Daniel Boduszek University of Huddersfield

d.boduszek@hud.ac.uk

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 d.boduszek@hud.ac.uk