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Testing for mediating and moderating effects with SAS
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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)
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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
Mediation
MediationMathieu 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, examplesMathieu 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
a b
c’
e3
e2
1) Y = i1 + cX + e1
2) Y = i2 + c’X + bM + e2
3) M = i3 + aX + e3
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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…
4) c is significant
5) a is significant
6) b is significant
7) c’ is smaller than c
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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
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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 http://quantpsy.org/sobel/sobel.htm
2222 ab baab
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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
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Mediation, examplePierce 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**
3 Control Techn .44 6.63** .20
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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
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Step 1
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Step 1
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 1 7.28598 7.28598 8.53 0.0038Error 245 209.38989 0.85465Corrected Total 246 216.67587
Root MSE 0.92447 R-Square 0.0336
Dependent Mean 2.80379 Adj R-Sq 0.0297
Coeff Var 32.97234
Parameter Estimates
Variable Label DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept Intercept 1 2.29459 0.18405 12.47 <.0001
coll_index collaboration index
1 0.06623 0.02268 2.92 0.0038
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Step 2
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Step 2
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 2 15.12033 7.56016 9.13 0.0002Error 241 199.51603 0.82787Corrected Total 243 214.63636
Root MSE 0.90987 R-Square 0.0704
Dependent Mean 2.80829 Adj R-Sq 0.0627
Coeff Var 32.39954
Parameter Estimates
Variable Label DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept Intercept 1 1.44424 0.33969 4.25 <.0001coll_index collaboration index 1 0.04314 0.02515 1.72 0.0876ks_index knowledge sharing index 1 0.05013 0.01785 2.81 0.0054
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Step 3
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Step 3
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 1 624.99711 624.99711 57.42 <.0001Error 262 2851.66696 10.88423Corrected Total 263 3476.66406
Root MSE 3.29912 R-Square 0.1798
Dependent Mean 20.71926 Adj R-Sq 0.1766
Coeff Var 15.92299
Parameter Estimates
Variable Label DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept Intercept 1 16.12346 0.63957 25.21 <.0001coll_index collaboration index 1 0.59563 0.07860 7.58 <.0001
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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 errorInput 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)
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Indirect effect & Sobel test
http://quantpsy.org/sobel/sobel.htm
Moderation
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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)
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Moderation
Homologizer:Error term is function of z, R square is dependent on zIf the sample is split into subgroups according to values of z, we observe
different R squares in the subgroupsPure 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 moderatorA. 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 SEd= SQRT (SEB12 + SEB2
2)
If |d| > 1.96* SEd, it is significant at p<.05
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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
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Moderation
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Moderation
Summary, first run MRA1. If xz- interaction is significant
1. If the main effect of z is significant -> quasi2. If the main effect of z is not significant -> pure
2. If xz- interaction is not significant1. If the main effect of z is significant ->predictor2. 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 regression1. If the R square is different in the subgroups -> homologizer2. 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
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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
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SAS example on moderation
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SAS example on moderation
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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 0Sort the data according to the variable male
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Main effects
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Model diagnostics & SAS code
PROC REG DATA=tiltu12.recodedsorted_tapPLOTS(ONLY)=ALL ;
Linear_Regression_Model: MODEL pituus = male isanpit aidipit/SELECTION=NONE SCORR1 SCORR2 TOL SPEC ;
RUN;
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Output
Number of Observations Read 127Number of Observations Used 124Number of Observations with Missing Values 3
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 3 6621.62294 2207.20765 124.57 <.0001Error 120 2126.15126 17.71793Corrected Total 123 8747.77419
Root MSE 4.20927 R-Square 0.7569Dependent Mean 171.33871 Adj R-Sq 0.7509Coeff Var 2.45669
Parameter Estimates
Variable Label DF
Parameter
Estimate
Standard
Error t ValuePr > |
t|
SquaredSemi-
partialCorr Type I
SquaredSemi-partialCorr Type II Tolerance
Intercept Intercept 1 15.39805 14.02663 1.10 0.2745 . . .male 1 12.07190 0.80985 14.91 <.0001 0.51938 0.45004 0.98437isanpit isanpit 1 0.35037 0.05908 5.93 <.0001 0.13520 0.07124 0.92602aidipit aidipit 1 0.54126 0.07613 7.11 <.0001 0.10237 0.10237 0.91214
Test of First and Second Moment Specification
DF Chi-Square Pr > ChiSq8 6.66 0.5734
Significant model, high R square, homoskedastic, all parameters significant, no collinearity
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Centering the data for interaction analysis
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Build the interaction variable
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Main effects with centered data
Root MSE 4.20927 R-Square 0.7569
Dependent Mean 171.33871 Adj R-Sq 0.7509
Coeff Var 2.45669
Parameter Estimates
Variable Label DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept Intercept 1 167.34719 0.46324 361.26 <.0001male 1 12.07190 0.80985 14.91 <.0001stnd_isanpit Standardized isanpit: mean = 0 1 0.35037 0.05908 5.93 <.0001stnd_aidipit Standardized aidipit: mean = 0 1 0.54126 0.07613 7.11 <.0001
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Test the significance of interaction using SAS code
PROC REG DATA=TILTU12.INTER_STD_TAPPLOTS(ONLY)=ALL ;
MODEL pituus = male stnd_isanpit stnd_aidipit;MODEL pituus = male stnd_isanpit stnd_aidipit mom_dad;test mom_dad=0;
RUN;
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Output: no interaction
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 4 6623.57123 1655.89281 92.76 <.0001Error 119 2124.20296 17.85045Corrected Total 123 8747.77419
Root MSE 4.22498 R-Square 0.7572Dependent Mean 171.33871 Adj R-Sq 0.7490Coeff Var 2.46586
Parameter Estimates
Variable Label DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept Intercept 1 167.30805 0.47983 348.68 <.0001male 1 12.08258 0.81352 14.85 <.0001stnd_isanpit Standardized isanpit: mean = 0 1 0.35227 0.05958 5.91 <.0001stnd_aidipit Standardized aidipit: mean = 0 1 0.54150 0.07642 7.09 <.0001mom_dad 1 0.00380 0.01150 0.33 0.7417
Test 1 Results for Dependent Variable pituus
Source DFMean
Square F Value Pr > FNumerator 1 1.94829 0.11 0.7417Denominator 119 17.85045
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Plot the interactionUse 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.independent variables mean std.dev. low value high value regr.coeff.Constant 167,308x1 0,346 0,478 -0,132 0,824 12,083x2 0 0 0 0 0x3 0 0 0 0 0x4 0 0 0 0 0x5 mom 0 5,22 -5,22 5,22 0,5415z1 dad 0 6,676 -6,676 6,676 0,3523x5z1-interaction 0,0038
z1 low z2 high160
162
164
166
168
170
172
174
176
178
x5 low
x5 high
level of z1
pre
dic
ted
y
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Subgroup analysis for sex
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Output: R square seems better for men and mom’s height more important for men
Number of Observations Read 83Number of Observations Used 83
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 2 1152.63304 576.31652 30.30 <.0001Error 80 1521.77660 19.02221Corrected Total 82 2674.40964
Root MSE 4.36145 R-Square 0.4310Dependent Mean 167.08434 Adj R-Sq 0.4168Coeff Var 2.61033
Parameter Estimates
Variable DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept 1 167.30507 0.48058 348.13 <.0001stnd_isanpit 1 0.32938 0.07119 4.63 <.0001stnd_aidipit 1 0.45014 0.09590 4.69 <.0001
Number of Observations Read 44Number of Observations Used 41Number of Observations with Missing Values 3
Analysis of Variance
Source DFSum of
SquaresMean
Square F Value Pr > FModel 2 1004.19848 502.09924 36.29 <.0001Error 38 525.70396 13.83431Corrected Total 40 1529.90244
Root MSE 3.71945 R-Square 0.6564Dependent Mean 179.95122 Adj R-Sq 0.6383Coeff Var 2.06692
Parameter Estimates
Variable DFParameter
EstimateStandard
Error t Value Pr > |t|Intercept 1 179.23734 0.59037 303.60 <.0001stnd_isanpit 1 0.42680 0.10251 4.16 0.0002stnd_aidipit 1 0.73150 0.11831 6.18 <.0001
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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 EstimatesSSE 6063.02261 DFE 121
MSE 50.10762 Root MSE 7.07867
SBC 848.67819 AIC 840.217345
MAE 5.97064578 AICC 840.417345
MAPE 3.46608523 HQC 843.654339
Durbin-Watson 0.7169 Regress R-Square 0.3069
Total R-Square 0.3069
Structural Change Test
TestBreak Point Num DF Den DF F Value Pr > F
Chow 83 3 118 77.80 <.0001
Parameter Estimates
Variable DF EstimateStandard
Error t ValueApproxPr > |t| Variable Label
Intercept 1 171.3387 0.6357 269.53 <.0001
stnd_isanpit 1 0.3306 0.0993 3.33 0.0012 Standardized isanpit: mean = 0
stnd_aidipit 1 0.6825 0.1270 5.37 <.0001 Standardized aidipit: mean = 0
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Is the effect of mom different for men and women?
d = bmen – bwomen
Standard error for difference SEd= SQRT (SE bmen 2 + SE bwomen
2)
Test value z= d/ SEd then compare z to standard normald= .73 - .45 = .28
SEd= sqrt (.1182 + .0962)= sqrt (.023)= .152Z= 1.84 < 1.96 not significant at 5% level
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