## Testing for Mediation and Moderation

**Posted on October 4, 2006**

Grayson N. Holmbeck (bio) provides tips for using multiple regression and SEM when testing for mediators and moderators.

**Q:**How do I determine whether a variable mediates or moderates an outcome, and what statistics should I use to test for mediation and moderation?

**A:**There has been a great deal of confusion regarding these two terms, and the literature is not always helpful because the terms are often used interchangeably. But it is essential to understand the difference between these two concepts, because it affects how you analyze and interpret your data. Both mediation and moderation can be tested through multiple regression and through structural equation modeling.

**MEDIATION**— How (by what means) does an effect occur? What accounts for the impact of A on C? A mediational model is a causal model, whereby it is hypothesized that A "causes" B and that B then "causes" C. What follows is based on Holmbeck (1997). See other more recent papers for additional detail (e.g., MacKinnon, et al., 2002).Four conditions must be met for B to be a mediator:

1) A (predictor) is significantly associated with C

2) A (predictor) is significantly associated with B

3) B is significantly associated with C (after controlling for A)

4) The impact of A on C is significantly less after controlling for B

**Multiple regression**

You can test the 4 conditions using 3 multiple regressions:

1) Condition 2: A-C

2) Condition 1: A-B

3) Condition 3: A and B as predictors with C as dependent variable. Use simultaneous entry to test Condition 3.

**SEM**

Note: SEM is appropriate to use when you have multiple measures for each of the constructs.

Step 1: assess the fit of A-C

Step 2: if fit of A-C is adequate, then test fit of overall A-B-C

Step 3: if overall model is adequate, then assess A-B, and B-C

Step 4: if all of the above are significant in the predicted directions, then test A-B-C under 2 conditions:

1)A-C path constrained to zero

2)A-C path not constrained

Step 5: does the non-constrained model fit significantly better than the constrained? (test difference between 2 model chi-squares).If the 2 models don't differ significantly, this means that there is significant mediation. Keep in mind that lower (nonsignificant) chi-square values indicate better fit.

Step 6: Report and compare the A-C path coefficients for when B is included and not included in the model. There is an important distinction between indirect and mediated effects; indirect effects may show A-B and B-C as significant, but will not show A-C as significant.

**MODERATION**— when (under what conditions) does the effect occur? Under what conditions of B is A significantly associated with C?

**Multiple regression**

You use a slightly different approach depending on whether it's a dichotomous or continuous variable. Effects may be difficult to detect if the independent variable and the moderator are highly correlated with each other.

Step 1: Center the continuous predictor and moderator variables to eliminate multicollinearity effects between the predictor and moderator, and the interaction terms (to center a variable: subtract the sample mean from all individuals' scores on the variable, getting a revised sample mean of 0 for that variable). For dichotomous variables, code as 0-1.

Step 2: Enter the predictor (A) and moderator (B) main effects (hierarchical, stepwise, or simultaneous depending on your conceptual framework).

Step 3: Enter the interaction between predictor (A) and moderator (B).

**Structural Equation Modeling**

SEM may be better than regression, which compounds measurement error when computing interaction terms. SEM is preferable when you have multiple measures for each of the constructs.

Example testing dichotomous variables: if you're testing the relationship between A (which has multiple measures) and C (also has multiple measures) to see if it varies as a function of B (i.e. gender), you assess the overall fit of the model under 2 conditions:

1) no constraints — allows the relationship to vary as function of gender

2) constrained — allows no variation as function of gender

You then test the significance of the difference between the goodness-of-fit chi-square values for the two models. If Model 1 fits the data better than Model 2, then there is a significant interaction effect. Keep in mind that lower (nonsignificant) chi-square values indicate better fit.

Holmbeck, G., (1997). Toward terminological, conceptual, and statistical clarity in the study of mediators and moderators: Examples from the child-clinical and pediatric psychology literatures. *Journal of Consulting and Clinical Psychology, 65*(4), 599-610.
MacKinnon, D., Lockwood, C., Hoffman, J. (2002). A comparison of methods to test mediation and other intervening variable effects. *Psychological Methods, 7*(1), 83-104.

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