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Mediation That is, Indirect Effects What is a Mediator? • An intervening variable. • X causes M and then M causes Y. 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 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). • 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. • 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. 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. Differences in Coefficients • Compare – The correlation between Y and X (ignoring M) – With the β for predicting Y from X (partialled for M) • The assumptions of this analysis are not reasonable. • Can lead to conclusion that M is mediator even when M is unrelated to Y. 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 M partialled for X – This product is the indirect effect of X through M on Y The Test Statistic (TS) TS • TS is usually evaluated by comparing it to the standard normal distribution (z) • There is more than one way to compute TS. Sobel’s (1982) first-order approximation • The standard error is computed as 2 2 2 2 • is bM.X or rM.X , 2 is its standard error • is bY.M(X) or Y.M(X) , 2 is its standard error Alternative Error Terms • Aroian’s (1944) second-order exact solution 2 2 2 2 2 2 • Goodman’s (1960) unbiased solution 2 2 2 2 2 2 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 Causal Steps • Attitude is significantly correlated with behavior, r = .525. • Attitude is significantly correlated with intention, r = .767. • 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. Product of Coefficients Coefficients Unstandardized Coefficients Model 1 B Standardized Coefficients Std. Error (Constant) .075 9.056 attitude .807 .414 1.065 .751 intent a. Dependent Variable: behav a Beta t Sig. .008 .993 .337 1.950 .056 .245 1.418 .162 Aroian’s second-order exact solution TS 2 2 2 2 2 2 . 423 (1 . 065 ) . 423 (. 751 ) 1 . 065 (. 046 ) . 046 (. 751 ) 2 2 2 1 . 3935 2 2 2 http://quantpsy.org/sobel/sobel.htm Or, Using Values of t Merde, short of statistical significance. 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. 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. Z Z 9 . 108 1 . 418 12 . 915 . • Significant ! Shrout and 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. 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. 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. • 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. 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: 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. DIRECT AND TOTAL EFFECTS Coeff s.e. b(YX) 1.2566 .2677 b(MX) .4225 .0464 b(YM.X) 1.0650 .7511 b(YX.M) .8066 .4137 t 4.6948 9.1078 1.4179 1.9500 Sig(two) .0000 .0000 .1617 .0561 INDIRECT EFFECT AND SIGNIFICANCE USING NORMAL DISTRIBUTION Value s.e. LL 95 CI UL 95 CI Z Sig(two) Sobel .4500 .3231 -.1832 1.0831 1.3929 .1637 BOOTSTRAP RESULTS FOR INDIRECT EFFECT Mean s.e. LL 95 CI Effect .4532 .2911 -.1042 UL 95 CI 1.0525 LL 99 CI -.2952 UL 99 CI 1.2963 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. Preacher & Kelley’s 2 • This statistic is the ratio of the indirect effect to the maximum value that the indirect effect could assume given the constraints imposed by variances and covariance of X, M, and Y. • It has the advantage of ranging from 0 to 1, as a proportion should. Preacher & Kelley’s 2 • Notice that for our data this statistic is significantly greater than zero. Preacher and Kelley (2011) Kappa-squared Effect Boot SE BootLLCI BootULCI INTENT 0.1416 0.0813 0.0094 0.3163 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. 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. Process Hayes %process (data=pmi2, vars=cond pmiZ importZ reactionZ, y=reactionZ, x=cond, m=importZ pmiZ, boot=10000, total=1,normal=1,contrast=1,model=4); Serial Multiple Mediation %process (data=pmi2, vars=cond pmiZ importZ reactionZ, y=reactionZ, x=cond, m=importZ pmiZ, boot=10000, total=1,normal=1,contrast=1,model=6); Moderated Mediation • Female attorney loses promotion because of sex discrimination. • Protest Condition: experimentally manipulated, attorney does (1) or does not (0) protest the decision. Independent Variable. • Response Appropriateness: Subjects’ rating of how appropriate the attorney’s response was. Mediator. 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. Moderator. Process Hayes %process (data=protest2, vars=protest RespapprZ SexismZ LikingZ, y=LikingZ, x=protest, w=SexismZ, m=RespapprZ, quantile=1,model=8, boot=10000); A Fly in the Ointment 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. a, b, and c are direct effects x, m, and y are autoregressive effects