1. Sociology 601 Class 28: December 8, 2009 Homework 10 Review –polynomials –interaction effects Logistic regressions –log odds as outcome –compared to linear model predicting p –odds ratios vs. log odds coefficients –inferential statistics: Wald test –maximum likelihood models and likelihood ratios –loglinear models for categorical variables: e.g. 5x5 mobility table –logit models for ordinal variables –event history models 1

2. Sociology 601 Class 28: December 8, 2009 Homework 10 (Thursday) Review: interaction effects F-tests –review: for full equation –for partial model Multicollinearity –example: state murder rates –not an issue for polynomials, multiplicative interactions Next class: review –email us topics you want reviewed! 2

3. Review: Regression with Interaction effects 3 Two approaches: separate regressions by groups (e.g., two regressions one for men and one for women) multiplicative interaction term in one regression same estimates for effects in either way multiplicative interaction term provides a significance test of difference multiplicative interaction term less easily interpreted Multiplicative interaction models types: categorical (e.g., gender, race) or interval (e.g., age) first, main concern: is interaction coefficient statistically significant? “component” coefficients are just estimates when the other component = zero plotting helps

4. Inferences: F-tests Comparing models 4 Comparing Regression Models, Agresti & Finlay, p 409: Where: R c 2 = R-square for complete model, R r 2 = R-square for reduced model, k = number of explanatory variables in complete model, g = number of explanatory variables in reduced model, and N = number of cases.

5. Example: F-tests Comparing models 5 Complete model: men’s earnings on age, age square, age cubed, education, and currently married dummy. Reduced model: men’s earnings on education and currently married dummy. F-test comparing model is whether age variables, as a group, have a significant relationship with earnings after controls for education and marital status

6. Example: F-tests Comparing models 6 Complete model: men’s earnings. regress conrinc age agesq agecu educ married if sex==1 Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 5, 719) = 45.08 Model | 1.1116e+11 5 2.2233e+10 Prob > F = 0.0000 Residual | 3.5461e+11 719 493199914 R-squared = 0.2387 -------------+------------------------------ Adj R-squared = 0.2334 Total | 4.6577e+11 724 643334846 Root MSE = 22208 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | 5627.049 8127.377 0.69 0.489 -10329.18 21583.27 agesq | -75.30909 210.0421 -0.36 0.720 -487.6781 337.0599 agecu |.1985975 1.768176 0.11 0.911 -3.272807 3.670003 educ | 3555.331 317.9738 11.18 0.000 2931.063 4179.599 married | 8664.627 1690.098 5.13 0.000 5346.51 11982.74 _cons | -127148.4 102508.3 -1.24 0.215 -328399.8 74103.01 ------------------------------------------------------------------------------ Note: none of the three age coefficients are, by themselves, statistically significant. R c 2 =.2387; k = 5.

7. Example: F-tests Comparing models 7 Reduced model: men’s earnings. regress conrinc educ married if sex==1 Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 2, 722) = 80.20 Model | 8.4666e+10 2 4.2333e+10 Prob > F = 0.0000 Residual | 3.8111e+11 722 527850916 R-squared = 0.1818 -------------+------------------------------ Adj R-squared = 0.1795 Total | 4.6577e+11 724 643334846 Root MSE = 22975 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- educ | 3650.611 328.1065 11.13 0.000 3006.454 4294.767 married | 10721.42 1716.517 6.25 0.000 7351.457 14091.38 _cons | -16381.3 4796.807 -3.42 0.001 -25798.65 -6963.944 ------------------------------------------------------------------------------ R r 2 =.1818; g = 2.

8. Inferences: F-tests Comparing models 8 F = ( 0.2387 – 0.1818) / (5 – 2)df 1 =5-2; df 1 =725-6 ( 1 -.2387) / (725 – 6) = 0.0569/3 0.7613/719 = 26.87, df=(3,719), p <.001 (Agresti & Finlay, table D, page 673)

9. Multicollinearity (A&F 14.3) 9 “ Redundant” variables large standard errors loss of statistical significance variable 1 is significant in Model 1 variable 2 is significant in Model 2 neither 1 nor 2 is significant in Model 3 including both variables. sometimes: strange sign of coefficient sometimes: magnitude jumps unrealistically problem is not enough cases high on 1 and low on 2 and vice-versa. Every case that is high on 1 is also high on 2. So, you can’t separate the two effects in this sample.

10. Multicollinearity: Solutions 10 Choose one (and footnote the other) Get a bigger or better sample If both variables are alternate measures of the same concept, make a scale.

11. Multicollinearity: Not a problem always 11 Only if you are trying to separate the effects of variable 1 and variable 2 what is the effect of variable 1 holding variable 2 constant? Not an issue if: polynomials multiplicative interaction effects

12. Next: Review for Final 12 Please email us any topics you want reviewed!