1. Sociology 601 Class 23: November 17, 2009 Homework #8 Review –spurious, intervening, & interactions effects –stata regression commands & output F-tests and inferences (A&F 11.4) 1

2. Review: Types of 3-variable Causal Models Spurious x 2 causes both x 1 and y e.g., age causes both marital status and earnings Intervening x 1 causes x 2 which causes y e.g., marital status causes more hours worked which raises annual earnings No statistical difference between these models. Statistical interaction effects: The relationship between x 1 and y depends on the value of another variable, x 2 e.g., the relationship between marital status and earnings is different for men and women. 2

3. Review: Causal Models with earnings & marital status bivariate relationship: 1.married earnings spuriousness: 2. married earnings age intervening: 3. married hoursearnings interaction effect: 4.married earnings gender 3

4. Review: Stata Commands describe summarize tab tab xcat, sum(yvar) drop if / keep if gen / replace ttest regress predict / predict, residuals histogram / scattergram graph box yvar, over(xvar) 4

5. Review: Regression models using Stata see: http://www.bsos.umd.edu/socy/vanneman/socy601/conrinc.do 5

6. Review: Regression models with Earnings, Marital status and Age bivariate relationship:. * association of earnings and marital status:. regress conrinc married 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] -------------+---------------------------------------------------------------- married | 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 ------------------------------------------------------------------------------. spuriousness (partial):. * age makes the marriage-earnings relationship partly spurious:. regress conrinc married age Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 2, 722) = 36.20 Model | 4.2454e+10 2 2.1227e+10 Prob > F = 0.0000 Residual | 4.2332e+11 722 586315863 R-squared = 0.0911 -------------+------------------------------ Adj R-squared = 0.0886 Total | 4.6577e+11 724 643334846 Root MSE = 24214 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- married | 8243.081 1840.613 4.48 0.000 4629.489 11856.67 age | 702.0977 111.7749 6.28 0.000 482.6551 921.5403 _cons | 8836.284 4387.025 2.01 0.044 223.4344 17449.13 ------------------------------------------------------------------------------ 6

7. Review: Regression models with Earnings, Marital status and Hours Worked Intervening variable relationship (hours worked):. * hours worked explains some of how marital status increases earnings:. regress conrinc married age hrs1 Source | SS df MS Number of obs = 664 -------------+------------------------------ F( 3, 660) = 25.02 Model | 4.4322e+10 3 1.4774e+10 Prob > F = 0.0000 Residual | 3.8970e+11 660 590458672 R-squared = 0.1021 -------------+------------------------------ Adj R-squared = 0.0980 Total | 4.3402e+11 663 654637868 Root MSE = 24299 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- married | 7328.527 1934.225 3.79 0.000 3530.551 11126.5 age | 631.5836 117.8463 5.36 0.000 400.1848 862.9824 hrs1 | 281.3472 71.47315 3.94 0.000 141.0051 421.6894 _cons | -232.1376 5465.426 -0.04 0.966 -10963.86 10499.58 ------------------------------------------------------------------------------ But: problem with N! Create new hours worked:. gen hrs=hrs1 (101 missing values generated). replace hrs=hrs2 if hrs1>=. (24 real changes made, 2 to missing). replace hrs=0 if hrs1>=. & wrkstat>=3 (101 real changes made) 7

8. Review: Regression models with Earnings, Marital status and Hours Worked Intervening variable relationship (revised hours worked):. regress conrinc married age hrs Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 3, 721) = 36.27 Model | 6.1081e+10 3 2.0360e+10 Prob > F = 0.0000 Residual | 4.0469e+11 721 561294582 R-squared = 0.1311 -------------+------------------------------ Adj R-squared = 0.1275 Total | 4.6577e+11 724 643334846 Root MSE = 23692 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- married | 7465.107 1805.967 4.13 0.000 3919.526 11010.69 age | 640.1643 109.891 5.83 0.000 424.4197 855.9089 hrs | 278.3368 48.31685 5.76 0.000 183.4783 373.1954 _cons | -493.7634 4587.79 -0.11 0.914 -9500.786 8513.259 ------------------------------------------------------------------------------ b(married) reduced to 7465.1 from 8243.1 (N= 725 for both) 8

9. Review: Regression models with Earnings Marital status, Age, and Hours worked. 9 Model 0Model 1Model 2xModel 2 Married10,383.4***8,243.1***7,328.5***7,465.1*** Age702.1***631.6***640.2*** Hours worked281.3***278.3*** Constant35,065.3***8,836.3*-232.1n.s.-493.8n.s. N725 664725 R-square0.0420.0910.1020.133

10. Review: Regression models with Earnings and Marital status, separately by Gender Statistical Interaction Effect:. * association of earnings and marital status for men:. regress conrinc married if sex==1 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] -------------+---------------------------------------------------------------- married | 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 ------------------------------------------------------------------------------. * association of earnings and marital status for women:. regress conrinc married if sex==2 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] -------------+---------------------------------------------------------------- married | 755.3387 1492.253 0.51 0.613 -2174.17 3684.848 _cons | 26201 1038.855 25.22 0.000 24161.57 28240.42 ------------------------------------------------------------------------------ 10

11. Inferences: F-tests of global model H o : β 1 = β 2 =... β k = 0 α or β 0 ? F-tests of H 0 : Calculate new test statistic, F ratio of “explained variance” / “unexplained variance” F-distribution: ratio of chi-square distributions df 1 (numerator); df 2 (denominator) if df 1 =1, then F = t 2 Table D, pages 671-673 Global F-test less useful (almost always significant unless you have a really bad model or very small N). Base for F-test comparing regression models (later) 11

12. F-test: Method 1, STATA output. regress conrinc married age hrs1 Source | SS df MS Number of obs = 725 -------------+------------------------------ F( 3, 721) = 36.27 Model | 6.1081e+10 3 2.0360e+10 Prob > F = 0.0000 Residual | 4.0469e+11 721 561294582 R-squared = 0.1311 -------------+------------------------------ Adj R-squared = 0.1275 Total | 4.6577e+11 724 643334846 Root MSE = 23692 ------------------------------------------------------------------------------ conrinc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- married | 7465.107 1805.967 4.13 0.000 3919.526 11010.69 age | 640.1643 109.891 5.83 0.000 424.4197 855.9089 hrs | 278.3368 48.31685 5.76 0.000 183.4783 373.1954 _cons | -493.7634 4587.79 -0.11 0.914 -9500.786 8513.259 ------------------------------------------------------------------------------ df 1 = 3 (= k = # parameters = β (married), β (age), β (hrs) ) df 2 = 721 [ = N – (k+1) = 725 – (3+1) ] F (3,721) = 2.60 (α =.05); 36.27 >> 2.60 12

13. F-test: Method 2, using R-square 13

14. F-test: Method 3, using SSE and Model SS 14 F = 2.0360e+10 / 561294582 = 36.27

15. Inferences: β i 15 H 0 : β i = 0 what we are usually most interested in test statistic:

16. Next: Regression with Dummy Variables 16 Agresti and Finlay 12.3 (skim 12.1-12.2 on analysis of variance) Example: marital status, 3 categories currently married never married widowed separated divorced