### The Basics and Introduction

```Moderation:
Introduction
David A. Kenny
What Is Moderation
The causal relationship from a causal
variable or X to an outcome or Y changes as
a function of a moderator or M.
– X and M interact to cause Y.
– Effect of stress on mood is moderated by
gender.
8
The Effect of Stress on Mood
Varies by Gender
7
6
Mood
5
Male
4
Female
3
2
1
0
Low
High
Stress
Causation and Correlation
• Need to know causal direction of the X to
Y relationship.
• If X is a manipulated variable, there should
be no relationship between X and M.
• Unlike mediation, there is no reason why
necessarily X and M should be correlated.
Timing of Measurement
• Typically M is measured before or at the
same time as X.
• Will discuss whether a moderator can be
caused by X when we discuss
Assumptions.
Statistical Estimation
• Typically estimated as the interaction
between X and M
• Y = aX + bM + cXM + E
a = “main effect” of X
b = “main effect” of M
c = interaction between X and M
• Important to include both X and M in the
model.
• Will discuss the Interpretation in another
webinar.
Linearity
• Using an product term implies a
linear relationship between M and
X to Y relationship.
• The effect of X on Y changes by a
constant amount as M increases
or decreases
• For example: the effect of Stress
on Marital Satisfaction changes
by the same amount for every
year married.
Statistical Estimation
• What was described has been
called moderated regression
analysis.
–One equation
–Moderation as an interaction
• Problematic alternatives
–Separate slopes
–Difference in correlations
–Median split
Separate Slopes?
• M is categorical.
• Slope computed for each group in
separate analyses.
• Difficult to test for moderation and
less power.
• Is the correct analysis when there
are heterogeneous errors.
Separate Correlations?
• Categorical moderator.
• Compute the correlation between
X and Y to determine moderation
within each category.
• Problem: Differences in X
variance.
–If men have more variance on X
than women, then women would
likely have a weaker correlation
(restriction in range).
Median Split?
• M is measured at the interval
level of measurement.
• A questionable way to measure
and test for moderation is to split
M at the mediation and compute
slopes for X to Y above and
below the split value.
• Big loss in power (see Aiken &
West).
Flipping X and M
• Because XM is an interaction, it
can be interpreted as the effect of
X depends on M or the that the
effect of M depends on X.
• Either a different discipline or a
rethinking might lead to treating X
as the moderator instead of M.
• Sometimes this is a useful
exercise.
Diagrams of Moderation
.5
.7
Key Resource
• Aiken, L. S., & West, S. G. (1991). Multiple
regression: Testing and interpreting interactions.
Newbury Park, CA: Sage.