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

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

Visualizing shapes of interaction patterns between two categorical

independent variables

Jane E. Miller, PhD

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

Overview

• Three general shapes of interactions• What do interaction patterns between two

categorical independent variable look like?• From three-way association to regression model

with interactions

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

Review: 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), known as the “modifier.”

• The presence of an interaction means that one can’t express the direction or size of the association between X1 and Y without also specifying the values of X2.

• 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, 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.

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

.)

<HS =HS >HS

Mother’s educational attainment

BlackWhite

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

Education main effect, but no race main effect or interaction

<HS =HS >HS

Mother’s educational attainment

BlackWhite

BW (g

.)

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

Education and race main effects, but no interaction

BW (g

.)

<HS =HS >HS

Mother’s educational attainment

BlackWhite

Size of black/white birth weight gap is same in each education group.

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

Education and race main effects,and interaction: Size of gap

BW (g

.)

<HS =HS >HS

Mother’s educational attainment

BlackWhite

Size of black/white birth weight gap varies across education groups.

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

Education and race main effects, and interaction: Direction of gap

BW (g

.)

<HS =HS >HS

Mother’s educational attainment

BlackWhite

Direction of black/white birth weight gap varies across education groups.

Black>white Black<white Black<white

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

Summary: Some possible patterns of race, education, and birth weight

BW

<HS =HS >HS

BlackWhite

BW

<HS =HS >HS

BW

<HS =HS >HS

<HS =HS >HS<HS =HS >HS

BWBW

Interaction: magnitude Interaction: direction & magnitude

Main effect race Main effect educ Main effects race & educ

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

From three-way associations to regression model with interactions

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

Using the three-way chart to plan your multivariate model

• 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, 2nd edition.

Using the three-way chart to verify your multivariate results

• 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, 2nd edition.

Next steps toward a model with interactions

• 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, 2nd edition.

Summary

• 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, 2nd edition.

Suggested resources• 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, 2nd edition.

Suggested online resources

• 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, 2nd edition.

Suggested practice exercises

• 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, 2nd edition.

Contact information

Jane E. Miller, PhDjmiller@ifh.rutgers.edu

Online materials available athttp://press.uchicago.edu/books/miller/multivariate/index.html