Path and Directionality Discovery in Individual Dynamic Models: A Regularized Unified Structural Equation Modeling Approach for Hybrid Vector Autoregression

Abstract

There recently has been growing interest in the study of psychological and neurological processes at an individual level. One goal in such endeavors is to construct person-specific dynamic assessments using time series techniques such as Vector Autoregressive (VAR) models. However, two problems exist with current VAR specifications: (1) VAR models are restricted in that contemporaneous relations are typically modeled either as undirected relations among residuals or directed relations among observed variables, but not both; (2) current estimation frameworks are limited by the reliance on stepwise model building procedures. This study adopts a new modeling approach. We first extended the current unified SEM (uSEM) framework, a widely used structural VAR model, to a hybrid representation (i.e., “huSEM”) to include both undirected and directed contemporaneous effects, and then replaced the stepwise modeling with a LASSO-type regularization for a global search of the optimal sparse model. Our simulation study showed that regularized huSEM performed uniformly the best over alternative VAR representations and/or modeling approaches, with respect to accurately recovering the presence and directionality of hybrid relations and reliably removing false relations when the data are generated to have two types of contemporaneous relations. The present study to our knowledge is the first application of the recently developed regularized SEM technique to the estimation of huSEM, which points to a promising future for statistical learning in psychometric models.

Appendices

Appendix A: Model Equivalency

We present here a transformation from an huSEM to an equivalent VAR. To start we provide the transformation from Equation 3 in the text:

$$\begin{aligned} Y_t = (I-A)^{-1}\Phi ^* Y_{t-1} + (I-A)^{-1}\zeta _t \end{aligned}$$

(11)

where I is a $p \times p$ identity matrix and all other values are as before. Note here that $\Phi ^*$ indicates the lagged matrix obtained in uSEM. It follows that the transformed $\Psi$ matrix becomes: $\Theta = (I-A)^{-1}\Psi (I-A)^{-1\prime }$. Following this transformation, we obtain an equivalent VAR representation (Equation 2). Note that $\Phi = (I-A)^{-1}\Phi ^*$ is the estimates one would expect from VAR analyses of this model. Here we use a numerical example to explicate the different values. Take the following data generating matrices for a hybrid uSEM:

$$\begin{aligned} A = \begin{bmatrix} 0 &{} 0 &{} 0 \\ .4 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{bmatrix},\quad \Phi ^* = \begin{bmatrix} .5 &{} 0 &{} 0 \\ 0 &{} .6 &{} 0 \\ 0 &{} 0 &{} .2 \\ \end{bmatrix},\quad \Psi ^* = \begin{bmatrix} .7 &{} 0 &{} .5 \\ 0 &{} .7 &{} 0 \\ .5 &{} 0 &{} .7 \\ \end{bmatrix}. \end{aligned}$$

(12)

Following transformation the matrices become:

$$\begin{aligned} \Phi = \begin{bmatrix} .5 &{} 0 &{} 0 \\ .2 &{} .6 &{} 0 \\ 0 &{} 0 &{} .2 \\ \end{bmatrix},\quad \Theta = \begin{bmatrix} 1 &{} .4 &{} .5 \\ .4 &{} 1.16 &{} .2 \\ .5 &{} .2 &{} 1 \\ \end{bmatrix} \end{aligned}$$

(13)

As can be seen by this toy example, a contemporaneous covariance emerges among variables 2 and 3: $\Theta _{2,3}=0.4$. When the variances of the errors are 1, the covariance value of residuals between two given variables in a transformed VAR model will equal that of the corresponding contemporaneous coefficient in the $\varvec{A}$ matrix of a huSEM. When the variance is less than one, this value will be smaller than the contemporaneous path; if it is larger, this value is higher for the covariance than for the directed value. Additionally, a lagged relation between variables 2 and 3 $\Phi _{2,3} = .2$ now exists in this transformed version. Since the diagonal of $\Phi$ will always be less than one for stability of the process to hold (one assumption of these time series) then the cross-lagged value in the transformed VAR will always be lower in absolute value than the value for the coefficient between the same two variables in original the $\varvec{A}$ matrix. Code is provided in the online supplement for simulating data and replicating these results.

Appendix B: Convergence Result

Nonconvergence occurs when the estimation algorithm fails to arrive at values which meet prescribed minimum criteria under the default settings of the program within a set number of iterations (Anderson & Gerbing, 1984). Since model misspecification is a possible source of nonconvergence, the low convergence rate of Reg-uSEM might result from a number of reasons, one of which might be the omission of the covariance parameters for the contemporaneous variables. This leads to a failure to return a function of estimable parameters that reasonably approximates the variance-covariance structure of the variables provided by the data (Table 2).

Table 2 Rate of Convergence for the models using regsem.

Appendix C: Supplemental Result

See Figs. 10, 11 and Table 3.

Fig. 10

Result of fMRI Study Person One. Note: (1) IL = left Insula, IR = right Insula, CM = right median cingulate, PR = right precentral gyrus, SMA = left supplementary motor area; (2) regsem does not provide a plotting function, we used qgraph to plot the result. Plot consists of solid lines on the left represents contemporaneous relations with two-headed curves represent covariance relations, plot with dotted lines on the right represents lagged effects. Green lines represent positive values, and red represent negative. Line width corresponds with coefficient estimate size. Plots only present $> .01$ relations, similar to the convention in programs like graphicalVAR (Color figure online).

Fig. 11

Result of fMRI Study Person One. Note: (1) IL = left Insula, IR = right Insula, CM = right median cingulate, PR = right precentral gyrus, SMA = left supplementary motor area; (2) regsem does not provide a plotting function, we used qgraph to plot the result. Plot consists of solid lines on the left represents contemporaneous relations with two-headed curves represent covariance relations, plot with dotted lines on the right represents lagged effects. Green lines represent positive values, and red represent negative. Line width corresponds with coefficient estimate size. Plots only present $> .01$ relations, similar to the convention in programs like graphicalVAR.

Table 3 Sensitivity result of supplemental simulation.