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In statistics, moderation and mediation can occur together in the same model. Moderated mediation, also known as conditional indirect effects, occurs when the treatment effect of an independent variable A on an outcome variable C via a mediator variable B differs depending on levels of a moderator variable D. Specifically, either the effect of A on the B, and/or the effect of B on C depends on the level of D.

Muller, Judd, & Yzerbyt (2005) model
Muller, Judd, and Yzerbyt (2005) were the first to provide a comprehensive definition of this process. The following regression equations are fundamental to their model of moderated mediation (note: X = independent variable A, Y = outcome variable C, Me = mediator variable B, and Mo = moderator variable D):

Y = β40 + β41X + β42Mo + β43XMo + ε4

This equation assesses moderation of the overall treatment effect of A on C.

Me = β50 + β51X + β52Mo + β53XMo + ε5

This equation assesses moderation of the treatment effect of A on the mediator B.

Y = β60 + β61X + β62Mo + β63XMo + β64Me + β65MeMo + ε6

This equation assesses moderation of the effect of the mediator B on C, as well as moderation of the residual treatment effect of A on C.

A fundamental equality exists among these equations, such that:

β43 – β63 = β64 β53 + β65 β51

In order to have moderated mediation, there must be an overall treatment effect of A on the outcome variable C (β41), which does not depend on the moderator (β43 = 0). Either the treatment effect of A on the mediator B depends on the moderator (β53 ≠ 0) and/or the effect of the mediator B on the outcome variable C depends on the moderator (β65 ≠ 0).

At least one of the products on the right side of the above equality must not equal 0 (i.e. if β53 ≠ 0, then β64 ≠ 0, or if β65 ≠ 0, then β51 ≠ 0). As well, since there is no overall moderation of the treatment effect of A on the outcome variable C (β43 = 0), this means that β63 cannot equal 0. In other words, the residual direct effect of A on the outcome variable C, controlling for the mediator, is moderated.

Additions by Preacher, Rucker, & Hayes (2007)
In addition to the three manners proposed by Muller and colleagues in which moderated mediation can occur, Preacher, Rucker, and Hayes (2007) proposed that the independent variable A itself can moderate the effect of the mediator B on the outcome variable C. They also proposed that a moderator variable D could moderate the effect of A on B, while a different moderator E moderates the effect of B on C.

Differences between moderated mediation and mediated moderation
Moderated mediation relies on the same underlying models (specified above) as mediated moderation. The main difference between the two processes is whether there is overall moderation of the treatment effect of A on the outcome variable C. If there is, then there is mediated moderation. If there is no overall moderation of A on C, then there is moderated mediation.

Testing for moderated mediation
In order to test for moderated mediation, some claim that it is best to examine a series of models, sometimes called a piecemeal approach, and look at the overall pattern of results. These researchers claim that a single overall test would be insufficient to analyze the complex processes at play in moderated mediation, and would not allow one to differentiate between moderated mediation and mediated moderation.

Bootstrapping has also been suggested as a method of estimating the sampling distributions of a moderated mediation model in order to generate confidence intervals. This method has the advantage of not requiring that any assumptions be made about the shape of the sampling distribution.

Preacher, Rucker and Hayes also discuss an extension of simple slopes analysis for moderated mediation. Under this approach, one must choose a limited number of key conditional values of the moderator that will be examined. As well, one can use the Johnson-Neyman technique to determine the range of significant conditional indirect effects.

Preacher, Rucker, and Hayes (2007) have created an SPSS macro that provides bootstrapping estimations as well as Johnson-Neyman results.