Specification errors for interaction models: Implications for the shape of the overall pattern

1. Specification errors for interaction models: Implications for the shape of the overall pattern Jane E. Miller, PhD The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

2. Overview • Review: Model specification with main effects and interaction terms • Implications of leaving the main effects terms out of a model intended to test for interactions • Repercussions for • An interaction between two categorical independent variables • An interaction between one categorical and one continuous independent variable The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

3. List of variables used in examples • Dependent variable = birth weight in grams (BW) • Independent variables: • Main effects terms: • Race • Two nominal categories (non-Hispanic black; non-Hispanic white is the reference category) • One main effect dummy variable: NHB • Coded 1 = non-Hispanic black, 0 = non-Hispanic white • Mother’s education • Three ordinal categories (< HS; = HS; > HS is the reference category) • Two main effects dummies: <HS, =HS • Each coded 1 = named category, 0 = all other values The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

4. List of variables, continued • Interaction between race and mother’s education • Two interaction term dummies: NHB_<HS; NHB_=HS • Each named using the “_” convention to link the names of the component variables. • Each coded 1 = named category, 0 = all other values • E.g., NHB_<HS = 1 for those who are both NHB and < HS, = 0 for all other combinations of race and education The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

5. Model specification with interactions: race and education • BW = f (race, education, race_education) • Birth weight is a function of race, education, and the race-by-education interaction • To specify a model that does not impose assumptions about the shape of the association, need ALL of the main effects and interaction term variables related to race and mother’s education • BW = f (NHB, <HS, =HS, NHB_<HS, NHB_=HS) • Yellow denotes the main effects terms • Green denotes the interaction terms The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

6. Some possible patterns of race, education, and birth weight BW BW BW < HS = HS > HS < HS = HS > HS < HS = HS > HS Main effect: education Main effects: race & education Main effect: race Black White BW BW < HS = HS > HS < HS = HS > HS Interaction: magnitude Interaction: direction & magnitude The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

7. What happens if the specification omits the main effects terms? • If we omit the main effects terms for the two independent variables involved in the interaction, the implied model is specified BW = f (NHB_<HS, NHB_=HS) • Then the estimated βs for those two variables compare those groups against everyone else • In this case all whites (regardless of mother’s educational attainment) plus blacks whose mothers have > HS • This implicitly assumes that those four groups all have equal mean birth weight, rather than testing for differences across those groups The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

8. Repercussions of misspecification “&” used to denote a group with that combination of characteristics, not an interaction term • Any differences among • NHB & > HS • NHW & < HS • NHW &= HS • and NHW & > HS will be overlooked because there are no terms in the model to test for such differences. • β0 (the constant or intercept term) will be a weighted average of birth weight for those four groups combined • βNHB_<HS and βNHB_=HS will estimate the difference in mean birth weight for those groups compared to that combined reference category The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

9. Implied pattern if main effects of race and education are omitted Implied reference category for specification BW = f (NHB_<HS, NHB_=HS) βNHB_<HS βNHB_=HS β0 < HS = HS > HS BW Non-Hispanic black Non-Hispanic white

10. Implied pattern if main effects of race and education are omitted βNHB_=HS BW = f (NHB_<HS, NHB_=HS) βNHB_<HS β0 Non-Hispanic black Non-Hispanic white BW < HS = HS > HS The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

11. Observed pattern based on model of NHANES III data with main effects and interaction terms BW = f (NHB, <HS, =HS, NHB_<HS, NHB_=HS) βNHB + β<HS+βNHB_<HS βNHB + β=HS+βNHB_=HS Model estimates separate levels (intercepts) for each combination of race and education β<HS β=HS βNHB β0 BW Non-Hispanic black Non-Hispanic white < HS = HS > HS The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

12. Interaction between a continuous and a categorical independent variable (IV) • Example: Race and income-to-poverty ratio (IPR) • Race is a two-category IV, specified with a dummy variable NHB, coded • 1 = non-Hispanic black • 0 = non-Hispanic white (the reference category) • IPR is a continuous variable calculated as annual family income (in dollars) divided by the Federal Poverty Level for a family of that size and age composition • The interaction between race and IPR is a continuous variable calculated as the product of the NHB dummy and IPR NHB_IPR =NHB × IPR The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

13. Model specification to test an interaction between continuous and categorical IVs • For a model with an interaction between two independent variables, need all of the ALL of the main effects and interaction term variables related to those two independent variables • E.g., for a model of birth weight by race and IPR, include the main effect and interaction terms related to race and family IPR: BW = f (NHB, IPR,NHB_IPR) The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

14. What happens if the specification omits the main effects terms? • If we omit the main effects terms for the two independent variables involved in the interaction, the implied model is specified BW = f (NHB_IPR) • Then the coefficient βNHB_IPRestimates the slope of the IPR/birth weight curve for blacks, but does not • Allow for a different intercept for blacks than for white • Test for a difference in slopes of the IPR/birth weight curves for blacks and for whites The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

15. Some 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 The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

16. Implied pattern based on NHANES III data if main effects of race and IPR are omitted BW = f (NHB_IPR) specification forces • The intercept to be the same for black and white infants • The slope of IPR/birth weight curve for white infants to be zero (flat) • The estimated slope of IPR/birth weight curve for black infants to be negative BW β0 White Black βNHB_IPR IPR The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

17. Observed pattern based on model of NHANES III data with main effects and interaction terms • BW = f (NHB, IPR, NHB_IPR) specification estimates • Different intercepts for blacks and for whites • Different slopes for blacks and for whites • Slopes for both racial/ethnic groups are positive BW = βIPR White Black β0 = βIPR + βNHB_IPR = β0 + βNHB IPR The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

18. Summary • Models intended to test for interactions should initially include allmain effects and interaction terms for the independent variables involved • Such a specification • Does not impose a priori assumptions about the shape of the association among the IVs and DV • Allows the data to reveal the shape and size of that pattern • Empirical criteria can be used to simplify the specification if βs for some term(s) are not statistically significantly different from one another The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

19. Suggested resources • Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. University of Chicago Press, chapter 16. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

20. Suggested online resources • Podcasts on • Visualizing shapes of interaction patterns • Creating variables and specifying models to test for interactions • Calculating the shape of an interaction pattern from regression coefficients • Two categorical independent variables • One categorical and one continuous independent variable • Testing whether a multivariate specification can be simplified The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

21. Suggested practice exercises • Using your own data, estimate the following models for an interaction between two categorical independent variables • Main effects only • Main effects and interactions • Interaction terms only (omit the associated main effects terms) • Using a spreadsheet, calculate and graph the implied overall pattern of the association between the two IVs involved in the interaction and your DV for EACH of the three specifications • See spreadsheet template • Repeat the exercise for an interaction between one categorical and one continuous independent variable

22. 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.