1. Mediation: Power and Effect Size David A. Kenny

2. 2 The Beginning Model

3. 3 The Mediational Model

4. 4 Key Effects Indirect effect: ab Direct effect: c′ Total effect: c or c′ + ab

5. 5 Power Estimation Assumes that the model is correctly specified. All variables are standardized making a, b, c, and c′ are beta coefficients. Power determined by the implied correlation between the two variables controlling for relevant variables. Joint significance method used for indirect effect. Use PowMedR to compute power.

6. 6 Power: a and b Medium Effect Sizes Let c′ = 0 a = b =.3 c = ab =.09 If N = 100, then power for the following tests are: a:.868 b:.832 ab:.723 c:.143

7. 7 Relative Power of a versus b Path b has less power than the test of path a, due to collinearity of due to path a. Path (c′ = 0) N for.80 power a b a b.1.1 782 790.3.3 84 92.5.5 28 36.7.7 13 23

8. 8 How the Power of b depends on a Path (c′ = 0) Power (N = 100) a b b.0.3.865.1.3.862.3.3.832.5.3.756.7.3.590.9.3.267

9. 9 Power of ab: Distal versus Proximal Mediation Distal mediators have more power than proximal mediators. Path Power (N = 100) a b ab ab.9.5444.49 104.8.6125.49 42.7.7.49 22.6125.8.49 18.5444.9.49 23

10. 10 Power of ab: Bigger is not Better with Proximal Mediation Path N for Power of.80 a b ab ab.1.1.01 1031.2.1.02 815.3.1.03 860.4.1.04 931.5.1.05 1042.6.1.06 1220

11. 11 Power of ab versus c with Complete Mediation (note ab = c) Path N for Power of.80 a b c ab c ab.1.1.01.01 78485 1031.2.2.04.04 4902 258.3.3.09.09 966 114.4.4.16.16 303 65.5.5.25.25 122 42 Note the relatively low power for c which means that the power for Step 1 can be weak.

12. 12 Power of ab versus c′ Power for the test of ab is generally more powerful than the test of c′. Path N for Power of.80 a b c′ ab c′ ab.1.1.01.01 78469 1031.2.2.04.04 4886 257.3.3.09.09 948 113.4.4.16.16 283 62.5.5.25.25 99 36

13. 13 Covariates Consider the effect of X on M: Power decreases slightly due to a loss of degrees of freedom. Power increases if the covariates explain variance of M. Power decreases if the covariates explain variance of X.

14. Effect Size Estimation See Preacher & Kelley (2011, Psychological Methods, 16, 93-115) for an extensive treatment of the topic. Simple approach taken here. Compute an effect size for a and b and multiply the together. Preacher and Kelley refer to this as the completely standardized indirect effect. 14

15. Effect Size for a and b For a use r unless X is a dichotomy and then use d. For b use a partial r controlling for X. If there are covariates, partial them out. Compute the product to obtain an effect size for ab. –rr or dr 15

16. Proposed Standards rr –Small.01 –Medium.09 –Large.25 dr –Small.02 –Medium.15 –Large.40 16

17. 17 Mediation Webinar PowMedR Acknowledgements: Work by Fritz, MacKinnon, Judd, and Hoyle