Visualizing shapes of interaction patterns with continuous independent variables

1. Visualizing shapes of interaction patterns with continuous independent variables Jane E. Miller, PhD

2. Overview • 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

3. 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.

4. Three general shapes of interaction patterns • Size: Theeffect of X1 on Y is larger for some values of X2 than for others; • Direction: the effect of X1 on Y is positive for some values of X2 but negative for other values of X2; • 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.

5. Possible patterns: Interaction betweenone 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.

6. 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. BW (g.) Income ($)

7. Income and race main effects,but no interaction Income/birth weight curves for blacks and whites have same slope (their curves are parallel) Butdifferent intercepts BW (g.) White Black Income ($)

8. Income main effect and interaction with race, but no race main effect Income/birth weight curves for blacks and whites have different slopes sameintercept BW (g.) White Black Income ($)

9. Income and race main effectsand interaction: Divergent curves Income/birth weight curves for blacks and whites have Different slopes anddifferentintercepts White Black BW (g.) Income ($)

10. Income and race main effects and interaction: Convergent curves Income/birth weight curves for blacks and whites have different slopes anddifferentintercepts BW (g.) White Black Income ($)

11. 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) anddifferentintercepts BW (g.) White Black Disordinal curves are those that cross in the observed range. Income ($)

12. Possible patterns among income, race, and birth weight White Black BW BW BW Income Income Income Income main effect Income & race main effects, and interaction: converging Income & race main effects BW BW BW Income Income Income Income & race main effects, and interaction: diverging from same intercept Income & race main effects, and interaction: disordinal Income & race main effects, and interaction: diverging from different intercepts

13. From three-way associations to regression model with interactions

14. 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).

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. Suggested online resources • Podcasts on • Introduction to interactions • Creating variables and specifying regression models to test for interactions • Calculating overall pattern from interaction coefficients

21. 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

22. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html