Analysing Moderation and Mediation Effects with Structural Equation Models

1. Analysing Moderation and Mediation Effects with Structural Equation Models • Volkmar Höfling, Helfried Moosbrugger, Augustin Kelava, Polina Dimitruk, & Karin Schermelleh-Engel • J. W. Goethe University, Frankfurt am Main, Germany SMABS – EAM Conference 2006, Budapest

2. Outline • Criteria of Mediation and Moderation • Mediated Moderation and Moderated Mediation • The Combined Moderation and Mediation Model • Monte-Carlo Study • Results • Conclusions

3. Criteria of Mediation and Moderation(c.f. Baron & Kenny, 1986) • Mediation: • a ≠ 0 • b ≠ 0 • The inclusion of a mediator results in:c ↓ (partial mediation)c = 0 (complete mediation) • Moderation: • a ≠ 0 • b ≠ 0 or b = 0 • c ≠ 0

4. Mediated Moderation • Analyses of regression models: • O is regressed on P, MO, and PxMO • ME is regressed on P, MO and PxMO • O is regressed on ME, P, MO and PxMO • If PxMO → O is weaker in 3) than in 1), this would indicate mediated moderation.

5. Moderated Mediation • Analyses of regression models: • O is regressed on P1, P2, and P1xME • ME is regressed on P1, P2 and P1xME • O is regressed on ME, P1, P2 and P1xME • If the mediational effects by ME vary across the levels of P1, this wouldindicate moderated mediation.

6. Methodological Problems of (nonlinear) Regression Models • Regression can lead to biased estimates of the regression coefficients as the measurement errors are usually not taken into account. • The reliability problem will even be aggravated by adding interaction terms to the linear regression equation, because the reliability of a product variable is always lower than the reliability of the single predictor variables. • Example for uncorrelated predictor variables:Rel(X1) = Rel(X2) = .80; Rel(X1X2) = .64 • → Solution: Analyses with nonlinear structural equation modeling

7. Y1 e1 The Combined Moderation and Mediation Model(c.f. Preacher, Rucker, & Hayes, submitted) l11 z1 X1 d1 Y2 l11 e2 x1 g11 l21 X2 d2 l21 Y3 h1 e3 l31 g21 l31 X3 d3 z2 21 Y4 e4 l42 g12 l42 l52 h2 Y5 X4 g22 e5 d4 x2 l62 l52 X5 d5 w112 w212 Y6 e6 l62 x1 x2 X6 d6 Reliability of indicator variables:Rel(Xn) = .75 Rel(Yn) = .75 True values of linear and nonlinear effects: g11 = b21 = .4g12 = g21 = g22 = .2w112 = w112 = .2

8. Monte-Carlo Study (1) • The Monte-Carlo study was carried out in order to test whether it is possible to analyse combined moderation and mediation models with structural equation modeling (simultaneously). • The Monte-Carlo study should furthermore investigate whether the estimated parameters correspond to the reference values of the true parameters both for complete and partial mediation. • Data were generated from a population with known parameter values:→ 500 samples of sample size 400 were drawn. • For each sample parameters of the investigated model were estimated by MPlus (Muthén & Muthén, 2006) with „Latent Moderated Structural Equations“ (LMS; Klein & Moosbrugger, 2000). • The means of 500 estimates for every parameter were computed.

9. Monte-Carlo Study (2) • In order to investigate the consequences of specified models without mediator variable each of the combined models was analysed with and without mediator variable.

10. Results (1): Partial Mediation with two Moderation Effects Table 1. Mean parameter estimates for a moderation model (2 moderation effects) with and without mediator variable. • The analysis of the partial mediation and moderation model results in salient unbiased parameter estimates. • The analysis of the moderation model without mediator variable results in different, but consistentparameter estimates.

11. Results (2): Estimated Standard Errors Table 2. Standard deviations (SD) and estimated standard errors (SE) for the moderation model with 2 moderation effects and with mediator variable. • The standard errors are estimated correctly. This is true for all analysed models.

12. Conclusions • The analyses of Moderation and (complete or partial) Mediation Models with MPlus and LMS result in unbiased parameter estimates and correct estimated standard errors. • The analyses result in unbiased parameter estimates with and without mediator variable, therefore analyses should be done based on theoretical deliberations and within a confirmatory approach. • A limitation of the analyses of Moderation and Mediation Models with MPlus and LMS is the lack of a model test for nonlinear latent models. • Future studies should investigate both different levels of moderation effects and intercorrelations between the predictor variables in Moderation and Mediation Models. • Furthermore should be examined if all types of nonlinear effects (quadratic and moderation effects) ought to be analysed simultaneously.

13. References • Baron, R.M. & Kenny, D.A. (1986). The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations. Journal of Personality and Social Psychology, 51 (6), 1173-1182. • James, L.R & Brett, J.M. (1984). Mediators, Moderators, and Tests for Mediation. Journal of Applied Psychology, 69 (2), 307-321. • Klein, A. & Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457-474. • Muller, D., Judd, C.M., & Yzerbyt, V.Y. (2005). When Moderation is Mediated and Mediation is Moderated. Journal of Personality and Social Psychology, 89 (6), 852-863. • Muthén, L.K. & Muthén, B.O. (2006). MPlus user‘s guide version 4.1. Los Angeles, CA: Muthén & Muthén. • Preacher, K.J., Rucker, D.D., & Hayes, A.F. (submitted). Suggested Procedures for Adressing Moderation Mediation Hypotheses. Multivariate Behavioral Research.