1. 1 Multiple Mediator Models  Most behaviors are affected by multiple mediators.  Straightforward extension of the single mediator case but interpretation can be more difficult.  The product of coefficients methods is the best way to evaluate models with multiple mediators but difference and causal step methods can work.

2. 2 MEDIATOR M1M1 INDEPENDENT VARIABLE X DEPENDENT VARIABLE Y MEDIATOR M2M2 M3M3 M4M4 c 1.The independent variable causes the dependent variable: Y = c X +   Step 1

3. 3 MEDIATOR M1M1 INDEPENDENT VARIABLE X DEPENDENT VARIABLE Y MEDIATOR M2M2 M3M3 M4M4 a1a1 a2a2 a3a3 a4a4 2. The independent variable causes the potential mediators: M 1 = a 1 X +  ,  M 2 = a 2 X +  , M 3 = a 3 X +  , M 4 = a 4 X +   Step 2

4. 4 MEDIATOR M1M1 INDEPENDENT VARIABLE X DEPENDENT VARIABLE Y MEDIATOR M2M2 M3M3 M4M4 a1a1 a2a2 a3a3 a4a4 b1b1 b2b2 b3b3 b4b4 c’ 3.The mediators must cause the dependent variable controlling for exposure to the independent variable: Y = c’ X + b 1 M 1 + b 2 M 2 + b 3 M 3 + b 4 M 4 +   Step 3

5. 5 Mediated effects = a 1 b 1, a 2 b 2, a 3 b 3, a 4 b 4 Standard error = Total mediated effect= a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 = c - c’ Direct effect= c’ Total effect= a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 +c’=c Test for significant mediation: z’=Compare to empirical distribution of the mediated effect a1b1a1b1 Measures of Mediation

6. 6 Measures of Relative Effect  Proportion Mediated = a i b i  c’+  a i b i  a i b i /c  Ratio of Mediated to Direct = a i b i /c’  Simulation studies suggest large samples are necessary for these values to be accurate for the single mediator model, e.g. 500 for the proportion and 1000 for the ratio, MacKinnon et al. (1995).  Absolute values do and squaring terms do not improve the situation.

7. 7 Expectancy effects on Achievement  Harris and Rosenthal (1985) meta-analysis of mediators of the relation between teacher expectancy and student performance.  Here is a hypothetical study (N=40) with two mediators. (M1) social climate and (M2) material covered. Y is a test of achievement and X is the randomly assigned student ability value for each student. It was hypothesized that the ability score invokes an expectancy which affects warmth and material covered which leads to greater achievement.

8. 8 SAS Program for Expectancy effects on Achievement Model proc reg; model y=x; model y=x m1 m2/covb; model m1=x; model m2=x;

9. 9 SPSS Program for Expectancy effects on Achievement Model Regression /variables= x y m1 m2 /dependent=y /enter=x. regression /variables= x y m1 m2 /dependent=y /enter=x m1 m2. regression /variables= x y m1 /dependent=m1 /enter= x. regression /variables x y m2 /dependent=m2 /enter= x.

10. 10 Two Mediator Model MEDIATOR M1M1 X Y M2M2.2219 (.1460) INDEPENDENT VARIABLE DEPENDENT VARIABLE.8401 (.1580).5297 (.1696).1122 (.2073).5690 (.1568)

11. 11 Mediated Effect Measures a 1 b 1 = (.8401) (.5690) =.4781 for mediation through social climate and a 2 b 2 = (.2219) (.5297) =.1175 for mediation through feedback. The total mediated effect of a 1 b 1 (.4781) plus a 2 b 2 (.1175) equals.5956 which is equal to c-c’ =.7078-.1122 =.5956. The a 1 b 1 mediated effect (s a1b1 =.1499) was statistically significant (t a1b1 = 3.183) and the a 2 b 2 mediated effect (s a2b2 =.0838) was not (t a2b2 = 1.403). The standard error of the total mediated effect is equal to.1717 yielding a z statistic of 3.468.

12. 12 Confidence Limits Mediation through social climate, Asymmetric LCL=.2079 and UCL =.8284. Using the delta standard error, LCL=.1654 and UCL =.7906. Mediation through feedback, Asymmetric LCL = -.0261 and UCL =.3106. Using the delta standard error, LCL= -.0510 and UCL =. 2861.

13. 13 Special Topic: Test of Equality of two Mediated Effects S a1b1-a2b2 = Add 2b 1 b 2 s a1a2 to the equation if there is a covariance between a 1 and a 2, s a1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small. The difference between the two mediated effects is equal to.3605 with a standard error of.1717 yielding a z statistic of 2.099. Contrasts can be used to test pairs of mediated effects in any model. See MacKinnon (2000) Contrasts in Multiple Mediator Models

14. 14 Multiple Mediator Model of Intent to Use Anabolic Steroids Group Intentions Knowledge of the effects of AAS use Team as inform- ation source Perceived risks of AAS use Reasons to use AAS.236 2.42 (.258).149.62 (.108).217.52 (.061).168.44 (.066) -.083 -.02 (.006) -.079 -.08 (.006) -.265 -.25 (.024).155.09 (.014).000.001 (.056)

15. 15 Mediated Effects EffectEstimateEstimate/LCLUCL (Std Error) SE Knowledge-.046-3.00-.075-.017 (.015) Team as-.041-2.97-.068-.014 Information(.014) Perceived Severity-.108-5.56-.145-.071 (.013) Reasons to use.0564.29.031.081 Anabolic Steroid(.031) Direct Effect of.0010.017-.109.111 Program on Intentions(.056)

16. 16 Contrasts of Mediated Effects Multiple mediator models introduce more than one mediated effect for each dependent variable. Contrasts may used to compare pairs of effects or two groups of mediated effects. The direct effect may be included in contrasts also. Any combination of effects may be compared as long as all effects have the same dependent variable – makes scaling of all effects the same and thus they may be directly compared to one another.

17. 17 Contrast Examples Group Intentions to use AAS Knowledge of the effects of AAS use Team as inform- ation source Perceived risks of AAS use Reasons to use AAS a1 a4 a2 a3 b1 b2 b3 b4 c’ a1b1-a2b2 2(a2b2) -(a3b3+a4b4) a2b2+c’ – 2(a4b4)

18. 18 Contrast Standard Errors Standard errors for contrasts are derived using the multivariate delta method. This is a general method for finding variances of functions (and is the technique used by Sobel (1982) to find the variance of the mediated effect). The standard error formula will vary according to the effects being compared. For a simple contrast of two mediated effects: S a1b1-a2b2 = Add 2b 1 b 2 s a1a2 to the equation if there is a covariance between a 1 and a 2, s a1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small.

19. 19 Pairwise Contrasts for the ATLAS program Effects Model EffectEstimateEstimate/LCLUCL (Std Error) SE Pairwise Contrast:-.005-0.22-.046.036 Of Knowledge vs.(.021) Team as Information Pairwise Contrast:-.0662.67-.115-.017 Of Team as (.025) Information vs. Perceived Severity From MacKinnon (2000) Contrasts in Multiple Mediator Models.

20. 20 Special Topic: Inconsistent Mediation Models Inconsistent mediation models are models where at least one of the mediated effects and direct effects have different signs (see MacKinnon, Krull, & Lockwood 2000). If the overall effect of X on Y is zero but there is a significant mediated effect, then it is an inconsistent mediation model. These effects are sometimes called suppressor effects. In these models the effect of X on Y actually increases when the mediator is included in the model. “one may be equally misled in assuming that an absence of relation between two variables is real, whereas it may be due.. to the intrusion of a third variable” (Rosenberg, 1968, p. 84).

21. 21 Inconsistent mediation in ATLAS Data REASONS TO USE AAS XMXM PROGRAM XY INTENTION TO USE AAS.573 (.105).073 (.014) -.181 (.056) Mediated effect =.042 Standard error =.011

22. 22 Mediators of null effect of status on perceived sexual harassment (Sheets & Braver,1999) Power Perceptions M1M1 X Y Social Dominance M2M2 + Organizational Status Harassment + - 0 +

23. 23 Mediators of the null effect of age on typing (Salthouse, 1984) Reaction Time M1M1 X Y Skill M2M2 + Age Typing Proficiency + + 0 -

24. 24 Mediation in Structural Equation Models  Many models have multiple dependent variables, multiple independent variables, and multiple mediators.  With more than one dependent variable, a more detailed modeling approach is required. The new method is called path analysis or covariance structure modeling.  Matrices are used to specify and estimate these models because matrices organize all the variables in the model. The number and type of mediated effects are increased in these models. Matrix equations are used to find mediated effects and their standard errors.

25. 25 Socioeconomic Status and Achievement  Duncan et al. (1972) presented data on achievement that have been used to illustrate methodological developments in mediation. The data are from 3214, 35-44 year old males measured during the March of 1962 as part of a large survey of the civilian labor force.  There are six variables: X1 father’s education, X2 father’s occupation, X3 number of siblings in the respondent’s family, Y1 respondent’s education, Y2 respondent’s occupational status, and Y3 respondent’s income.  Many types of mediated effects

26. 26 Father Occupation X1 Father Education X2 No. of Siblings X3 Resp. Education  1 Resp. Income  3 Resp. Occupation  2 B32.0704 (.0045) B21 4.3767 (.1202) B31.1998 (.0364)  23 -.4631 (.1231)  33 -.0373 (.0314)  13 -.2281 (.0176)  11.0385 (.0025)  31.0114 (.0045)  21.1352 (.0175)  12.1707 (.0156)  22.0490 (.1082)  32.0712 (.0275)

27. 27 Father Occupation X1 Resp. Education  1 Resp. Occupation  2 B21 4.3767 (.1202)  11.0385 (.0025) X 1 –>η 1 –> η 2 γ 11 β 21 (.0385) (4.3747) =.1685 s γ11β21 = Square Root[ (.0385) 2 (.1202) 2 + (4.3747) 2 (.0025) 2 ]=.0118

28. 28 Mediated Effects EffectParameters Estimate SE FEDUC -> REDUC -> ROCC X 1 –>η 1 –> η 2 γ 11 β 21.1685.0118 FEDUC -> ROCC -> RINC X 1 –> η 2 –> η 3 γ 21 β 32.0095.0014 FEDUC -> REDUC -> RINC X 1 –> η 1 –> η 3 γ 11 β 31.0077.0015 FEDUC -> REDUC ->ROCC -> RINC X 1 –> η 1 –> η 2 –> η 3 γ 11 β 21 β 32.0119.0011 FOCC -> REDUC -> ROCC X 2 –> η 1 -> η 2 γ 12 β 21.7473.0713

29. 29 Father Occupation X1 Resp. Education  1 Resp. Income  3 Resp. Occupation  2 B32.0704 (.0045) B21 4.3767 (.1202)  11.0385 (.0025)

30. 30 Three Path Mediated Effect X M1M1 M2M2 Y b1b1 b2b2 b3b3 b4b4 Mediated effect = b 1 b 2 b 3 Var(b 1 b 2 b 3 ) = b 1 2 b 2 2 s b3 2 + b 1 2 b 3 2 s b2 2 + b 2 2 b 3 2 s b1 2 + 2 b 1 b 2 b 3 2 s b2b1 2 + 2 b 1 b 2 2 b 3 s b1b3 2 + 2 b 1 2 b 2 b 3 s b2b3 2 Standard Error(b 1 b 2 b 3 )=

31. 31 Father Occupation X1 Resp. Education  1 Resp. Income  3 Resp. Occupation  2 B32.0704 (.0045) B21 4.3767 (.1202) B31.1998 (.0364)  11.0385 (.0025)  21.1352 (.0175)

32. 32 LISREL and EQS Total Mediated Effects for the SES Model The keyword EF command on the OUTPUT line in LISREL requests output of total mediated effects and their standard errors. The keyword EFFECTS=YES on the /PRINT line has EQS print out total mediated effects and standard errors. These programs print the total mediated effect of X on Y. For example,with this model the total mediated effect of X 1 on  2 is the same as the specific mediated effect, X 1 ->  1, ->  2, =.1683. The total mediated effect of X 1 on  3 equals X 1 ->  2 ->  3 plus X 1 ->  1 ->  3, plus X 1 ->  1 ->  2 ->  3 or the sum of three specific indirect effects. You will need to apply the formulas above to find specific mediated effects and their standard errors.

33. 33 EQS Total Mediated effects for the SES Model DECOMPOSITION OF EFFECTS WITH NONSTANDARDIZED VALUES PARAMETER INDIRECT EFFECTS __________________________ INC1961 =V1 =.308*V3 +.148*V4 +.029*V5 +.090*V6.021.014.002.012 14.403 10.286 13.186 7.413.070 E2 +.508 E3.004.031 15.682 16.601 OCC1962 =V2 =.998*V4 +.168*V5 +.747*V6 + 4.377 E3.082.012.071.120 12.197 14.281 10.492 36.402

34. 34 LISREL Total Mediated effects for the SES Model Indirect Effects of X on Y FATHOCC FATHEDUC NUMSIB ________ ________ ________ EDUC _ _ _ _ _ _ OCC1962 0.1683 0.7473 0.9982 (0.0118) (0.0713) (0.0819) 14.2746 10.4868 12.1916 INC1961 0.0291 0.0902 0.1485 (0.0022) (0.0121) (0.0143) 13.3260 7.4621 10.3749

35. 35 Mplus 3.0 (2004) Indirect Effect Capabilities Mplus 3.0 will compute bias-corrected bootstrap confidence intervals. Specify the number of bootstrap samples, BOOTSTRAP =500 and include CINTERVAL on the OUTPUT line. Mplus 3.0 now computes standard errors and confidence intervals for tests of specific indirect effects with the “MODEL INDIRECT” statement! MODEL INDIRECT INC1961 IND FATHOCC; Requests the three indirect effects from father’s occupation to income in 1961. INC1961 IND EDUC FATHEDUC; Requests specific indirect effect from father’s education to 1961 income.

36. 36 Latent Variable Mediation Model X M Y a b c’ X1X1 X2X2 X3X3 M1M1 M2M2 M3M3 Y1Y1 Y2Y2 Y3Y3

37. 37 Latent Variable Mediation Models Equations for standard errors of mediated effects are more complicated because they include the measurement models for the variables in the model. Covariance between a and b may be nonzero so use formula that includes covariance between a and b. SEM programs compute the values of total mediated effect and Mplus 3.0 will compute specific mediated effects that include appropriate covariances in the standard error calculations. Resampling methods can also be used to obtain confidence intervals such as in Mplus 3.0 by specifying the number of bootstrap samples, BOOTSTRAP =500 and CINTERVAL on the OUTPUT line.

38. 38 Summary of Multiple Mediators There are methods to incorporate multiple mediators and latent variables in mediator models. These models require a covariance structure analysis program to estimate the models. Standard errors of mediated effects can contrasts among mediated effects can be evaluated. However, remember the assumptions of the single mediator model apply to the multiple mediator model. The additional variables address the omitted variable assumption. But other assumptions still apply. Specificity of significant mediation paths improve interpretation. The results from a multiple mediator model may shed light on the true underlying mechanisms but there are alternative explanations of results. Remember that the path relating the mediators to Y is correlation.