Moderation & Mediation in Regression. Pui-Wa Lei, Ph.D Professor of Education Department of Educational Psychology, Counseling, and Special Education

Transcription

1 Moderation & Mediation in Regression Pui-Wa Lei, Ph.D Professor of Education Department of Educational Psychology, Counseling, and Special Education

2 Introduction Mediation and moderation are used to understand the mechanism(s) or process(es) by which an effect works and establish contingencies (Hayes, 2012). Mediation addresses how it determines whether/how a variable affects outcome through other variables or mediators. Moderation addresses when it determines whether the magnitude or direction of effect changes depending on other variables or moderators (i.e., interaction). They can be used in combination: e.g., moderated mediation, mediated moderation, conditional process modeling.

3 Relevant concepts Total effect Direct effect Indirect effect Conditional effect (i.e., simple slope) Conditional direct effect Conditional indirect effect

4 Simple Mediation Model Modeling steps: a M b 1. M = i M + ax + e M X X c c Y Y 2. Y = i Y + bm + c X + e Y Direct effect = c Indirect effect = ab Total effect = c + ab = c (in second model)

5 Does social behavior intervention have a long-term effect on Math achievement via immediate post intervention Math achievement? W1Math 20.2* 1.6* Cond Controls 83.1 Y5Math

6 Parallel Multiple Mediation Model Modeling steps: X M 1 a 1 b 1 c a 2 b 2 M 2 Y 1. M 1 = i M1 + a 1 X + e M1 2. M 2 = i M2 + a 2 X + e M2 3. Y = i Y + b 1 M 1 + b 2 M 2 + c X + e Y Direct effect = c Indirect effect via M 1 = a 1 b 1 Indirect effect via M 2 = a 2 b 2 Total indirect effect = a 1 b 1 + a 2 b 2 Total effect = c + a 1 b 1 + a 2 b 2 = c

7 Reid, et al. (2009). Gender, Language, and Social Influence: A Test of Expectation States, Role Congruity, and Self-Categorization Theories.65* Perceived similarity to speaker.49* Speaker assertive language style.31* Agreement with speaker regarding lowering drinking age.69* Perceived competence of speaker.35*

8 Warner & Vroman (2011). Happiness Inducing Behaviors in Everyday Life: An Empirical Assessment of The How of Happiness.45* Positive/Proactive behaviors.24* Extraversion.44* Happiness.33*.13* Spiritual behaviors.08.16* Physical health behaviors

9 Serial Multiple Mediation Model Modeling steps: M 1 a 3 M 2 1. M 1 = i M1 + a 1 X + e M1 2. M 2 = i M2 + a 2 X + a 3 M 1 + e M2 3. Y = i Y + b 1 M 1 + b 2 M 2 + c X + e Y a 1 a 2 b b 2 1 c X Y Direct effect = c Indirect effect via M 1 = a 1 b 1 Indirect effect via M 2 = a 2 b 2 Indirect effect via M 1 & M 2 = a 1 a 3 b 2 Total indirect effect = a 1 b 1 + a 2 b 2 +a 1 a 3 b 2 Total effect = c + a 1 b 1 + a 2 b 2 + a 1 a 3 b 2 = c

10 Van Jaarsveld, et al. (2010). The Role of Job Demands and Emotional Exhaustion in the Relationship Between Customer and Employee Incivility Job demands 1.66* Emotional exhaustion.16*.57* * Perceived customer incivility.50* Employee incivility

11 Simple Moderation Model M X c 1 M c 2 Y X Y c 3 XM Y = i + c 1 X + c 2 M + c 3 XM + e = i + c 2 M + (c 1 + c 3 M)X + e Conditional effect of X = c 1 + c 3 M (i.e., simple slope for X)

12 Moderating the negative effect of Age on Endurance by Exercise Age -.26* Exercise Exercise.97* Endurance Age Endurance.05* Age X Exercise

13 Multiple Additive Moderators M W X M Y X Y W XM XW Y = i + c 1 X + c 2 M + c 3 W + c 4 XM + c 5 XW + e Conditional effect of X = c 1 + c 4 M + c 5 W

14 Multiple Multiplicative Moderators M M X XMW M Y X Y W XM XW MW Y = i + c 1 X + c 2 M + c 3 W + c 4 XM + c 5 XW + c 6 MW + c 7 XMW + e Conditional effect of X = c 1 + c 4 M + c 5 W + c 7 MW Conditional XM interaction = c 4 + c 7 W

15 Moderated Mediation (1 st stage) {or Mediated Moderation} W M a 1 a 2 M X a 3 b X 1. M = i M + a 1 X + a 2 W + a 3 XW + e M Y W XW c 3 c 2 c 1 Y 2. Y = i Y + bm + c 1 X + c 2 W + c 3 XW + e Y Conditional indirect effect of X via M = (a 1 + a 3 W)b Conditional direct effect of X = c 1 + c 3 W {In mediated moderation: Indirect effect of XW via M = a 3 b}

16 Does the indirect effect of social behavior intervention via immediate Math posttest on delayed Math posttest moderate by receipt of supplemental service? W1Math 22.0* SuppSer W1Math Cond -59.4* * Cond Y5Math SuppSer Cond x SuppSer Y5Math

17 Moderated Mediation (2 nd stage) W M M a b 1 X Y X c Y b 2 b 3 1. M = i M + ax + e M W MW 2. Y = i Y + b 1 M + b 2 W + b 3 MW + c X + e Y Conditional indirect effect of X via M = a(b 1 + b 3 W)

18 There are many more possibilities Hayes ( ) documents 76 model templates (

19 Follow-up procedures What are the next steps after fitting the regression or path models? oprobing significant interactions otesting indirect effects

20 Probing Significant Interactions Purposes: testing conditional effects & understanding patterns of interactions. Methods ( Pick a point approach: Select points of interest on the moderator(s) Calculate conditional effects based on the estimated regression functions Standard errors depend on variances and covariances among the regression coefficients Conditional effect estimate/appropriate SE ~ t (df=n-k-1) Johnson-Neyman approach: search for two points on the moderator that define the regions of significance, within which all conditional effects are statistically significant.

21 Testing indirect effects Indirect effects involve products of regression coefficients. Indirect effects are generally not normally distributed. Sobel s (1982) test is simple but relies on normal approximation ( Bootstrapping (standard errors and/or confidence intervals) with bias correction. Monte Carlo method (

22 PROCESS (Hayes, ): a SAS/SPSS macro PROCESS is a helpful tool in conducting these follow-ups ( It uses a regression-based path analytic framework (OLS or logistic) for estimating: direct and indirect effects in single and multiple mediator models, 2-3 way interactions in moderation models with simple slopes & regions of significance, conditional indirect effects in moderated mediation models (single/multiple mediators/moderators), indirect effects of interactions in mediated moderation models (single/multiple mediators).

23 Example: Age (centerx) x Exercise (centerz) on Endurance (yendu) SPSS Macro: Age Exercise Endurance 1. Open Macro syntax Run all; then close syntax. 2. Open data file. 3. New Syntax (type the following syntax and run all): Process VARS = yendu centerx centerz /Y = yendu /X = centerx /M = centerz /MODEL = 1 /COVCOEFF=1 /JN = 1 /PLOT = Copy/paste and run plot syntax (from output) to get interaction figure.

24 Model = 1 Y = yendu X = centerx M = centerz Sample size 245 ************************************************************************** Outcome: yendu Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant centerz centerx int_ Product terms key: int_1 centerx X centerz R-square increase due to interaction(s): R2-chng F df1 df2 p int_

25 Conditional effect of X on Y at values of the moderator (M) centerz Effect se t p LLCI ULCI

26 R Example 1. Get model estimates and covariance matrix from SPSS. Covariance matrix of regression parameter estimates constant centerz centerx int_1 constant centerz centerx int_ Go to and type in appropriate values Calculate and Submit to Rweb.

27 Region of Significance ======================================================= Z at lower bound of region = Z at upper bound of region = (simple slopes are significant *outside* this region.) Simple Intercepts and Slopes at Conditional Values of Z ======================================================= At Z = cv1... simple intercept = (0.9332), t= , p=0 simple slope = (0.0931), t= , p=0 At Z = cv2... simple intercept = (0.6466), t= , p=0 simple slope = (0.064), t= , p= At Z = cv3... simple intercept = (0.902), t= , p=0 simple slope = (0.0931), t= , p=0.697 ======================================================= Line for cv1: From {X=-29.18, Y= } to {X=32.82, Y=5.2581} Line for cv2: From {X=-29.18, Y= } to {X=32.82, Y= } Line for cv3: From {X=-29.18, Y= } to {X=32.82, Y= }

28 Simple Mediation Example W1Math Cond Y5Math Controls SPSS Macro: 1. Open Macro syntax Run all; then close syntax. 2. Open data file. 3. New Syntax (type the following syntax and run all): process vars = Y5grade Y5Math Cond Gender SpED SuppSer White W1Math /y = Y5Math /x = Cond /m = W1Math /model = 4 /total = 1 /effsize = 1 /boot=5000 /normal=1 /conf=95.

29 MODEL = 4 Y = Y5MATH X = COND M = W1MATH STATISTICAL CONTROLS: CONTROL= Y5GRADE GENDER SPED SUPPSER WHITE SAMPLE SIZE 466 Outcome: W1Math Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant Cond Y5grade Gender SpED SuppSer White ************************************************************************** Outcome: Y5Math Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant W1Math Cond Y5grade Gender SpED SuppSer White ************************** TOTAL EFFECT MODEL **************************** Outcome: Y5Math Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant Cond Y5grade Gender SpED SuppSer White

30 *************** TOTAL, DIRECT, AND INDIRECT EFFECTS ************ Total effect of X on Y Effect SE t p LLCI ULCI Direct effect of X on Y Effect SE t p LLCI ULCI Indirect effect of X on Y Effect Boot SE BootLLCI BootULCI W1Math Partially standardized indirect effect of X on Y Effect Boot SE BootLLCI BootULCI W1Math Normal theory tests for indirect effect Effect se Z p

31 Example: Moderated mediation (1 st stage) SuppSer W1Math Cond Y5Math Process Vars = Y5grade Y5Math Cond Gender Sped Suppser White W1math /Y = Y5Math /X = Cond /M = W1math / W=suppser /Model = 8 /Boot=5000 /Conf=95 /Seed=34421.

32 Outcome: W1Math Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant Cond SuppSer int_ Y5grade Gender SpED White Product terms key: int_1 Cond X SuppSer ************************************************************************** Outcome: Y5Math Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant W1Math Cond SuppSer int_ Y5grade Gender SpED White Product terms key: int_2 Cond X SuppSer

33 ******************** DIRECT AND INDIRECT EFFECTS ************************* Conditional direct effect(s) of X on Y at values of the moderator(s): SuppSer Effect SE t p LLCI ULCI Conditional indirect effect(s) of X on Y at values of the moderator(s): Mediator SuppSer Effect Boot SE BootLLCI BootULCI W1Math W1Math ******************** INDEX OF MODERATED MEDIATION ************************ Mediator Index SE(Boot) BootLLCI BootULCI W1Math When the moderator is dichotomous, this is a test of equality of the conditional indirect effects in the two groups.

34 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), Bauer, D. J., & Curran, P. J. (2005). Probing interactions in fixed and multilevel regression: inferential and graphical techniques. Multivariate Behavioral Research, 40, Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, Hayes, A. F. (2012). PROCESS: A versatile computational tool for observed variable mediation, moderation, and conditional process modeling [White paper]. Retrieved from Hayes, A. F. (2013). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. Guilford. Hayes ( ). Model templates for PROCESS for SPSS and SAS. Retrieved from Hayes ( ). PROCESS: A SAS/SPSS macro. Available from Reid, S. A., Palomares, N. A., Anderson, G. L., & Bondad-Brown, B. (2009). Gender, Language, and Social Influence: A Test of Expectation States, Role Congruity, and Self-categorization Theories. Human Communication Research, 35, Van Jaarsveld, D. D., Walker, D. D., & Skarlicki, D. P. (2010). The Role of Job Demands and Emotional Exhaustion in the Relationship between Customer and Employee Incivility. Journal of Management, 36, Warner, R. M. & Vroman, K. G. (2011). Happiness Inducing Behaviors in Everyday Life: An Empirical Assessment of The How of Happiness. Journal of Happiness Studies, 12,