1. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating interaction effects from OLS coefficients: Interaction between 1 categorical and 1 continuous independent variable Jane E. Miller, PhD

2. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview General equation for a model with main effects and interactions Coding of main effects and interaction terms Solving for the interaction pattern based on estimated coefficients – Intercept – Slope Graphical depiction of the sum of coefficients for particular combinations of the independent variables

3. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Review: Contingency of coefficients in an interaction model Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 1 _ X 2, Inclusion of the interaction term X 1 _ X 2 means that the β i s on the main effects terms X 1 and X 2 no longer apply to all values of X 1 and X 2. – The main effects and interactions β i s for X 1 and X 2 are contingent upon one another and cannot be considered separately.

4. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Review: Implications for interpreting main effects and interaction βs Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 1 _ X 2, In the interaction model: – β 1 estimates the effect of X 1 on Y when X 2 = 0, – β 2 estimates the effect of X 2 on Y when X 1 = 0, – β 3 must also be considered in order to calculate the shape of the overall pattern among X 1, X 2, and Y. E.g., when X 1 and X 2 take on other values.

5. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Review: Some possible patterns of association between IPR, race, and birth weight IPR BW IPR BW IPR BW IPR BW White Black No racial difference in IPR/BW relation: intercept and slope same for blacks & whites. Blacks & whites have same intercept but different slope of IPR/BW curves Blacks & whites have different slope and intercepts of IPR/BW curves Blacks & whites have same slope but different intercepts of IPR/BW curves

6. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. General equation for predicted value of DV based on an interaction model The general equation to calculate the predicted value of the dependent variable includes – main effects coefficients – interaction term coefficients – values of the independent variables = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR)

7. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating overall effect of interaction for specific case characteristics = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR) Each coefficient is multiplied by the value of the associated variable for cases with the characteristics of interest. To see which coefficients pertain to which cases, fill in values of variables for different combinations of race and the income-to-poverty ratio (IPR).

8. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Estimated coefficients β Intercept 3,106 Main effect terms Non-Hispanic black (NHB) –177 Income-to-poverty ratio (IPR) 23 Interaction term NHB_IPR –5–5 IPR = family income ($) / Federal Poverty Level for a family of that size and age composition. Reference category: Non-Hispanic whites.

9. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting the intercept The intercept β 0 from an OLS model is an estimate of the level of the dependent variable when continuous variables take the value 0, for infants in the reference category for all categorical variables. In a model where – The dependent variable is birth weight in grams. – The reference category is specified to be non-Hispanic white infants. β 0 is an estimate of birth weight when IPR = 0, for non- Hispanic white infants.

10. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Review: Coding of main effect and interaction term variables: race and income Case characteristics – SELECTED VALUES Variables Main effects termsInteraction term NHBIPRNHB_IPR Non-H white & IPR = 0.000.00 Non-H white & IPR = 0.500.50 Non-H white & IPR = 1.001.00 For a two-category race variable (non-Hispanic white = reference category). E.g., IPR = 0.5 means family income is half the Federal Poverty Level (FPL); IPR = 2.0 means family income is twice the FPL. Reference category

11. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the value of the intercept for one group NHBIPRNHB_IPR Non-H white & IPR = 0.000.0 = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR ) The intercept for non-Hispanic whites is calculated: = β 0 + (β NHB × 0) + (β IPR × 0.0) + (β NHB_IPR × 0.0) = β 0 Thus, the intercept for non-Hispanic white infants (when IPR = 0) collapses to include only β 0 because all of the other coefficients in the formula are multiplied by a value of 0.

12. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting the IPR/birth weight pattern IPR is a continuous variable – The coefficient is an estimate of the effect on the dependent for a 1-unit increase in the continuous IV, with categorical variables set to their reference category values. So β IPR estimates the increment in birth weight for every one-unit increase in IPR (e.g., from family income at the poverty line to twice the poverty line) – It is the slope of the IPR/birth weight curve for infants in the reference category, in this case, non-Hispanic white infants.

13. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating values for the IPR/birth weight curve for white infants NHBIPRNHB_IPR Non-H white & IPR = 1.501.50.0 = β 0 + (β NHB × 0) + (β IPR × 1.5) + (β NHB_IPR × 0) = β 0 + (β IPR × 1.5) Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (β IPR ) because the other coefficients are multiplied by 0. = β 0 + (β IPR × IPR) = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR)

14. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating values for the IPR/birth weight curve for white infants NHBIPRNHB_IPR Non-H white & IPR = 3.003.00.0 = β 0 + (β NHB × 0) + (β IPR × 3.0) + (β NHB_IPR × 0) = β 0 + β IPR × 3.0 = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR)

15. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting the race main effect The main effect β NHB estimates the difference in birth weight between non-Hispanic black infants and those in the reference category (non- Hispanic whites), when continuous variables are set at the value 0. It is an estimate of the difference in intercept between black and white infants when IPR is 0.

16. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the intercept for different values of the categorical variable NHBIPRNHB_IPR Non-H white & IPR = 0.000.0 NHBIPRNHB_IPR Non-H black & IPR = 0.010.0 As we saw a moment ago, for the intercept for non-Hispanic whites is calculated: = β 0 + (β NHB × 0) + (β IPR × 0.0) + (β NHB_IPR × 0.0) = β 0 For non-Hispanic blacks, the intercept is calculated: = β 0 + (β NHB × 1) + (β IPR × 0.0) + (β NHB_IPR × 0.0) = β 0 + β NHB

17. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. More on the race main effect It is an estimate of the difference in intercept between black and white infants when IPR is 0. = β 0 + β NHB = 3,106 + (– 177) = 2,929 In other words, black infants born to families with an IPR of zero have a predicted birth weight of 2,929 grams. – or 177 grams LOWER than that of their white counterparts.

18. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating values for the IPR/birth weight curve for white infants Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (β IPR ) because the other coefficients are multiplied by 0. = β 0 + (β NHB × NHB) + (β IPR × IPR) + (β NHB_IPR × NHB_IPR) = β 0 + (β NHB × 0) + (β IPR × IPR) + (β NHB_IPR × 0) = β 0 + (β IPR × IPR)

19. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating values for the IPR birth weight curve for black infants NHBIPRNHB_IPR Non-H black & IPR = 1.511.5 = β 0 + (β NHB × 1) + (β IPR × 1.5) + (β NHB_IPR × 1.5) For Non-Hispanic blacks, the equation includes all three terms (β NHB, β IPR, and β NHB_IPR ) because each of those coefficients is multiplied by a non-zero value.

20. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting the coefficient on the interaction between race and IPR The slope – for blacks = β IPR + β NHB_IPR = 23 + (–5) = 18 – for whites = β IPR = 23 The race_IPR coefficient tests whether the slope of the IPR/birth weight pattern is different for non-Hispanic black infants than for their non- Hispanic white counterparts. – β NHB_IPR is thus the estimated difference in slope for blacks compared to whites.

21. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. More on the race/IPR interaction The estimated coefficients mean that each 1- unit increase in IPR is associated with  23 grams more birth weight among non-Hispanic white infants.  18 grams more birth weight among non-Hispanic black infants.  Thos values are the slopes of the respective IPR/BW curves for the two racial/ethnic groups.

22. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Preparing to graph the slope of IPR/birthweight by race For infants in the reference category (non- Hispanic white), – Multiply selected values of IPR by β IPR and add to β 0 to obtain predicted birth weight at interesting values of IPR. For non-Hispanic black infants, – Multiply selected values of IPR by β IPR + β NHB_IPR then add to β 0 + β NHB.

23. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculated birth weight by race for selected values of IPR IPR (family income in multiples of the FPL) Non-Hispanic whiteNon-Hispanic black FormulaResultFormulaResult 0 = β 0 + 0 × β IPR = 3,106 + 0×23 3,106 = β 0 + β NHB + 0 × (β IPR + β NHB_IPR ) = 3,106 – 177 + 0 × (23 – 5) 2,929 1 = β 0 + 1× β IPR = 3,106 + 1×23 = 3,106 + 23 3,129 = β 0 + β NHB + 1 × (β IPR + β NHB_IPR ) = 3,106 – 177 + 1 × (23 – 5) = 2,929 + 1 × (18) = 2,929 + 18 2,947 … 6 = β 0 + 6 × β IPR = 3,106 + 6×23 = 3,106 + 390 3,244 = β 0 + β NHB + 6 × (β IPR + β NHB_IPR ) = 3,106 – 177 + 6 × (23 – 5) = 2,929 + 6 × (18) = 2,929 + 108 3,037 β 0 = 3,106; β IPR = 23; β NHB = –177; β NHB_IPR = –5

24. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Use a spreadsheet to calculate and graph the interaction Spreadsheets can – Store The estimated coefficients The input values of the independent variables The correct generalized formula to calculate the predicted values for many combinations of the IVs involved in the interaction – Graph the overall pattern See spreadsheet template and voice-over explanation

25. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. 3,200 3,100 3,000 2,900 2,800 * Ref cat = Reference category = non-Hispanic white infants. = β 0 + β NHB = 3,106 + ( – 177) = 2,929 = intercept for black infants 142 3,300 Birth weight (grams) IPR 0 = β 0 = intercept = 3,106 = predicted BW for ref cat * = β IPR = 23 = slope of IPR/ BW curve for ref cat * = β IPR + β NHB_IPR = 23 – 5 = 18 = slope of IPR/ BW curve for non-Hispanic black infants Predicted birth weight by race/ethnicity and IPR 6

26. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overall shape of the race/IPR/ birth weight pattern Based on this set of βs, black infants have – a lower birth weight than whites at all IPR levels. Negative coefficient on the NHB main effect yields a lower intercept for blacks than for whites. – a slower rate of birth weight increase as IPR rises. Negative coefficient on NHB_IPR, which yields a shallower slope of the IPR/birth weight curve for blacks than for whites. Thus the deficit in birth weight for blacks widens with increasing IPR.

27. 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.

28. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Summary An interaction between a continuous and a categorical independent variable will yield differences in the intercept and/or slope of the association between the continuous IV and the DV. Calculating the overall shape of an interaction requires adding together the pertinent main effects and interaction term βs for combinations of the categorical IV and selected values of the continuous IV in the interaction. – A spreadsheet can be helpful for storing and organizing the βs, input values, and formulas.

29. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested resources Chapters 9 and 16 of Miller, J.E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Chapters 8 and 9 of Cohen et al. 2003. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition. Florence, KY: Routledge.

30. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Supplemental online resources Podcasts – Introduction to interactions – Creating variables to test for interactions – Specifying models to test for interactions – Interpreting multivariate regression coefficients Spreadsheet template for calculating overall effect of an interaction between a categorical and a continuous independent variable.

31. 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. – Question #4 in the problem set for Chapter 16 – Suggested course extensions for Chapter 16 “Applying statistics and writing” exercise #2.

32. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html