Transcript Document

Workshop
Moderated
Regression Analysis
EASP summer school 2008, Cardiff
Wilhelm Hofmann
2
Overview of the workshop
Introduction to moderator effects
 Case 1: continuous  continuous variable
 Case 2: continuous  categorical variable
 Higher-order interactions
 Statistical Power
 Outlook 1: dichotomous DVs
 Outlook 2: moderated mediation analysis

3
Main resources





The Primer: Aiken & West (1991). Multiple regression:
Testing and interpreting interactions. Newbury Park, CA:
Sage.
Cohen, Aiken, & West (2004). Regression analysis for
the behavioral sciences, [Chapters 7 and 9]
West, Aiken, & Krull (1996). Experimental personality
designs: Analyzing categorical by continuous variable
interactions. Journal of Personality, 64, 1-48.
Whisman & McClelland (2005). Designing, testing, and
interpreting interactions and moderator effects in family
research. Journal of Family Psychology, 19, 111-120.
This presentation, dataset, syntaxes, and excel
sheets available at Summer School webpage!
4
What is a moderator effect?


Effect of a predictor variable (X) on a criterion
(Z) depends on a third variable (M), the
moderator
Synonymous term: interaction effect
M
X
Y
5
Examples from social psychology
Social facilitation: Effect of presence of
others on performance depends on the
dominance of responses (Zajonc, 1965)
 Effects of stress on health dependent on
social support (Cohen & Wills, 1985)
 Effect of provocation on aggression
depends on trait aggressiveness (Marshall
& Brown, 2006)

6
Simple regression analysis
Yˆ  b0  b1 X
X
Y
7
Simple regression analysis
Yˆ  b0  b1 X
Y
b1
b0
X
8
Multiple regression with additive
predictor effects
Yˆ  b0  b1 X  b2 M
X
M
Y
9
Multiple regression with additive
predictor effects
Yˆ  b0  b1 X  b2 M
Yˆ  (b  b M )  b X
0
2
1
intercept
High M

Medium M
Y
Low M
b1
b0
X
b2

The intercept of
regression of Y on X
depends upon the
specific value of M
Slope of regression of Y
on X (b1) stays constant
10
Multiple regression including
interaction among predictors
Yˆ  b0  b1 X  b2 M  b3 X  M
X
M
XM
Y
11
Multiple regression including
interaction among predictors
Yˆ  b0  b1 X  b2 M  b3 X  M
Yˆ  (b  b M )  (b  b M )  X
0
2
1
intercept
3
slope

Y
High M
Medium M

Low M
X
The slope and intercept
of regression of Y on X
depends upon the
specific value of M
Hence, there is a
different line for every
individual value of M
(simple regression line)
12
Regression model with interaction:
quick facts
Yˆ  b  b X  b M  b X  M
0


2
3
The interaction is carried by the XM term, the product of
X and M
The b3 coefficient reflects the interaction between X and
M only if the lower order terms b1X and b2M are included
in the equation!


1
Leaving out these terms confounds the additive and
multiplicative effects, producing misleading results
Each individual has a score on X and M. To form the
XM term, multiply together the individual‘s scores on X
and M.
13
Regression model with interaction
Yˆ  b  b X  b M  b X  M
0




2
3
There are two equivalent ways to evaluate whether an
interaction is present:


1
Test whether the increment in the squared multiple correlation (R2)
given by the interaction is significantly greater than zero
Test whether the coefficient b3 differs significantly from zero
Interactions work both with continuous and categorical
predictor variables. In the latter case, we have to agree on
a coding scheme (dummy vs. effects coding)
Workshop Case I: continous  continuous var interaction
Workshop Case II: continuous  categorical var interaction
14
Case 1: both predictors (and the
criterion) are continuous
X: height
M: age
Y: life satisfaction

Does the effect of
height on life
satisfaction depend
on age?
height
age
height
age
Life Sat
15
The Data
(available at the summer school homepage)
16
Descriptives
17
Advanced organizer for Case 1


I) Why median splits are not an option
II) Estimating, plotting, and interpreting the
interaction
 Unstandardized
solution
 Standardized solution


III) Inclusion of control variables
IV) Computation of effect size for interaction
term
18
I) Why we all despise median splits:
The costs of dichotomization
For more details, see Cohen, 1983; Maxwell & Delaney, 1993; West, Aiken, & Krull, 1996)

So why not simply split both X and M into two
groups each and conduct ordinary ANOVA to
test for interaction?
 Disadvantage
#1: Median splits are highly sample
dependent
 Disadvantage #2: drastically reduced power to detect
(interaction) effects by willfully throwing away useful
information
 Disadvantage #3: in moderated regression, median
splits can strongly bias results
19
II) Estimating the unstandardized
solution
Unstandardized = original metrics of
variables are preserved
 Recipe

 Center
both X and M around the respective
sample means
 Compute crossproduct of cX and cM
 Regress Y on cX, cM, and cX*cM
20
Why centering the continuous
predictors is important


Centering provides a meaningful zero-point for X and M (gives
you effects at the mean of X and M, respectively)
Having clearly interpretable zero-points is important because, in
moderated regression, we estimate conditional effects of one
variable when the other variable is fixed at 0, e.g.:
Yˆ  (b0  b2 M )  (b1  b3 M )  X
Yˆ  b0  b1  X




when
M 0
Thus, b1 is not a main effect, it is a conditional effect at M=0!
Same applies when viewing effect of M on Y as a function of X.
Centering predictors does not affect the interaction term, but all
of the other coefficients (b0, b1, b2) in the model
Other transformations may be useful in certain cases, but mean
centering is usually the best choice
21
SPSS Syntax
*unstandardized.
*center height and age (on grand mean) and compute interaction term.
DESC var=height age.
COMPUTE heightc = height - 173 .
COMPUTE agec = age - 29.8625.
COMPUTE heightc.agec = heightc*agec.
REGRESSION
/STATISTICS = R CHA COEFF
/DEPENDENT lifesat
/METHOD=ENTER heightc agec
/METHOD=ENTER heightc.agec.
22
SPSS output
b0
b1
b2
b3
ˆ  5.016 .034 X  .017 M  .008 X  M
Y
Do not interpret
betas as given by
SPSS, they are
wrong!
Test of significance
of interaction
23
Plotting the interaction



SPSS does not provide a straightforward module
for plotting interactions…
There is an infinite number of slopes we could
compute for different combinations of X and M
Minimum: We need to calculate values for high
(+1 SD) and low (-1 SD) X as a function of high
(+1 SD) and low (-1 SD) values on the
moderator M
24
Unstandardized Plot
6
5.5
Life Satisfaction
Compute values for the plot either by hand…
5
ˆ  5.016 (.034 X)  (.017 M)  (.008 X  M)
Y
SD (height)  9.547
SD (age)  4.963
4.5
4
3.5
Low Height
High Height
Effect of height on life satisfaction
 1 SD below the mean of age (M)
-1 SD of height:
+1 SD of height:

ˆ  5.016 (.034 (9.547))  (.017 (4.963))  (.008 (9.547)  (4.963))  4.2280
Y
ˆ  5.016 (.034 (9.547))  (.017 (4.963))  (.008 (9.547)  (4.963))  5.6352
Y
1 SD above the mean of age (M)
-1 SD of height:
+1 SD of height:
ˆ  5.016 (.034 (9.547))  (.017 (4.963))  (.008 (9.547)  (4.963))  5.1548
Y
ˆ  5.016 (.034 (9.547))  (.017 (4.963))  (.008 (9.547)  (4.963))  5.0459
Y
25
… or let Excel do the job!
Adapted from Dawson, 2006
26
Interpreting the unstandardized plot: Effect of height moderated by age
Intercept; LS at mean of
height and age (when both are centered)
6
Life Satisfaction
5,5
Simple slope of
height at mean
age
b = .034
Change in the slope of height for a
each
1 SD
one-unit increase
increase
in age in age
5
b = .034+(-.008*4.9625) = -.0057
Simple slope of age at mean height (difficult to
illustrate)
4,5
Low Age
4
Mean Age
High Age
3,5
163
Low Height
173
Mean
Height
183
High Height
27
Interpreting the unstandardized plot: Effect of age moderated by height
Simple slope of
age at mean height
Intercept; LS at mean of
age and height (when centered)
6
b = .017+(-.008*9.547) = -.059
height
Life Satisfaction
Change
in in
the
slope
ofof
age
Change
the
slope
age
5,5
forfor
a1
SD
increase
in
height
each one-unit increase in
5
b = .017
Simple slope of height at mean age (difficult to
illustrate)
4,5
Low Height
Mean Height
4
High Height
3,5
Low Age
Mean Age
High Age
28
Estimating the proper standardized
solution

Standardized solution (to get the betaweights)
 Z-standardize
X, M, and Y
 Compute product of z-standardized scores for
X and M
 Regress zY on zX, zM, and zX*zM
 The unstandardized solution from the output
is the correct solution (Friedrich, 1982)!
29
Why the standardized betas given
by SPSS are false


SPSS takes the z-score of the product (zXM)
when calculating the standardized scores.
Except in unusual circumstances, zXM is
different from zxzm, the product of the two zscores we are interested in.
zY  1 z X  2 zM  3 z XM


zY  1 z X  2 zM  3 z X zM
Solution (Friedrich, 1982): feed the predictors
on the right into an ordinary regression. The Bs
from the output will correspond to the correct
standardized coefficients.
30
SPSS Syntax
*standardized.
*let spss z-standardize height, age, and lifesat.
DESC var=height age lifesat/save.
*compute interaction term from z-standardized scores.
COMPUTE zheight.zage = zheight*zage.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight zage
/METHOD=ENTER zheight.zage.
31
SPSS output
Side note: What happens if we do not standardize Y?
→Then we get so-called half-standardized regression coefficients
(i.e., How does one SD on X/M affect Y in terms of original units?)

32
Standardized plot
1
Life Satisfaction
0,5
 = .240
Change in the beta of height for a 1 SD
increase in age
 = .240+(-.270*1) = -.030
0
Low Height
Mean Height
High Height
Low Age
-0,5
Mean Age
High Age
-1
33
Simple slope testing
Test of interaction term: Does the
relationship between X and Y reliably
depend upon M?
 Simple slope testing: Is the regression
weight for high (+1 SD) or low (-1 SD)
values on M significantly different from
zero?

34
Simple slope testing


Best done for the standardized solution
Simple slope testing for low (-1 SD) values of M


Simple slope test for high (+1 SD) values of M


Add +1 (sic!) to M
Subtract -1 (sic!) from M
Now run separate regression analysis with each
transformed score
Add 1 SD -1 SD
original scale
(centered)
0
-1 SD
+1 SD
0
-1 SD
Subtract 1 SD
+1 SD
0
+1 SD
35
SPSS Syntax
***simple slope testing in standardized solution.
*regression at -1 SD of M: add 1 to zage in order to shift new zero point one sd below the
mean.
compute zagebelow=zage+1.
compute zheight.zagebelow=zheight*zagebelow.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight zagebelow
/METHOD=ENTER zheight.zagebelow.
*regression at +1 SD of M: subtract 1 to zage in order to shift new zero point one sd
above the mean.
compute zageabove=zage-1.
compute zheight.zageabove=zheight*zageabove.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight zageabove
/METHOD=ENTER zheight.zageabove.
36
Simple slope testing: Results
37
Illustration
1
 = .509, p = .003
Life Satisfaction
0,5
 = -.030, p = .844
0
Low Height
Mean Height
High Height
Low Age
-0,5
Mean Age
High Age
-1
38
III) Inclusion of control variables




Often, you want to control for other variables
(covariates)
Simply add centered/z-standardized continuous
covariates as predictors to the regression
equation
In case of categorical control variables, effects
coding is recommended
Example: Depression, measured on 5-point
scale (1-5) with Beck Depression Inventory
(continuous)
39
SPSS
COMPUTE deprc =depr – 3.02.
REGRESSION
/DEPENDENT lifesat
/METHOD=ENTER heightc agec deprc
/METHOD=ENTER agec.heightc.
40
A note on centering the control variable(s)

If you do not center the control variable, the intercept will be affected
since you will be estimating the regression at the true zero-point
(instead of the mean) of the control variable.
Depression
centered
Depression
uncentered
(intercept estimated
at meaningless
value of 0 on the
depr. scale)
41
IV) Effect size calculation
Beta-weight () is already an effect size
statistic, though not perfect
 f2 (see Aiken & West, 1991, p. 157)

42
Calculating f2
rY2. AI  rY2. A
f 
1  rY2. AI
2
rY2. AI : Squared multiple correlation resulting from combined prediction of Y by the
additive set of predictors (A) and their interaction (I) (= full model)
rY2. A : Squared multiple correlation resulting from prediction by set A only (= model
without interaction term)


In words: f2 gives you the proportion of systematic variance
accounted for by the interaction relative to the unexplained variance
in the criterion
Conventions by Cohen (1988)
f2 = .02: small effect
 f2 = .15: medium effect
 f2 = .26: large effect

43
Example
2
2
r

r
.110 .054
2
Y . AI
Y.A
f 

 .063
2
1  rY . AI
1  .110
 small to medium effect
44
Case 2: continuous  categorical
variable interaction (on continous DV)

Ficticious example
 X: Body height (continuous)
 Y: Life satisfaction (continuous)
 M: Gender (categorical: male vs.

female)
Does effect of body height on life satisfaction
depend on gender?
Our hypothesis: body height is more important
for life satisfaction in males
45
Advanced organizer for Case 2


I) Coding issues
II) Estimating the solution using dummy coding



III) Estimating the solution using unweighted effects
coding





Unstandardized solution
Standardized solution
(Unstandardized solution)
Standardized solution
IV) What if there are more than two levels on categorical
scale?
V) Inclusion of control variables
VI) Effect size calculation
46
Descriptives
47
I) Coding options

Dummy coding (0;1):

Allows to compare the effects of X on Y between the reference group
(d=0) and the other group(s) (d=1)
 Definitely preferred, if you are interested in the specific regression
weights for each group

Unweighted effects coding (-1;+1): yields unweighted mean effect of
X on Y across groups

Preferred, if you are interested in overall mean effect (e.g., when
inserting M as a nonfocal variable); all groups are viewed in comparison
to the unweighted mean effect across groups
 Results are directly comparable with ANOVA results when you have 2 or
more categorical variables

Weighted effects coding: takes also into account sample size of
groups

Similar to unweighted effects coding except that the size of each group
is taken into consideration
 useful for representative panel analyses
48
II) Estimating the unstandardized
solution using dummy coding

Unstandardized solution
 Dummy-code
M (0=reference group; 1=comparison
group)
 Center X  cX
 Compute product of cX and M
 Regress Y on cX, M, and cX*M
49
SPSS Syntax
*Create dummy coding.
IF (gender=0) genderd = 0 .
IF (gender=1) genderd = 1 .
*center height (on grand mean) and compute interaction term.
DESC var=height.
COMPUTE heightc =height - 173 .
*Compute product term.
COMPUTE genderd.heightc = genderd*heightc.
*Regress lifesat on heightc and genderd, adding the interaction term.
REGRESSION
/DEPENDENT lifesat
/METHOD=ENTER heightc genderd
/METHOD=ENTER genderd.heightc.
50
SPSS output
ˆ  (b  b M)  (b  b M)  X
Y
0
2
1
3
ˆ  (b )  (b )  X when M  0
Y
0
1
ˆ  (b  b )  (b  b )  X when M  1
Y
0
2
1
3
b0
b1
b2
b3
51
Estimating the standardized
solution using dummy coding

Standardized solution
 Dummy-code
M (0=reference group;
1=comparison group)
 Z-standardize X and Y
 Compute crossproduct of zX and M
 Regress zY on zX, M, and zX*M
 The unstandardized solution from the output
is the correct solution (Friedrich, 1982)!
52
SPSS Syntax
*compute z-scores of all continuous varialbes involved and then
compute interaction term.
DESC var=lifesat height/save.
COMPUTE genderd.zheight = genderd*zheight.
EXECUTE .
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight genderd
/METHOD=ENTER genderd.zheight.
53
SPSS output standardized solution
.507 = estimated difference in regression weights between groups
54
Correct regression equations
Yˆ  4.966 .005 X  .199 M  .073 X  M
zYˆ  .007 .036 zX  .146 M  .507 zX  M
55
Plotting the interaction

Convention: calculate predicted values for
high (+1 SD) and low (-1 SD) values of X
in both groups of M
56
Unstandardized Plot
Yˆ  4.966 .005 X  .199 M  .073 X  M
SD(height)  9.547

Females (reference group; M=0)
 -1
SD:
 +1 SD:

Yˆ  4.966 .005 (9.547)  .199 (0)  .073 (9.547 0)  4.918
Yˆ  4.966 .005 (9.547)  .199 (0)  .073 (9.547 0)  5.014
Males (M=1)
SD: Yˆ  4.966 .005 (9.547)  .199 (1)  .073 (9.5471)  4.022
 +1 SD: Yˆ  4.966 .005 (9.547)  .199 (1)  .073 (9.5471)  5.512
 -1
57
Excel spreadsheet
Adapted from Dawson, 2006
58
Interpreting the unstandardized plot
Intercept for reference group
at mean of height (when height is centered)
5.6
5.4
Life Satisfaction
5.2
Slope of height for
reference group
Change in the slope when „going“
from reference group to other group
5
4.8
Difference in intercept
between reference and
comparison group
at mean of height
4.6
4.4
4.2
Women
4
163
173
183
3.8
Low Height (-1 SD)
Mean
Height
High Height (+1 SD)
Men
59
Interpreting the standardized plot
Intercept for reference group
at mean of height (when height is centered)
Z-Standardized Life Satisfaction
1
0.5
Slope of height for
reference group
Difference in the slope when „going“
from reference group to other group
0
Low Height (-1 SD)
-0.5
High Height (+1 SD)
Difference in intercept
between both groups
at mean of height
Women
-1
Men
60
Simple slope testing
Test of interaction term answers the
question: Are the two regression weights
in group A and B significantly different
from each other?
 Simple slope testing answers: Is the
regression weight in group A (or B)
significantly different from zero?

61
Simple slope testing


Use dummy coding
Simple slope test of the reference group
(women)
 Is
already given in SPSS output as the test of the
conditional effect for M!

Simple slope test of the comparison group (men)
 Easiest
way: recode M such that group B is now the
reference group (0). Then do regression analysis all
over again.
62
*Simple slopes comparison group
*(recode men=0; women=1).
IF (gender=0) genderd2 = 1.
IF (gender=1) genderd2 = 0.
COMPUTE genderd2.zheight = genderd2*zheight.
REGRESSION
/MISSING LISTWISE
/DEPENDENT zlifesat
/METHOD=ENTER zheight genderd2
/METHOD=ENTER genderd2.zheight.
Z-Standardized Life Satisfaction
1
 = .544, p = .003
0.5
 = .036, p = .807
0
Low Height (-1 SD)
High Height (+1 SD)
-0.5
Women
-1
The effect of height on life satisfaction is significant for
men, but not for women.
Men
63
III) Estimating the unstandardized solution
using unweighted effects coding

Unstandardized solution
 Effect-code
 Center
M (-1 = group A; 1 =group B)
X
 Compute crossproduct of centered Xc and M
 Regress Y on Xc, M, and Xc*M
 Interpret the unstandardized solution from the output
64
Estimating the standardized solution using
unweighted effects coding

Standardized solution (to get the beta-weights)
 Effect-code
M (-1 = group A; 1 =group B)
 Z-standardize X and Y
 Compute crossproduct of z-standardized scores for X
and M
 Regress zY on zX, M, and zX*M
 Again, the unstandardized solution from the output is
the correct (standardized) solution (Friedrich, 1982)!
65
SPSS Syntax
(standardized solution only)
IF (gender=0) gendere = -1.
IF (gender=1) gendere = 1.
COMPUTE gendere.zheight = gendere*zheight.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight gendere
/METHOD=ENTER gendere.zheight.
66
Interpreting the standardized plot
1.00
Dependent variable
Unweighted grand mean of both groups at
mean of height (when height is centered)
0.50
Unweighted mean
slope across both
groups
Deviation of the slope for the
group coded 1 from the
unweighted mean slope
0.00
Low Height
-0.50
High Height
Difference in intercept
between group coded 1
from the unweighted grand mean
Women
Mean Effect
Men
-1.00
67
To sum up and compare
Dummy coding




Unweighted effects coding
In dummy coding, the contrasts are with the reference group (0)
In unweighted effects coding, the contrasts are with the unweighted
mean of the sample
Regression weights for unweighted effects coding equal exactly half
of the weights for dummy coding.
Dummy/effects coding does not change the significance test of the
interaction (and the simple slope tests)
68
Further issues
V) What if there are more than 2 groups?
 VI) Adding control variables
 VII) Computing the effect size for the
interaction term

69
V) What if there are more than 2
groups?


Coding systems can be easily extended to N levels of categorical
variable
Example: 3 groups (dummy coding) give you 3 possibilities:
Group 1
Group 2
Group 3




Group 1 as Base Group 2 as Base Group 3 as Base
D1
D2
D1
D2
D1
D2
0
0
1
0
1
0
1
0
0
0
0
1
0
1
0
1
0
0
You need N-1 dummy variables
Include each dummy and its interaction with other predictor in
equation
Interpretation: each dummy captures difference between reference
group and group coded 1
Statistical evaluation of overall interaction effect: R2 change
70
V) What if there are more than 2
groups?

Example: 3 groups using effects coding:
Group 1
Group 2
Group 3


Option1
C1
1
0
-1
C2
0
1
-1
Option2
C1
1
-1
0
C2
0
-1
1
Option3
C1
-1
1
0
C2
-1
0
1
Interpretation: each coding var captures the difference between group
coded 1 and unweighted grand mean
Statistical evaluation of overall interaction effect: R2 change
71
VI) Adding control variables

Simply add centered covariates as
predictors to the unstandardized
regression equation
(or z-standardized covariates to the
standardized regression equation).
72
VII) Effect size calculation

Again, f2 should be used:
rY2. AI  rY2. A
f 
1  rY2. AI
2
rY2. AI : Squared multiple correlation resulting from combined prediction of Y by the
additive set of predictors (A) and their interaction (I) (= full model)
rY2. A : Squared multiple correlation resulting from prediction by set A only (= model
without interaction term)
73
Higher-order interactions
Higher-order interactions: interactions
among more than 2 variables
 All basic principles (centering, coding,
probing, simple slope testing, effect size)
generalize to higher-order interactions
(see Aiken & West, 1991, Chapter 4)

74
Example






Y: Life satisfaction (continuous)
X: Body height (continuous)
M1: Age (continuous)
M2: Gender (categorical: male vs. female)
Is the moderator effect of age and height
different in males and females?
Important: Include all lower-level (e.g., two-way)
interactions before inserting the higher-order
(e.g., three-way) term!
75
Syntax
*Standardized solution
*compute z-scores of all continuous varialbes involved and then compute twoway and three way interaction term(s).
*two-way.
COMPUTE genderd.zheight = genderd*zheight.
COMPUTE genderd.zage = genderd*zage.
COMPUTE zheight.zage = zheight*zage.
*three-way.
COMPUTE genderd.zheight.zage = genderd*zheight*zage.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight zage genderd
/METHOD=ENTER zheight.zage genderd.zheight genderd.zage
/METHOD=ENTER genderd.zheight.zage.
76
SPSS output
Three-way interaction:
p = .090
effect size f 2 
.185  .151
 .042
1  .185
77
SPSS output (cont‘d)
Slope of height in females at
mean of age
Change in slope of height for
males at mean of age
Difference in slope of height for
males at mean of age as
compared to males 1 SD above
the mean of age
78
Plotting the interaction






Plot first-level moderator effect (e.g., height  age) at
different levels of the third variable (e.g., gender)
It is best to use separate graphs for that
There are 6 different ways to plot the three-way
interaction…
Best presentation should be determined by theory
In the case of categorical vars it often makes sense to
plot the separate graphs as a function of group
The logic to compute the values for different
combinations of high and low values on predictors is the
same as in the two-way case
79
Excel sheet for three-way IA
Adapted from Dawson, 2006
80
Plotting the
three-way
interaction
Males
1
0.5
=.029
0
Low Height
High Height
-0.5
-1
Low Age
High Age
Z-Standardized Life Satisfaction
Z-Standardized Life Satisfaction
Females
1
=.029+.346 =.375
0.5
=.375 -.435
= -.06
0
Low Height
High Height
-0.5
-1
Low Age
High Age
81
Simple slope tests
This syntax estimates the beta of the steep slope of
height for males low in age (see previous slide):
*recode group membership.
IF (gender=0) genderd2 = 1 .
IF (gender=1) genderd2 = 0 .
*transform age.
COMPUTE zagebelow=zage+1.
*compute new product terms.
COMPUTE zheight.zagebelow=zheight*zagebelow.
COMPUTE genderd2.zheight = genderd2*zheight.
COMPUTE genderd2.zagebelow = genderd2*zagebelow.
COMPUTE zheight.zagebelow = zheight*zagebelow.
COMPUTE genderd2.zheight.zagebelow = genderd2*zheight*zagebelow.
REGRESSION
/DEPENDENT zlifesat
/METHOD=ENTER zheight zagebelow genderd2
/METHOD=ENTER zheight.zagebelow genderd2.zheight
genderd2.zagebelow
/METHOD=ENTER genderd2.zheight.zagebelow.
82
Output simple slope test
Slope of height in
males one SD below
the mean of age
83
The challenge of statistical power
when testing moderator effects

If variables were measured without error, the following
sample sizes are needed to detect small, medium, and
large interaction effects with adequate power (80%)




Busemeyer & Jones (1983): reliability of product term of
two uncorrelated variables is the product of the
reliabilites of the two variables



Large effect (f2 = .26):
N = 26
Medium effect (f2 = .13): N = 55
Small effect (f2 = .02):
N = 392
.80 x .80 = .64
Required sample size is more than doubled (trippled)
when predictor reliabilites drop from 1 to .80 (.70) (Aiken
& West, 1991)
Problem gets even worse for higher-order interactions
84
Outlook 1: Dichotomous DV


What if the DV is dichotomous (e.g., group
membership, voting decision etc.)?
Use moderated logistic regression (Jaccard,
2001)
Logit( )  b0  b1X  b2M  b3X  M
85
Outlook 2:
Moderated Mediation Analysis
M
X
M
N
Y
Z
86
Outlook 2: Moderated mediated
regression analysis

Preacher, K. J., Rucker, D. D., & Hayes, A. F.
(2007). Assessing moderated mediation hypotheses:
Theory, methods, and prescriptions. Multivariate
Behavioral Research, 42, 185-227.

Check out http://www.comm.ohio-
state.edu/ahayes/SPSS%20programs/modmed.
htm, for a copy of the paper and a convenient spss
macro that does all the computations
87
End of presentation

Thank you very much for your attention!
88
Appendix
89
Some don‘ts for Case II
Useful procedures to get a first feel for the data, but not appropriate
tests for interaction:
a) Testing the difference in subgroup correlations
- confound true moderator effects with difference in predictor variance
(Whisman & McClelland, 2005)
- does not control for possible interdependence among predictor and
moderator
- loss of power
b) Splitting the file and regressing Y on X separately by the two groups
- does not control for possible interdependence among predictor and
moderator
- does not test for difference in regression weights
Difference in regression weights: .428
90
Dummy coding: Standardized Plot
zYˆ  .007 .036 zX  .146 M  .507 zX  M
SD(height)  1

Females (reference group; M=0)
 -1
SD:
 +1 SD:

Yˆ  .007 .036 (1)  .146 (0)  .507 (1 0)  0.043
Yˆ  .007 .036 (1)  .146 (0)  .507 (1 0)  0.029
Males (M=1)
SD: Yˆ  .007 .036 (1)  .146 (1)  .507 (11)  0.696
 +1 SD: Yˆ  .007 .036 (1)  .146 (1)  .507 (11)  0.39
 -1
91
Nonlinear interactions
Change in slopes is monotonic and linear
 Can also be modelled to be nonlinear
(e.g., curvilinear)
 See Aiken & West, chapter 5

92
Taken from Preacher, K. J. (2007). Median splits and extreme groups
93
Dummy coding
ˆ  (b  b M)  (b  b M)  X
Y
0
2
1
3
ˆ  (b )  (b )  X when M  0
Y
0
1
ˆ  (b  b )  (b  b )  X when M  1
Y
0
2
1
3
94
Unweighted effects coding
ˆ  (b  b M)  (b  b M)  X
Y
0
2
1
3
ˆ  (b  b )  (b  b )  X when M  1
Y
0
2
1
3
ˆ  (b  b )  (b  b )  X when M  1
Y
0
2
1
3