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→Regression equations for moderated mediation and mediated moderation
==Regression equations for moderated mediation and mediated moderation==
==Regression equations for moderated mediation and mediated moderation==
Muller, Judd, and Yzerbyt (2005)<ref name="Muller"/> 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 ''ε<sub>i</sub>'' represents the measurement error of each regression equation.
Muller, Judd, and Yzerbyt (2005)<ref name="Muller"/> 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 ''ε<sub>i</sub>'' represents the measurement error of each regression equation.
[[File:Mediation.jpg|434px|right|A simple statistical mediation model.]]
[[File:Mediation.jpg|434px|right|A simple statistical mediation model.]]
'''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).
'''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).
: <math>Y=\beta_{40} +\beta_{41}X +\beta_{42}Mo +\beta_{43}XMo + \varepsilon_4</math>
:<math>Y=\beta_{40} +\beta_{41}X +\beta_{42}Mo +\beta_{43}XMo + \varepsilon_4</math>
* To establish overall moderation, the ''β''<sub>43</sub> regression weight must be significant (first step for establishing mediated moderation).
* To establish overall moderation, the ''β''<sub>43</sub> regression weight must be significant (first step for establishing mediated moderation)
* Establishing moderated mediation requires that there be no moderation effect, so the ''β''<sub>43</sub> regression weight must not be significant.
* Establishing moderated mediation requires that there be no moderation effect, so the ''β''<sub>43</sub> regression weight must not be significant.
'''Step 2''': Moderation of the relationship between the independent variable and the mediator (path ''A'').
'''Step 2''': Moderation of the relationship between the independent variable and the mediator (path ''A'').
: <math>Me=\beta_{50} +\beta_{51}X +\beta_{52}Mo +\beta_{53}XMo + \varepsilon_5</math>
: <math>Me=\beta_{50} +\beta_{51}X +\beta_{52}Mo +\beta_{53}XMo + \varepsilon_5</math>
* If the ''β''<sub>53</sub> regression weight is significant, the moderator affects the relationship between the independent variable and the mediator.
* If the ''β''<sub>53</sub> regression weight is significant, the moderator affects the relationship between the independent variable and the mediator.
'''Step 3''': Moderation of both the relationship between the independent and dependent variables (path ''A'') and the relationship between the mediator and the dependent variable (path ''B'').
'''Step 3''': Moderation of both the relationship between the independent and dependent variables (path ''A'') and the relationship between the mediator and the dependent variable (path ''B'').
: <math>Y=\beta_{60} +\beta_{61}X +\beta_{62}Mo +\beta_{63}XMo +\beta_{64}Me +\beta_{65}MeMo + \varepsilon_6</math>
: <math>Y=\beta_{60} +\beta_{61}X +\beta_{62}Mo +\beta_{63}XMo +\beta_{64}Me +\beta_{65}MeMo + \varepsilon_6</math>
* If both ''β''<sub>53</sub> in step 2 and ''β''<sub>63</sub> in step 3 are significant, the moderator affects the relationship between the independent variable and the mediator (path ''A'').
* If both ''β''<sub>53</sub> in step 2 and ''β''<sub>63</sub> in step 3 are significant, the moderator affects the relationship between the independent variable and the mediator (path ''A'').
* If both ''β''<sub>53</sub> in step 2 and ''β<sub>65</sub>'' in step 3 are significant, the moderator affects the relationship between the mediator and the dependent variable (path ''B'').
* If both ''β''<sub>53</sub> in step 2 and ''β<sub>65</sub>'' 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.
==Causal mediation analysis==
==Causal mediation analysis==
===Fixing versus conditioning===
===Fixing versus conditioning===