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

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

The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Review: What is an interaction? 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 ), known as the “modifier.” 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. Three general shapes of interaction patterns 1.Size: The effect of X 1 on Y is larger for some values of X 2 than for others; 2.Direction: the effect of X 1 on Y is positive for some values of X 2 but negative for other values of X 2 ; 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 Income ($) No racial difference in income/birth weight relation: slope and intercept same for blacks and whites. BW (g.)

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 ($) BW (g.)

The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Income main effect and interaction with race, but no race main effect White Black Income/birth weight curves for blacks and whites have different slopes same intercept Income ($) BW (g.)

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

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

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

The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Possible patterns among income, race, and birth weight Income BW Income BW Income BW Income BW Income BW Income BW White Black Income main effect Income & race main effects Income & race main effects, and interaction: converging Income & race main effects, and interaction: diverging from same intercept Income & race main effects, and interaction: diverging from different intercepts Income & race main effects, and interaction: disordinal

The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. From three-way associations to regression model with interactions

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. 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, 2 nd 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, 2 nd 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, 2 nd 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, 2 nd edition. Suggested resources Chapter 16 of Miller, J. E The Chicago Guide to Writing about Multivariate Analysis, 2nd edition. Jaccard, J. J., and R. Turrisi Interaction Effects in Multiple Regression. 2nd ed. Berkeley Hills, CA: Sage Publications. Chapters 8 and 9 of Cohen et al 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. 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, 2 nd 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, 2 nd edition. Contact information Jane E. Miller, PhD Online materials available at