关键词: 潜变量, 交互效应, 乘积指标法, 潜调节结构方程, 贝叶斯法, 标准化估计

Abstract:

Analyzing the interaction effect of latent variables has become an important topic in both theoretical and empirical studies. Standardized estimation plays an important role in model interpretation and effect comparison. Although Wen et al. (2010) has formulated the appropriate standardized estimation for the latent interaction effects, there is no popular commercial software that provides the appropriate standardized estimation before the launch of Mplus 8.2 in 2019.
Previous comparisons of methods for estimating latent interaction were based on the original estimation. In this study, through a simulation experiment, the appropriate standardized estimation of latent interaction effects is obtained respectively by four methods: the product indicator (PI) approach, Latent Moderated Structural Equations (LMS), Bayesian method without prior information (BN), and Bayesian method with prior information (BI). Then these estimations are compared in terms of the bias of estimation, the bias of standard error, type Ⅰ error rate and statistical power.
The true model in the simulation is based on the structural equation $\eta=0.4 \xi_{1}+0.4 \xi_{2}+\gamma_{3} \xi_{1} \xi_{2}+\zeta$ where the latent variables $\eta$,$\xi_{1}$,$\xi_{2}$ each had three indicators with a standardized factor loading of 0.7. Experiment factors include the distribution of two exogenous latent variables (normal, non-normal), correlation $\phi_{12}$ between two exogenous latent variables (0, 0.3 and 0.7), interaction effect $\gamma_{3}$ (0, 0.2), sample size N (100, 200, and 500) and estimation method (PI, LMS, BN, BI).
There are five main findings. (1) the proportion of proper solution of LMS and the two Bayesian methods were close to 100% in all treatments, while PI was almost fully proper when N = 500. (2) Under the normal condition, the bias of standardized estimation of latent interaction obtained by LMS, BI and BN was ignorable, and PI was acceptable when N = 500. Under the non-normal condition, the bias of LMS and Bayesian methods inflated seriously with increasing correlation of two exogenous latent variables, but PI was still acceptable when N = 500. (3) Under both distribution conditions, the bias of standard error of standardized estimation of latent interaction obtained by LMS and BN was small and acceptable, while PI was acceptable when N = 500, and BI tended to overestimate the standard error. (4) Under normal conditions, the type I error rates of LMS were acceptable only when the sample size was large, while the other methods were acceptable in all conditions. Under the non-normal condition, the type I error rates of PI were still acceptable, while the other methods were acceptable only when the sample size was small or the correlation between two exogenous latent variables was low. (5) The statistical power of latent interaction obtained by PI was lower than that by any other method, and a large sample size (e.g., N = 500) was required to ensure the PI with statistical power over 80%; LMS and BN had higher statistical power, while BI had the highest one in all conditions.
For the latent interaction, the results of comparing different methods in standardized estimation are quite similar to those in the original estimation. Under the normal condition, it is recommended to use LMS to estimate the interaction effect of latent variables, with the caution of Type I error rate and effect size for inference. If accurate prior information can be obtained, Bayesian method is preferred, especially in the case of a small sample. When the variables are not normally distributed, the unconstrained product indicator approach is recommended, which is more robust than the other methods, but the sample size should be large enough (N = 500 or above). If the correlation between exogenous latent variables is low (it can be estimated and tested by confirmatory factor analysis), Bayesian method without prior information can be considered for small samples.

Key words: latent variable, interaction effect, product indicator, Latent Moderated Structural Equations, Bayesian estimation, standardized estimation