Two-Stage Path Analysis With Definition Variables: A General Framework for Modeling With Categorical Indicators

• In Event: Composite/Factor Scores Analyses With Bias Correction: Good Alternatives to Full Structural Equation Modeling?

Abstract

When estimating path coefficients among latent constructs measured with errors, full structural equation modeling (SEM), which simultaneously estimates the measurement and structural parameters, is generally regarded as the gold standard. However, especially in small samples, full SEM may give unstable or nonconverged results. In practice, researchers commonly rely instead on computing composite scores or factor scores and use those as observed variables in a path analysis for simplifying the model or avoiding model convergence issues. Whereas recent approaches such as reliability adjustment methods (Hsiao et al., 2018) and factor score regression (e.g., Devlieger et al., 2016) has been proposed to mitigate the bias induced by omitting measurement errors in composite/factor scores with continuous indicators (Cole & Preacher, 2014), those approaches are not yet applicable to models with categorical indicators, which, unlike the continuous indicators case, result in measurement error variance that varies across observations. Categorical indicators also present computational challenges to full SEM, as maximum likelihood estimation requires using numerical integration which becomes infeasible with more than a few latent variables (Embretson & Reise, 2000), and weighted least squares estimation requires a relatively large sample size for stable results (Li, 2016).

In this paper, we introduce the two-stage path analysis (2S-PA) with definition variable as a general framework for path modeling to adjust for measurement errors that follow different distributions across observations. The 2S-PA framework separates the estimations of factor scores and path coefficients, and thus allows for different estimation methods in the measurement and the structural path models and easier diagnoses of model assumptions. We conduct three simulation studies. In Study 1, we compare the performance of 2S-PA with full SEM and other reliability adjustment methods when there were measurement error on the latent predictor measured by categorical indicators, and find that while 2S-PA gives slightly smaller path coefficient estimates with small sample sizes, it performs similarly to full SEM in other conditions and has better convergence rates and control of SE bias and Type I error rates, especially in small samples. In Study 2, when both the latent predictor and the latent outcome are measured with error, 2S-PA outperforms full SEM in terms of convergence rates, bias, efficiency, and control of Type I error rates in small to moderate samples. In Study 3, with a mediation path model, we find that 2S-PA again provides better control on Type I error and coverage rates of 95% confidence intervals for the indirect effect. We also illustrate 2S-PA using data from a national data set, and show how researchers can implement it in Mplus and OpenMx.

The 2S-PA framework represents a highly flexible framework that allows researchers to use different estimation methods and psychometric models for scoring the different components in a path model, and adjust for the measurement errors of those factor scores. Given the computational challenges and numerical instability of full SEM in small to moderate samples, 2S-PA is a viable alternative for accurately estimating structural relations among constructs.

Authors

• Hok Chio (Mark) Lai, University of Southern California

• Yu-Yu Hsiao, University of New Mexico