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1 2559 Outline 1. Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) 1.1 Similarities and differences between MRA with dummy variables and ANOVA 1.2 MRA and path analysis (PA) 2. Mediator and basic mediation model 3. Misspecification of mediation model 4. Analysis of mediation model 5. Interaction effects and moderating effects 6. Moderator and basic moderation model 7. Misspecification of mediation model 8. Analysis of moderation model 9. Moderated mediation and mediated moderation model Mediation and moderation models 22,

2 1. Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) ANOVA Research question using ANOVA - Are there any differences in means of a DV among groups of IV? - Is there any intervention effects (IV) on an expected outcome (DV)? - What is the direct effect of a causal variable (IV) on a DV? ANOVA design - Randomized control group - Causal survey Yij j SST SSB SSW 2 SSB / SST ij Mediation and moderation models 22, Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) SRA Research question using SRA - Is there any change in a DV due to a change in an IV? - What is the direct effect of a cause (IV) on a DV? - What are the predicted value of the DV given the known value of IV? SRA design - Randomized control group - Causal survey Y i (Xi) SST = SSReg. + SSRes. R 2 SSReg / SST i Mediation and moderation models 22,

3 1. Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) Type of SRA 1. Simple and multiple regression analysis models Y = b 0 + b 1 X 1 +e Y = b 0 + b 1 X 1 + b 2 X e 2. Polynomial regression analysis model Y = b 0 + b 1 X + b 2 X 2 + b 3 X e 3. Multiple regression analysis model with dummy variables Y = b 0 + b 1 D 1 + b 2 D D m-1 + e 4. Multiple regression analysis model with interaction term Y = b 0 + b 1 X + b 2 Z + b 3 XZ +. + e 5. Multiple regression with centering IV Y = b 0 + b 1 (X - ) e Mediation and moderation models 22, Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) SRA with dummy variable (regression approach to ANOVA) SRA design - Randomized control group - Causal survey Y 1(D 1) (D 2 2 SST = SSReg. + SSRes. R 2 SSReg / SST i ) ij Form of regression equation 1. Raw score form Y = b 0 + b 1 X + e Y = b 0 + b 1 X 2. Standard score form z Y = z X +z e z Y = z X 3. Deviation score form (Y - Y) = b1 (X - X) + e 4. Centering IV form Y = Y + b1 (X - X) + e 5. Form with dummy variables Mediation and moderation models 22,

4 1.1 Similarities and differences between MRA with dummy variables and ANOVA Y i (Xi) SST = SSReg. + SSRes. R 2 SSReg / SST i Y 1(D 1) (D 2 2 SST = SSReg. + SSRes. R 2 SSReg / SST i ) ij ANOVA MRA with dummy var. DV Metric variable Metric variable IV s Non-metric variable Non-metric variable relation Linear or non-linear Linear or non-linear Explained variances Eta square R square Mediation and moderation models 22, Similarities and differences between MRA with dummy variables and ANOVA Definition of dummy variable (or indicator variable) Dummy variable = a dichotomous variable created to represent the attribute, usually coded 1 representing the present of the attribute, and 0 to represent its absence Type of dummy variable 1. Dummy coding: the coding is [0, 1] 2. Effects coding: the coding is [1, 0, -1]. The intercept (a) will equal to the grand mean, and each of the b s will equal to the group effects 3. Orthogonal coding: the coding are the contrasts which yield the coded vectors to be orthogonal or uncorrelated See Note: Copy 10 pages Mediation and moderation models 22,

5 1.2 MRA and path analysis (PA) Conceptual framework MRA PA See Note: Copy 10 pages Mediation and moderation models 22, Mediator and basic mediation model Mediator (mediating variable, or intervening variable) A third variable that intervenes between the effect of a predictor and the outcome variable; it carries or transmits the effect(s) of the antecedent predictor to the outcome or dependent variable Form of mediation models Me = a 0 +ax+ e Y = b 0 +c X + bme + e c = direct effect of X on Y a = direct effect of X on Me b = direct effect of Me on Y ab = indirect effect of X on Y c + ab = total effect of X on Y r = correlation coeff. of X and Y r = (c + ab) + error Mediation and moderation models 22,

6 2. Mediator and basic mediation model Type of mediation models (Muthen, Muthen & Asparouhov, 2016) Prototypical mediation model Mediation model with a control var. C Multiple mediation model Sequential multiple mediation model Full/complete mediation model Partial mediation model Mediation and moderation models 22, Mediator and basic mediation model Mediation usage (Morgan-Lopez, A. O. & MacKinnon, D. P., 2006) Mediation processes guide the development and evaluation of preventive intervention trials. In etiological studies, mediation analyses help identify links between risk factors and outcomes. In program evaluation, mediation analyses provide practical information about the success or failure of action theory and the conceptual theory used in the development of the program. Action theory refers to the relation between program components and the mediator(s) that the program is designed to change. The conceptual theory refers to the relation between the mediator(s) and the outcome variable. Through mediation analysis, researchers can evaluate whether or not a program was successful in changing the mediating variable that it was designed to change (action theory) and whether or not the mediating variable changed the outcome variable (conceptual theory). Mediation and moderation models 22,

7 3. Misspecification of mediation model Principles Mediation, or an indirect effect, is said to occur when the causal effect of an independent variable (X) on a dependent variable (Y ) is transmitted by a mediator (Me). In other words, X affects Y because X affects Me, and Me, in turn, affects Y. Mediation effect and indirect effect are often used interchangeably (Preacher, Rucker & Hayes, 2007). Complete or full mediation is the case in which variable X no longer affects Y after M has been controlled and so path c' is zero. Partial mediation is the case in which the path from X to Y is reduced in absolute size but is still different from zero when the mediator is controlled (Baron & Kenny, 1986; Kenny, 2009). Whereas mediation analyses can provide information about mediation processes, they cannot provide information about whether or not these processes differ across subpopulations (Morgan-Lopez, & MacKinnon, 2006). Mediation and moderation models 22, Misspecification of mediation model Principles The mediator can be chosen too close to the outcome and with a distal mediator path b is large and path a is small. Ideally in terms of power, standardized a and b should be comparable in size. The power of the test of ab is maximal when b is somewhat larger than a. So distal mediators result in somewhat greater power than proximal mediators. The mediator can be too close in time or in the process to the initial variable and so path a would be relatively large and path b relatively small. An example of a proximal mediator is a manipulation check. The use of a very proximal mediator creates multicollinearity (Kenny, 2009). Mediation analysis also makes all of the standard assumptions of the general linear model (i.e., linearity, normality, homogeneity of error variance, and independence of errors). It is strongly advised to check these assumptions before conducting a mediation analysis (Kenny, 2009). Mediation and moderation models 22,

8 3. Misspecification of mediation model Principles If Me is a successful mediator, it is necessarily correlated with X due to path a. This correlation, called collinearity, affects the precision of the estimates of the last set of regression equations. If X were to explain all of the variance in Me, then there would be no unique variance in Me to explain Y. Given that a is nonzero, the power of the tests of the coefficients b and c' is compromised. The effective sample size for these tests is approximately N(1 - r 2 ) where N is the total sample size and r is the correlation between the initial variable and the mediator. So if Me is a strong mediator (path a is large), to achieve equivalent power the sample size would have to be larger than what it would be if Me were a weak mediator. Thus, multicollinearity is to be expected in a mediation analysis and it cannot be avoided (Kenny, 2009). Mediation and moderation models 22, Misspecification of mediation model Mediation is a hypothesis about a causal network. The conclusions from a mediation analysis are valid only if the causal assumptions are valid Therefore, if those assumptions are either not chcked or are invalid, then the mediation analysis results is invalid (Kenny, 2009). Measurement Error in the Mediator If the mediator is measured with less than perfect reliability, then the effects (b and c') are likely biased. The effect b is likely underestimated and the effect of the initial variable on the outcome (path c') is likely over-estimated if ab is positive (which is typical). The over-estimation of c' is exacerbated to the extent to which path a is large. To remove the biasing effect of measurement error: a) multiple indicators of the mediator can be used to tap a latent var. b) use instrumental var. estimation assuming that c' is zero. c) fix the error variance at the value or one minus the reliability times the variance of the measure. Mediation and moderation models 22,

9 3. Misspecification of mediation model Reverse Causal Effects The mediator (Me) may be caused by the outcome variable (Y). When the initial variable is a manipulated variable, it cannot be caused by either the Me or the Y. But because both the Me and Y are not manipulated variables, they may cause each other. Often it is advisable to interchange the ME and the Y and have the outcome Y "cause" the mediator Me. If the results look similar to the specified mediation pattern (i.e., the c' and b are about the same in the two models), one would be less confident in the specified model. Reverse causal effects can be theoretically ruled out. Ideally, the mediator should be measured temporally before the outcome variable. Smith s approach: Both the Me and Y are treated as outcome variables, and they each may mediate the effect of the other. Smith s approach uses a different variable (instrumental var.) that is known to cause each of them but not the other. Thus, mediation can be estimated and tested with models of feedback. Mediation and moderation models 22, Misspecification of mediation model Omitted Variables In this case, there is a variable that causes both variables in the equation. For example, at Step 3, there is a variable that causes both the mediator and the outcome. This is the most difficult specification error to solve. Although there has been some work on the omitted variable problem, the only complete solution is to specify and measure such variables and control for their effects. Note that if the initial variable is randomized, then omitted variables do not bias the estimates at Steps 1 and 2. Even, if X is manipulated, path c' is biased which implies there is an omitted variable that causes M and Y. Sometimes the source of correlation between the mediator and the outcome is a common method effect. For instance, the measuring scale of the two variables is the same. Ideally, efforts should be made to ensure that the two variables do not share method effects (e.g., both are self-reports from the same person). A latent variable analysis might be used to remove the effects of correlated measurement error. Mediation and moderation models 22,

10 4. Analysis of mediation model 4.1 Baron & Kenny Steps in Mediation Analysis 1. Estimate and test path c (must be sig.) 2. Estimate and test path a (must be sig.) 3. Estimate and test path b (must be sig.) 4. Estimate and test path c' (c' = 0: complete med., c'< c: partial med.) Mediation and moderation models 22, Analysis of mediation model 4.2 Mediation analysis with PROCESS Berger (2015); and Hayes (2014, 2015) Mediation and moderation models 22,

11 4. Analysis of mediation model 4.3 SEM using LISREL or Mplus Variety of mediation model Antecedent var. Mediator Dependent var. Metric var. Metric var. Metric var. Metric var. Non- metric var. Metric var. Metric var. Metric var. Non-metric var. Metric var. Non- metric var. Non-metric var. Non-metric var. Metric var. Metric var. Non-metric var. Non- metric var. Metric var. Non-metric var. Metric var. Non-metric var. Non-metric var. Non- metric var. Non-metric var. Mediation and moderation models 22, References Muthen, B. O., Muthen,L.K. & Asparouhov,T. (2016). Regression and mediation analysis using Mplus. Los Angees, CA: Muthen & Muthen. Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality & Social Psychology, 51, Berger, D. (2015). Using Correlation and Regression: Mediation, Moderation, and More. Part 4: Application Demonstrations. Session 2b: Demonstration of Applications of Multiple Regression 1. Available at Dale Burger s Statistics website: Hayes, A. F. (2014).Frequently asked questions about my MACROS. Available online at ac.th/+csco a2f2f6e73756e6c72662e70627a++/macrofaq.html Hayes, A. F. (2015) An index and test of linear moderated mediation, Multivariate Behavioral Research, 50, 1-22, DOI: / James, L. R., & Brett, J. M. (1984). Mediators, moderators, and tests for mediation. Journal of Applied Psychology, 69, Kenny, D. A. (2009, 2011, 2015). Mediation. David A. Kenny homepage. Available at net/cm/mediate.htm Morgan-Lopez, A. O. & MacKinnon, D. P. (2006). Demonstration and evaluation of a method for assessing mediated moderation. Behavioral Research Methods, 38, Preacher, K. J., Rucker, D. D. & Hayes, A. F.(2007). Addressing moderated mediation hypothesis: Theory, methods and prescription. Multivariate Behavioral Research, 42, Mediation and moderation models 22,

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