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

Abstract

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.

Keywords:

• exact discrete model
• latent difference scores
• latent differential equation modeling
• stochastic differential equations

Notes

¹ The derivative represents the Wiener process (a.k.a. Brownian motion). The Wiener process produces errors that are mutually independent, and the increments between two occasions are stationary and normally distributed (Arnold, 1974; Gardiner, 2009). The variance of the Wiener process depends on the time over which it is integrated (i.e., dt) such that the accumulation of errors over longer periods of time produces larger variances. Premultiplication by the constant G, which is a matrix of estimated parameters, allows the variance to be scaled for the variable of interest.

² Exogenous variables could be included in Figure 1b. An intercept term, for example, can be added to Figure 1b by adding a mean model, and allowing the mean of the latent to be estimated.

³ The Voelkle and Oud (2013) code includes a trapezoidal approximation of the A matrix——as opposed to the rectangular approximation based on as discussed in Singer (2012, Equation 17).

⁴ In the script used, values of 0 and 1 are treated as fixed, and therefore true values of 0 and 1 were perturbed by 0.0001. Earlier simulations used the full analytic solution based on eigenvalue decomposition (e.g., Voelkle et al., 2012); this method did not converge for a very large percentage of data sets (> 50%).

⁵ Initial simulations examined convergence of InSDE models. Two-construct, two-time-point data were simulated with a wide range of autoeffects (−1 to 0) and cross-effects (−0.5 to 0.5). Convergence problems were observed for models with poor starting values and a large number of latent steps; improved convergence occurred when the number of intermediate latent steps was increased sequentially from small values to larger values, with prior converging model results used as starting values. Both autoeffects and cross-effects were observed to converge on their true values as the number of latent steps increased, with autoeffects and cross-effects farther from zero (i.e., small autoregressive effects or large cross-lag effects in discrete time) requiring more latent steps for unbiased estimation. With fewer than 50 intermediate latent steps the bias can be substantial (2% or more), but from 100 to 150 intermediate latent steps the bias fell to less than 1% for all combinations of conditions examined. Plots of the estimated parameters over a series of intermediate latent steps might be useful for identifying whether estimates are converging to a reasonable value, or failing to converge. In a very small number of cases (0.1%) extremely poor estimates were produced, and the SEM convergence did not flag these poor models; all data sets producing extremely poor estimates converged on reasonable parameter estimates as the number of intermediate latent steps increased to 90 and above. The simulation suggested low bias results can be achieved with as few as 100 to 150 intermediate latent variables across a wide range of conditions. The computing requirements to run a series of 12 models from 10 to 150 latent steps was relatively reasonable (< 4 GB, < 15 min); computations were run on a single processor (2.60 GHz) Linux machine using OpenMx (v1.4-3059; Boker et al., 2011; R 2.15.3 64-bit R, 2007). Inefficiencies in the original code with regard to matrix inversion suggest that the time required to run these models can be reduced substantially (> 50%).

⁶ For the EDM, nonconvergence and implausible autoeffect estimates (< −4.0) were produced in 3 and 43 data sets, respectively. At the final estimation of 150 intermediate latent steps, InSDE did not converge eight times, produced an implausible autoeffect estimate(s) 74 times, and produced a nonpositive definite expected matrix 148 times. For the cases with nonpositive definite matrices, convergence could likely be achieved in most cases by more gradually increasing the number of intermediate latent steps.

⁷ The calculation of the expected covariance matrix is deceptively difficult from a programming perspective. Programs such as OpenMx (Boker et al., 2011) allow one to extract matrices used to calculate the expected covariance matrix, with all optimized values substituted into the matrices. The expected covariance matrix, including all latent steps, can then be calculated; for example, with RAM notation, one would solve for ; that is, removing the filter matrix so as to produce the expected covariance matrix between all latent variables.

⁸ Due to an improper solution for the covariance between IS and ED (correlation > 1), the covariances for the initial latent variables produced a nonpositive definite matrix, which is required for mvrnorm(). The correlation was set to 0.999 (closest allowable value) to allow for estimation of trajectories.