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

kNR

kRF

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 Y

Moderated 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 causality

a 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

XHighLow

Z High

Z Medium

Z 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

XHighLow

Z = 10

Z = 0

b weight of X with interactionterm in model

Scale Invariance

Low

High

Z = 10

Y

XHighLow

Z = 0

Z = -10

b 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 Slopes

Y’ = 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

Model1

2

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)

positaff

negaff

(Constant)

positaff

negaff

paxna

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 lines

Y’ = 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 os it ive A ff ect

Low NA

High NA

Med NA

Interaction between PA and NA in predicting WFC

0

2

4

6

8

10

12

14

16

18

20

1 2 3

Positive Affect

Wo

rk-F

am

ily C

on

flic

t

Low NA

High NA

Med NA

Interaction Between Age and FIW

0

2

4

6

8

10

12

14

1 2 3 4

Family Interfering with Work

Ph

ysic

al S

ymp

tom

s

Age 37.4

Age 26.0

Age 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

Model1

2

R R SquareAdjustedR Square

Std. Error ofthe Estimate

R SquareChange F Change df1 df2 Sig. F Change

Change Statistics

Durbin-Watson

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 4.510

4146.392 499

1924.149 2 962.075 215.166 .000b

2222.243 497 4.471

4146.392 499

Regression

Residual

Total

Regression

Residual

Total

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 3.942

1.148 .137 .919 8.368 .000 .878

-3.53E-02 .015 -.254 -2.312 .021 -.065

(Constant)

Publications

(Constant)

Publications

Publictions Squared

Model1

2

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig. Lower Bound

95% Confidence Interval for B

Dependent Variable: Interviewsa.

0 2 4 6 8 10 12 14 16 18 205

6

7

8

9

10

11

12

13

14

Publications

Pre

dic

ted

Nu

mb

er

of I

nte

rvie

ws

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

Scatterplot

Dependent 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

Scatterplot

Dependent 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