1. Multiple Regression 3 Sociology 5811 Lecture 24 Copyright © 2005 by Evan Schofer Do not copy or distribute without permission

2. Announcements Schedule: –Today: More multiple regression Interaction terms, nested models –Next Class: Multiple regression assumptions and diagnostics Reminder: Paper deadline coming up soon! Questions about the paper?

3. Review: Dummy Variables Question: How can we incorporate nominal variables (e.g., race, gender) into regression? Option 1: Analyze each sub-group separately –Generates different slope, constant for each group Option 2: Dummy variables –“Dummy” = a dichotomous variables coded to indicate the presence or absence of something –Absence coded as zero, presence coded as 1.

4. Review: Dummy Variables Strategy: Create a separate dummy variable for all nominal categories Example: Race –Coded 1=white, 2=black, 3=other (like GSS) Make 3 dummy variables: –“DWHITE” is 1 for whites, 0 for everyone else –“DBLACK” is 1 for Af. Am., 0 for everyone else –“DOTHER” is 1 for “others”, 0 for everyone else Then, include two of the three variables in the multiple regression model.

5. Dummy Variables: Interpretation Visually: Women = blue, Men = red INCOME 100000800006000040000200000 HAPPY 10 9 8 7 6 5 4 3 2 1 0 Overall slope for all data points Note: Line for men, women have same slope… but one is high other is lower. The constant differs! If women=1, men=0: The constant (a) reflects men only. Dummy coefficient (b) reflects increase for women (relative to men)

6. Dummy Variables: Interpretation Ex: Job Prestige DBLACK coefficient of -2.66 indicates that African Americans have, on average, 2.6 points less job prestige compared to the reference group (in this case, white respondents).

7. Dummy Variables Comments: 1. Dummy coefficients shouldn’t be called slopes –Referring to the “slope” of gender doesn’t make sense –Rather, it is the difference in the constant (or “level”) 2. The contrast is always with the nominal category that was left out of the equation –If DFEMALE is included, the contrast is with males –If DBLACK, DOTHER are included, coefficients reflect difference in constant compared to whites.

8. Interaction Terms Question: What if you suspect that a variable has a totally different slope for two different sub- groups in your data? Example: Income and Happiness –Perhaps men are more materialistic -- an extra dollar increases their happiness a lot –If women are less materialistic, each dollar has a smaller effect on income (compared to men) Issue isn’t men = “more” or “less” than women –Rather, the slope of a variable (income) differs across groups

9. Interaction Terms Issue isn’t men = “more” or “less” than women –Rather, the slope of a variable coefficient (for income) differs across groups Again, we want to specify a different regression line for each group –We want lines with different slopes, not parallel lines that are higher or lower.

10. Interaction Terms Visually: Women = blue, Men = red INCOME 100000800006000040000200000 HAPPY 10 9 8 7 6 5 4 3 2 1 0 Overall slope for all data points Note: Here, the slope for men and women differs. The effect of income on happiness (X1 on Y) varies with gender (X2). This is called an “interaction effect”

11. Interaction Terms Examples of interaction: –Effect of education on income may interact with type of school attended (public vs. private) Private schooling has bigger effect on income –Effect of aspirations on educational attainment interacts with poverty Aspirations matter less if you don’t have money to pay for college Question: Can you think of examples of two variables that might interact? Either from your final project? Or anything else?

12. Interaction Terms Interaction effects: Differences in the relationship (slope) between two variables for each category of a third variable Option #1: Analyze each group separately Look for different sized slope in each group Option #2: Multiply the two variables of interest: (DFEMALE, INCOME) to create a new variable –Called: DFEMALE*INCOME –Add that variable to the multiple regression model.

13. Interaction Terms Consider the following regression equation: Question: What if the case is male? Answer: DFEMALE is 0, so b 2 (DFEM*INC) drops out of the equation –Result: Males are modeled using the ordinary regression equation: a + b 1 X + e.

14. Interaction Terms Consider the following regression equation: Question: What if the case is female? Answer: DFEMALE is 1, so b 2 (DFEM*INC) becomes b 2 *INCOME, which is added to b 1 –Result: Females are modeled using a different regression line: a + (b 1 +b 2 ) X + e –Thus, the coefficient of b 2 reflects difference in the slope of INCOME for women.

15. Interpreting Interaction Terms Interpreting interaction terms: A positive b for DFEMALE*INCOME indicates the slope for income is higher for women vs. men –A negative effect indicates the slope is lower –Size of coefficient indicates actual difference in slope Example: DFEMALE*INCOME. Observed b’s: –Income: b =.5 –DFEMALE * INCOME: b = -.2 Interpretation: Slope is.5 for men,.3 for women.

16. Interpreting Interaction Terms Example: Interaction of Race and Education affecting Job Prestige: DBLACK*EDUC has a negative effect (nearly significant). Coefficient of -.576 indicates that the slope of education and job prestige is.576 points lower for Blacks than for non-blacks.

17. Continuous Interaction Terms Two continuous variables can also interact Example: Effect of education and income on happiness –Perhaps highly educated people are less materialistic –As education increases, the slope between between income and happiness would decrease Simply multiply Education and Income to create the interaction term “EDUCATION*INCOME” And add it to the model.

18. Interpreting Interaction Terms How do you interpret continuous variable interactions? Example: EDUCATION*INCOME: Coefficient = 2.0 Answer: For each unit change in education, the slope of income vs. happiness increases by 2 –Note: coefficient is symmetrical: For each unit change in income, education slope increases by 2 Dummy interactions effectively estimate 2 slopes: one for each group Continuous interactions result in many slopes: Each value of education*income yields a different slope.=

19. Interpreting Interaction Terms Interaction terms alters the interpretation of “main effect” coefficients Including “EDUC*INCOME changes the interpretation of EDUC and of INCOME See Allison p. 166-9 –Specifically, coefficient for EDUC represents slope of EDUC when INCOME = 0 Likewise, INCOME shows slope when AGE=0 –Thus, main effects are like “baseline” slopes And, the interaction effect coefficient shows how the slope grows (or shrinks) for a given unit change.

20. Dummy Interactions It is also possible to construct interaction terms based on two dummy variables –Instead of a “slope” interaction, dummy interactions show difference in constants Constant (not slope) differs across values of a third variable –Example: Effect of of race on school success varies by gender African Americans do less well in school; but the difference is much larger for black males.

21. Dummy Interactions Strategy for dummy interaction is the same: Multiply both variables –Example: Multiply DBLACK, DMALE to create DBLACK*DMALE –Then, include all 3 variables in the model –Effect of DBLACK*DMALE reflects difference in constant (level) for black males, compared to white males and black females You would observe a negative coefficient, indicating that black males fare worse in schools than black females or white males.

22. Interaction Terms Comments: 1. If you make an interaction you should also include the component variables in the model: –A model with “DFEMALE * INCOME” should also include DFEMALE and INCOME There are rare exceptions. But when in doubt, include them 2. Sometimes interaction terms are highly correlated with its components That can cause problems (multicollinearity – which we’ll discuss next week).

23. Interaction Terms 3. Make sure you have enough cases in each group for your interaction terms –Interaction terms involve estimating slopes based on sub-groups in your data (e.g., black females). If you there are hardly any black females in the dataset, you can have problems.

24. Interaction Terms 4. Interaction terms are confusing at first… but they are VERY important –Example: Race, class, gender. –Most sociologists argue that they operate interactively The experience of black lower-class females is different from black upper-class females or white lower-class females Interaction terms are a powerful way of identifying such intersections in quantitative data –In short: Make the effort to consider how variables interact… it is a very useful way of thinking.

25. Nested Models It is common to conduct a series of multiple regressions, not just one –Each model adds variables or sets of variables This is useful for several reasons: –1. To show how coefficients change when new variables are added Which suggests which variables mediate others –2. To examine whether additional variables increase model fit (r-square).

26. Nested Models Question: Do the new variables substantially improve the model? Idea #1: Look for increase in Adjusted R-Square That gives you a sense of whether new variables “improve” the model Idea #2: Conduct a formal F-test –Recall that F-tests allow comparisons of variance (e.g., SSbetween to SSwithin) –This allows you to test the hypothesis that the added variables improve the model overall.

27. Nested Models F-tests require “nested models” –Models are the same, except for addition of new variables –You can’t compare totally different models this way Tests following Hypotheses: H0: Two models have the same R-square H1: Two models have different R-square

28. Nested Models SPSS can conduct an F-test between two regression models A significant F-test indicates: –The second model (with additional variables) is a significant improvement (in R-square) compared to the first.