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 controlLook at actual dataLook at actual data (jitter)Gore vs. ClintonGore vs. partyBack to the basic data3D Relationship3D Linear Relationship3D Relationship: Clinton3D Relationship: partyThe Linear Relationship between Three VariablesThe Slope CoefficientsThe Slope Coefficients More SimplyThe InterceptThe Matrix formWhat Difference Does This Make?Consider 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 OutputDrinking and Greek Life ExampleDependent variable: Times Drinking in Past 30 DaysSlide 36Slide 37Three RegressionsThe PictureAccounting for the effects of frat house living and Greek membership on drinkingMultiple Regression17.871Spring 2006Gore Likeability Example•Suppose:–Gore’s* likeability is a function of Clinton’s likeability and not directly a function of party–Clinton’s likeability is a function of one’s partisan identification plus other factors–What would the regression of Gore likeability on Clinton likeability look like?Party IDClinton LikeabilityGoreLikeabilitye1e2*This example probably works better if we’re predicting the likeability of Socks the cat.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 control•Remove confounding effects•Improve efficiencyPartyClintonlikeabilityGorelikeabilityeePartyClintonlikeabilityGorelikeabilityeClinton thermometerLook at actual dataGore thermometer01000 1000 100Clintonthermometer0100-1 1-11Party (3 pointscale)graph7 clinton gore party3, matrixLook at actual data (jitter)Gore thermometer01000 1000 100Clintonthermometer0100-1 1-11Party (3 pointscale)graph7 gore clinton party3, matrix jitter(5)Gore vs. ClintonGore thermometerGraphs by Party (3 point scale)Clinton thermometer Gore thermometer pyp Fitted valuesparty3==-10100party3==00 100party3==10 1000100OverallWithin partyRep.Ind.Dem.Gore vs. partyGore thermometerGraphs by clinton_levelParty (3 point scale) Gore thermometer Fitted values py_party3_allclinton_level==-10100clinton_level==0-1 1clinton_level==1-1 10100OverallWithin partyClinton lowClinton med.Clinton highBack to the basic dataGore thermometer01000 1000 100Clintonthermometer0100-1 1-11Party (3 pointscale)3D Relationship3D Linear Relationship3D Relationship: Clinton0501003D Relationship: partyRepIndDemThe 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 Intercept2211022110 ....rearrange.you if that Note- XXYXXYNote: Add “hats” (^) over all the Greek lettersThe Matrix formy1y2…yn1 x1,1x2,1… xk,11 x1,2x2,2… xk,21 … … … …1 x1,nx2,n… xk,n ( )X X X y1What Difference Does This Make?One Regression vs. a separate regression for each independent variableConsider 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(ˆ211clintonpartyclintonclintonclin tongoreclintonclin tongoreMMB. 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
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