Moderation:
Assumptions
David A. Kenny
What Are They?
Causality
Linearity
Homogeneity of Variance
No Measurement Error
Causality
• X and M must both cause Y.
• Ideally both X and M are manipulated
variables and measured before Y. Of
course, some moderators cannot be
manipulated (e.g., gender).
Causal Direction
• Need to know causal direction of the X to
Y relationship.
• As pointed out by Irving Kirsch, direction
makes a difference!
Surprising Illustration
• Judd & Kenny (2010, Handbook
of Social Psychology), pp. 121-2
(see Table 4.1).
• A dichotomous moderator with
categories A and B
• The X  Y effect can be stronger
for the A’s than the B’s.
• The Y  X effect can be stronger
for the B’s than the A’s.
Direction of Causality Unclear
• In some cases, causality is
unclear or the two variables may
not even be a direct causal
relationship.
• Should not conduct a moderated
regression analysis.
• Tests for differences in variances
in X and Y, and if no difference,
test for differences in correlation.
Crazy Idea?
• Assume that either X  Y or Y 
X.
• Given parsimony, moderator
effects should be relatively weak.
• Pick the causal direction by the
one with fewer moderator effects.
Proxy Moderator
• Say we find that Gender
moderates the X  Y
relationship.
• Is it gender or something
correlated with gender: height,
social roles, power, or some other
variable.
• Moderators can suggest possible
mediators.
Graphing
• Helpful to look for violations of
linearity and homogeneity of
variance assumptions.
• M is categorical.
• Display the points for M in a
scatterplot by different symbols.
• See if the gap between M
categories change in a nonlinear
way.
Linearity
• Using a product term implies a
linear relationship between M and
X to Y relationship: linear
moderation.
–The effect of X on Y changes by
a constant amount as M
increases or decreases.
• It is also assumed that the X  Y
effect is linear: linear effect of X.
Alternative to Linear
Moderation
• Threshold model: For X to cause
Y, M must be greater (lesser)
than a particular value.
• The value of M at which the effect
of X on Y changes might be
empirically determined by
adapting an approach described
by Hamaker, Grasman, and
Kamphuis (2010).
Second Alternative to
Linear Moderation
• Curvilinear model: As M
increases (decreases), the effect
of X on Y increases but when M
gets to a particular value the
effect reverses.
Testing Linear Moderation
• Add M2 and XM2 to the regression
equation.
• Test the XM2 coefficient.
–If positive, the X  Y effect
accelerates as M increases.
–If negative, then the X  Y effect
de-accelerates as M increases.
• If significant, consider a
transformation of M.
The Linear Effect of X
• Graph the data and look for
nonlinearities.
• Add X2 and X2M to the regression
equation.
• Test the X2 and X2M coefficients.
• If significant, consider a
transformation of X.
Nonlinearity or Moderation?
• Consider a dichotomous
moderator in which not much
overlap with X (X and M highly
correlated).
• Can be difficult to disentangle
moderation and nonlinearity
effects of X.
Nonlinear
Relationship
Y
X
Moderation
Y
X
Homogeneity of Variance
• Variance in Moderation
Analysis
–X
–Y (actually the errors in Y)
Different Variance in X for
Levels of M
• Not a problem if regression
coefficients are computed.
• Would be a problem if the
correlation between X and Y
were computed.
–Correlations tend to be
stronger when more
variance.
Equal Error Variance
• A key assumption of moderated
regression.
• Visual examination
– Plot residuals against the predicted
values and against X and Y
• Rarely tested
– Categorical moderator
• Bartlett’s test
– Continuous moderator
• not so clear how to test
Violation of Equal Error
Variance Assumption:
Categorical Moderator
• The category with the smaller
variance will have too weak a
slope and the category with the
larger variance will too strong a
slope.
• Separately compute slopes for
each of the groups, possibly using
a multiple groups structural
equation model.
Violation of Equal Error
Variance Assumption:
Continuous Moderator
• No statistical solution that I am
aware of.
• Try to transform X or M to
create homogeneous
variances.
Variance Differences as a
Form of Moderation
• Sometimes what a moderator does is
not so much affect the X to Y
relationship but rather alters the
variances of X and Y.
• A moderator may reduce or increase
the variance in X.
–Stress  Mood varies by work
versus home; perhaps effects the
same, but much more variance in
stress at work than home.
Measurement Error
• Product Reliability (X and M have a
normal distribution)
–Reliability of a product: rxrm(1 + rxm2)
–Low reliability of the product
–Weaker effects and less power
• Bias in XM Due to Measurement
Error in X and M
• Bias Due to Differential X Variance for
Different Levels of M
Differential Reliability
• categorical moderator
• differential variances in X
• If measurement error in X, then
reliability of X varies, biasing the two
slopes differentially.
• Multiple groups SEM model should be
considered
Additional
Webinars
• Effect Size and Power
• ModText
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