Abstract: Moderation and mediation are frequently used in the research of psychology and behavior. Moderation occurs when the effect of an independent variable on a dependent variable varies according to the level of a third variable, termed as moderator, which interacts with the independent variable. Moderation focuses on factors that influence the strength and/or direction of the relation between the independent variable and dependent variable. Mediation indicates that the effect of an independent variable on a dependent variable is transmitted through a third variable, which is called mediator. Mediation addresses how that effect is produced. It is not uncommon for hypotheses about moderation and mediation relationships to occur in the same context. Models in which interaction effects are hypothesized to be mediated are appearing with increasing frequency. When a moderating effect is transmitted through a mediator, the effect is termed mediated moderation and the model is mediated moderator model. For example, stressful life events moderated the effect of sensation seeking on tobacco and alcohol use, and this moderation effect is mediated by affiliation with deviant peers. There are at least five methods for testing mediated moderation models. But some of the methods are difficult to be understood and explained. After discussing the merits and demerits of different methods, we propose a procedure for testing mediated moderation models. The newly proposed procedure is likely to be better than any single testing method in terms of the sum of type 1 and type 2 error rates. The procedure is summarized as below: (1) Regress Y on X, U, and UX. A significant coefficient (c3) associated with UX implies that U is the moderator of the relation between Y and X. Stop if the coefficient (c3) associated with UX is not significant. (2) Regress W on X, U, and UX. Regress Y on X, U, W, UX and UW. The moderated effect of U on the relation between Y and X is mediated by W if the coefficient (a3) from UX to W and the coefficient (b1) from W to Y are both significant, or/and the coefficient (a3) from UX to W and the coefficient (b2) from UW to Y are both significant. U indirectly moderates the effect of X on Y by moderating the effect of W on Y if the coefficient (a1) from X to Y and the coefficient (b2) from UW to Y are both significant. In other cases, we should go to step (4). (3) The moderated effect is completely mediated if the coefficient ( ) from UX to Y is not significant in step (2). The moderated effect is partially mediated if the coefficient ( ) from UX to Y is still significant in step (2). Testing is over. (4) Compute the confidence intervals of a3b1, a3b2 and a1b2, by using bias-corrected percentile Bootstrap method (if no prior information is available) or Markov chain Monte Carlo (MCMC) method (if prior information is available). The mediated effect of W is significant if at least one of the confidence intervals of a3b1, a3b2 and a1b2 does not contain 0. In this case, we should go to step (3). Otherwise, the mediated moderation effect is not significant and testing is over. As an illustration, the procedure is applied to an empirical study in which stressful life events moderated the effect of sensation seeking on tobacco and alcohol use, and this moderation effect is mediated by affiliation with deviant peers. The procedure can be extended to the situation where latent variables are used. When a standardized solution is pursued, we should use appropriate standardized solution rather than a standardized solution obtained directly from software output, which is usually inappropriate when a moderation effect is involved in the model.

Key words: moderation, mediation, mediated moderation, Bootstrap method, MCMC method