Visualizing shapes of interaction patterns with continuous independent variables
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Three general shapes of interactions
• What do interaction patterns between categorical and one continuous independent variable look like?
• From three-way association to regression model with interactions
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• The association between one independent variable (X
1
) and the dependent variable (Y) differs depending on the value of a second independent variable (X
2 as the “modifier.”
), known
• The presence of an interaction means that one can’t express the direction or size of the association between
X
1 and Y without also specifying the values of X
2
.
• In the lingo of “generalization, example, exception”
(GEE), interactions are an exception to a general pattern among those variables.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

1. Size: The effect of X
1 values of X
2 on Y is than for others; larger for some
2. Direction: the effect of X
1 some values of X of X
2
;
2 on Y is positive for but negative for other values
3. The effect of X
1 on Y is non-zero (either positive or negative) for some values of X
2 but is not statistically significantly different from zero for other values of X
2
.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd 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, 2 nd edition.

Income main effect, but no race main effect or interaction with income
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, 2 nd edition.

Income and race main effects, but no interaction
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, 2 nd edition.

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, 2 nd edition.

Income and race main effects and interaction: Divergent curves
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, 2 nd edition.

Income and race main effects and interaction: Convergent curves
Income/birth weight curves for blacks and whites have different slopes and different intercepts
Income ($)
White
Black
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

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.
Income ($)
White
Black
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

BW
Possible patterns among income, race, and birth weight
BW BW
White
Black
BW
Income
Income main effect
BW
Income
Income & race main effects
Income
Income & race main effects, and interaction: converging
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, 2 nd edition.

The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Create a three-way chart of the association
• To gain a sense of the shape of the relationship among your variables, graph the three-way association.
• E.g., the clustered bar charts was created based on differences in means of the DV (birth weight) according to the cross-tabulated categorical values of the two IVs (race and education).
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Check it against theory and previous studies.
• Does it make sense?
• Anticipate which main effects and interaction terms are needed in the specification.
• See which of the charts shown here best characterize the pattern.
• Note that other shapes of patterns are also possible.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Check the pattern calculated from the estimated coefficients against the simple three-way chart.
• If the shapes are wildly inconsistent with one another, probably reflects an error in either
– How you specified the model, or
– How you calculated the overall pattern from the coefficients.
• Small changes in the shape or size of the pattern may occur due to controlling for other variables in your multivariate model.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• The next module will show how to
• Create variables needed for interaction
• Specify the model to formally test for interaction effects
• Later modules will explain how to calculate the overall shape of an interaction from the estimated coefficients.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Real-world examples of interactions can take many forms, including various combinations of main effect and interactions.
• Interactions can occur in terms of
– Direction
– Magnitude
• A three-way chart can help identify which of the many theoretically possible shapes characterize the relationship among your IVs and DV.
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Chapter 16 of Miller, J. E. 2013. The Chicago
Guide to Writing about Multivariate Analysis,
2nd edition.
• Jaccard, J. J., and R. Turrisi. 2003. Interaction
Effects in Multiple Regression. 2nd ed. Berkeley
Hills, CA: Sage Publications.
• 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, 2 nd edition.

• Podcasts on
– Introduction to interactions
– Creating variables and specifying regression models to test for interactions
– Calculating overall pattern from interaction coefficients
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
– Questions #1 and 2 in the problem set for Chapter 16
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

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, 2 nd edition.