Introduction to moderator effects
Hierarchical Regression analysis with continuous moderator
Hierarchical Regression analysis with categorical moderator
Effect of a predictor variable (X) on a criterion (Y) depends on a third variable (M) - the moderator
Synonymous term: interaction effect
X
M
Y
A significant interaction between the moderator and the IV means that the effect of the IV on the DV changes depending on the level of the moderator
In multiple regression, the simple slope of the IV on the DV changes depending on the level of the moderator, and with continuous moderators we generally compare “high” levels of the moderator (+1 standard deviation above the mean) to “low” levels (-1 SD below the mean)
X
M Y
XM
X
Y
XMbbMbbY
MXbMbXbbY
)()(ˆ
ˆ
3120
3210
Low M
Medium M
High M
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)
Does the effect of criminal identity on criminal thinking depend on extraversion?
Unstandardized = original metrics of variables are preserved
Procedure
Center both X and M around the respective sample means
Compute crossproduct of cX and cM (create interaction terms)
Regress Y on cX, cM, and cX*cM (hierarchical multiple regression with interaction terms)
Centering
• Find the mean of the variable you want to center
• Go to “Transform” then “Compute”
• In the box that says “target
variable” rename the variable you
want to center
Subtract the mean from
this variable, so if the mean
for Extraversion is 4.22 you
would have an expression
that looks like this: Extra –
4.22
Click OK
You should see a new
variable “CentExtraversion”
in your dataset
Do the same with Central,
Affect, and Ties
Go to Transform then Compute In the box that says target variable create a
name for your interaction term
In the box that says Target Variable create a name for your interaction term (e.g., CxE)
Place the variables that you want to create an interaction term. Multiply your variables together.
For example, if you want to create an interaction between centrality*extraversion
Click OK, and you should see a new variable term representing your interaction (CxE)
Do the same with other predictors
So why not simply split both predictor and moderator 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
The purpose of centering is to reduce the correlations between the interaction terms and predictors, so that the effect of predictors are distinguishable from the interactions
Centering provides a meaningful zero-point for X (predictor) and M (moderator) - 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
Centering predictors does not affect the interaction term, but all of the other coefficients in the model
From the menu at the top of the screen click Analyze, then select Regression, then Linear
Choose you continuous DV (Criminal Thinking) and move it into the Dependent box
In Bock 1 of your regression, place your main effects (your independent variables that made up your interaction) in the box that says Independent(s)
Click Next
Block 2 –place your interaction terms
Click on the Statistics button. Select the following:
Estimates
Covariance Matrix
Model fit
R square change
Descriptives
Part and Partial correlations
Collinearity diagnostics
Click Continue
Click on the Options button.
In the Missing Values section, click on Exclude cases pairwise.
Click on Continue.
And OK
Coefficients table gives Tolerance and Variance Inflation Factor (VIF)
Tolerance value less than.10 – possible multicollinearity
VIF value above 10 – possible multicollinearity
If you exceed these values, you should remove one of the IVs
Step 2: Evaluating the model
Check the R Square in the Model Summary box. Variables entered in Block 1 explained 29% of the variance (.29 x 100) in DV.
After Block 2 interaction terms have been included, the model as a whole explained 35% of variance in DV.
In the column labelled R Square Change (on the line marked Model 2) – Criminal Identity explained additional 6 % of the variance in DV.
This is significant contribution, as indicated by Sig. F Change value for this line (.000)
The ANOVA table indicates that the model as a whole (which includes both blocks of variables) is significant
F (7, 295) = 22.79, p < .0005
Do not interpret
betas as given by
SPSS, they are
wrong!
Test of significance
of interaction
Change in the
slope of in-group
affect for each
one-unit increase
in extraversion
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
Use ModGraph - download from http://www.victoria.ac.nz/psyc/paul-jose-
files/modgraph/download.php
Click on Continuous
Data Entry
Input information is taken from
the regression analysis output.
1. enter the unstandardized
regression coefficient (B)
2. the mean (should be 0.00)
and the standard deviation
3. B for the interaction term and
the constant
Click Calculate
And See Chart
In interpreting the meaning of a figure, it is often important to know the values of the simple slopes, and to know whether these slopes differ significantly from zero
So, after the figure has been generated, go to the “Continuous Slopes Computations”
This page brings forward relevant information already entered in the data entry page And asks for additional information to be supplied After these critical items are entered, simply click on “Calculate” and simple slopes, standard errors, the degrees of freedom, t-values, and associated p-values are displayed. Results
Standardized solution (to get the beta-weights) Z-standardize X (predictor), M (moderator), and Y
(criterion variable)
Compute product of z-standardized scores for X and M (create interaction terms)
Regress zY on zX, zM, and zX*zM (hierachical moderated regression)
The unstandardized solution from the output is the correct solution (Friedrich, 1982)!
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
Click on Continuous
Data Entry
Input information is taken from
the regression analysis output.
1. enter the unstandardized
regression coefficient (B)
2. the mean (should be 0.00)
and the standard deviation
3. B for the interaction term and
the constant
Click Calculate
And See Chart
Change in the beta of affect for a 1
SD increase in extraversion
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?
In interpreting the meaning of a figure, it is often important to know the values of the simple slopes, and to know whether these slopes differ significantly from zero
So, after the figure has been generated, go to the “Continuous Slopes Computations”
This page brings forward relevant information already entered in the data entry page And asks for additional information to be supplied After these critical items are entered, simply click on “Calculate” and simple slopes, standard errors, the degrees of freedom, t-values, and associated p-values are displayed. Results
= -.03, p > .05
= .248, p < .05
= .526, p < .05
Beta-weight () is already an effect size statistic, though not perfect f2 (see Aiken & West, 1991, p. 157)
:
:
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)
2
.
2
.
2
.2
1 AIY
AYAIY
r
rrf
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
.
AY
AIY
r
r .35
.29 .35 - .29 / 1 - .35 = .06 / .65 = .09
Variables X: Criminal Friend Index (continuous) Y: Recidivism (continuous) Moderator: Location (categorical: Urban vs. Rural – scored
1/0) Does effect of criminal friends on recidivistic
behaviour depend on location?
Our hypothesis: Associations with criminal friends is more important for development of recidivistic behaviour in urban areas.
Unstandardized solution
Dummy-code moderator (0=reference group; 1=comparison group)
Center predictor X cX
Compute product of cX and M (interaction term)
Regress Y on cX, M, and cX*M
From the menu at the top of the screen click Analyze, then select Regression, then Linear
Choose you continuous DV (Level of Recidivism) and move it into the Dependent box
In Block 1 of your regression, place your main effects (your independent variables that made up your interaction) in the box that says Independent(s)
Click Next
Block 2 –place your interaction term
Click on the Statistics button. Select the following:
Estimates
Covariance Matrix
Model fit
R square change
Descriptives
Part and Partial correlations
Collinearity diagnostics
Click Continue
Click on the Options button.
In the Missing Values section, click on Exclude cases pairwise.
Click on Continue.
And OK
Use ModGraph
Click on Categorical
Data Entry
Input information is taken from
the regression analysis output.
1. enter the unstandardized
regression coefficient (B)
2. the mean (should be 0.00)
and the standard deviation
3. B for the interaction term and
the constant
Click Calculate
And See Chart
Change in the slope
when „going“
from reference
group to other group
The simplest way to do this within SPSS is to set up a scatter plot of the independent by the dependent variable, using the categorical predictor to set markers for cases.
To see the regression lines for urban and rural superimposed on this plot, we need to edit the scatter plot
To start the SPSS Chart Editor, right click on the scatter plot
Select Edit Content In Separate Window from the top-level menu in the Chart Editor
Dialog window, select Elements from the pull-down menu, choose Fit Line at
Subgroups
Standardized solution Dummy-code M (0=reference group;
1=comparison group)
Z-standardize X and Y
Compute crossproduct of zX and M (create interaction terms)
Regress zY on zX, M, and zX*M (hierachical moderated regression)
The unstandardized solution from the output is the correct solution (Friedrich, 1982)!
-.351 = estimated difference in regression weights between groups
-.089 = difference in intercept between both groups at mean of criminal friend index
.561 = simple slope for reference group - the effect of criminal friend index on
recidivism is significant for participants from rural areas (to check for the other group –
recode your categorical variable 1=0 and 0=1 and re-run regression analysis)
.210 = simple slope for reference group (this time urban is reference category because
we have recoded our moderator) - the effect of criminal friend index on recidivism is
significant for participants from urban areas
Input information is taken from
the regression analysis output.
1. enter the unstandardized
regression coefficient (B)
2. the mean for main effect
(should be 0.00) and the
standard deviation
3. B for the interaction term and
the constant
Click Calculate
And See Chart
Difference in the slope when
„going“ from reference
group to other group
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?
In interpreting the meaning of a figure, it is often important to know the values of the simple slopes, and to know whether these slopes differ significantly from zero
So, after the figure has been generated, go to the “Categorical Slopes Computations”
This page brings forward relevant information already entered in the data entry page And asks for additional information to be supplied After these critical items are entered, simply click on “Calculate” and simple slopes, standard errors, the degrees of freedom, t-values, and associated p-values are displayed. Results
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 D2
Group 1 0 0 1 0 1 0
Group 2 1 0 0 0 0 1
Group 3 0 1 0 1 0 0
Group 1 as Base Group 2 as Base Group 3 as Base
Simply add centered covariates as predictors to the unstandardized regression equation (or z-standardized covariates to the standardized regression equation).
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)
What if the DV is dichotomous (e.g., group membership, voting decision etc.)?
Use moderated logistic regression (Jaccard, 2001)
Daniel Boduszek [email protected]