Interaction
Lyytinen & Gaskin
Interaction – Definition
• In factorial designs, interaction effects are the
joint effects of two predictor variables in
addition to the individual main effects.
• This is another form of moderation (along
with multi-grouping) – i.e., the XY
relationship changes form (gets stronger,
weaker, changes signs) depending on the
value of another explanatory variable (the
moderator)
Hair et al 2010 pg. 347
2
Interaction Effects
• Interactions represent non-additive effects
• i.e., when the joint effects X and Z on Y are
more or less than their additive effects.
– e.g., Diet * Exercise = greater Weight loss
– e.g., Chocolate * Cheese = Yucky!
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Additive versus Interaction effects
Additive
Effect
Negative
Interaction
Positive
Interaction
X=0 X=1
Z=0 1 3
Y
Z=1 3 5
Diff 2 = 2
X=0 X=1
Z=0 1 3
Y
Z=1 3 4
Diff 2 > 1
X=0 X=1
Z=0 1 3
Y
Z=1 3 6
Diff 2 < 3
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Simple interaction example
Diet
X
The interaction
term is used for
testing the
moderating effect
of Exercise
Diet x
Exercise
Weight Loss
Y
Exercise
Z
Exercise
Diet
Weight Loss
5
high
high
Weight
loss
Weight
loss
low
high
low
high
Diet
Exercise
high
Weight
loss
low
high
Diet x Exercise
H6 2O
Weight loss example
Additive
Effect
Diet=0 Diet=1
Exer=0 1 WL 3
Exer=1 3
5
Diff
2 = 2
i.e., Exercising does
not alter the
effectiveness of dieting
Negative
Interaction
Positive
Interaction
Diet=0 Diet=1
Diet=0 Diet=1
Exer=0 1
3
Exer=0 1 WL 3
WL
Exer=1 3
4
Exer=1 3
6
Diff
2 > 1
Diff
2 < 3
i.e., Exercising makes
dieting less effective
i.e., Exercising makes
dieting more effective
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Other examples
• Interaction between adding sugar to coffee and stirring the
coffee. Neither of the two individual variables has much
effect on sweetness but a combination of the two does.
• Interaction between adding carbon to steel and quenching.
Neither of the two individually has much effect on strength
but a combination of the two has a dramatic effect.
• Interaction between smoking and inhaling asbestos fibers:
Both raise lung carcinoma risk, but exposure to asbestos
multiplies the cancer risk in smokers and non-smokers.
• Interaction between genetic risk factors for type 2 diabetes
and diet (specifically, a "western" diet). The western dietary
pattern has been shown to increase diabetes risk for
subjects with a high "genetic risk score", but not for other
subjects.
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Interaction in literature
• Interaction (a form of moderation) is central to
research in the organizational and social sciences.
• Interaction is involved in research demonstrating:
– the effects of motivation on job performance are
stronger among employees with high abilities (Locke
& Latham, 1990),
– the effects of distributive justice on employee
reactions are greater when procedural justice is low
(Brockner & Wiesenfeld, 1996),
– the effects of job demands on illness are weaker when
employees have control in their work environment
(Karasek, 1979; Karasek & Theorell, 1990).
Edwards 2009
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Why interaction
• Interactions enable more precise explanation
of causal effects by providing a method for
explaining not only how X effects Y, but also
under what circumstances the effect of X
changes depending on the moderating
variable of Z.
10
Interaction vs. Multi-group
• Literature makes little distinction between
interaction and multi-group.
• Interaction is often treated like multi-group
– High vs Low values of age, income, size etc.
• The interpretation is much the same, but the
method is different.
– Multi-group: categorical variables, split dataset,
constrained paths
– Interaction: continuous variables, whole dataset,
interaction variables
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The statistical side of it
• A regular regression equation involving two
independent variables:
Y = b 0 + b 1X + b 2Z + e
• In ordinary least squares (OLS) regression, the
product of two variables can be used to represent
the interactive effect:
Y = b0 + b1X + b2Z + b3XZ + e
• where, XZ is the product term that represents the
interaction effect, and b3 is the change in the
slope of the regression of XY when Z changes
by one unit.
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The statistical side of it (cont)
• Essentially, the interaction regression equation
specifies that the slope of the line relating X to Y
changes at different levels of Z, or equivalently,
that the slope of the line relating Z to Y changes
at different levels of X.
• Saunders (1956) first demonstrated that a
product term accurately reflects a continuous
variable interaction. Similarly, natural polynomial
or powered variables (X2, X3, etc.) can be used to
represent higher order nonlinear effects of a
variable such as a quadratic or cubic trend of age
or time.
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Significance of interaction effects
• Are slopes of regression lines for XY
significantly different at differing values of Z?
• e.g., is the slope of the relationship between
diet and weight loss significantly different
between those who exercise very little and
those who exercise a lot?
• The way to determine the significance is to
calculate a p-value for the regression of XZY
(AMOS handles this)
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Significance of interaction part 2
• One may also desire to know whether the
change in the XY relationship with and
without the interaction effect is significant.
• This can be done through a rather complex method
available at: http://www.people.ku.edu/~preacher/interact/mlr2.htm
• He has a tool for it.
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Range of Significance
• The region of significance defines the specific values
of z at which the regression of y on x moves from nonsignificance to significance (see Preacher 2007).
• There are lower and upper bounds to the region. In
many cases, the regression of y on the focal predictor is
significant at values of the moderator that are less than
the lower bound and greater than the upper bound,
and the regression is non-significant at values of the
moderator falling within the region.
• However, there are some cases in which the opposite
holds (e.g., the significant slopes fall within the region).
• We will not calculate this, but there are ways to do so
(see Preacher 2007).
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Statistical Interaction Considerations
• Multicollinearity
– Interaction terms can be highly collinear with
constituent IVs – this can be addressed by centering
the means (Edwards 2009)
• Non-normal distribution handling
– If IVs are not normally distributed, the product term
(interaction term) will likely result in biased
estimations
– Neither is it true that if the IVs are normally
distributed their product will be normally distributed
– This can sometimes be fixed through transformation
of either or both the IVs and the interaction term
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Statistical Interaction Considerations
• Reliability
– low reliability of either the IVs or the moderator
(especially) is likely to increase either type I or
type II errors (Edwards 2009)
• Isolating unique effects
– If the effect of XZY is significant, both XY and
ZY must remain in the path model, even if nonsignificant, in order to isolate the unique effects of
the interaction term
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Mean Centering
• One way to fix multicollinearity issues
inherent in the use of interaction terms is to
mean-center all involved variables.
• This involves subtracting the variable mean
from each response, thus placing the new
mean at zero.
• Similarly, you can standardize the variable,
which is simply replacing the variable values
with their corresponding z-scores (mean=0,
sd=1)
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Benefits of Centering
• Centering can make otherwise uninterpretable
regression coefficients meaningful, and
• Centering reduces multicollinearity among
predictor variables.
• Centering has no effect on linear regression
coefficients (except b0)
20
Mean Centering vs. Standardizing
Original value Mean Centered Standardized
Mean
5
0
0
Std Dev
2.73
2.73
1
A
1
-4
-1.46
B
2
-3
-1.10
C
3
-2
-0.73
D
4
-1
-0.37
E
5
0
0.00
F
6
1
0.37
G
7
2
0.73
H
8
3
1.10
I
9
4
1.46
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Higher order interactions
• We work with 2-way interactions, but interactions are
not limited to X and Z.
• A three-way interaction looks like this:
Y = b0 + b1X + b2Z + b3W + b4XZ + b5XW + b6ZW + b7XZW
• Where W is a second interaction term
• There are no mathematical limits on the number of
interacting terms, but there are certainly practical limitations.
• One challenge is testing significance, as there are approaches
to do this with 3 variables but not with more
• Another challenge is the interpretation of the interactions and
what they truly mean theoretically
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How To
1. In SPSS create new variables by standardizing all
variables in the model (except categorical ones) and
then computing a product variable
2. Use like an IV in AMOS.
3. Trim model, starting with interaction effects
1. Don’t trim paths from constituent IVs unless the parent
interaction is deleted due to insignificance (this is
because all the paths and their coefficients need to be
interpreted together with the interaction term)
4. Adjust per Model Fit issues
5. If interactions are significant, plot them (there is
software for it)
6. Interpret the interaction/moderation effects
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1a. Standardize
Standardize all variables in the
model (unless categorical)
Result
24
1b. Compute product variable
Go here
Type new name (no
spaces or mathematical
symbols allowed)
Type or click the product
expression
Hit OK
Result
25
2. Put in AMOS
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3. Trim
Start here
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4. Attend to Model Fit
• Just normal model fit stuff
• Don’t forget about:
– SRMR
– Chi-square/df
– CFI
– RMSEA
– Modification indices
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5. Plot interaction
(if the interaction is significant)
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6. Interpret
• atrust is a stronger predictor of vallong (i.e., has a
larger slope value – Beta) for cases of high ctrust.
• Thus, ctrust positively moderates (amplifies) the
effect of atrust on vallong.
• For small differences
like this one, the
moderation is
significant if the ZY
Beta is significant
(in AMOS output)
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Plotting approach
• To ease interpretation of interaction, we treat
them somewhat like multi-group variables…
• Select high and low values of moderator
– High: one standard deviation above the mean
– Low: one standard deviation below the mean
-1
0
+1
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More on Interpretation
Exercise positively moderates (amplifies) the
relationship between diet and weight loss.
i.e., Exercising
makes dieting
more effective
Exercise negatively moderates (dampens) the
relationship between diet and weight loss.
i.e., Exercising
makes dieting less
effective
Exercise does not moderate the relationship
between diet and weight loss.
i.e., Exercising
does not alter the
effectiveness of
dieting
Exercise inversely moderates the relationship
between diet and weight loss.
i.e., All or nothing
moderation,
32 do
both or do neither
Additional Resources
• This is a site hosted by Kristopher Preacher (as in
the Preacher and Hayes articles), and is very
informative regarding interactions:
– http://www.people.ku.edu/~preacher/interact/intera
ctions.htm
• If you are interested in calculating the range of
significance interaction values, refer to this
somewhat complex (yet simplified) tool:
– http://www.people.ku.edu/~preacher/interact/mlr2.h
tm
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