Testing for mediating and moderating effects with SAS

Contingency / elaboration / 3rd variable models One best management practice vs. contingency perspective Failure to find main effects -> use of moderators More than 50% of empirical strategy research have a contingency element nowadays −Venkatraman 1989 main types: −Interaction moderation −Subgroup moderation −Mediation −Configurations, gestalt (cluster analysis) Footer

Contingency / elaboration / 3rd variable models Fairchild et al 2007, Annual Review of Psychology 58: Third variable could be -Mediator x-> z -> y -Confounding variable x y (lead to spurious x-y relationship) -Covariate x -> y x -> y -Moderator / interaction Footer

Mediation

Mediation Mathieu et al 2008, Org. Res. Meth. −X -> M -> Y −Underlying mechanism through which X predicts Y −Baron & Kenny (1986) Journal Of Personality and Social Psych., 51,

Mediation, examples Mathieu et al 2008, Org. Res. Meth. −Structure – strategy – performance (IO paradigm) −Strategy – structure – performance (Chandler) −Theory of reasoned action (Ajzen) −Technology adoption model (Davis) −RBV

Mediation Independent variable X Mediating variable M Dependent variable Y ab c’ e3e3 e2e2 1)Y = i 1 + cX + e 1 2)Y = i 2 + c’X + bM + e 2 3)M = i 3 + aX + e 3

Mediation Causal steps (Baron & Kenny 1986): 1)Y = i 1 + cX + e 1 2)Y = i 2 + c’X + bM + e 2 3)M = i 3 + aX + e 3 Full of partial mediation exists when… 1)c is significant 2)a is significant 3)b is significant 4)c’ is smaller than c 9

Mediation, assumptions 1)Residuals in eq 2 and 3 are independent 2)M and residual in eq 2 are independent 3)No XM interaction in eq 2 4)No misspecification 1)Causal order x->m->y not y->m->x 2)Causal direction m y 3)Unmeasured variables 4)Measurement error 10

Size of Mediation, indirect effect total effect = direct effect + indirect effect c = c’ + ab You can calculate either c – c’ from equations 1 and 2 or ab from equations 2 and 3 and test for significance using z-distribution Standard error for the indirect effect by Sobel 1982, works ok with samples n>100, but is very conservative (low power) Sobel test tool in web

12 Mediation examples Pierce et al. (2004) Work environment structure and psychological ownership: the mediating effects of control. The journal of social psychology, 144(5): Linear regression Gassenheimer & Manolis (2001) The influence of product customization and supplier selection on future intentions: the mediating effects of salesperson and organizational trust. Journal of managerial issues, 13(4): LISREL

Mediation, example Pierce et al 2004 Hypothesis A: control mediates the relationship between WES and ownership Hypothesis B: control mediates the relationship between tech and ownership 13 stepCriterionPredictorbtR2R2 1Ownership YWES X **.12 2Ownership YWES X *.24 Control M ** 3Control MWES X **.21 1OwnershipTechn **.10 2ownershipTechn control ** 3ControlTechn **.20

Mediation, example with SAS Assign the library TILTU12 Open the dataset Data_med_mod Test a model, where knowledge sharing is expected to mediate the effect of collaboration on innovative performance -Use the Baron & Kenny causal steps to estimate the model -Use the Sobel test calculator to test the significance of the indirect effect 14

Step 1 Footer

Step 1 Footer Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 coll_indexcollaboration index

Step 2 Footer

Step 2 Footer Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 coll_indexcollaboration index ks_indexknowledge sharing index

Step 3 Footer

Step 3 Footer Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 coll_indexcollaboration index <.0001

Indirect effect & Sobel test Footer From the SAS output you get a=.596, b=.05, c=.066 and c’=.043 Input the a value from step 3 and its std error Input the b value from step 2 and its std error The calculator shows -the test statistic z = ab / std error of ab -std error of ab -Significance test that ab differs from zero -Note: the calculator does not show the value of ab (.596 *.05 in this case)

Indirect effect & Sobel test Footer

Moderation

Moderation A predictor has a differential effect on the outcome variable depending on the level of the moderator variable Guidelines for testing in Sharma et al (1981) JMR 18(3): Venkatraman 1989, AMR 14: Footer Related to x and/or yNot related to x and y No interaction with xIntervening, exogenous, antecedent, suppressor, predictor Homologizer (influences strength of x-y relationship) Interaction with xQuasi moderator (influences form of x-y relationship) Pure moderator (influences form of x-y relationship)

Moderation Homologizer: Error term is function of z, R square is dependent on z If the sample is split into subgroups according to values of z, we observe different R squares in the subgroups Pure and Quasi moderator: The regression coefficient of x is a function of z Pure y = a + b 1 x + b 2 xz or y = a + (b 1 + b 2 z)x Quasi y = a + b 1 x + b 3 z + b 2 xz -> either x or z can be the moderator A. Subgroup analysis Split the sample into subgroups based on the moderator (z) and run the x- y model separately in each subgroup Compare the R squares (and/or parameter estimates) of the subgroups, Chow test can be used for testing the significance of the difference in R squares Difference in parameter estimates d= B1 – B2 Standard error of the difference SE d = SQRT (SE B1 2 + SE B2 2 ) If |d| > 1.96* SE d, it is significant at p<.05 Footer

Moderation B: MRA (interaction) The variables should (maybe, see Echambadi & Hess 2004) be mean-centered (or residual-centered, see Lance 1988) to avoid collinearity 1.Y = a + b 1 x 2.Y = a + b 1 x + b 2 z 3.Y = a + b 1 x + b 2 z + b 3 xz Interpretation: Z is a predictor if b 3 = 0 and b 2 ≠ 0 Z is a pure moderator if b 2 = 0 and b 3 ≠ 0 Z is a quasi moderator if b 2 ≠ 0, ja b 3 ≠ 0 Use graphics to help interpretation of results 26

Moderation 27

Moderation Summary, first run MRA 1.If xz- interaction is significant 1.If the main effect of z is significant -> quasi 2.If the main effect of z is not significant -> pure 2.If xz- interaction is not significant 1.If the main effect of z is significant ->predictor 2.If the main effect of z is not significant, and z is unrelated with x -> split into subgroups based on z and run x-y regression 1.If the R square is different in the subgroups -> homologizer 2.If the R square is not different in the subgroups -> z plays no role Examples: Wiklund & Shepherd (2005) Entrepreneurial orientation and small business performance: a configurational approach. Journal of business venturing, 20(1):71-91 Rasheed (2005) Foreign entry mode and performance: The moderating effects of environment. Journal of small business management, 43(1):

Footer

31 SAS example on moderation -Dataset TAPDATA -Examine the relationships between an individual’s sex, height, and the parents’ heights -Main effects -Interaction effect of parents’ heights? -Is sex a moderator, and what type of moderator? -First assign the library and then open the data and create a scatterplot

32 SAS example on moderation

33 SAS example on moderation

34 Data transformations Create a new file into your library selecting only variables you will need (sukup, pituus, isanpit, aidipit) Add a computed column called male, where you have recoded sukup= 2 as 0 Sort the data according to the variable male

35 Main effects

36 Model diagnostics & SAS code PROC REG DATA=tiltu12.recodedsorted_tap PLOTS(ONLY)=ALL; Linear_Regression_Model: MODEL pituus = male isanpit aidipit /SELECTION=NONE SCORR1 SCORR2 TOL SPEC; RUN;

37 Output Number of Observations Read 127 Number of Observations Used 124 Number of Observations with Missing Values 3 Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standar d Errort ValuePr > |t| Squared Semi-partial Corr Type I Squared Semi-partial Corr Type I ITolerance Intercept male < isanpit < aidipit < Test of First and Second Moment Specification DFChi-SquarePr > ChiSq Significant model, high R square, homoskedastic, all parameters significant, no collinearity

38 Centering the data for interaction analysis

39 Build the interaction variable

40 Main effects with centered data Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 male <.0001 stnd_isanpit Standardized isanpit: mean = <.0001 stnd_aidipit Standardized aidipit: mean = <.0001

41 Test the significance of interaction using SAS code PROC REG DATA=TILTU12.INTER_STD_TAP PLOTS(ONLY)=ALL; MODEL pituus = male stnd_isanpit stnd_aidipit; MODEL pituus = male stnd_isanpit stnd_aidipit mom_dad; test mom_dad=0; RUN;

42 Output: no interaction Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableLabelDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 male <.0001 stnd_isanpit Standardized isanpit: mean = <.0001 stnd_aidipit Standardized aidipit: mean = <.0001 mom_dad Test 1 Results for Dependent Variable pituus SourceDF Mean SquareF ValuePr > F Numerator Denominator

43 Plot the interaction Use the file interaktio_simple.xls Standard deviations are for dad and for mom (both means are 0) Mean value for Male is.346 unstd. independent variablesmeanstd.dev.low valuehigh valueregr.coeff. Constant167,308 x10,3460,478-0,1320,82412,083 x x x x5 mom05,22-5,225,220,5415 z1 dad06,676-6,6766,6760,3523 x5z1-interaction0,0038

44 Subgroup analysis for sex

45 Output: R square seems better for men and mom’s height more important for men Number of Observations Read 83 Number of Observations Used 83 Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 stnd_isanpit <.0001 stnd_aidipit <.0001 Number of Observations Read 44 Number of Observations Used 41 Number of Observations with Missing Values 3 Analysis of Variance SourceDF Sum of Squares Mean SquareF ValuePr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates VariableDF Parameter Estimate Standard Errort ValuePr > |t| Intercept <.0001 stnd_isanpit stnd_aidipit <.0001

46 Chow test proves that models for men and women are different (data must be sorted!) PROC AUTOREG DATA=TILTU12.INTER_STD_TAP PLOTS(ONLY)=ALL; MODEL pituus = stnd_isanpit stnd_aidipit /CHOW=(83) ; RUN; Ordinary Least Squares Estimates SSE DFE 121 MSE Root MSE SBC AIC MAE AICC MAPE HQC Durbin-Watson Regress R-Square Total R-Square Structural Change Test Test Break PointNum DFDen DFF ValuePr > F Chow <.0001 Parameter Estimates VariableDFEstimate Standard Errort Value Approx Pr > |t|Variable Label Intercept <.0001 stnd_isanpit Standardized isanpit: mean = 0 stnd_aidipit <.0001Standardized aidipit: mean = 0

47 Is the effect of mom different for men and women? d = b men – b women Standard error for difference SE d = SQRT (SE b men 2 + SE b women 2 ) Test value z= d/ SE d then compare z to standard normal d= =.28 SE d = sqrt ( )= sqrt (.023)=.152 Z= 1.84 < 1.96 not significant at 5% level