Addressing Alternative Explanations: Multiple RegressionGore Likeability ExampleDemocratic pictureIndependent pictureRepublican pictureCombined data pictureCombined data picture with regressionCombined data picture with “true” regression lines overlaidTempting yet wrong normalizationsSummary: Why we controlGore vs. Clinton3D Relationship3D Linear RelationshipThe Linear Relationship between Three VariablesThe Slope CoefficientsThe Slope Coefficients More SimplyThe Matrix formConsider two regression coefficientsSeparate regressionsWhy did the Clinton Coefficient change from 0.62 to 0.51The CalculationsAccounting for total effectsAccounting for the total effectAccounting for the total effects in the Gore thermometer exampleThe OutputOther approaches to addressing confounding effects?Drinking and Greek Life ExampleDependent variable: Times Drinking in Past 30 DaysSlide 29Slide 30Three RegressionsThe PictureAccounting for the effects of frat house living and Greek membership on drinkingAddressing Alternative Explanations: Multiple Regression17.871Spring 2007Gore Likeability ExampleDid Clinton hurt Gore in the 2000 election?How would you test this?What alternative explanations would you need to address?Other examples of alternative explanations based on omitted variables?Democratic pictureClinton thermometerIndependent pictureClinton thermometerRepublican pictureClinton thermometerCombined data pictureClinton thermometerCombined data picture with regressionClinton thermometerCombined data picture with “true” regression lines overlaidClinton thermometerTempting yet wrong normalizationsClinton thermometerClinton thermometerSubtract the Goretherm. from theavg. Gore therm. scoreSubtract the Clintontherm. from theavg. Clinton therm. scoreSummary: Why we controlAddress alternative explanations by removing confounding effectsImprove efficiencyGore vs. ClintonGore thermometerGraphs by Party (3 point scale)Clinton thermometer Gore thermometer pyp Fitted valuesparty3==-10100party3==00 100party3==10 1000100OverallWithin partyRep.Ind.Dem.3D Relationship3D Linear RelationshipThe Linear Relationship between Three VariablesiiiiXXY,22,110The Slope CoefficientsniiniiiniiniiiniiniiiniiniiiXXXXXXXXXXYYXXXXXXXXXXYY12,221,22,11112,221,12212,111,22,11212,111,111)())((ˆ- )())((ˆand )())((ˆ- )())((ˆThe Slope Coefficients More Simply)var(),cov(ˆ- )var(),cov(ˆand)var(),cov(ˆ- )var(),cov(ˆ22112221212111XXXXYXXXXXYXThe Matrix formy1y2…yn1 x1,1x2,1… xk,11 x1,2x2,2… xk,21 … … … …1 x1,nx2,n… xk,n ( )X X X y1Consider two regression coefficients)var(),cov(ˆ- )var(),cov(ˆ vs.)var(),cov(ˆ1212111111XXXXYXXYXMMBWhen does ? Obviously, when 0)var(),cov(ˆ1212XXXMMB11ˆˆSeparate regressions(1) (2) (3)Intercept 23.1 55.9 28.6Clinton 0.62 -- 0.51Party -- 15.7 5.8Why did the Clinton Coefficient change from 0.62 to 0.51. corr gore clinton party,cov(obs=1745) | gore clinton party3-------------+--------------------------- gore | 660.681 clinton | 549.993 883.182 party3 | 13.7008 16.905 .8735The Calculations5122.01105.06227.0182.883905.167705.5182.883993.549)var(),cov(ˆ)var(),cov(ˆ6227.0182.883993.549)var(),cov(ˆ211clintonpartyclintonclin t onclintongoreclin t onclintongoreMMB. corr gore clinton party,cov(obs=1745) | gore clinton party3-------------+--------------------------- gore | 660.681 clinton | 549.993 883.182 party3 | 13.7008 16.905 .8735Accounting for total effectsMMMBMMBMMXXXXYX21211212111212111ˆˆ ˆˆ- ˆˆ)var(),cov(ˆ- )var(),cov(ˆ(i.e., regression coefficientwhen we regress X2 (as dep. var.)on X1 (as ind. var.)Accounting for the total effect21211ˆˆ ˆMMB21Total effect = Direct effect + indirect effectYX1X2M2ˆM1ˆ Accounting for the total effects in the Gore thermometer exampleEffect Total Direct IndirectClinton 0.62 0.51 0.11Party 15.7 5.8 9.9The Output. reg gore clinton party3 Source | SS df MS Number of obs = 1745-------------+------------------------------ F( 2, 1742) = 1048.04 Model | 629261.91 2 314630.955 Prob > F = 0.0000 Residual | 522964.934 1742 300.209492 R-squared = 0.5461-------------+------------------------------ Adj R-squared = 0.5456 Total | 1152226.84 1744 660.68053 Root MSE = 17.327------------------------------------------------------------------------------ gore | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- clinton | .5122875 .0175952 29.12 0.000 .4777776 .5467975 party3 | 5.770523 .5594846 10.31 0.000 4.673191 6.867856 _cons | 28.6299 1.025472 27.92 0.000 26.61862 30.64119------------------------------------------------------------------------------Other approaches to addressing confounding effects?ExperimentsDifference-in-differences designsOthers?Is regression the best approach to addressing confounding effects?ProblemsDrinking and Greek Life ExampleWhy is there a correlation between living in a fraternity/sorority house and drinking?Greek organizations often emphasize social gatherings that have alcohol. The effect is being in the Greek organization itself, not the house.There’s something about the House environment itself.Dependent variable: Times Drinking in Past 30 Days. infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493weight99 494-512 using da3818.dat,clear(14138 observations read). recode timespast30 timeshs (1=0) (2=1.5) (3=4) (4=7.5) (5=14.5) (6=29.5) (7=45)(timespast30: 6571 changes made)(timeshs: 10272 changes made). replace timespast30=0 if screen<=3(4631 real changes made). tab timespast30timespast30 | Freq. Percent Cum.------------+----------------------------------- 0 | 4,652 33.37 33.37 1.5 | 2,737 19.64
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