Sociology 602 Martin Week 10, April 9, 2002

1. Sociology 602 (Martin)Week 10, April 9, 2002 Adding polynomial variables to a model NKNW 7.7 Adding interaction terms to a model NKNW 7.8 Criteria for building a model general approaches NKNW 8.1 all-possible regressions procedures for NKNW 8.3 model building forward stepwise regression NKNW 8.4

2. Polynomial models (NKNW 7.7) Form of a polynomial model: Yhat = bo + b1X1 + b2X12 + b2X12 + b3X13 + … When to use a polynomial model: when you have a theoretical presumption that the response function is a polynomial function (example: the distance an object falls as a function of time). when the response function is complex or unknown, but it fits pretty well to a polynomial function. (example: death rates as a function of age)

3. Second-order polynomials Second-order polynomials are the commonest kind; they include only a squared term: Yhat = bo + b1X1 + b2X12 Example: girls’ height in inches as a function of age, for ages 2-12 Yhat = 20 + 3*X1 – 0.2X12 Predict the height of a girl at age 2, 5, 8, and 11.

4. Graphing second-order polynomials Second-order polynomials always reflect a response function with a single curve. The linear (first order) term describes the general trend as upward or downward, for values of X near 0. The squared (second-order) term describes the curvature as upward or downward. Sketch these examples Yhat = 20 + 3*X1 + 0.2X12 Yhat = 20 + 3*X1 – 0.2X12 Yhat = 20 – 3*X1 + 0.2X12 Yhat = 20 – 3*X1 – 0.2X12

5. Graphing higher-order polynomials Third-order polynomials describe response functions where the curvature changes over time. Yhat = 20 + 3*X1 - 0.2X12 + 0.07X13 Higher-order polynomials describe response functions where the change in the curvature changes over time, as in bimodal distributions. Yhat = 20 + 3*X1 - 0.2X12 + 0.07X13 - 0.01X14 You rarely see high order polynomials in social research.

6. Warnings for polynomial regression Polynomial regression can be a good way to explain error related to important control variables. However, polynomial regression creates at least four problems: 1.) It is difficult to interpret any of the coefficients related to the variable with the polynomial specification. 2.) Each “order” in a polynomial regression eats up a degree of freedom. 3.) Polynomial terms tend to be highly collinear. 4.) The model becomes highly unstable at extreme x-values, and don’t even think of extrapolating.

7. Model building with polynomial regression The standard order for adding polynomial terms to a model is to start with the first order term, add the second order term if necessary, and so on. Never use a model that has a higher order term for X, but is missing a lower-order term for X, even if an F-test indicates otherwise.

8. Interaction regression models (NKNW 7.8) Form of an interaction model: Yhat = b0 + b1X1 + b2X2 + b3X1X2 When to use an interaction model: When it appears that the effect of X1on Y varies with the value of X2. When you wish to test whether the effect of X1on Y varies with the value of X2.

9. Dichotomous interaction terms The easiest type of interaction occurs when both X1 and X2 are scaled as dichotomous variables Example: what is the effect of college attendance and gender on income? Data coding: X1(male) X2(college) X1*X2 female, college 0 1 0*1=0 female, no college 0 0 0*0=0 male, college 1 1 1*1=1 male, no college 1 0 1*0=0 Note: three coefficients for four categories (which category does the intercept describe?)

10. Dichotomous interaction coefficients In the income example with X1 and X2 both dichotomous, Yhat = b0 + b1X1 + b2X2 + b3X1X2 We can interpret the coefficients using a 2X2 table: