Mediation

1. Mediation That is, Indirect Effects

2. What is a Mediator? • An intervening variable. • X causes M and then M causes Y.

3. MacKinnon et al., 2002 • 14 different ways to test mediation models • Grouped into 3 general approaches • Causal Steps (Judd, Baron, & Kenney) • Differences in Coefficients • Product of Coefficients

4. Causal Steps • X must be correlated with Y. • X must be correlated with M. • M must be correlated with Y, holding constant any direct effect of X on Y. • When the effect of M on Y is removed, X is no longer correlated with Y (complete mediation) or the correlation between X and Y is reduced (partial mediation).

5. First you demonstrate that the zero-order correlation between X and Y (ignoring M) is significant. • Next you demonstrate that the zero-order correlation between X and M (ignoring Y) is significant.

6. Now you conduct a multiple regression analysis, predicting Y from X and M. The partial effect of M (controlling for X) must be significant. • Finally, you look at the direct effect of X on Y. This is the Beta weight for X in the multiple regression just mentioned. For complete mediation, this Beta must be (not significantly different from) 0. For partial mediation, this Beta must be less than the zero-order correlation of X and Y.

7. Criticisms • Low power. • Should not require that X be correlated with Y • X could have both a direct effect and an indirect effect on Y • With the two effects being opposite in direction but equal in magnitude.

8. Differences in Coefficients • Compare • The correlation between Y and X (ignoring M) • With the β for predicting Y from X (partialledfor M) • The assumptions of this analysis are not reasonable. • Can lead to conclusion that M is mediator even when M is unrelated to Y.

9. Product of Coefficients • The best approach • Compute the indirect path coefficient for effect of X on Y through M • The product of • rXM and • β for predicting Y from X partialledfor M • This product is the indirect of X through M on Y

10. The Test Statistic (TS) • TS is usually evaluated by comparing it to the standard normal distribution (z) • There is more than one way to compute TS.

11. Sobel’s (1982) first-order approximation • The standard error is computed as •  is bM.X or rM.X , 2 is its standard error •  is bY.M(X) or Y.M(X), 2 is its standard error

12. Alternative Error Terms • Aroian’s (1944) second-order exact solution • Goodman’s (1960) unbiased solution

13. Ingram, Cope, Harju, and Wuensch (2000) • Theory of Planned Behavior -- Ajzen & Fishbein (1980) • The model has been simplified for this lesson. • The behavior was applying for graduate school. • The subjects were students at ECU

15. Causal Steps • Attitude is significantly correlated with behavior, r = .525. • Attitude is significantly correlated with intention, r = .767.

16. The partial effect of intention on behavior, holding attitude constant, falls short of statistical significance,  = .245, p = .16. • The direct effect of attitude on behavior (removing the effect of intention) also falls short of statistical significance,  = .337, p = .056. • No strong evidence of mediation.

17. Product of Coefficients

18. Aroian’s second-order exact solution

19. http://quantpsy.org/sobel/sobel.htm

20. Or, Using Values of t Merde, short of statistical significance.

21. Mackinnon et al. (1998) • TS is not normally distributed • Monte Carlo study to find the proper critical values. • For a .05 test, the proper critical value is 0.9 • Wunderbar, our test is statistically significant after all.

22. Mackinnon et al. (1998) Distribution of Products • Find the product of the t values for testing  and  • Compare to the critical value, which is 2.18 for a .05 test. • Significant !

23. Shroutand Bolger (2002) • With small sample sizes, best to bootstrap. • If X and Y are temporally proximal, good idea to see if they are correlated. • If temporally distal, not a good idea, because • More likely that X  Y has more intervening variables, and • More likely that the effect of extraneous variables is great.

24. Opposite Direct and Indirect Effects • X is the occurrence of an environmental stressor, such as a major flood, and which has a direct effect of increasing • Y, the stress experienced by victims of the flood. • M is coping behavior on part of the victim, which is initiated by X and which reduces Y.

25. Partial Mediation ? • X may really have a direct effect upon Y in addition to its indirect effect on Y through M. • X may have no direct effect on Y, but may have indirect effects on Y through M1 and M2. If, however, M2 is not included in the model, then the indirect effect of X on Y through M2 will be mistaken as being a direct effect of X on Y.

26. There may be two subsets of subjects. In the one subset there may be only a direct effect of X on Y, and in the second subset there may be only an indirect effect of X on Y through M.

27. Causal Inferences from Nonexperimental Data? • I am very uncomfortable making causal inferences from non-experimental data. • Sure, we can see if our causal model fits well with the data, • But a very different causal model may fit equally well. • For example, these two models fit the data equally well:

29. Bootstrap Analysis • Shrout and Bolger recommend bootstrapping when sample size is small. • They and Kris Preacher provide programs to do the bootstrapping. • I’ll illustrate Preacher’s SPSS macro. • He has an SAS macro too.

31. Direct, Indirect, and Total Effects • IMHO, these should always be reported, and almost always standardized. • the direct effect of attitude is .337 • The indirect effect is (.767)(.245) = .188. • The total effect = .337 + .188 = .525. • rxy =.525: we have partitioned that correlation into two distinct parts, the direct effect and the indirect effect.

32. Parallel Multiple Mediation • Experimental Manipulation: Subjects told article they are to read will be (1) on the front page of newspaper or (0) in an internal supplement. • Importance: Subjects’ rating of how important the article is. Mediator. • Influence: Subjects’ rating how influential the article will be. Mediator.

33. Parallel Multiple Mediation (2) • The article was about an impending sugar shortage. • Reaction: Subjects’ intention to modify their own behavior (stock up on sugar) based on the article. Dependent variable.

34. Process Hayes %process (data=pmi2, vars=condpmiZimportZreactionZ, y=reactionZ, x=cond, m=importZpmiZ, boot=10000, total=1,normal=1,contrast=1,model=4);

36. Serial Multiple Mediation %process(data=pmi2, vars=condpmiZimportZreactionZ, y=reactionZ, x=cond, m=importZpmiZ, boot=10000, total=1,normal=1,contrast=1,model=6);

38. Moderated Mediation • Female attorney loses promotion because of sex discrimination. • Protest Condition: experimentally manipulated, attorney does (1) or does not (0) protest the decision. • Response Appropriateness: Subjects’ rating of how appropriate the attorney’s response was. Putative mediator.

39. Moderated Mediation (2) • Liking: Subjects’ ratings of how much they like the attorney. Dependent variable. • Sexism: Subjects’ ratings of how pervasive they think sexism is.

40. Process Hayes %process(data=protest2, vars=protest RespapprZSexismZLikingZ, y=LikingZ, x=protest, w=SexismZ, m=RespapprZ, quantile=1,model=8, boot=10000);

43. A Fly in the Ointment

44. Cross-Sectional Data • Most published tests of mediation models have used data where X, M, and Y were all measured at the same time and X not experimentally manipulated. • But what we really need is longitudinal data. • Mediation tests done with cross-sectional data produce biased results.

45. a, b, and c are direct effectsx, m, and y are autoregressive effects