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

X

M

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

X Y

XbbY 10ˆ

7

Simple regression analysis

XbbY 10ˆ

X

Yb1

b0

8

Multiple regression with additive predictor effects

X

M Y

MbXbbY 210ˆ

9

Y

XbMbbY

MbXbbY

120

210

)(ˆ

ˆ

Multiple regression with additive predictor effects

X

b1

b0

Low M

Medium M

High M

b2

intercept

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

X

M Y

XM

MXbMbXbbY 3210ˆ

11

X

Y

Multiple regression including interaction among predictors

Low M

Medium M

High M

XMbbMbbY

MXbMbXbbY

)()(ˆ

ˆ

3120

3210

intercept slope

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

MXbMbXbbY 3210ˆ

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! 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

MXbMbXbbY 3210ˆ

There are two equivalent ways to evaluate whether an interaction is present: 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 Life Sat

heightage

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

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

For more details, see Cohen, 1983; Maxwell & Delaney, 1993; West, Aiken, & Krull, 1996)

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 cMRegress 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.:

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

0ˆ

)()(ˆ

10

3120

MwhenXbbY

XMbbMbbY

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

MX008.M017.X034.016.5Y

Do not interpret betas as given by SPSS, they are wrong!

Test of significance of interaction

b0

b1

b2

b3

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

Effect of height on life satisfaction 1 SD below the mean of age (M)-1 SD of height:

+1 SD of height:

1 SD above the mean of age (M) -1 SD of height:

+1 SD of height:

Unstandardized Plot

Compute values for the plot either by hand…

2280.4))963.4()547.9(008.())963.4(017(.))547.9(034(.016.5Y

6352.5))963.4()547.9(008.())963.4(017(.))547.9(034(.016.5Y

1548.5))963.4()547.9(008.())963.4(017(.))547.9(034(.016.5Y

963.4)age(SD

547.9)height(SD

)MX008.()M017(.)X034(.016.5Y

0459.5))963.4()547.9(008.())963.4(017(.))547.9(034(.016.5Y

3.5

4

4.5

5

5.5

6

Low Height High Height

Lif

e S

atis

fact

ion

25

… or let Excel do the job!

Adapted from Dawson, 2006

26

3,5

4

4,5

5

5,5

6

Low Height High Height

Lif

e S

atis

fact

ion

Low Age

Mean Age

High Age

Interpreting the unstandardized plot: Effect of height moderated by age

Mean Height

Intercept; LS at mean of height and age (when both are centered)

Simple slope of height at mean age

163 173 183

Simple slope of age at mean height (difficult to illustrate)

Change in the slope of height for eachone-unit increase in age

b = .034

b = .034+(-.008*4.9625) = -.0057

Change in the slope of height for a 1 SDincrease in age

27

3,5

4

4,5

5

5,5

6

Low Age Mean Age High Age

Lif

e S

atis

fact

ion

Low Height

Mean Height

High Height

Interpreting the unstandardized plot: Effect of age moderated by height

Intercept; LS at mean of age and height (when centered)Simple slope of

age at mean height

Simple slope of height at mean age (difficult to illustrate)

Change in the slope of age for each one-unit increase in height

Change in the slope of age for a 1 SD increase in height

b = .017+(-.008*9.547) = -.059

b = .017

28

Estimating the proper standardized solution Standardized solution (to get the beta-

weights)Z-standardize X, M, and YCompute product of z-standardized scores for

X and MRegress zY on zX, zM, and zX*zMThe 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 z-scores we are interested in.

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.

XMMXY zzzz 321 MXMXY zzzzz 321

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

-0,5

0

0,5

1

Low Height Mean Height High Height

Lif

e S

atis

fact

ion

Low Age

Mean Age

High Age

Change in the beta of height for a 1 SDincrease in age

= .240

= .240+(-.270*1) = -.030

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

Add +1 (sic!) to M

Simple slope test for high (+1 SD) values of M Subtract -1 (sic!) from M

Now run separate regression analysis with each transformed score

0-1 SD +1 SD

0-1 SD +1 SD

0-1 SD +1 SD

original scale(centered)

Add 1 SD

Subtract 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

-0,5

0

0,5

1

Low Height Mean Height High Height

Lif

e S

atis

fact

ion

Low Age

Mean Age

High Age

= .509, p = .003

= -.030, p = .844

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

SPSSCOMPUTE 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

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

2.

2.

2.2

1 AIY

AYAIY

r

rrf

:

:

2.

2.

AY

AIY

r

r Squared multiple correlation resulting from combined prediction of Y by the additive set of predictors (A) and their interaction (I) (= full model)

Squared multiple correlation resulting from prediction by set A only (= model without interaction term)

43

Example

063.110.1

054.110.

1 2.

2.

2.2

AIY

AYAIY

r

rrf

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

Unstandardized solution Standardized solution

III) Estimating the solution using unweighted effects coding (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

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

1MwhenX)bb()bb(Y

0MwhenX)b()b(Y

X)Mbb()Mbb(Y

3120

10

3120

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 YCompute crossproduct of zX and MRegress zY on zX, M, and zX*MThe 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

MXMXY 073.199.005.966.4ˆ

MzXMzXYz 507.146.036.007.ˆ

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

Females (reference group; M=0) -1 SD: +1 SD:

Males (M=1) -1 SD: +1 SD:

Unstandardized Plot

547.9)(

073.199.005.966.4ˆ

heightSD

MXMXY

918.4)0547.9(073.)0(199.)547.9(005.966.4ˆ Y

022.4)1547.9(073.)1(199.)547.9(005.966.4ˆ Y

014.5)0547.9(073.)0(199.)547.9(005.966.4ˆ Y

512.5)1547.9(073.)1(199.)547.9(005.966.4ˆ Y

57

Excel spreadsheet

Adapted from Dawson, 2006

58

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

Low Height (-1 SD) High Height (+1 SD)

Lif

e S

atis

fact

ion

Women

Men

Interpreting the unstandardized plot

Mean Height

Intercept for reference group at mean of height (when height is centered)

Slope of height forreference group

163 173 183

Difference in intercept between reference and comparison groupat mean of height

Change in the slope when „going“ from reference group to other group

59

-1

-0.5

0

0.5

1

Low Height (-1 SD) High Height (+1 SD)

Z-S

tan

dar

diz

ed L

ife

Sat

isfa

ctio

n

Women

Men

Interpreting the standardized plot

Intercept for reference group at mean of height (when height is centered)

Slope of height forreference group

Difference in intercept between both groups at mean of height

Difference in the slope when „going“ from reference group to other group

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.

The effect of height on life satisfaction is significant for men, but not for women.

-1

-0.5

0

0.5

1

Low Height (-1 SD) High Height (+1 SD)

Z-S

tan

dar

diz

ed L

ife

Sat

isfa

ctio

n

Women

Men

= .544, p = .003

= .036, p = .807

63

III) Estimating the unstandardized solution using unweighted effects coding

Unstandardized solution Effect-code M (-1 = group A; 1 =group B) Center 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

-1.00

-0.50

0.00

0.50

1.00

Low Height High Height

Dep

end

ent

vari

able

Women

Mean Effect

Men

Interpreting the standardized plot

Unweighted grand mean of both groups at mean of height (when height is centered)

Unweighted mean slope across both groups

Difference in intercept between group coded 1 from the unweighted grand mean

Deviation of the slope for the group coded 1 from the unweighted mean slope

67

To sum up and compare

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)

Dummy coding Unweighted effects coding

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:

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

D1 D2 D1 D2 D1 D2Group 1 0 0 1 0 1 0Group 2 1 0 0 0 0 1Group 3 0 1 0 1 0 0

Group 1 as Base Group 2 as Base Group 3 as Base

70

V) What if there are more than 2 groups? Example: 3 groups using effects coding:

Interpretation: each coding var captures the difference between group coded 1 and unweighted grand mean

Statistical evaluation of overall interaction effect: R2 change

C1 C2 C1 C2 C1 C2Group 1 1 0 1 0 -1 -1Group 2 0 1 -1 -1 1 0Group 3 -1 -1 0 1 0 1

Option1 Option2 Option3

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:

2.

2.

2.2

1 AIY

AYAIY

r

rrf

:

:

2.

2.

AY

AIY

r

r Squared multiple correlation resulting from combined prediction of Y by the additive set of predictors (A) and their interaction (I) (= full model)

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 two-way

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

042.185.1

151.185.2

fsizeeffect

Three-way interaction:

p = .090

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

Females

-1

-0.5

0

0.5

1

Low Height High Height

Z-S

tan

dar

diz

ed L

ife

Sat

isfa

ctio

n

Low Age High Age

Males

-1

-0.5

0

0.5

1

Low Height High Height

Z-S

tan

dar

diz

ed L

ife

Sat

isfa

ctio

n

Low Age High Age

=.029+.346 =.375=.375 -.435

= -.06=.029

81

Simple slope testsThis 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%) Large effect (f2 = .26): N = 26 Medium effect (f2 = .13): N = 55 Small effect (f2 = .02): N = 392

Busemeyer & Jones (1983): reliability of product term of two uncorrelated variables is the product of the reliabilites of the two variables .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)

MXbMbXbb)(Logit 3210

85

Outlook 2: Moderated Mediation Analysis

X Y

M

Z

MN

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

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End of presentation

Thank you very much for your attention!

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Appendix

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Some don‘ts for Case II

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

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

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Females (reference group; M=0) -1 SD: +1 SD:

Males (M=1) -1 SD: +1 SD:

Dummy coding: Standardized Plot

043.0)01(507.)0(146.)1(036.007.ˆ Y

029.0)01(507.)0(146.)1(036.007.ˆ Y

696.0)11(507.)1(146.)1(036.007.ˆ Y

1)(

507.146.036.007.ˆ

heightSD

MzXMzXYz

39.0)11(507.)1(146.)1(036.007.ˆ Y

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Nonlinear interactions

Change in slopes is monotonic and linear Can also be modelled to be nonlinear

(e.g., curvilinear) See Aiken & West, chapter 5

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Taken from Preacher, K. J. (2007). Median splits and extreme groups

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Dummy coding

1MwhenX)bb()bb(Y

0MwhenX)b()b(Y

X)Mbb()Mbb(Y

3120

10

3120

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Unweighted effects coding

1MwhenX)bb()bb(Y

1MwhenX)bb()bb(Y

X)Mbb()Mbb(Y

3120

3120

3120