ISSN 0439-755X
CN 11-1911/B

中国科学院心理研究所

• 研究报告 •

### 潜变量交互效应标准化估计：方法比较与选用策略

1. 华南师范大学心理学院/心理应用研究中心, 广州 510631
• 收稿日期:2021-08-09 出版日期:2022-01-25 发布日期:2021-11-26
• 通讯作者: 温忠麟 E-mail:wenzl@scnu.edu.cn
• 基金资助:
国家自然科学基金项目(32171091);国家自然科学基金项目(31771245)

### Standardized estimates for latent interaction effects: Method comparison and selection strategy

WEN Zhonglin(), OUYANG Jinying, FANG Junyan

1. School of Psychology & Center for Studies of Psychological Application, South China Normal University, Guangzhou 510631, China
• Received:2021-08-09 Online:2022-01-25 Published:2021-11-26
• Contact: WEN Zhonglin E-mail:wenzl@scnu.edu.cn

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.