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1、Interaction and MediationInteraction and Mediation Kenneth Law (罗胜强)() Department of Management The Chinese University of Hong Kongygg 香港中文大学管理系 Kenneth Law 同济大学 2010 1 Interaction and MediationInteraction and Mediation Moderator Interaction effects MediatorMediator Moderated mediation Kenneth Law 同
2、济大学 2010 2 Moderators ModeratorsModerators M XY Kenneth Law 同济大学 2010 3 ModeratorModerator A moderator variable is any variable which, when systematically varied, “causes” the relationship between two other variables to change. (Stone, p.26)(Stone, p.26) Moderator variable is a variable whose differ
3、ent values d tithtf thl tihi b ttdetermine the nature of the relationship between two other variables. (Schmitt however, work-family conflict relates to career satisfaction only for men in latercareer satisfaction only for men in later career. Kenneth Law 同济大学 2010 14 Interaction EffectsInteraction
4、Effects Interaction effects means that the total effect of twoInteraction effects means that the total effect of two predictor variables does not equal to the sum of their linear effects on the outcome variable. Job Enrichment x1 y b1 Health care costs Frequent exercise x2 b2 = b0+ b1x1+ b2x2+ inter
5、action effects y Kenneth Law 同济大学 2010 15 b0+ b1x1+ b2x2 + interaction effectsy Interaction EffectsInteraction Effects Interaction effects means that the total effect of two dii bldlhf h ipredictor variables does not equal to the sum of their linear effects on the outcome variable. Job Enrichment X1
6、 Frequent exercise Y Health care costs Frequent exercise Y b2 Job Enrichment Health care costs Y b1 X2 21 = b0+ b1x1+ b2x2+ interaction effects y Kenneth Law 同济大学 2010 16 b0+ b1x1+ b2x2 + interaction effectsy Interaction EffectsInteraction Effects Job x1 = b0+ b1x1+ b2x2 + interaction effects y Job
7、Enrichment x2 y b1 Health care costs Frequent exercise x2 b2 1. There is no need for a direct effect for moderation. 2Interaction means bilateral moderation2. Interaction means bilateral moderation. 3. Conceptual, subtle differences. Kenneth Law 同济大学 2010 17 Interaction EffectsInteraction Effects Co
8、lella, A., 2 Main effects as average effect of X on Y at X1.29*.25* X2.44*.39* 2.Main effects as average effect of X1on Y at all levels of X2 X1 * X2(interaction).18* Model R2.39*.46* Significant interactionSignificant interaction Kenneth Law 同济大学 2010 28 Interpreting main effects with interactionpg
9、 X1Relationship between X2andYX1Relationship between X2and Y 5Y = -0.5 + 1.5 X2 20Y0 01 0 X20Y = 0.0 + 1.0 X2 80Y = -0.1 + 0.5 X2 Overall Relationship between X2and Y Y = 0 15 + 0 9 XY = 0.15 + 0.9 X2 Main effect Kenneth Law 同济大学 2010 29 Main effect Post hoc analysis Hi h XLX median High X1Low X1 me
10、an ddLXHi h Xs.d.s.d.Low X1High X1 Kenneth Law 同济大学 2010 30 Sub-group analysis Fl Career Satisfaction (Y) y = .24 + .36 X1 Female 2841 X Male y = .28 - .41 X1 Work-family conflict (X) Note: this is not a good test of moderating effects Work-family conflict (X) Career Satisfaction (Y) Gender Kenneth
11、Law 同济大学 2010 31 (X) (Y) Moderation vs. Interaction Gender (M) Work-family conflict (X1) Career Satisfaction (Y) () Y = b0+ b1X1+ b2M + b3X1*M Subordinate disability (X1) Leader Member Exchange (Y) Subordinate performance (X2) Leader Member Exchange (Y) Kenneth Law 同济大学 2010 32 Y = b0+ b1X1+ b2X2+ b
12、3X1*X2 Testing interactions in SEMTesting interactions in SEM Cortina, J.M., Chen, G., is independent of ;x1is independent of x2; and X is multivariate normal with zero mean, then 111xxx x x x x x x x 222 12 xxx x Xxx x1 x2 38 Ping (1995)Ping (1995) 12 ()()Var XVar xx 1122 1212 () () xXxxXx X Var Va
13、r 1212 121212 2 () ()()() ()()() xxXxx xxXxxxx Var VarVarVar VVV 2 1212 2 ()()() () () xxXxx XXX VarVarVar Var XVar 2 122 ( )() () zz Var ZVar zzVar 2 2 121 and ()( XxxXx whereVarVar 2)x 122 ( )()() zz 39 Kenneth Law 同济大学 2010 Ping (1995)g () 121122 ( ,)(,) ()()()() xxxzzz Cov x zCov CovCovCovCov 12
14、121212 12 12 (,)(,)(,)(,) (,) (,) xxzzxxzxzzxz xxzz xzxz CovCovCovCov Cov Cov 1212 11122122 (,)(,) ( ,)( ,)(,)(,) Cov X ZCov xxzz Cov x zCov x zCov x zCov x z 12 (,) xzxz 11122122 11122122 11122122 ()()()() (,)(,)(,)(,) (,) xzxzxzxzxzxzxzxz xzxzxzxzxz CovCovCovCov Cov 112212xzzxzz 1212 (,) (,) ()()
15、xz xxzzxz Cov Cov Cov X ZCov (,)(,) xzxz Cov X ZCov 40 Kenneth Law 同济大学 2010 Ping (1995)Ping (1995) According to Kenny but it does not imply any overall moderation of the treatment effect. M llD J dd CM the might prime led to the expectation that the partner is competitive Main effect: Why would pri
16、ming affect the cooperatives of the participants? Interaction effect: Why would the effect of priming on Social value orientation the might prime led to the expectation that the partner is competitive Prosocial vs. proself pg cooperation depend on ones social value orientation? Priming Cooperation o
17、rientation Kenneth Law 同济大学 2010 77 Morality vs. Might Silent or betray Mediated moderationMediated moderation Prosocial peopleProself people Social value orientation Morality Might Cooperative Competitive Priming Morality Might Cooperative Competitive Interaction effect: Why would the effect of pri
18、ming on Social value orientation cooperation depend on ones social value orientation? CooperationPriming Expected cooperativeness Prosocial people expect the other party would cooperate under morality priming. Kenneth Law 同济大学 2010 78 Example ofExample of Moderated Mediation Kenneth Law 同济大学 2010 79
19、 Moderated MediationModerated Mediation (1) Classical conditioning: when one feels good, one has a higher chance of liking/agreeing with the message (i.e. attitude) (2) Mood can exert a main effect on attitudes by biasing the thoughts that come to mind as a person thinks about the persuasive communi
20、cation. Subjects were asked to recall a recent event that made them feel very positive whether one is persuaded by a persuasive argument Positive Mood Positive thoughts Persuasion feel very positive Kenneth Law 同济大学 2010 80 Moderated MediationModerated Mediation (1) Classical conditioning: when one
21、feels good, one has a higher chance of liking/agreeing with the message (i.e. attitude) (2) Mood can exert a main effect on attitudes by biasing the thoughts that come to mind as a person thinks about the persuasive communicationas a person thinks about the persuasive communication. Need for cogniti
22、on Subjects were asked to recall a recent event that made them feel very positive whether one is persuaded by a persuasive argument Positive Mood Positive thoughts Persuasion feel very positive Kenneth Law 同济大学 2010 81 The Mathematics of Mediated moderation &Mediated moderation & Moderated mediation
23、 Kenneth Law 同济大学 2010 82 Classical Mediation Me XY 2132 XY 11 22120 11110 (2) (1) XMe XY 31 3323130 22120 (3) ( ) MeXY Sub (2) into (3) and compare with (1), we have 32213111 32213111 * * ( )( )p( ) Total effect = direct effect + indirect effect Kenneth Law 同济大学 2010 83 32213111 Classical Moderatio
24、nClassical Moderation M XY 11 Mo XY 41 42 43 1141 (4) 443424140 XMoMoXY Kenneth Law 同济大学 2010 84 Mediated ModerationMediated Moderation (3) (2) (1) 3323130 22120 11110 MeXY XMe XYClassical Mediation (5) (4) * 443424140 32213111 XMMXM XMoMoXY Classical Moderation (6) (5) 6656463626160 553525150 MeMoM
25、eXMoMoXY XMoMoXMe Sub (5) into (6) and compare with (4), we have * * 516553646343 Kenneth Law 同济大学 2010 85 Mediated moderation (5) (4) 553525150 443424140 XMoMoXMe XMoMoXY (6) into (5) of Substitute (6) 55352515063626160 6656463626160 XMoMoXXMoMoXY Me MeMoMeXMoMoXY 22 564645364526451645063626160 655
26、352515065 5535251506463626160 XMoMoXXMoMoX MoXMoMoX XMoMoXXMoMoXY 55646 2 6553 2 65526551636453 6550645262645161645060 65 2 6553 2 655265516550 MoXMoMoXMo MoX MoXMoMoXMoMo :have we, 0 and 0 and 0 that assume and parameter, same theestimating be should equation above theand (4) Since 5564665536552655
27、1636453 Cov(Mo,X),Mo N(E(X) Kenneth Law 同济大学 2010 86 516553646343 Mediated Moderation (2) (1) 22120 11110 XMe XY (5) (4) (3) ( ) 443424140 3323130 22120 XMoMoXMe XMoMoXY MeXY (6) (5) 6656463626160 553525150 MeMoMeXMoMoXY XMoMoXMe 41Overall treatment effect on Y at the average level of Mo 42Moderator
28、 effect on Y on average across the two treatment levels42Moderator effect on Y on average across the two treatment levels 43Change in overall treatment effect on Y as Mo increases 51Treatment effect on Me at the average level of Mo 52Moderator effect on Me on average across the two treatment levels
29、53Change in treatment effect on Me as Mo increases53Change in treatment effect on Me as Mo increases 61Residual direct treatment effect on Y at the average level of Mo 62Moderator effect on Y on average within the two treatment levels and at the average level of Me 63Change in residual direct treatm
30、ent effect on Y as Mo increases 64Mediator effect on Y on average within the two treatment levels and at the average level of Mo Kenneth Law 同济大学 2010 87 64Mediator effect on Y on average within the two treatment levels and at the average level of Mo 65Change in mediator effect on Y as Mo increase.
31、What does that mean? How should we interpret this equality? 436364536551 * * Kenneth Law 同济大学 201088 What does it mean? 404142434 (4) (5) YXMoXMo MeXMoXMo 505152535 6061626364656 (5) (6) MeXMoXMo YXMoXMoMeMeMo Mo Me Mo Y Me X 51+53Mo 64+65Mo Mo X 51 Me Y Mo 61 64 53 65 61+63Mo Mo 63 Total moderation
32、 effect = Direct moderation effect + Indirect moderation effect 616351536465 2 61516463516553645365 *MoMoMo MoMo Total moderation effect = Direct moderation effect + Indirect moderation effect Kenneth Law 同济大学 2010 89 61516463516553645365 What does it mean?What does it mean? (4)YXMoXMo Mo Me Mo 5 40
33、4142434 505152535 6061626364656 (4) (5) (6) YXMoXMo MeXMoXMo YXMoXMoMeMeMo X 5 1 Y Mo 6 1 6 4 6 3 5 3 6 5 6351655364 51 Me Mo 65 Mo Me 64 53 Me Mo X 51 Y 65 XY 64 XY Mo 63 Direct moderation (No mediation) Second-stage moderation First-stage moderation Kenneth Law 同济大学 2010 90 M di t ddtiMediated mod
34、eration Mo Me Mo 64 Me XY 53 65 51 404142434 (4) (5) YXMoXMo MXMXM 505152535 6061626364656 (5) (6) MeXMoXMo YXMoXMoMeMeMo 436364 536551 * * Kenneth Law 同济大学 2010 91 Mediated moderationMediated moderation Mo 64 Me Mo 53 65 51 404142434 (4)YXMoXMo XY 505152535 6061626364656 (5) (6) MeXMoXMo YXMoXMoMeM
35、eMo 436364 536551 * * Kenneth Law 同济大学 2010 92 Mediated moderationedtedodet o (4) 443424140 XMoMoXY * * (6) (5) 516553646343 6656463626160 553525150 MeMoMeXMoMoXY XMoMoXMe To demonstrate mediated moderation in a sample of data, one estimates Model 4 through 6. In Model 4, we would expect 43to be sig
36、nificant, indicating overall treatment moderation. gg In Model 5 and 6, either (or both) of two patterns should exist: (53and 64) are significant, or (51and 65)are significant. And as a result, the moderation of the residual treatment effect, 63, should be reduced in magnitude (or nonsignificant for
37、 full mediation) compared to the moderation of the overall treatment effect , 43. Kenneth Law 同济大学 2010 93 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), p. 856. Moderated mediation vs. Medi
38、ation moderation It became apparent to us, as our ideas evolved, that we had much more to say at a theoretical level about the inherent similarities between the two processes. They both rely on the same analytic models, they both imply moderated indirect effects of the treatment variable on the outc
39、ome, and they both imply that the overall moderation of the treatment effect is altered once the (moderated) mediating process is controlled. At the same time, we would not want to claim that moderated mediation and mediated moderation are one and the same. By defining the prototypic moderated media
40、tion case, we can clearly differentiate between them. Additionally, mediated moderation clearly does imply that the overall moderation of a treatment effect is reduced once the (moderated) p y() mediating process is controlled. That said, however, we also have argued that if one allows for the possi
41、bility of moderated mediation in cases where there is overall moderation of the treatment effect, then the dbhbf hldistinction between the two processes can become more a matter of theoretical preference than anything else. The analytic models are clear cut and the two processes can be defined in th
42、e prototypic cases But these as we have already said represent probably two sides of the same coin In Kenneth Law 同济大学 2010 94 cases. But these, as we have already said, represent probably two sides of the same coin. In talking about that coin, we can either concentrate on describing each side in tu
43、rn, or we can recognize that they both define the common coin. Edwards & Lambert (2007)Edwards & Lambert (2007) Methods for Integrating Moderation andMethods for Integrating Moderation and Mediation: A General Analytical Framework Using Moderated PathAnalysisUsing Moderated Path Analysis. Jeffrey R. Edwards & Lisa Schurer Lambert Psychological Methods 2007 Vol 12(1) 1 22Psychological Methods, 2007, Vol.12(1), 1-22. Kenneth Law 同济大学 2010 95 Mediated moderation or moderated mediation M Z M XY Kenneth Law 同济大学 2010 96 Piece ApproachPiece Approach Testin