Workshop on Statistical Mediation and Moderation: Statistical Mediation

1. Workshop on Statistical Mediation and Moderation:Statistical Mediation Paul Jose Victoria University of Wellington 27 March, 2008 SASP Conference

2. What do you want to know? • Let’s briefly have each person state what he or she would like to learn this morning. • Also, what is your level of statistical knowledge/experience? • Okay, let me tell you what I’m planning to cover.

3. What am I doing today? • I want to define mediation and moderation • How are they similar or different? • Basic mediation and moderation • Advanced mediation and moderation • Questions and answers

4. Where does one start? • I began to be interested in mediation and moderation because I found that I was increasingly using these approaches in understanding “process” among variables. • I found that there was little about these techniques in traditional statistics textbooks—I mostly obtained information through word-of-mouth. • . . . and I was confused. I don’t like being confused, so I did something about it. I educated myself on these techniques. And now I can pass on what I’ve learned. Let me list what I consider to be the main sources of confusion.

5. Five major sources of confusion • Moderation and mediation sound alike. It makes it seem that they are very similar, and or they derive from the same origin. They are somewhat similar (cousins), but they don’t come from the same place. • Second, statistics textbooks typically do not do a very good job of explaining these two approaches. Exception: Howell (2006). • Third, reports of moderation and mediation in the research literature are not always clear or accurately performed.

6. More confusion • Both are special cases of two separate broad statistical approaches: mediation is a special case of semi-partial correlations (path modeling) and moderation is a special case of statistical interactions (from ANOVA). Both are included under GLM, but this is not usually appreciated. • It’s not entirely clear what distinguishes a moderating variable from a mediating variable. Can one a priori define mediating and moderating variables?

7. One last stumbling block • Problem: there are no easily used statistics programmes that compute mediation and moderation. Can do analyses in SPSS and other programmes that do regression, but there is no graphing capability dedicated to either mediation or moderation (except ModGraph and MedGraph). • What we have here is a case of the users “getting ahead” of the statisticians in the sense that researchers frequently use mediation and moderation but many statisticians aren’t even familiar with the terms.

8. Background and history • Most people’s awareness of this area comes from this article: Baron, Reuben M. & Kenny, David 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. • Cited about 6,500 times by PsychInfo’s count. And that’s just in Psychology. • Most people are unclear about what they said and how to perform the techniques.

9. Let’s get started:Similarities and differences Similarities: • They both involve three variables; • You can use regression to compute both; • You wish to see how a third variable affects a basic relationship (IV to DV). Differences: • You create a product term in moderation; not in mediation; • You don’t have to centre anything in mediation; • Moderation can be used on concurrent or longitudinal data, but mediation is best used on longitudinal data. • Graphing is critical for moderation; helpful for mediation.

10. How do you know if you have a moderator or a mediator? What’s the diff? • Moderators tend to be variables that are relatively immune to change over time (personality trait, gender, ethnic group, etc.). • Mediators tend to be variables that change in relation to other variables (anxiety, helpfulness, honesty, mood). • However, there is a class of variables (e.g., coping efforts/strategies) that might be examined in both ways. These two categories are not mutually exclusive.

11. So let’s focus on mediation first • Definition: “A mediating variable is one which specifies how (or the mechanism by which) a given effect occurs between an independent variable (IV) and a dependent variable (DV).” (Holmbeck, 1997, p. 599). • The question you wish to answer is whether the effect of the IV on the DV is at least partially mediated by a third variable (MV). • You can answer this question with two regressions (and a correlation matrix). • Let’s consider a specific example.

12. An example from my research Stressor intensity Depression Rumination

13. The theories • Susan Nolen-Hoeksema believes that an individual who ruminates more ends up more depressed. X => Y. Notice that it’s a causal statement. • I don’t disagree with her, but I think that this simple effect should be embedded within the stress and coping context. • We know that stress leads to depression. The question I want to ask is whether at least part of the effect of stress on depression occurs because certain individuals ruminate about stressful events, and this rumination leads to depression.

14. The basic relationship Stressor intensity Depression One must have a significant correlation between the IV and DV (in fact among all 3 variables). The essential question is whether by adding a third variable, one can at least partially explain the basic relationship. Let’s look at some real data.

15. The two steps Step 1 .45*** Stressor intensity Depression Step 2 Stressor intensity .45*** Depression (.29**) .51*** .46*** Rumination (.32**)

16. Baron & Kenny’s 4 criteria • IV to MV must be significant • IV to DV must be significant • MV to DV must be significant (when entered with the IV) • “The effect of the IV on the DV must be less in the third equation than the second. Perfect mediation holds if the IV has no effect when the mediator is controlled.” • “must be less” is measured with the Sobel formula (see following pages) • “Perfect mediation” occurs when the original relationship goes to zero. This never happens in psychology. I have a proposal for how to deal with this issue, presented below.

17. What changed? • Note that the beta weight from IV to DV changed: from .45 to .29. • What does that tell us? • According to Baron and Kenny (1986), if one obtains a significant drop in beta for this relationship, then one has obtained significant mediation. • How can one test whether this is significant or not? (It is not simply whether it goes from significant to non-significant.) One needs to compute the Sobel’s test: z-value = a*b/SQRT(b2*sa2 + a2*sb2)

18. Who ‘ya gonna call? • Many people have been using a web-site by Preacher and Leonardelli, and it’s quite useful for computing the Sobel’s statistic: http://www.psych.ku.edu/preacher/sobel/sobel.htm • Let me show you how to use the site. It is generally very helpful. • I have invented my own programme to do what P & L’s site does, and MORE. Let’s check it out too.

19. Preparatory work • Before we run off to use these, please know that you have to obtain some statistical information first: • Compute a correlation matrix of the 3 variables; • Perform a multiple regression of the IV on the mediating variable; and • Perform a multiple regression of the IV and mediator on the DV (simultaneous inclusion).

20. Correlation matrix

21. Results from the two regressions 1st regression (Stress on Rumination): B 7.501 (unstand regression coefficient) se .938 (standard error) 2nd regression (Stress, Rumination on Depression): You select the B and se for the mediating variable here: B .069 se .016 new beta for Stress .288 new beta for Rumination .317 (P & L web-site needs the first four values.)

22. Okay, go to the programmes • It is necessary to have written down the pertinent statistical output, or to have printed off the relevant sections. • Can do both programmes on the internet. • If you’re away from the internet you can download the Excel macro of MedGraph and run it whenever you want.

23. MedGraph output

24. Comparison of web-sites • Preacher’s site has been around longer, it allows variations on the Sobel formula, and gives you an alternate way to compute the Sobel’s t. • My site results in a graphical presentation of results, I think it’s harder to make mistakes with my programme, and it has/will have information about the type of mediation.

25. My criteria for type of mediation • At present my programme stipulates: • None: non-significant Sobel’s z-value • Partial: significant Sobel’s and significant basic relationship in the 2nd regression (IV to DV) • Full: significant Sobel’s and non-significant basic relationship in the 2nd regression (IV to DV) • Dave Kenny argues against this (see his web-site), and I tend to agree with him now. My new approach is on the following page.

26. What kind of mediation? • None: non-significant Sobel’s z-value • Partial: significant Sobel’s and ratio < .80. (ratio is indirect/total; in this case it’s .161/.449) • Full: significant Sobel’s and ratio > .80 ------------------------------------------------------ • In the present case we have a significant Sobel’s and ratio = .36. Thus, we have partial mediation. Notice that I don’t use the term “perfect mediation”. There is no consensus on the partial/full mediation issue.

27. Causal finding? • Many researchers would be keen to argue from this result that the experience of stress leads to rumination, which in turn partially leads to depressive symptoms, i.e., a causal argument. Is this merited? • Cole and Maxwell (2003) argue strenuously that concurrent mediation CANNOT support a causal statement. They argue that few concurrent mediation results actually turn out to hold up in longitudinal data. What do they mean?

28. Shared and unique variance Stress Depressive symptoms Basic relationship is just a correlation between two variables.

29. Three variables: mediation Direct effect Stress Depressive symptoms Indirect effect Rumination The green area indicates the degree of shared variance among the three variables: that’s the size of the “indirect effect”. It is hard to argue that these relationships are causal with these data: they arethe size of shared and unique variance.

30. Warnings! • One must have all three correlations be significant before launching this. K now suggests that 1st one may be optional. • Be sure that you do the regressions correctly, and that you are taking the proper statistical information from the print-outs (B vs. b). • Some people make causal arguments from these results. They are shaky at best. • Types of specification error: 1) ordering of variables, 2) variables with/without error, and 3) “third variable problem” • Longitudinal data are best. • Bootstrapping is best with small N samples. • Path models involving more than three variables is the general case—don’t do a bunch of three-variable mediation analyses when you can do one path model.

31. Specification error • Major boogeyman in path model analytic work: have you correctly specified your model? • Several issues here: • Temporal order of variables • Variables measured with error • Missing variable?

32. Why is your proposed model the best? Rumination Stress intensity Depressive symptoms There are exactly 6 combinations of any three variables—why is your proposed model the best? Why not test all of them? I have, and in the present case I find six instances of partial mediation. Which is correct? They all tell us something useful about shared and unique variance.

33. Variables measured with error • One can obtain biased estimates of the indirect effect if the MV is measured with significant error. (Same is true of the IV and DV too, by the way.) • Answer? Do mediation in a latent variable path model in SEM. Possible but not easy. Also, a lot of the times one doesn’t have a sufficient N or multiple indicators of the variables (3 indicators per variable). Would look like this:

34. Latent variable path model Stress intensity Depression .30*** (.20***) .40*** .24*** Rumination Indirect effect = .10; direct effect = .20; ratio = .33 (.36 in MR)

35. Missing variable? • This is the old “third variable problem”, but in this case we might wish to call it the “fourth variable problem”. • My student, Kirsty Weir, suggests that anxiety/worry might “explain” the relationship between rumination and depression. Graph is on the following page. • One can never completely resolve this question: include the likely candidates and try to reject them.

36. The road from stress to depression Note that the Rum to Dep path was removed because it was non- significant when we added the 4th variable (control). Is the 3-variable mediation pattern wrong then?

37. Bootstrapping • David MacKinnon and others have argued that typical multiple regression analysis is unbiased only for large samples. (present case N = 575) • They suggest: • Large sample: use MR • Small sample: use bootstrapping • What is bootstrapping?

38. Wave of the future • Bootstrapping is a compilation of regression results from many subsets of the original dataset. • The programme selects a subset of the data (e.g., 50 from 100 participants), runs the regression analysis, stores the result, does it again and again up to a predetermined number of times, and then compiles the results of the repeated analyses. • Baron & Kenny didn’t mention this—wasn’t used in 1986 very much at all. It is performed now, but infrequently. It is the wave of the future.

39. So how does one do this? • If you toddle off to SPSS to do this, you will be disappointed. Although it can perform bootstrapping, it is not set up to do mediation bootstrapping. • Preacher and Hayes (see the Preacher web-site on mediation) offers two different macros: SAS and SPSS. Download it and use it within SPSS. (not easy) • Let’s look at the results of the SPSS macro.

40. Macro output Run MATRIX procedure: DIRECT AND TOTAL EFFECTS Coeff s.e. t Sig(two) b(YX) .3934 .0288 13.6685 .0000 b(MX) 1.0412 .0691 15.0779 .0000 b(YM.X) .1369 .0165 8.3200 .0000 b(YX.M) .2508 .0322 7.8002 .0000 INDIRECT EFFECT AND SIGNIFICANCE USING NORMAL DISTRIBUTION Value s.e. LL 95 CI UL 95 CI Z Sig(two) Sobel .1426 .0196 .1042 .1810 7.2723 .0000 BOOTSTRAP RESULTS FOR INDIRECT EFFECT Mean s.e. LL 95 CI UL 95 CI LL 99 CI UL 99 CI Effect .1434 .0239 .1001 .1939 .0879 .2113 SAMPLE SIZE 575 NUMBER OF BOOTSTRAP RESAMPLES 2000 It’s telling us that the indirect effect was significant—agrees with the multiple regression result, but this is an unbiased estimate. (z = 3.80 before)

41. Mediation with longitudinal data • . . . is very complicated but is very illuminating. • Much of structural equation modelling (SEM) is devoted to trying to understand mediational models. • Path modelling with longitudinal data is hard to do but will generate very interesting and interpretable results. • One should obtain the same variables at different times of measurement to allow residualisation.

42. Hierarchical multiple regression Time 1 Time 2 Rum 2nd step Dep Dep 1st step This is N-H’s hypothesis: Rum1 should explain unique variance in Dep2 after Dep1 is entered, i.e., explaining new variance in the residual.

43. Back to Venn diagrams,but with a difference Dep2 Dep1 Stability coefficient: typically medium to large. The purple area is the residual variance. It represents the change in depression over this time period. The overlapping area refers to the stability of depression over this time period.

44. Does Rum1 predict any of the residual? Dep1 Dep2 Rum1 The red area is the amount of variance in Dep2 explained by Rum1, i.e., the degree to which Rum1 explains change in depression over time.

45. So what’s the answer? .72*** • Perform a hierarchical regression: IVDV • Dep1 Dep2 • Rum1 I found that N-H’s hypothesis was not supported: Rum1 did not explain any of the residual of Dep2 after Dep1 was entered. .05ns

46. This is what it looks like Dep1 Dep2 Rum1 Although Dep1 and Rum1 are significantly correlated, Rum1 doesn’t explain much new variance in Dep2 above and beyond what Dep1 can do.

47. The other direction .64*** IVDV • Rum1 Rum2 • Dep1 This result suggests that depression may contribute to rumination over a 3-month period of time, but not the other way around. It is recommended that you perform a path analysis in SEM for this type of analysis: allows for concurrent correlation (see next page). .08*

48. Two time points Time 1 Time 2 Rum Rum Dep Dep SEM computes all of these relationships simultaneously, allowing one to identify the unique relationships. Enact in LISREL, EQS, AMOS, etc. What did I find?

49. Same basic results Time 1 Time 2 .63*** Rum Rum .08* .47** .43** Dep Dep .74*** But you get model fit indices, modification indices, and so forth . . . I deleted the Rum1 to Dep2 path because it was non-significant.

50. Three time points and three variables Time 1 Time 2 Time 3 Stress Stress Stress my hypoth ? Rum. Rum. Rum. N-H MR Dep. Dep. Dep.