1. Sociology 601 Class 26: December 1, 2009 (partial) Review –curvilinear regression results –cubic polynomial Interaction effects –example: earnings on married and gender –example: earnings on marital statuses and gender –example: earnings on age and gender –example: earnings on age and education F-tests comparing models Article example 1

2. Review: Regression with Curvilinearity 2

3. Example 1: Regression with Interaction, step 0 Regress earnings on gender and married/not married y i = β 0 + β 1 gender + β 2 married + e i both gender and married are dummy variables easier calculations if all dummy variables are 0/1 no interaction: assumes marriage has same association with (higher) earnings for both men and women. regress conrinc gender mar1 Source | SS df MS Number of obs = 1474 -------------+------------------------------ F( 2, 1471) = 82.24 Model | 8.5661e+10 2 4.2830e+10 Prob > F = 0.0000 Residual | 7.6612e+11 1471 520817334 R-squared = 0.1006 -------------+------------------------------ Adj R-squared = 0.0993 Total | 8.5178e+11 1473 578263951 Root MSE = 22821 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gender | -13867.11 1191.798 -11.64 0.000 -16204.91 -11529.3 mar1 | 5465.959 1192.441 4.58 0.000 3126.894 7805.025 _cons | 37785.12 1073.949 35.18 0.000 35678.49 39891.75 ------------------------------------------------------------------------------ married people (m&f) earn $5466 more than non married women (gender=1) earn $13,867 less than men 3

4. Example 1: Regression with Interaction, step 1 Separate regressions of earnings on married, by gender:. regress conrinc mar1 if gender==0 /* men */ Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 1, 723) = 31.29 Model | 1.9321e+10 1 1.9321e+10 Prob > F = 0.0000 Residual | 4.4645e+11 723 617501240 R-squared = 0.0415 -------------+------------------------------ Adj R-squared = 0.0402 Total | 4.6577e+11 724 643334846 Root MSE = 24850 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- mar1 | 10383.4 1856.279 5.59 0.000 6739.057 14027.74 _cons | 35065.27 1380.532 25.40 0.000 32354.94 37775.6 ------------------------------------------------------------------------------. regress conrinc mar1 if gender==1 /* women */ Source | SS df MS Number of obs = 749 -------------+------------------------------ F( 1, 747) = 0.26 Model | 106732224 1 106732224 Prob > F = 0.6129 Residual | 3.1118e+11 747 416578779 R-squared = 0.0003 -------------+------------------------------ Adj R-squared = -0.0010 Total | 3.1129e+11 748 416164546 Root MSE = 20410 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- mar1 | 755.3387 1492.253 0.51 0.613 -2174.17 3684.848 _cons | 26201 1038.855 25.22 0.000 24161.57 28240.42 ------------------------------------------------------------------------------ looks like marriage is associated with higher earnings more for men (+$10,383, p<001) than for women (+$755, n.s.) 4

5. Example 1: Regression with Interaction, step 2 to test whether the male and female coefficients are significantly different, we must calculate an interaction model: y i = β 0 + β 1 gender i + β 2 married i + β 3 gender i *married i + e i. gen byte margen=gender*mar1 (1 missing value generated). regress conrinc gender mar1 margen Source | SS df MS Number of obs = 1474 -------------+------------------------------ F( 3, 1470) = 60.89 Model | 9.4145e+10 3 3.1382e+10 Prob > F = 0.0000 Residual | 7.5764e+11 1470 515399826 R-squared = 0.1105 -------------+------------------------------ Adj R-squared = 0.1087 Total | 8.5178e+11 1473 578263951 Root MSE = 22702 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gender | -8864.271 1710.548 -5.18 0.000 -12219.65 -5508.897 mar1 | 10383.4 1695.885 6.12 0.000 7056.784 13710.01 margen | -9628.059 2372.993 -4.06 0.000 -14282.87 -4973.246 _cons | 35065.27 1261.245 27.80 0.000 32591.24 37539.3 ------------------------------------------------------------------------------ t(b 3 ) = -4.06; p<001; so marriage has different associations with earnings for men and women 5

6. Example 1: Regression with Interaction, step 2b results for the interaction model: y hat = $35,065 - $8,864*gender + $10,383*married - $9,628 *gender i *married Calculate average earnings for different types: The marriage effect: The marriage effect for men is 45448-35065 = 10383 = b 2 The marriage effect for women is 26956-26201 = 755 = b 2 + b 3 The gender effect: The gender effect for the not married is 26201-35065= -8864 = b 1 The gender effect for the married is 26956-45448 = -18492 = b 1 + b 3 b 3 = the difference in the marriage effect between men & women b 3 = the difference in the gender effect between the married & unmarried 6 constantgendermarriedmargentotal unmarried men =35065-8864*0+10383*0-9628*0*035065 unmarried women35065-8864*1+10383*0-9628*1*026201 married men =35065-8864*0+10383*1-9628*0*145448 married women =35065-8864*1+10383*1-9628*1*126956