Testing for mediating and moderating effects with SAS

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

http://davidakenny.net/cm/mediate.htm

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

http://quantpsy.org/sobel/sobel.htm

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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


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



Coeff Var 32.39954

Parameter Estimates


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


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



Coeff Var 15.92299

Parameter Estimates


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


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



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)

http://davidakenny.net/cm/moderation.htm

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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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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


Source DFSum of

SquaresMean


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



Coeff Var 2.45669

Parameter Estimates


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


Source DFSum of

SquaresMean



Parameter Estimates


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


Source DFSum of

SquaresMean



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


Source DFSum of

SquaresMean



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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Testing for mediating and moderating effects with SAS