Structural Equation Modeling of Latent Interactions Without Using the Mean Structure
潜变量交互效应建模:告别均值结构

Estimating the interaction between variables is a particularly important theoretical, substantive, and empirical issue in psychology, as well as in many other social and behavioral sciences. Interactions between (multiple indicator) latent variables are rarely used because of the implementation complexity especially when the mean structure is known as a necessary part of any latent interaction model. There are four types of parameters related to the mean structure, which are namely, the intercepts of the y-measurement equations, the intercepts of the x-measurement equations, the intercepts of the structural equations, and the means of the exogenous latent variables. In this article, it is shown that the mean structure in the latent interaction model comes from the non-zero mean of the latent interaction construct ξ_1ξ_2 (the product of the two first terms). Thus, the means of the exogenous latent variables and the intercepts of the y-measurement equations are always necessary even if all indicators are mean-centered when the traditional latent interaction construct is used. By building a new latent interaction construct so that its mean is zero, we obtain a structural equation model of latent interaction in which the mean structure is no longer necessary and the parameters of main and interaction effects are unchanged. A simulation study comparing the estimated parameters and goodness of fit indices of the two latent interaction models with and without the mean structure by using the matched-pair product indicators and the unconstrained approach is demonstrated. The simulation results are consistent with the theoretical predictions. This research unambiguously shows that the mean structure problem which has unduly deterred the applied researchers for a long time can now be solved.