A simple statistical mediation model.

Direct vs. indirect effects

In the diagram shown above, assuming linear relationships, the indirect effect is the product of paths coefficients A and B. In general, including nonlinear models, the total effect is equal to the difference between the direct effect and the indirect effect of a unit decrease in the independent variable^[3]. In contrast, the indirect effect (sometimes referred to as mediated effect) refers to the extent to which the dependent variable changes when the independent variable is held fixed and the mediator variable changes to the level it would have attained had the independent variable increased by one unit^[3].

Complete vs. partial mediation

When the direct effect between the independent variable and the dependent variable (path C in the diagram above) is no longer statistically different from zero fixing the mediator variable, the mediation effect is said to be complete.

If, however, the absolute size of the direct effect between the independent variable and the dependent variable is reduced after controlling for the mediator variable, but the direct effect is still significantly different from zero, the mediation effect is said to be partial. In all cases, the operation of "fixing a variable" must be distinguished from that of "controlling for a variable", which has been inappropriately used in the literature ^[1][2][4].

Suppression

Suppression is defined as "a variable which increases the predictive validity of another variable (or set of variables) by its inclusion in a regression equation.^[5]. For instance, if you are set to examine the effect of a treatment (e.g. medication) on an outcome (e.g. healing from a disease), a suppression would mean that instead of the drop that you would see from the direct effect of the treatment on the outcome when the mediator is introduced, the opposite happens. The inclusion of the mediating variable into the equation increases the relation between the treatment and outcome rather accounts for (decreases in terms of the size of the statistical relation).

Suppression is a contentious issue and continues to be debated in the literature . However, it was suggested recently that suppression should be viewed as adding interest to the results , rather than as a confound or problem. It has been also suggested though that testing for suppression should be based on a priori assumptions about the theoretical relation between the variables and the role of the mediating variable as a suppressor^[5][6].

Pearl (2000, page 139)^[4] has argued that "suppression" is an illusionary effect emanating from confusing causal and associational relationships, as in Simpson's paradox.

Moderated Mediation

Mediation and moderation can co-occur in statistical models. It is possible to mediate moderation and moderate mediation.

Moderated mediation is when the effect of the treatment effect "A" on the mediator "B", and/or when the partial effect of "B" on "C", depends on levels of another variable (D). This definition has been outlined by Muller, Judd, and Yzerbyt (2005) ^[7] and Preacher, Rucker, and Hayes (2007)^[8]

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 -> B path or at the B ->C. 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 later there is no moderation but the effect of either the treatment (A) on the mediator (B) is moderated or the effect of the mediator (B) on the outcome (C) is moderated.^[7]

Mediator variable

A mediator variable (or mediating variable) in statistics is a variable that describes how rather than when effects will occur by accounting for the relationship between the independent and dependent variables. A mediating relationship is one in which the path relating A to C is mediated by a third variable (B).

For example, a mediating variable explains the actual relationship between the following variables. Most people will agree that older drivers (up to a certain point), are better drivers. Thus:

aging \to better driving

But what is missing from this relationship is a mediating variable that is actually causing the improvement in driving: wisdom. The mediated relationship would look like the following:

aging \to increased wisdom \to better driving

(And yes, it could also be argued that instead of increased wisdom it is increased responsibility for one's actions, but separating those two is difficult to do.)

Mediating variables are often contrasted with moderating variables, which pinpoint the conditions under which an independent variable exerts its effects on a dependent variable. A moderating relationship can be thought of as an interaction. It occurs when the relationship between variables A and B depends on the level of C.

In fact, the best explanation of the above relationship is a combination of mediating and moderating variables. Increased age does often lead to increased wisdom, but it also requires the maintenance of good reflexes. Thus, there is an interaction between age and reflexes that serves as a moderating variable. Increasing age only improves driving until reflexes begin to suffer, at which point the moderator variable (age x reflexes) results in worse driving. Or:

aging \to increased wisdom \to better driving \to worse driving

(until) aging \to decline in reflexes \nearrow

Significance of mediation

Bootstrapping http://www.comm.ohio-state.edu/ahayes/sobel.htm http://www.psych.ku.edu/preacher/sobel/sobel.htmhttp://www.comm.ohio-state.edu/ahayes/SPSS%20programs/indirect.htm is becoming the most popular method of testing mediation because it does not require the normality assumption to be met, and because it can be effectively utilized with smaller sample sizes (N<25). However, mediation continues to be (perhaps inappropriately ) most frequently determined using the (1) the logic of Baron and Kenny http://davidakenny.net/cm/mediate.htm or (2) the Sobel test.

External links

• Free Sobel test calculator for the significance of mediation
• Free indirect effect calculator for mediation models
• Free interactive statistics tests for mediation and interactions
• summary of mediation methods at PsychWiki
• SPSS and SAS macros for bootstrapping indirect effects

References

1. ↑ ^{a b} Kaufman, J. S., MacLehose R. F., Kaufman S (2004). A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation. Epidemiology Innovations and Perspectives, 1:4.
2. ↑ ^{a b} Robins, J. M., Greenland, S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2):143-55.
3. ↑ ^{a b c} Pearl, J. (2001) Direct and indirect effects. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, 411-420.
4. ↑ ^{a b}
5. ↑ ^{a b} MacKinnon, D. P., Krull, J. L., Lockwood, C. M. (2000). Equivalence of the Mediation, Confounding and Suppression Effect. Prevention Science, 1(4), 173-181.
6. ↑ Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: new procedures and recommendations. Psychological Methods, 7(4), 422-445.
7. ↑ ^{a b} Muller, D., Judd, C. M., Yzerbyt, V. Y. (2005). When moderation is mediated and mediation is moderated. Journal of Personality and Social Psychology, 89(6), 852-863.
8. ↑ Preacher, K. J., Rucker, D. D. & Hayes, A. F. (2007). Assessing moderated mediation hypotheses: Strategies, methods, and prescriptions. Multivariate Behavioral Research, 42, 185-227.

• Baron, R. M. and Kenny, D. A. (1986) The Moderator-Mediator Variable Distinction in Social Psychological Research - Conceptual, Strategic, and Statistical Considerations, Journal of Personality and Social Psychology, Vol. 51(6), pp. 1173-1182.

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