regression analyses ii mediation & moderation
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Regression Analyses II
Mediation & Moderation
Review of RegressionMultiple IVs but single DVY’ = a+b1X1 + b2X2 + b3X3...bkXkWhere k is the number of predictorsFind solution where Sum(Y-Y’)2 minimized
Interactions & RegressionY’ = a + b1X + b2Z + b3XZ
Curvilinearity & RegressionY’ = a + b1X + b2X2 + b3X3
FR R k k
R N ky y
y
( ) / ( )( ) /
. .
.
122
22
1 2
122
11 1
1/)1(/
2
2
total
effect
kNRkR
F
Testing Significance of R2
With df = k1-k2 & N - k1 - 1
k2 is subset of k1
Significance of R2
Significance of Increment in R2
With df = keff and N - ktot - 1
• Researchers often confuse • Two completely different processes and
analytical approaches• Mediation - effect of an IV on DV occurs
through another variable• Moderation - effect of IV on DV depends
on the level of another variable
Mediation and Moderation
Mediation and Moderation
Indirect causal – Z is a mediator of X and Y
X YZ
X YModerated causal – Z is moderator of X and Y
Z
• Implies processes or mechanisms by which an IV influences a DV
• Often interpreted as “causal” mechanisms• Direct effects: Effect of IV on DV outside the
mediator (c)• Indirect effects: Effect of IV on the DV through the
mediator (a * b)• Total effects: Effect of IV on the DV through both
indirect and direct effects (c + (a*b))
Mediation
Mediation• IV related to mediator (X and M: path a)• Mediator related to DV (M and Y: path b)• IV related to DV (X and Y: path c)• Relationship between the IV and DV is
weakened or n.s. when mediator is controlled (X not related to Y when controlling for M: no path c when controlling for paths a and b)
b
c
a MX Y
(1) Regress mediator on IV (test for path a)IV must be related to mediatorbeta = direct effect of X on M
(2) Regress DV on IV (test for path c)IV must relate to DVbeta = total effect of X on Y
(3) Regress DV on both IV and mediatorMediator must affect DV after controlling for IVFull mediation if effect of IV disappears (beta n.s.)Partial mediation if effect of X remains but beta is reduced but significant
Steps to Test for Mediationb
c
a MX Y
Example of Mediationb
c
aPositive Affect
Job SatWorkSE
Regression #1. PA predicts WSEBeta PA = .279*
Regression #2. PA predicts JSBeta PA = .463*
Regression #3. PA and WSE predict JSBeta PA = .345* (compare to beta from regression #2)Beta WSE = .372*
Is there evidence of mediation?
Mediation: Order of Causality– With three variable systems, difficult to
determine proper causal order– Use issues of timing, logic, and theory to help
determine causal order– If data collected at single point in time, not a
test of causalitya bX M Y
a bM X Y
Moderation• A test of moderation is a test of interaction• Multiplicative effects of IVs on a DV
Low
High
Y
X HighLow
Z High
Z MediumZ Low
Moderated & Curvilinear Effects• Enter main effects first
– Significance test for increment in R2
– Interpretation of must occur in this step– Can enter main effects all at once or one at a time
• Enter curvilinear or interaction terms second– Significance test for increment in R2 – Interpretation of must occur at point when interactions
entered– Can enter in any order, but lower order must precede higher
order interactions– Two-way interactions must be entered before testing three-
way interactions– Curvilinear effects and interaction effects may be confounded
when IVs are intercorrelated
Testing Moderation(1) Create cross-product of two IVs; Compute XM = X * M(2) Partial main effects first; Interpreted at Step 1
• When interaction term not included, b weights for main effects indicate “general effects”• When interaction term included, b weights for main effects indicate effect of one variable on Y when the other is zero
Slopes
(1) Y’ = a + b1X + b2M + b3XM
Rewritten: (2) Y’ = a + b1X + b3XM + b2M
Rewritten:(3) Y’ = a + (b1 + b3M)X + b2M
*You can clearly see that the value for b1 is the value when M = 0 (no moderation) so that b3M cancels out.
Scale Invariance
Low
High
Z = 20
Y
X HighLow
Z = 10
Z = 0
b weight of X with interactionterm in model
Scale Invariance
Low
High
Z = 10
Y
X HighLow
Z = 0
Z = -10b weight of X with interactionterm in model
Now subtract 10 points fromall Z scores
Lack of Scale Invariance• Why main effects are not scale invariant(1) Y’ = a + b1X + b2M + b3XM• Now, let’s subtract a constant c from X and a
constant f from M and rewrite equation 1:(2a) Y’ = a + b1(X - c) + b2(M - f) + b3(X - c)(M - f)
Solving:(2b) Y’ = (a - b1c - b2f + b3cf) + (b1 - b3f)X + (b2 - b3c)M + b3XM
Simple SlopesY’ = a + b1X + b2M + b3XM
Rewritten: Y’ = a + b1X + b3XM + b2M
Rewritten: Y’ = a + (b1 + b3M)X + b2M
You can now compute a slope for X at a given value of M. This is known as a simple slope. If you choose meaningful points for M, then you can interpret the simple slopes. That’s what the graph does for you visually.
Moderation - Interpretation
• Interpretation of interactions– Simple slopes– Plotting
• Plotting interactions– Continuous (pick -1 SD, mean, +1 SD)– Categorical (codes representing group)
Moderation - Interpretation
Low
High
Y
X HighLow
Z HighZ MediumZ Low
Group 1
Low
High
X HighLow
Group 2Y
Moderation - Issues
• Predictors and interaction terms will be highly correlated unless centered– The high correlation does not create problems
with collinearity or interpretation (unless extremely high) b/c partial main effects first and findings are scale invariant
– But if you did not partial main effects first, it would screw up the regression weights & they would not be interpretable
• Measurement error influences detection and interpretation of moderating effects– Low reliability has complex influences on tests of
moderation• Testing for moderation often has low power
(unreliability, error heterogeneity, etc.)• This is particularly true in field research
– Small effect size (1% to 3% of variance)– Requires “X” pattern of the moderating IVs– Testing for moderation and curvilinearity together
requires “filled” pattern of the moderating IVs
Moderation - Issues
Do positive and negative affect interact in predicting work-family conflict?REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA CHA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT wfc /METHOD=ENTER positaff negaff
/METHOD=ENTER paxna .
Model Summary
.328a .107 .094 3.33723 .107 8.055 2 134 .000
.386b .149 .130 3.27088 .042 6.491 1 133 .012
Model12
R R SquareAdjustedR Square
Std. Error ofthe Estimate
R SquareChange F Change df1 df2 Sig. F Change
Change Statistics
Predictors: (Constant), negaff, positaffa.
Predictors: (Constant), negaff, positaff, paxnab.
Coefficientsa
5.857 1.769 3.310 .001.038 .043 .073 .878 .382.182 .045 .335 4.014 .000
13.396 3.430 3.906 .000-.197 .101 -.382 -1.944 .054-.256 .178 -.471 -1.442 .152.014 .005 .864 2.548 .012
(Constant)positaffnegaff(Constant)positaffnegaffpaxna
Model1
2
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: wifa.
Following up on the interactionStep 1. Regression Equation from Final StepY’ = 13.396 + (-.197*PA) + (-.256*NA) + (.014*NA*PA)Step 2. Moderator take mean, +1SD, -1SD
PA Mean = 33.06PA +1SD = 39.82PA -1SD = 26.30
Step 3. Insert points from last step to create 3 regression linesMean - 13.396+(-.197*33.06)+(-.256*NA)+(.014* NA *33.06)+1SD - 13.396+(-.197*39.82)+ (-.256*NA)+(.014* NA *39.82)-1SD - 13.396+(-.197*26.30)+(-.256*NA)+(.014* NA * 26.30)Reduces:Y’ = 6.88 + (.207*NA)Y’ = 5.55 + (.557*NA)Y’ = 8.22 + (.368*NA)
• Plot the following linesY’ = 6.88 + (.207*NA) [Mean]Y’ = 5.55 + (.557*NA) [+1]Y’ = 8.22 + (.368*NA) [-1]
• Useful to choose several points on line• Interpret
Following up on the interaction
I nter ac ti on between P A and NA i n pr edi c ti ng WFC
0
2
4
6
8
10
12
14
16
18
20
1 2 3
P osit ive A ff ect
Low NA
High NA
Med NA
Interaction between PA and NA in predicting WFC
0
2
4
6
8
10
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14
16
18
20
1 2 3
Positive Affect
Wor
k-Fa
mily
Con
flict
Low NAHigh NAMed NA
Interaction Between Age and FIW
0
2
4
6
8
10
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14
1 2 3 4
Family Interfering with Work
Phys
ical
Sym
ptom
s
Age 37.4Age 26.0Age 48.4
• Presence of interactions qualifies the interpretation of main effects• Presence of higher order interactions qualifies the interpretation of lower order interactions• df are used up quickly as more potential interactions are added• Interpreting interactions with more than 3 variables is very difficult
Interaction Effects
Moderation/Mediation Models
• Testing mediation & moderation models together
• If Z is a categorical variable, can test model using multiple groups analysis in SEM
• If Z is continuous, can test model using special SEM models, but very difficult
X M Y
Z
Moderation/Mediation Models
(1) Enter X, enter Z, enter XZ, enter M, enter MZ– If increment R2 for neither XZ nor MZ significant, no evidence for
moderation– If MZ significant, suggests moderated mediation (MM)– If MZ not significant after controlling for XZ, but XZ is signif, then
suggests XZ has direct moderating effect (not mediated through MZ)
(2) Enter M, enter Z, enter MZ, enter X, enter XZ– If MZ significant, and XZ was signif in step 1 but no longer signif
here, suggests MM– If MZ not signif, no evidence for MM
X M Y
Z
Curvilinearity
• Curvilinearity can be considered moderation of a variable by itself
• Tested the same way as moderation• Most of the same issues regarding
moderation apply to curvilinearity
Low
High
Y
X HighLow
X High
X Medium
X Low
• Nonlinear relationships.
• The quadratic effect of Publications: Publications2
• This new variable would be tested after original variable, Publications, had been entered.
• Publications2 is just the product of Publications with itself. It looks like any other product used to test an interaction. How can this variable be interpreted as an interaction?
Non-linear regression
Model Summaryc
.677a .458 .457 2.124 .458 421.311 1 498 .000
.681b .464 .462 2.115 .006 5.345 1 497 .021 2.006
Model12
R R SquareAdjustedR Square
Std. Error ofthe Estimate
R SquareChange F Change df1 df2 Sig. F Change
Change StatisticsDurbin-W
atson
Predictors: (Constant), Publicationsa.
Predictors: (Constant), Publications, Publictions Squaredb.
Dependent Variable: Interviewsc.
What would a significant quadratic effect of publications mean in addition to a significant linear effect?
ANOVAc
1900.250 1 1900.250 421.311 .000a
2246.142 498 4.5104146.392 4991924.149 2 962.075 215.166 .000b
2222.243 497 4.4714146.392 499
RegressionResidualTotalRegressionResidualTotal
Model1
2
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), Publicationsa.
Predictors: (Constant), Publications, Publictions Squaredb.
Dependent Variable: Interviewsc.
Coefficients
4.959 .201 24.672 .000 4.565.845 .041 .677 20.526 .000 .764
4.497 .283 15.902 .000 3.9421.148 .137 .919 8.368 .000 .878
-3.53E-02 .015 -.254 -2.312 .021 -.065
(Constant)Publications(Constant)PublicationsPublictions Squared
Model1
2
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig. Lower Bound95% Confidence Interval for B
Dependent Variable: Interviewsa.
0 2 4 6 8 10 12 14 16 18 205
6
7
8
9
10
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12
13
14
Publications
Pre
dict
ed N
umbe
r of I
nter
view
s
A quadratic effect indicates that the linear relation between a variable and the outcome changes slope across levels of the variable.
REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF R CHA ANOVA COLLIN TOL /NOORIGIN /DEPENDENT y /METHOD=ENTER x /RESIDUALS /CASEWISE ALL ZRESID SRESID LEVER COOK /SCATTERPLOT=(*RESID ,y) (*RESID, x) (*ZRESID,*ZPRED )
Syntax to Examine Residuals
ScatterplotDependent Variable: Y
X
43210-1-2-3-4
Reg
ress
ion
Res
idua
l
100
80
60
40
20
0
-20
-40-60
Output from Syntax
Syntax Polynomial Regression
REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF R CHA ANOVA COLLIN TOL /NOORIGIN /DEPENDENT y /ENTER x /ENTER x2 /RESIDUALS /CASEWISE ALL ZRESID SRESID LEVER COOK /SCATTERPLOT=(*RESID ,y) (*RESID, x) (*ZRESID,*ZPRED )Block Number 2. Method: Enter X2
Variable(s) Entered on Step Number 1.. XMultiple R .11435R Square .01308 R Square Change .01308Adjusted R Square -.00748 F Change .63597Standard Error 40.06215 Signif F Change .4291
Block Number 2. Method: Enter X2Variable(s) Entered on Step Number 2.. X2Multiple R .98170R Square .96374 R Square Change .95066Adjusted R Square .96220 F Change 1232.26856Standard Error 7.76022 Signif F Change .0000
ScatterplotDependent Variable: Y
X
43210-1-2-3-4
Reg
ress
ion
Res
idua
l
20
10
0
-10
-20
Residuals with X2 in the equation
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