between two regulatory behaviors.

Abstract: Permutation Tests of Coupled Latent Differential Equations

Abstract: Permutation Tests of Coupled Latent Differential Equations

The reservoir model: A differential equation model of psychological regulation.

Differential equation models can be used to describe the relationships between the current state of a system of constructs (e.g., stress) and how those constructs are changing (e.g., based on variable-like experiences). The following article describes a differential equation model based on the concept of a reservoir. With a physical reservoir, such as one for water, the level of the liquid in the reservoir at any time depends on the contributions to the reservoir (inputs) and the amount of liquid removed from the reservoir (outputs). This reservoir model might be useful for constructs such as stress, where events might "add up" over time (e.g., life stressors, inputs), but individuals simultaneously take action to "blow off steam" (e.g., engage coping resources, outputs). The reservoir model can provide descriptive statistics of the inputs that contribute to the "height" (level) of a construct and a parameter that describes a person's ability to dissipate the construct. After discussing the model, we describe a method of fitting the model as a structural equation model using latent differential equation modeling and latent distribution modeling. A simulation study is presented to examine recovery of the input distribution and output parameter. The model is then applied to the daily self-reports of negative affect and stress from a sample of older adults from the Notre Dame Longitudinal Study on Aging.

Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes

We examine emotion self-regulation and coregulation in romantic couples using daily self-reports of positive and negative affect. We fit these data using a damped linear oscillator model specified as a latent differential equation to investigate affect dynamics at the individual level and coupled influences for the 2 partners in each couple. Results indicate an absence of damping of relationship-specific affect within individuals in the sample. When both positive and negative affect are modeled at the individual level, the influence of positive affect is greater than that of negative affect. At the dyad level, the findings indicate coupled influences in both positive and negative affect between partners. With regard to positive affect, females are sensitive to their partners' overall displacement from average as well as their rate of change; males are sensitive only to their partners' displacement from average. For negative affect both partners are sensitive to each other's displacement from average, yet there are no coupled influences for rates of change in this dimension. We interpret the influence of the parameters on the system by examining the expected behavior of the system as a function of varying parameter values.

Reply to Steele & Ferrer: Modeling Oscillation, Approximately or Exactly?

This article addresses modeling oscillation in continuous time. It criticizes Steele and Ferrer's article “Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes” (2011), particularly the approximate estimation procedure applied. This procedure is the latent version of the local linear approximation procedure based on Boker (2001) and Boker, Neale, and Rausch (2004). It furthermore presents two exact alternative estimation procedures, one using filter techniques and the other using structural equation modeling.

Response to Oud & Folmer: Randomness and Residuals

This article presents our response to Oud and Folmer's “Modeling Oscillation, Approximately or Exactly?” (2011), which criticizes aspects of our article, “Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes” (2011). In this response, we present a conceptual explanation of the derivative-based estimation that we implemented in our original article, as well as the exact discrete model promoted by Oud & Folmer. We describe relevant differences between each of the two methods, highlight some of their benefits and limitations, and offer justifications for our choice.