Comparison Between Frequentist And Bayesian Estimation In Latent Moderated Structural Equations

In the fields of psychology,education,sociology and so on,an increasing number of studies have focused on models that integrate moderation and mediation in order to explain the complex psychological phenomena.There are two kinds of integrated models: moderated mediation model and mediated moderation model.Considering wider coverage for the concept of moderated mediation model,this study chooses it as the entry point to explore integrated model.Latent moderated structural equations(LMS)approach is a commonly used method for the analysis of moderated mediation model under a structural equation modeling framework.Compared with multiple linear regression and product indicators approach,LMS has the following advantages:(1)it treats the interaction term as a latent variable so that estimates are unattenuated by measurement error;(2)there is no need to construct product indicators,avoiding the problems in the process of product indicator generation;(3)it is easy to implement in commercial desktop software,Mplus and is convenient for applied researchers to use in practice.Researchers have discussed the performance of LMS based on frequentist estimation method.However,there are some limitations in frequentist estimation method.In recent years,researchers have paid attention to the Bayesian estimation method.The advantage of Bayesian estimation relies on the proper setting of prior distribution,and the setting of prior distribution and its associated hyperparameters are affected subjectively.Therefore,in the Bayesian framework,it is an important research topic to investigate the influence of prior distribution and how to assign the appropriate prior distribution.In addition,non-normal data is very common in practical research.The generalization of research results will be restricted if nonnormality of observed indicators is ignored.The purpose of this study includes:(1)based on the moderated mediation model,we investigate the influence of frequentist estimation(maximum likelihood estimation with the Satorra-Bentler correction for robust standard errors,MLR)and Bayesian estimation on the performance of parameter estimation under different data distributions(normal distribution,symmetrically moderated kurtosis distribution,symmetrically high kurtosis distribution and slightly skewed distribution).With Bayesian estimation,the settings of prior distribution cover different informative level(variance)and accuracy(the degree to which the mean deviates from the truth value).(2)An empirical example is provided to demonstrate the procedure of LMS based on different estimation methods in the analysis of the moderated mediation model.Simulation studies show that Bayesian estimation with weakly/strongly informative and correct prior distribution(BWC and BSC)perform best,which can provide accurate parameter estimation,excellent estimation precision,coverage rate and power.The performance of MLR in relative bias and coverage rate is similar to BWC and BSC,but MLR has problems in estimation precision and power.The incorrect setting of the mean in prior distribution will decrease the performance of Bayesian estimation,but it can be remedied by increasing the ratio of sample and item.Bayesian estimation with default prior distribution(BD)has problems in model convergence,especially with non-normally distributed data.Then when indicators are symmetrically high kurtosis distribution and slightly skewed distribution,BD has abnormal relative bias and root mean-square error.The empirical example focuses on the mechanism between mother’s over-reactivity and internet addiction in adolescents,in order to explore the mediation effect of self-control and moderation effect of mindfulness.This part systematically demonstrates the process of analyzing moderated mediation model by LMS based on MLR and Bayesian estimation methods.It covers theoretical model construction,data analysis,and result interpretation,which can be used as a guide for applied researchers.Through the above simulation studies and empirical example,it is hoped to provide guidance for researchers on how to choose appropriate estimation methods when exploring the moderated mediation model.Thus,the application of LMS can be extended in various fields.If the researchers can obtain prior information about unknown parameters,Bayesian estimation with informative prior distribution is recommended;otherwise,the recommendation is MLR.