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## A Note on the Specification of Error Structures in Latent Interaction Models (2015)

### BibTeX

@MISC{Mao15anote,

author = {Xiulin Mao and Jeffrey R Harring and Gregory R Hancock and Xiulin Mao},

title = {A Note on the Specification of Error Structures in Latent Interaction Models},

year = {2015}

}

### OpenURL

### Abstract

Abstract Latent interaction models have motivated a great deal of methodological research, mainly in the area of estimating such models. Product-indicator methods have been shown to be competitive with other methods of estimation in terms of parameter bias and standard error accuracy, and their continued popularity in empirical studies is due, in part, to their straightforward implementation and relative ease of estimation in mainstream structural equation modeling software. In recent years, the impact of different specifications of the mean structure of the structural model has been the focus of a fair amount of investigation in this area. Yet the effects of misspecification of the error structure of the observed variables implied by the model have not been investigated. In this study, the authors demonstrate algebraically the ramifications of misspecifying these error structures for the unconstrained product-indicator approach. Recommendations to practitioners based on these results are discussed. Keywords error structures, latent interaction models, unconstrained method, model misspecification A latent interaction model is a type of nonlinear structural equation model in which at least one exogenous latent variable enters the structural model in a nonlinear fashion The aforementioned latent interaction model includes both latent endogenous and latent exogenous variables, as denoted in Equation 1, using the LISREL notation where h is a latent criterion variable; g 0 is an intercept term; j 1 and j 2 are latent predictor variables; j 1 j 2 is the interaction term; and g 1 , g 2 , and g 3 are the regression coefficients corresponding to the linear predictors and the interaction term, respectively. The disturbance term z represents the regression residual. Methods for estimating the interaction model in Equation 1 are numerous The unconstrained approach is a simplification of constrained product-indicator approaches 1 The basic assumptions underlying the unconstrained approach with three indicators for each latent variable are as follows: 1. j 1 and j 2 are bivariate normal with means k 1 and k 2 , respectively; 2. d k ;N 0, u k ð Þ, for k = 1, . . . , 6 and are independent of j j for j = 1, 2; 3. e l ;N 0, u ð Þ, for l = 1, . . . , 3 and are independent of d k for k = 1, . . . , 6; 4. z;N 0, c ð Þ, and is independent of both d k and j j for k = 1, . . . , 6 and j = 1, 2; and In a series of articles, Marsh and colleagues Mao et al. 7 at PENNSYLVANIA STATE UNIV on October 6, 2016 epm.sagepub.com Downloaded from covariances either. Therefore, a ''patch'' is technically necessary to make the unconstrained model and methods based on the unconstrained model properly specified. In this article, we demonstrate algebraically the ramifications of misspecifying these error structures for the unconstrained product-indicator approach and implications this may have on drawing inferences regarding the interaction effect. To support the findings from the analytic derivations, we further investigate the consequences of misspecification of the error structure and misspecification due to assuming that the joint distribution of the observed and latent variables is multivariate normal by conducting a small population analysis in which we quantify the degree to which these misspecifications adversely affect the interact effect. Mathematical Derivation of Elements in the Error Covariance Matrix As a starting point, for the model depicted in The covariance can be specified as , and the Cov(d 2 , t x 25 ) = 0, On the basis of Defining Dl = l x 2 l x 5 À l x 25 , on the basis of Equation 2, the first piece of u 82 , Cov½d 2 , (l x 2 l x 5 À l x 25 )j 1 j 2 ), can be reexpressed as Moreover, given distributional assumptions about d and j Educational and Psychological Measurement 75 Similarly, and Therefore, by substitution, Now, on the basis of Bohrnstedt and Goldberger, if d 2 , j 1 , j 2 are multivariate normal, all third moments equal zero. Therefore, If indicator variables are centered, t x 5 = 0; thus, u 82 = 0. However, if j 1 and j 2 are not assumed to be bivariate normally distributed, as claimed to be allowed by the unconstrained approach, the third moment will not vanish: Furthermore, when the broader assumption of multivariate normality is released, the nonnormality applies not only to the first-order exogenous latent variables but also to the error terms (d 1 , . . . , d 6 ). Therefore, at PENNSYLVANIA STATE UNIV on October 6, 2016 epm.sagepub.com Downloaded from Thus, as the above mathematical derivations show, when the multivariate normality assumption is not satisfied, the error covariance elements are not zero, even when the matched-pair strategy is used to create the product indicators in the mean-centered unconstrained model proposed by Empirical Population Analysis Notably different from a simulation study, a population analysis is not focused on the behavior of statistics from repeated samplings but rather on the behavior of the parameter(s) in the population (see, e.g., The design of this population analysis was based on crossing four manipulated independent variables, yielding 36 conditions overall: 3 (observed indicator variable reliability levels) 3 3 (latent variable distributions) 3 2 (error term distributions for the observed variables) 3 2 (centering or noncentering). The population size for each cell was set to be N = 500,000. Indicator Reliability For the fully latent interaction model in which the structural model and measurement model are both accommodated in the larger model and estimated simultaneously, unreliability of the indicators may pose fewer threats Latent Variable and Error Distributions To determine and quantify the impact of the misspecification on parameter estimation, different nonnormal conditions were used. 10 Educational and Psychological Measurement 75 Data-Generating Model The latent portion of the model follows the regression structure in Equation 1 with population data generated from h = 10 + 2j 1 + 3j 2 + 0:127j 1 j 2 + z, where, as and followed the distributional assumptions previously outlined. Following Study Outcomes Four models were fit to the 18 sets of population data respectively: the unconstrained model proposed by Mao et al. 11 at PENNSYLVANIA STATE UNIV on October 6, 2016 epm.sagepub.com Downloaded from centered and uncentered; the correlated error (CE) models with the six error covariances specified in the model, centered and uncentered (see mathematical derivation of latent covariances between exogenous predictors and interaction term in Appendix B). Mplus 6.12 (Muthén & Muthén, 2010) was used to estimate each of the models. Because there are 18 sets of population data, 36 models were run (An Mplus input file can be found in Appendix C). The estimate for the population regression coefficient for the interaction effect,ĝ 3 , was the focus of the study, although the following conventional structural equation modeling fit indices were also mentioned: comparative fit index (CFI), root mean square error of approximation (RMSEA), and standardized root mean square residual (SRMR). It should be noted that the discrepancy between values in Results and Discussion 12 Educational and Psychological Measurement 75 13 at PENNSYLVANIA STATE UNIV on October 6, 2016 epm.sagepub.com Downloaded from 0.127, ranging from 0.122 to 0.130, and the estimates of the interaction regression coefficient averaged across the 18 conditions was 0.1263. Considering the random generation of the population data, this fluctuation was reasonable, and the estimates were very stable across all the conditions. The reliability of the indicators and the degree of nonnormality actually did not exert an influence on the estimation when error covariances were taken into account. For UE models, however, in which the error covariances were not specified, when the reliability of the indicators was high (0.95), the latent parameter estimates deviated from their respective population values of 3, 2, and 0.127 to a large extent across all the conditions. At the same time, the estimates for the interaction parameter showed a consistent downward bias, but the estimates for the first-order regression coefficient tended to show an upward bias. The model fit indices did not support a good fit of the UE models, as the CFI was much lower than the common benchmark value 0.95, and the RMSEA was also much higher than the common benchmark value of 0.05. The SRMR, though, still performed relatively well. When the reliability of the indicators was low (0.45) and medium (0.75), 12 conditions were not able to converge. When the reliability was 0.75, the estimates for the interaction coefficient were consistently lower than the population value, while they were consistently higher when the reliability of the indicators was low. The estimates for the regression coefficients for the first-order effects deviated from the true values to an even larger degree. The model fit indices also showed that UE models fit poorly because the error covariances were not properly specified. This also demonstrated that when there was model misspecification (in this case, the error covariance structures was missing), the reliability exerted a strong impact on the estimation of the model and would tend to cause convergence issues as the reliability of the indicators decreased. CE models, however, were stable and minimally affected by suboptimal reliability of the indicators. This result, therefore, illustrates the contention made previously that the unreliability of the indicators poses fewer problems to the fully latent model estimation, but this should be confined to the conditions in which the model, including error structure, was properly specified. Summary and Conclusions Both the mathematical derivation and empirical population analysis demonstrate that if the multivariate normality assumption is violated, the error covariances between