Moderation:
Interpretation
David A. Kenny
The Moderated
Regression Equation
• 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.
Interpretation of the Two Main
Effects
a: “main effect” of X
The effect of X when M is zero.
b: “main effect” of M
The effect of M when X is zero.
For these effects to be meaningful, zero
must be a meaningful value.
Centering
• Make sure zero in interpretable
for X and M.
• If not then center: Compute the
grand mean of X and M and
subtract it (or something close).
Interpretation of the Interaction
Effect (Moderation)
c = the effect of XM
For every one unit change of M,
the effect of X (i.e., increases
(or decreases) by c units.
Path c represents then how
much a changes.
The Moderated Effect of
X on Y
• Y = aX + bM + cXM + E
• Y = (a + cM)X + bM + E
• The effect of X on Y is equal to a +
cM.
• If we set a + cM to zero, then X has
no effect on Y when M = -a/c.
Brief Example
stress level or S
happiness or H
socio-economic status (C)
H = -2S + 1C + 0.1SC + E
For every one unit change of socioeconomic
status, the effect of Stress levels on
Happiness weakens by 0.1 units. Note
that for C = 20, the effect of stress on
happiness is zero.
Other Strategies for
Interpretation
• Computing simple effects
• Graphing the moderation
Simple Effects
• The effect that X has on Y at
different levels of M.
• Sometimes called “pick a point.”
• Two ways to compute:
• Plug into a + cM the value for M
and solve.
• Complicated to obtain a standard
error for this value.
Re-centering
• Mʹ is desired value of M for which
a simple effect is desired.
• Re-estimate the moderated
regression equation using M - Mʹ
for M.
• The new value of a will be the
simple effect of X when M = Mʹ.
• Note this gives a p value and a
confidence interval.
What To Do If M Is
Continuous?
• Pick two values
• Typically one standard deviation
above and below the mean of M.
• Make sure the values are possible
values.
Graphing
Better than simple effects because it
displays all of the effects.
How?
Compute the least-squares means
for each of the four combinations of
two values of X
two values of M
Place in the graph.
Connect the sets of two points with
a common value of M.
Graphing Programs
Paul Jose’s ModGraph
http://www.victoria.ac.nz/psyc/pauljose-files/modgraph/
Andrew Hayes’s
ModProbehttp://www.afhayes.com/
spss-sas-and-mplus-macros-andcode.html#modprobe
Regions of Significance
• Johnson-Neyman
• Potthoff extension
Regions of Significance
• Useful if M is measured at the
interval level of measurement.
• Tells you at what intervals of M is
the X  Y effect significant.
• It might tell you the when M is at
least 5, the effect is positive and
significant, but when M is less
than 3, the effect is negative and
significant.
• Can flip X and M.
Additional
Webinars
• Assumptions
• Effect Size and Power
• ModText
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