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: 593-614 Third variable could be - Mediator x-> z -> y - Confounding variable x <- z -> y (lead to spurious x-y relationship) - Covariate x -> y <- z or z -> x -> y - Moderator / interaction Footer Mediation Mediation Mathieu et al 2008, Org. Res. Meth. http://davidakenny.net/cm/mediate.htm − X -> M -> Y − Underlying mechanism through which X predicts Y − Baron & Kenny (1986) Journal Of Personality and Social Psych., 51, 1173-1182 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 e3 Mediating variable M a b e2 Independent variable X Dependent variable Y c’ 1) 2) 3) Y = i1 + cX + e1 Y = i2 + c’X + bM + e2 M = i3 + aX + e3 Mediation Causal steps (Baron & Kenny 1986): 1) Y = i1 + cX + e1 2) Y = i2 + c’X + bM + e2 3) M = i3 + aX + e3 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) 2) 3) 4) Residuals in eq 2 and 3 are independent M and residual in eq 2 are independent No XM interaction in eq 2 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) ab ab b a 2 2 2 2 Sobel test tool in web http://quantpsy.org/sobel/sobel.htm 11 Mediation examples Pierce et al. (2004) Work environment structure and psychological ownership: the mediating effects of control. The journal of social psychology, 144(5):507-534 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):418-435 LISREL 12 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 step Criterion Predictor b t R2 1 Ownership Y WES X .35 5.59** .12 2 Ownership Y WES X .17 2.60* .24 Control M .39 5.57** 3 Control M WES X .46 7.76** .21 1 Ownership Techn .31 4.41** .10 2 ownership Techn .13 1.76 .24 control .42 5.71** Techn .44 6.63** 3 Control .20 13 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 Source Model Error Corrected Total Analysis of Variance Sum of Mean DF Squares Square 1 7.28598 7.28598 245 209.38989 0.85465 246 216.67587 F Value 8.53 Root MSE 0.92447 R-Square 0.0336 Dependent Mean 2.80379 Adj R-Sq 0.0297 Coeff Var Pr > F 0.0038 32.97234 Parameter Estimates Variable Label Intercept Intercept coll_index collaboration index Parameter Standard DF Estimate Error t Value Pr > |t| 1 2.29459 0.18405 12.47 <.0001 1 0.06623 0.02268 2.92 0.0038 Footer Step 2 Footer Step 2 Source Model Error Corrected Total Analysis of Variance Sum of Mean DF Squares Square 2 15.12033 7.56016 241 199.51603 0.82787 243 214.63636 F Value 9.13 Root MSE 0.90987 R-Square 0.0704 Dependent Mean 2.80829 Adj R-Sq 0.0627 Coeff Var Variable Intercept coll_index ks_index Pr > F 0.0002 32.39954 Parameter Estimates Parameter Label DF Estimate Intercept 1 1.44424 collaboration index 1 0.04314 knowledge sharing index 1 0.05013 Footer Standard Error 0.33969 0.02515 0.01785 t Value 4.25 1.72 2.81 Pr > |t| <.0001 0.0876 0.0054 Step 3 Footer Step 3 Analysis of Variance Sum of Squares 624.99711 2851.66696 3476.66406 Source Model Error Corrected Total DF 1 262 263 Root MSE Dependent Mean Coeff Var 3.29912 R-Square 20.71926 Adj R-Sq 15.92299 Variable Intercept coll_index Mean Square 624.99711 10.88423 F Value 57.42 Pr > F <.0001 0.1798 0.1766 Parameter Estimates Parameter Standard Label DF Estimate Error t Value Pr > |t| Intercept 1 16.12346 0.63957 25.21 <.0001 collaboration index 1 0.59563 0.07860 7.58 <.0001 Footer Indirect effect & Sobel test http://quantpsy.org/sobel/sobel.htm 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) Footer Indirect effect & Sobel test http://quantpsy.org/sobel/sobel.htm Footer Moderation Moderation http://davidakenny.net/cm/moderation.htm 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):291-300 Venkatraman 1989, AMR 14:423-444 Related to x and/or y Not related to x and y No interaction with x Intervening, exogenous, antecedent, suppressor, predictor Homologizer (influences strength of x-y relationship) Interaction with x Quasi moderator (influences form of x-y relationship) Pure moderator (influences form of x-y relationship) Footer 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 + b1 x + b2 xz or y = a + (b1 + b2 z)x Quasi y = a + b1 x + b3z + b2xz -> either x or z can be the moderator A. Subgroup analysis Split the sample into subgroups based on the moderator (z) and run the xy 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 SEd= SQRT (SEB12 + SEB22) If |d| > 1.96* SEd, 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 + b1 x 2. Y = a + b1 x + b2 z 3. Y = a + b1 x + b2 z + b3 xz Interpretation: Z is a predictor if b3 = 0 and b2 ≠ 0 Z is a pure moderator if b2 = 0 and b3 ≠ 0 Z is a quasi moderator if b2 ≠ 0, ja b3 ≠ 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):41-54 28 Footer Footer 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 31 SAS example on moderation 32 SAS example on moderation 33 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 34 Main effects 35 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; ; 36 Output Number of Observations Read Number of Observations Used Number of Observations with Missing Values 127 124 3 4.20927 R-Square 0.7569 Root MSE Dependent Mean 171.33871 Adj R-Sq 0.7509 2.45669 Coeff Var Analysis of Variance Sum of Mean DF Squares Square F Value Pr > F Source 3 6621.62294 2207.20765 Model 120 2126.15126 17.71793 Error Corrected Total 123 8747.77419 Test of First and Second Moment Specification DF Chi-Square Pr > ChiSq 124.57 <.0001 8 6.66 0.5734 Parameter Estimates Variable Label DF 1 Intercept Intercep Parameter Estimate Squared Standar Squared Semi-partial d Semi-partial Corr Type I Error t Value Pr > |t| Corr Type I I Tolerance 15.39805 14.02663 12.07190 0.35037 0.54126 0.80985 0.05908 0.07613 1.10 0.2745 . . . 14.91 <.0001 5.93 <.0001 7.11 <.0001 0.51938 0.13520 0.10237 0.45004 0.07124 0.10237 0.98437 0.92602 0.91214 t male isanpit aidipit isanpit aidipit 1 1 1 Significant model, high R square, homoskedastic, all parameters significant, no collinearity 37 Centering the data for interaction analysis 38 Build the interaction variable 39 Main effects with centered data 4.20927 R-Square 0.7569 171.33871 Adj R-Sq 0.7509 Root MSE Dependent Mean 2.45669 Coeff Var Variable Intercept male stnd_isanpit stnd_aidipit Label Parameter Estimates Parameter Standard DF Estimate Error t Value Pr > |t| Intercept Standardized isanpit: mean = 0 Standardized aidipit: mean = 0 1 1 1 1 167.34719 12.07190 0.35037 0.54126 0.46324 0.80985 0.05908 0.07613 361.26 14.91 5.93 7.11 <.0001 <.0001 <.0001 <.0001 40 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; 41 Output: no interaction 4.22498 R-Square 0.7572 Root MSE 171.33871 Dependent Mean Adj R-Sq 0.7490 2.46586 Coeff Var Analysis of Variance Sum of Mean DF Squares Square F Value Pr > F Source 4 6623.57123 1655.89281 Model 119 2124.20296 17.85045 Error Corrected Total 123 8747.77419 Variable Intercept male stnd_isanpit stnd_aidipit mom_dad 92.76 <.0001 Label Parameter Estimates Parameter Standard DF Estimate Error t Value Pr > |t| Intercept Standardized isanpit: mean = 0 Standardized aidipit: mean = 0 1 1 1 1 1 167.30805 12.08258 0.35227 0.54150 0.00380 0.47983 0.81352 0.05958 0.07642 0.01150 348.68 14.85 5.91 7.09 0.33 <.0001 <.0001 <.0001 <.0001 0.7417 Test 1 Results for Dependent Variable pituus Mean Source DF Square F Value Pr > F 1 1.94829 0.11 0.7417 Numerator Denominator 119 17.85045 42 Plot the interaction Use the file interaktio_simple.xls Standard deviations are 6.676 for dad and 5.220 for mom (both means are 0) Mean value for Male is .346 unstd. mean 0,346 0 0 0 0 0 std.dev. low value high value regr.coeff. 167,308 0,478 -0,132 0,824 12,083 0 0 0 0 0 0 0 0 0 0 0 0 5,22 -5,22 5,22 0,5415 6,676 -6,676 6,676 0,3523 0,0038 178 176 174 172 predicted y independent variables Constant x1 x2 x3 x4 x5 mom z1 dad x5z1-interaction 170 x5 low 168 x5 high 166 164 162 160 z1 low level of z1 z2 high 43 Subgroup analysis for sex 44 Output: R square seems better for men and mom’s height more important for men Number of Observations Read Number of Observations Used Number of Observations with Missing Values Number of Observations Read 83 Number of Observations Used 83 Source Model Error Corrected Total Analysis of Variance Sum of Mean DF Squares Square F Value Pr > F 2 1152.63304 576.31652 80 1521.77660 19.02221 82 2674.40964 30.30 <.0001 4.36145 R-Square 0.4310 Root MSE 167.08434 Dependent Mean Adj R-Sq 0.4168 2.61033 Coeff Var Variable Intercept stnd_isanpit stnd_aidipit 167.30507 0.32938 0.45014 0.48058 0.07119 0.09590 348.13 4.63 4.69 Analysis of Variance Sum of Mean DF Squares Square F Value Pr > F 2 1004.19848 502.09924 38 525.70396 13.83431 40 1529.90244 36.29 <.0001 3.71945 R-Square 0.6564 Root MSE 179.95122 Dependent Mean Adj R-Sq 0.6383 2.06692 Coeff Var Parameter Estimates Parameter Standard DF Estimate Error t Value Pr > |t| 1 1 1 Source Model Error Corrected Total 44 41 3 <.0001 <.0001 <.0001 Variable Intercept stnd_isanpit stnd_aidipit Parameter Estimates Parameter Standard DF Estimate Error t Value Pr > |t| 1 1 1 179.23734 0.42680 0.73150 0.59037 0.10251 0.11831 303.60 4.16 6.18 <.0001 0.0002 <.0001 45 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 6063.02261 DFE 121 SSE 50.10762 Root MSE 7.07867 MSE 848.67819 AIC 840.217345 SBC 5.97064578 840.417345 MAE AICC 3.46608523 HQC 843.654339 MAPE 0.7169 Regress R-Square 0.3069 Durbin-Watson 0.3069 Total R-Square Test Chow Structural Change Test Break Point Num DF Den DF F Value Pr > F 83 3 Variable Intercept stnd_isanpit stnd_aidipit 118 77.80 <.0001 Parameter Estimates Standard Approx DF Estimate Error t Value Pr > |t| Variable Label 1 1 1 171.3387 0.3306 0.6825 0.6357 0.0993 0.1270 269.53 3.33 5.37 <.0001 0.0012 Standardized isanpit: mean = 0 <.0001 Standardized aidipit: mean = 0 46 Is the effect of mom different for men and women? d = bmen – bwomen Standard error for difference SEd= SQRT (SE 2 2 bmen + SE bwomen ) Test value z= d/ SEd then compare z to standard normal d= .73 - .45 = .28 SEd= sqrt (.1182 + .0962)= sqrt (.023)= .152 Z= 1.84 < 1.96 not significant at 5% level 47