Overview of categorical by
continuous 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 family income in dollars
• X2 is a dummy variable for black race
– coded 1 for black infants
– 0 for white infants, the reference category
• Birth weight = β0 + β1Income + β2Black implies that
the slope of the income/birth weight curve is the
same for black as for white infants
– the income/birth weight (X1/Y) association is not modified
by race
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Birth weight (grams)
Main effects of race and income,
but no interaction with income
Income/birth weight curves for blacks and whites have
same slope (their curves are parallel)
But different intercepts
White
Black
Income ($)
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., age_income would be the interaction between the two
variables “age” and “income.”
• E.g., for case #1, if
– age (X1) = 27,
– income (X2) =$10,000,
– interaction term age_income = 27 × 10,000 = 27,000
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Contingency of coefficients
in an interaction model
Y = β0 + β1X1 + β2X2 + β3X1 _ X2,
• Inclusion of the interaction term X1_ X2 means
that the βis on the main effects terms X1 and X2
no longer apply to all values of X1 and X2.
– The main effects and interactions βis for X1 and X2
are contingent upon one another and cannot be
considered separately.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Implications for interpreting main
effects and interaction coefficients
Y = β0 + β1X1 + β2X2 + β3X1_X2
• In the interaction model:
– β1 estimates the effect of X1 on Y when X2 = 0,
– β2 estimates the effect of X2 on Y when X1 = 0,
– β3 must also be considered in order to calculate the
shape of the overall pattern among X1, X2, and Y.
• E.g., when X1 and X2 take on other values.
• See podcast on calculating the shape of an
interaction pattern.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Example: Interaction model
BW = β1Income + β2Black + β3Income_black,
• If β3 is statistically significantly different from zero, the slope
of the income/birth weight curve is different for black than for
white infants.
• β1 estimates the association between income and birth
weight among whites (e.g., when Black = 0)
• β2 estimates the difference in birth weight for blacks
compared to whites at income = 0.
• β3 estimates how predicted birth weight deviates from the
value implied by β1 and β2 alone, for different combinations of
race and income.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Birth weight (grams)
Main effects of race and income,
and interaction of race with income
Birth weight/income curves for blacks and whites have
Different slopes
and
Different intercepts
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Possible patterns: Interaction between
one categorical and one continuous
independent variable
• Example: Race and income as predictors of birth
weight:
– Birth weight (BW) in grams is the dependent variable;
– The focal independent variable, annual family income, is
a continuous variable in $;
– The modifier, race, is a nominal independent variable.
• An interaction means that the association between
income and birth weight differs by race.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Income main effect, but no race main
effect or interaction with income
BW (g.)
No racial difference in income/birth weight relation:
slope and intercept same for blacks and whites.
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Income and race main effects,
but no interaction
BW (g.)
Income/birth weight curves for blacks and whites have
same slope (their curves are parallel)
But different intercepts
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
BW (g.)
Income main effect and interaction
with race, but no race main effect
Income/birth weight curves for blacks and whites have
different slopes
same intercept
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Income and race main effects
and interaction: Divergent curves
Income/birth weight curves for blacks and whites have
Different slopes
and different intercepts
BW (g.)
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Income and race main effects
and interaction: Convergent curves
BW (g.)
Income/birth weight curves for blacks and whites have
different slopes
and different intercepts
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
BW (g.)
Income and race main effects
and interaction: Disordinal curves
Income/birth weight curves for blacks and whites have
different slopes
(in this case, opposite-signed slopes)
and different intercepts
Disordinal curves are
those that cross in the
observed range.
White
Black
Income ($)
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Possible patterns among
income, race, and birth weight
BW
BW
Income
BW
Income
Income main effect
White
Black
Income
Income & race main effects,
and interaction: converging
Income & race main effects
BW
BW
BW
Income
Income & race main effects,
and interaction: diverging
from same intercept
Income
Income & race main effects,
and interaction: diverging
from different intercepts
Income
Income & race main effects,
and interaction: disordinal
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 continuous 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.