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Mediated Moderation
Mediated moderation is a variant of both moderation and mediation. This is where there is initially overall moderation and the direct effect of the moderator variable on the outcome is mediated either at the A path in the diagram above, between the independent variable and the moderating variable, or at the B path, between the moderating variable and the dependent variable. The main difference between mediated moderation and moderated mediation is that for the former there is initial moderation and this effect is mediated and for the latter there is no moderation but the effect of either the treatment on the mediator (path A) is moderated or the effect of the mediator on the outcome (path B) is moderated.

In order to establish mediated moderation, one must first establish moderation, meaning that the direction and/or the strength of the relationship between the independent and dependent variables (path C) differs depending on the level of a third variable (the moderator variable). Researchers next look for the presence of mediated moderation when they have a theoretical reason to believe that there is a fourth variable that acts as the mechanism or process that causes the relationship between the independent variable and the moderator (path A) or between the moderator and the dependent variable (path C).

Example

The following is a published example of mediated moderation in psychological research. Participants were presented with an initial stimulus (a prime) that made them think of morality or made them think of might. They then participated in the Prisoner’s Dilemma Game (PDG), in which participants pretend that they and their partner in crime have been arrested, and they must decide whether to remain loyal to their partner or to compete with their partner and cooperate with the authorities. The researchers found that prosocial individuals were affected by the morality and might primes, whereas proself individuals were not. Thus, social value orientation (proself vs. prosocial) moderated the relationship between the prime (independent variable: morality vs. might) and the behaviour chosen in the PDG (dependent variable: competitive vs. cooperative).

The researchers next looked for the presence of a mediated moderation effect. Regression analyses revealed that the type of prime (morality vs. might) mediated the moderating relationship of participants’ social value orientation on PDG behaviour. Prosocial participants who experienced the morality prime expected their partner to cooperate with them, so they chose to cooperate themselves. Prosocial participants who experienced the might prime expected their partner to compete with them, which made them more likely to compete with their partner and cooperate with the authorities. In contrast, participants with a pro-self social value orientation always acted competitively.

Models of Mediated Moderation

There are five possible models of mediated moderation, as illustrated in the diagrams below.


 * 1) In the first model the independent variable also mediates the relationship between the moderator and the dependent variable.
 * 2) The second possible model of mediated moderation involves a new variable which mediates the relationship between the independent variable and the moderator (the A path).
 * 3) The third model of mediated moderation involves a new mediator variable which mediates the relationsip between the moderator and the dependent variable (the B path).
 * 4) Mediated moderation can also occur when one mediating variable affects both the relationship between the independent variable and the moderator (the A path) and the relationship between the moderator and the dependent variable (the B path).
 * 5) The fifth an final possible model of mediated moderation involves two new mediator variables, one mediating the A path and the other mediating the B path.

Regression Equations for Moderated Mediation and Mediated Moderation
Muller, Judd, and Yzerbyt (2005) outline three fundamental models that underlie both moderated mediation and mediated moderation. Mo represents the moderator variable(s), Me represents the mediator variable(s), and εi represents the measurement error of each regression equation.

Step 1: Moderation of the relationship between the independent variable (X) and the dependent variable (Y), also called the overall treatment effect (path C in the diagram at the top of this page).
 * $$Y=\beta_{40} +\beta_{41}X +\beta_{42}Mo +\beta_{43}XMo + \varepsilon_4$$


 * To establish overall moderation, the β43 regression weight must be significant (first step for establishing mediated moderation).
 * Establishing moderated mediation requires that there be no moderation effect, so the β43 regression weight must not be significant.

Step 2: Moderation of the relationship between the independent variable and the mediator (path A in the diagram at the top of this page).
 * $$Me=\beta_{50} +\beta_{51}X +\beta_{52}Mo +\beta_{53}XM_o + \varepsilon_5$$


 * If the β53 regression weight is significant, the moderator affects the relationship between the IV and the mediator.

Step 3: Moderation of both the relationship between the independent and dependent variables (path A at the top of this page) and the relationship between the mediator and the dependent variable (path B at the top of this page).
 * $$Y=\beta_{60} +\beta_{61}X +\beta_{62}Mo +\beta_{63}XMo +\beta_{64}Me +\beta_{65}MeMo + \varepsilon_6$$


 * If both β53 in step 2 and β64 in step 3 are significant, the moderator affects the relationship between the independent variable and the mediator (path A).
 * If both β51 in step 2 and β65 in step 3 are significant, the moderator affects the relationship between the mediator and the dependent variable (path B).
 * Either or both of the conditions above may be true.