A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

Abstract

The bivariate Stable Trait, AutoRegressive Trait, and State (STARTS) model provides a general approach for estimating reciprocal effects between constructs over time. However, previous research has shown that this model is difficult to estimate using the maximum likelihood (ML) method (e.g., nonconvergence). In this article, we introduce a Bayesian approach for estimating the bivariate STARTS model and implement it in the software Stan. We discuss issues of model parameterization and show how appropriate prior distributions for model parameters can be selected. Specifically, we propose the four-parameter beta distribution as a flexible prior distribution for the autoregressive and cross-lagged effects. Using a simulation study, we show that the proposed Bayesian approach provides more accurate estimates than ML estimation in challenging data constellations. An example is presented to illustrate how the Bayesian approach can be used to stabilize the parameter estimates of the bivariate STARTS model.

Alleviating estimation problems in small sample structural equation modeling—A comparison of constrained maximum likelihood, Bayesian estimation, and fixed reliability approaches.

“…Given that the implementations of constrained ML and the Bayesian approach without additional prior information imposed practically the same restrictions on parameter space, this effect can be traced back to the mere difference in the MCMC-based as compared with the ML-based estimation procedure. As delineated in Lüdtke et al (2021) comparing penalized ML-estimation to MCMC-based estimation, in this context, a major difference between the two estimation methods lies in the summary statistic employed as the point estimate. Note that under uniform priors, the penalized ML estimate is equal to the constrained ML estimate (Rindskopf, 2012).…”

“…Recall that for implementing the Bayesian approach, we leveraged the stabilizing effects of choosing a parameterization in which the model parameters (i.e., standardized loadings, latent correlations) are bounded (see also Lüdtke et al, 2021). Regression coefficients were then derived from latent correlations.…”

“…Note that under uniform priors, the penalized ML estimate is equal to the constrained ML estimate (Rindskopf, 2012). While penalized ML uses the multivariate mode of the joint posterior distribution, in MCMC-based estimation, the univariate mean (EAP; as in the present study), mode, or median of the marginal posterior distribution is employed (Lüdtke et al, 2021). The EAP is known to be pulled toward the prior mean (i.e., shrinkage effect, see also Choi et al, 2011), especially under conditions with small samples.…”

“…To aid the estimation of parameters of interest, the problem of estimating an unbounded parameter may be transformed into the problem of estimating a (naturally) bounded parameter (see Lüdtke et al, 2021). We implement this by estimating standardized factor loadings and latent correlations rather than unstandardized factor loadings and latent regression parameters.…”

“…Parameterizations and specifications may heavily impact the performance of different estimation approaches (Chen et al, 2014; Choi & Levy, 2017; Gerbing & Anderson, 1987; Lüdtke et al, 2021; Rindskopf, 1983, 1984). Further, the performance of Bayesian SEM has been found to be highly dependent on chosen prior distributions, especially under small sample conditions (Miočević et al, 2021; van Erp et al, 2018).…”