Workshop Moderated Regression Analysis

1
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

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
7
Simple regression analysis
Y
b1
b0
X
8
Multiple regression with additive predictor
effects
X
M
Y
9
Multiple regression with additive predictor
effects
intercept
High M

• 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

Medium M
Y
b2
10
Multiple regression including interaction among
predictors
X
M
Y
X?M
11
Multiple regression including interaction among
predictors
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)

Y
High M
Medium M
Low M
X
12
Regression model with interaction quick facts

• 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 individuals
  scores on X and M.

13
Regression model with interaction

• 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
height?age
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 cXcM

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 M0!
• 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 varheight age.
• COMPUTE heightc height - 173 .
• COMPUTE agec age - 29.8625.
• COMPUTE heightc.agec heightcagec.
• REGRESSION
• /STATISTICS R CHA COEFF
• /DEPENDENT lifesat
• /METHODENTER heightc agec
• /METHODENTER heightc.agec.

22
SPSS output
Do not interpret betas as given by SPSS, they are
wrong!
b0 b1 b2 b3
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 PlotCompute values for the plot
either by hand

• 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

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)
Simple slope of height at mean age
b .034
Change in the slope of height for eachone-unit
increase in age
Change in the slope of height for a 1 SDincrease
in age
b .034(-.0084.9625) -.0057
Simple slope of age at mean height (difficult to
illustrate)
163
173
183
Mean Height
27
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
b .017(-.0089.547) -.059
Change in the slope of age for a 1 SD increase in
height
Change in the slope of age for each one-unit
increase in height
b .017
Simple slope of height at mean age (difficult to
illustrate)
28
Estimating the proper standardized solution

• Standardized solution (to get the beta-weights)
• Z-standardize X, M, and Y
• Compute product of z-standardized scores for X
  and M
• Regress zY on zX, zM, and zXzM
• 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 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.

?
30
SPSS Syntax

• standardized.
• let spss z-standardize height, age, and lifesat.
• DESC varheight age lifesat/save.
• compute interaction term from z-standardized
  scores.
• COMPUTE zheight.zage zheightzage.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight zage
• /METHODENTER 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
? .240
Change in the beta of height for a 1 SDincrease
in age
? .240(-.2701) -.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

Add 1 SD
original scale(centered)
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 zagebelowzage1.
• compute zheight.zagebelowzheightzagebelow.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight zagebelow
• /METHODENTER 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 zageabovezage-1.
• compute zheight.zageabovezheightzageabove.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight zageabove
• /METHODENTER zheight.zageabove.

36
Simple slope testing Results
37
Illustration
? .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
SPSS

• COMPUTE deprc depr 3.02.
• REGRESSION
• /DEPENDENT lifesat
• /METHODENTER heightc agec deprc
• /METHODENTER 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
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)

• 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
? 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 (01)
• Allows to compare the effects of X on Y between
  the reference group (d0) and the other group(s)
  (d1)
• Definitely preferred, if you are interested in
  the specific regression weights for each group
• Unweighted effects coding (-11) 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 (01)
• Allows to compare the effects of X on Y between
  the reference group (d0) and the other group(s)
  (d1)
• Definitely preferred, if you are interested in
  the specific regression weights for each group
• Unweighted effects coding (-11) 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 (0reference group 1comparison
  group)
• Center X ? cX
• Compute product of cX and M
• Regress Y on cX, M, and cXM

49
SPSS Syntax

• Create dummy coding.
• IF (gender0) genderd 0 .
• IF (gender1) genderd 1 .
• center height (on grand mean) and compute
  interaction term.
• DESC varheight.
• COMPUTE heightc height - 173 .
• Compute product term.
• COMPUTE genderd.heightc genderdheightc.
• Regress lifesat on heightc and genderd, adding
  the interaction term.
• REGRESSION
• /DEPENDENT lifesat
• /METHODENTER heightc genderd
• /METHODENTER genderd.heightc.

50
SPSS output
b0 b1 b2 b3
51
Estimating the standardized solution using dummy
coding

• Standardized solution
• Dummy-code M (0reference group 1comparison
  group)
• Z-standardize X and Y
• Compute crossproduct of zX and M
• Regress zY on zX, M, and zXM
• 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 varlifesat height/save.
• COMPUTE genderd.zheight genderdzheight.
• EXECUTE .
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight genderd
• /METHODENTER genderd.zheight.

53
SPSS output standardized solution
.507 estimated difference in regression weights
between groups
54
Correct regression equations
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

• Females (reference group M0)
• -1 SD
• 1 SD
• Males (M1)
• -1 SD
• 1 SD

57
Excel spreadsheet
Adapted from Dawson, 2006
58
Interpreting the unstandardized plot
Intercept for reference group at mean of height
(when height is centered)
Slope of height forreference group
Change in the slope when going from reference
group to other group
Difference in intercept between reference and
comparison groupat mean of height
163
173
183
Mean Height
59
Interpreting the standardized plot
Intercept for reference group at mean of height
(when height is centered)
Difference in the slope when going from
reference group to other group
Slope of height forreference group
Difference in intercept between both groups at
mean of height
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 men0 women1).
• IF (gender0) genderd2 1.
• IF (gender1) genderd2 0.
• COMPUTE genderd2.zheight genderd2zheight.
• REGRESSION
• /MISSING LISTWISE
• /DEPENDENT zlifesat
• /METHODENTER zheight genderd2
• /METHODENTER genderd2.zheight.

? .544, p .003
? .036, p .807

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

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 XcM
• 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 zXM
• Again, the unstandardized solution from the
  output is the correct (standardized) solution
  (Friedrich, 1982)!

65
SPSS Syntax (standardized solution only)

• IF (gender0) gendere -1.
• IF (gender1) gendere 1.
• COMPUTE gendere.zheight genderezheight.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight gendere
• /METHODENTER gendere.zheight.

66
Interpreting the standardized plot
Unweighted grand mean of both groups at mean of
height (when height is centered)
Unweighted mean slope across both groups
Deviation of the slope for the group coded 1
from the unweighted mean slope
Difference in intercept between group coded 1
from the unweighted grand mean
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
• 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
• 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

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 genderdzheight.
• COMPUTE genderd.zage genderdzage.
• COMPUTE zheight.zage zheightzage.
• three-way.
• COMPUTE genderd.zheight.zage genderdzheightzag
  e.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight zage genderd
• /METHODENTER zheight.zage genderd.zheight
  genderd.zage
• /METHODENTER genderd.zheight.zage.

76
SPSS output
Three-way interaction p .090
77
SPSS output (contd)
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
?.029.346 .375
?.375 -.435 -.06
?.029
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 (gender0) genderd2 1 .
• IF (gender1) genderd2 0 .
• transform age.
• COMPUTE zagebelowzage1.
• compute new product terms.
• COMPUTE zheight.zagebelowzheightzagebelow.
• COMPUTE genderd2.zheight genderd2zheight.
• COMPUTE genderd2.zagebelow genderd2zagebelow.
• COMPUTE zheight.zagebelow zheightzagebelow.
• COMPUTE genderd2.zheight.zagebelow
  genderd2zheightzagebelow.
• REGRESSION
• /DEPENDENT zlifesat
• /METHODENTER zheight zagebelow genderd2

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)

85
Outlook 2 Moderated Mediation Analysis
X
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/SP
  SS20programs/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 donts 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

• Females (reference group M0)
• -1 SD
• 1 SD
• Males (M1)
• -1 SD
• 1 SD

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
94
Unweighted effects coding