Abstract:

Mediated moderation (meMO) accounts for the moderating effect of a moderator (W) on the relationship between an independent variable (X) and a dependent variable (Y) transmitted through a mediator (M). It has been widely applied in psychological studies. However, the traditional meMO (also known as the first type of meMO, or meMO-I) is difficult to interpret, and thus, researchers often misapply it. In this regard, the second type of meMO (meMO-II), which has been in the foreground in recent years, can better articulate the definition of meMO. meMO-II indicates that the moderating effect of W on X→Y is mediated by an intervening variable, M, which is affected by W initially. An important theoretical implication of meMO-II is that it reveals how W moderates the effect of X on Y, for which meMO-I has limited applicability.

We compare and contrast meMO-II with other models that combine moderation and mediation. Then, we demonstrate the analytical procedure behind meMO-II. We conducted this analysis in three steps. The first step was to test whether there was a total moderating effect of W on X and Y. With Y as the dependent variable, we performed a hierarchical regression analysis in which the independent variable X and moderator W are entered first into the regression, followed by the interaction term XW. If there is a total moderating effect, this will be confirmed by XW's regression coefficient being statistically significant and ΔR² being high enough (e.g., ≥ 0.02 or even ≥ 0.03). The second step was to test whether there was an indirect moderating effect. If the regression coefficient a₁of W from the regression of M to W, and the coefficient b₂from interaction term XM from the regression of Y to X, W, M, XW and XM, are significant, respectively, then the indirect moderating effect is significant (causal steps approach). If at least one coefficient was insignificant, we would apply the bootstrap method with higher power to test the product coefficient a₁b₂ directly. If the 95% bootstrap confidence interval of a₁b₂did not contain 0, the indirect moderating effect was significant (product of coefficients approach). The third step was to conduct a simple slope analysis. In accordance with the relation between the moderators W and M, we fixed the two variables at a set level to investigate the difference in the effect of X on Y at different points. This method is known as the pick-a-point technique. The Johnson-Neyman (J-N) technique can also be adopted to identify the range of W values for which the main effect between the X and Y is significant. Furthermore, we applied the above procedure to an empirical study as an illustration.

Then, we present two analytical methods of meMO-II based on latent variables: latent moderated structural equations (LMS) and the factor score approach (FS). Several recent advances in modeling approach of meMO-II, including variable system (VS) and two-level mediated moderation (2meMO) are also introduced. As for VS, the product of path coefficients from XW→XM→Y is employed as the index of indirect moderation by multiplying X to the regression of M to W. Thus the disadvantages of endogeneity, incoherent paths and confusion about meMO-II, and the second-stage moderated mediation model in meMO-II, can be mitigated, whereas 2meMO uses error term partitioning in a multi-level model as a reference, applying the random effect of X on Y to the meMO-II model. This saves the trouble of verifying the homoscedasticity assumption.

Additionally, this paper also presents several effect sizes concerning indirect moderation. These include the ratio of the indirect moderating effect divided by the total moderating effect, υ (i.e., the square of a₁b₂), and an effect size based on the variance decomposition of coefficients ϕ_{MO_ind}(i.e., the ratio of the variance of a₁b₂ to the total variance of the path coefficients from X→Y).

Finally, we present the variant models of meMO-II, including multiple indirect moderation, suppression effect in moderation, multi-level indirect moderation, indirect moderation of categorical variables, moderated indirect moderation and indirectly moderated mediation.

Key words: the second type of mediated moderation, moderating mechanisms, two-level mediated moderation, variable system, latent variables