Overview of categorical by
categorical interactions:
Part I: Concepts, definitions, and shapes
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
What is an interaction?
• The association between one independent
variable (X1) and the dependent variable (Y)
differs depending on the value of a second
independent variable (X2).
• Can be thought of as an exception to a general
pattern:
– X1 is associated with Y in one way when X2 = 1, but
in a different way when X2 = 2.
• X1 is sometimes termed the “focal predictor”
• X2 is referred to as the “modifier” or “modifying
variable.”
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Statistical interactions defined
• When X1 and X2 not only potentially have
separate effects on Y, but also have a joint
effect that is different from the simple sum of
their respective individual effects.
– The association between X1and Y is conditional on
X2.
– The specific combinations of values of X1 and X2
determine the value of Y.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Three general
shapes of interaction patterns
1. Size: The effect of X1 on Y is larger for some
values of X2 than for others;
2. Direction: the effect of X1 on Y is positive for
some values of X2 but negative for other values
of X2;
3. The effect of X1 on Y is non-zero (either positive
or negative) for some values of X2 but is not
statistically significantly different from zero for
other values of X2.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Synonyms for “interaction”
• Terminology for interactions varies by
discipline.
• Common synonyms include:
– Effects modification
– Moderating effect
– Modifying effect
– Joint effect
– Contingency effect
– Conditioning effect
– Heterogeneity of effects
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Specifying an interaction model
• Multivariate regression specifications to test
for interactions include a combination of
“main effects terms” and “interaction terms.”
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Main-effects-only specification
• A main-effects-only model implies that
controlling for other covariates (Xi),
– the effect of X1 on Y is the same for all values of X2,
– and the effect of X2 is the same for all values of X1.
• Its specification can be written:
Y = β0 + β1X1 + β2X2, where
X1 is the main effect term for the first independent
variable (IV)
X2 is the main effect term for a second IV
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Example: Main-effects-only model
• If Y is birth weight in grams
• X1 is dummy variable for <HS
– coded 1 for mothers with less than complete high school
– 0 for all other infants, the reference category
• X2 is a dummy variable for black race
– coded 1 for black infants
– 0 for white infants, the reference category
• Birth weight = β0 + β1<HS + β2Black implies that the
shape of the education/birth weight association is
the same for black as for white infants
– the education/birth weight (X1/Y) association is not
modified by race
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education and race main effects only
Size of black/white birth weight gap
is same in each education group.
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Interaction specification
• A model with interactions implies that
controlling for other covariates,
– the effect of X1 on Y is different for different values
of X2.
Y = β0 + β1X1 + β2X2 + β3X1 _ X2, where
X1 is the main effect term for the focal IV in the
interaction,
X2 is the main effect term for the modifying IV,
X1 _X2 is the interaction term between the focal and
modifying IVs.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Interaction term
• The value of the interaction term variable is defined as
the product of the two component variables:
X1_ X2 = X1 × X2
• When naming an interaction term variable, I often use
an “_” to connect the names of the two component
variables.
– E.g., <HS_black would be the interaction between the two
variables “<HS” and “black.”
• E.g., for case #1, if
– <HS (X1) = 1
– Black (X2) = 1
– The interaction term <HS_black = 1 × 1 = 1
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Contingency of coefficients
in an interaction model
Y = β0 + β1X1 + β2X2 + β3X1 _ X2,
• Inclusion of the interaction term X1_ X2 means
that the βis on the main effects terms X1 and X2
no longer apply to all values of X1 and X2.
– The main effects and interactions βis for X1 and X2
are contingent upon one another and cannot be
considered separately.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Implications for interpreting main
effects and interaction coefficients
Y = β0 + β1X1 + β2X2 + β3X1_X2
• In the interaction model:
– β1 estimates the effect of X1 on Y when X2 = 0,
– β2 estimates the effect of X2 on Y when X1 = 0,
– β3 must also be considered in order to calculate the
shape of the overall pattern among X1, X2, and Y.
• E.g., when X1 and X2 take on other values.
• See podcast on calculating the shape of an
interaction pattern.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Example: Interaction model
BW = β1<HS + β2Black + β3<HS_black,
• If β3 is statistically significantly different from zero, the shape
of the education/birth weight association is different for black
than for white infants.
• β1 estimates the association between education and birth
weight among whites (e.g., when Black = 0)
• β2 estimates the difference in birth weight for blacks
compared to whites when <HS = 0 (e.g., for more educated
mothers).
• β3 estimates how predicted birth weight deviates from the
value implied by β1 and β2 alone, for different combinations of
education and race.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education and race main effects,
and interaction: Size of gap
Size of black/white birth weight gap
varies across education groups.
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Possible patterns: Interaction between
two categorical independent variables
• Example: Race and mother’s education as predictors of
birth weight
– Birth weight (BW) in grams is the dependent variable;
– The focal independent variable, mother’s educational
attainment, is an ordinal categorical variable;
– The modifier, race, is a nominal independent variable.
• An interaction means that the association between
mother’s education and birth weight differs by race .
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Race main effect, but no education
main effect or interaction
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education main effect, but no race
main effect or interaction
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education and race main effects,
but no interaction
Size of black/white birth weight gap
is same in each education group.
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education and race main effects,
and interaction: Size of gap
Size of black/white birth weight gap
varies across education groups.
BW (g.)
Black
White
<HS
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Education and race main effects,
and interaction: Direction of gap
Direction of black/white birth weight gap
varies across education groups.
BW (g.)
Black
White
Black>white
<HS
Black<white
Black<white
=HS
>HS
Mother’s educational attainment
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Summary: Some possible patterns of
race, education, and birth weight
BW
BW
<HS
=HS
BW
<HS
>HS
=HS
>HS
<HS
Main effect: education
Main effect: race
BW
=HS
>HS
Interaction: magnitude
>HS
Main effects: race & education
Black
White
BW
<HS
=HS
<HS
=HS
>HS
Interaction: direction & magnitude
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Continue on to Part II
• Information on
– Creating variables
– Specifying models
– Calculating overall shape of an interaction pattern
from regression coefficients
For a categorical by categorical interaction
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested resources
• Chapter 16, Miller, J. E. 2013. The Chicago
Guide to Writing about Multivariate Analysis,
2nd Edition.
• Chapters 8 and 9 of Cohen et al. 2003. Applied
Multiple Regression/Correlation Analysis for
the Behavioral Sciences, 3rd Edition. Florence,
KY: Routledge.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested exercises
• Study guide to The Chicago Guide to Writing
about Multivariate Analysis, 2nd Edition.
– Problem set for chapter 16
– Suggested course extensions for chapter 16
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Contact information
Jane E. Miller, PhD
jmiller@ifh.rutgers.edu
Online materials available at
http://press.uchicago.edu/books/miller/multivariate/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.