Multiple Regression 2
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1 Overvew Multple Regresson Dr Tom Ilvento Department of Food and Resource Economcs We wll contnue our dscusson of Multple Regresson wth a specal focus on lookng at Resduals Resdual Analyss looks at the error terms n the model The error term, or resdual, shows the part not explaned by our model Resduals can often pont to problems n our model We wll revst dummy varables n the model when there are other ndependent varables n the model We wll look at a short, but complete analyss to gve you an dea of how multple regresson can mprove our nsghts nto models Resduals n Regresson A Probablstc has a determnstc component and a random error component, denoted as e or! The model s our expectaton of the relatonshp. Ths s the determnstc part of the model The error component s very mportant Observed n populaton/sample Predcted from model Y! = " + " X + E( Y ) =! +! X Y = " + " X +! ) Y =! o +! X! = Y " ) The dfference between what we observe and what we predct Y Resduals or s The error terms wll always have a mean of zero Some wll be postve and some negatve We want all the pattern n the data to be represented by the model Excel or other software wll generate resduals and Predcted Y We can plot resduals versus any X n the model Or aganst Predcted Y It s usually best to work wth Standardzed Resduals (also called Studentzed) These are normed to have a mean of zero and standard devaton of Standardzed Resduals should be wthn Bvarate Ft of PRICE By PRICE Lnear Ft 5 5 ± standard devatons 4
2 Resdual Analyss The resduals are an mportant part of regresson Shows the ft of the model s the overall varance of the regresson Factors nto the standard errors for coeffcents Resdual Analyss shows the ft of the model and possble volatons n assumptons We want a random pattern wth standardzed resduals that are no more than ± standard devatons Look for patterns n resduals to gve clues Bad ft Outlers that don t ft the model well Volaton of assumptons Resdual Analyss can be thought of as a dagnostc tool to assess the aptness of your model 5 Std Resduals Std Resduals Let s Look at Some Resduals from s Remember, we want to see a random pattern Standardzed Resduals Versus Predcted LNSALES Predcted LNSALES Standardzed Resduals Versus Predcted Y Predcted Y Std Resduals Std Resduals vs Predcted Y Predcted Y Standardzed Resduals versus Predcted Prce Good!!! Data wth a tme element often show a resdual pattern Data wth a tme element - Tme Seres - often have error terms that are correlated wth each other The resdual n tme t+ s related to the error term n tme t Std Resdual - - Tme Seres Resdual Plot Predcted Y 7 Catalog Sales Data We want to look at the relatonshp between Sales and Salary We know there s a strong relatonshp, but the varance ncreases wth ncreasng salary, We also know there are extreme values for salary and for sales As a result, resduals look bad and show heteroscadastcty Look what happens when we transform the data usng the natural log of both varables Sometmes a transformaton can solve the problem of a pattern n resduals 8
3 Let s revst the Apartment Data The model to the left shows the regresson results that were dscussed the prevous lecture The resdual plot below and the R =.98 shows we have a very good ft to ths model Next we wll add dummy varables to reflect the condton of the apartment buldng Std Resd By Predcted PRICE Studentzed Resd PRICE Predcted PRICE Response PRICE Actual by Predcted Plot PRICE Actual PRICE Predcted P<. RSq=.98 RMSE=.8 Adj Root #APTS AREA PARKING AGE Parameter s Std *.*.74.6.* 9 Dummy varables n Multple Regresson We can also nclude dummy varables nto the multple regresson model When other varables are n the model, the ntercept now captures more than just the reference group But the nterpretaton of the coeffcents for each dummy varable s stll the same - how s that group dfferent from the reference group? The test test for the dummy varables s a test to see f there s a sgnfcant dfference of that group wth the reference group, holdng all other factors constant Controllng for other varables makes ths test more rgorous Apartment Data Let s see how our model changes when we add dummy varables of the condton of the apartment Ths s an expert judgement that the apartment condton s Excellent (DUME) Good (DUMG) Far we wll let Far be the reference category Now PRICE (per $,) s a functon of the followng attrbutes of the apartment buldng #APTS AREA PARKING AGE DUMG DUME Ths model has a good ft and the coeffcents are n the expected drecton and PARKING are nsgnfcant and not mportant Regresson of PRICE on Apartment Attrbutes Actual by Predcted Plot PRICE Actual PRICE Predcted P<. RSq=.98 RMSE=.8 Adj Root #APTS AREA PARKING AGE Parameter s Std Compare s *.*.74.6.* R mproved slghtly to.988 PARKING s now margnally sgnfcant DUME s postve and sgnfcant PRICE ncludng Condton Actual by Predcted Plot PRICE Actual PRICE Predcted P<. RSq=.99 RMSE=8.7 Adj Root #APTS AREA PARKING AGE DumG DumE Parameter s Std *.6* *.5* *
4 Compare the coeffcents Table : Comparson of Two s of PRICE on Apartment Attrbutes #APTS 4.45** *** AREA.55***.***..4 PARKING ** AGE -.85** -.885** DUMG 7.4 DUME 56.8** R * p <.; ** p<.5; ***p<. ) Y = $ $4.4(# APTS) + $.55(AREA) + $.() + $.696(PARKING)! $.85(AGE) ) Y = $7.89 +$5.645(# APTS) + $.(AREA) + $.() + $.99(PARKING)! $.89(AGE) + $56.(Excellent) + $7.4(Good) s the frst model There s a very good ft to ths data AREA, #APTS, and AGE are sgnfcant n the model adds n the Condton Dummy Varables R s larger at.988 #APTS coeffcent became larger and s more sgnfcant AREA s reduced slghtly PARKING s larger and sgnfcant DUME s postve and sgnfcant and adds about $56k n value The F-Test Hypothess Test for a Regresson Ho: Ha: Assumptons Test Statstc Rejecton Regon Concluson: Ho:! =! =! =!4 =!5 =!6 =!7 = Ha: At least one! " Equal varances, normal dstrbuton F* = 9.86 p <. F.5, 7, 7 d.f. =. We can see somethng s gong on n the model F* > F.5, 7, 7 d.f. or p <. Reject Ho: Ho:! =! =! =!4 =!5 =!6 =!7 = And we should explore the ndvdual t-tests for the coeffcents 4 The Hypothess Test for a Regresson Coeffcent An Analyss from start-to-fnsh Ho: Ha: Assumptons Test Statstc Rejecton Regon Concluson: Ho:!DUME = Ha:!DUME " Equal varances, normal dstrbuton t* =.68 p =.58 t.5/, 7 =. t* > t.5, 7 or p =.58 Reject Ho:!DUME = Addng n Condton adds somethng to the model The coeffcent for DUME s sgnfcantly dfferent from zero. Apartments consdered n Excellent condton get about $56, more n value compared wth those n Far condton, holdng everythng else constant. 5 A drector of a tranng program for an nsurance company wants to test to see f dfferent methods of teachng produce dfferent results. She randomly dvdes students nto three groups - a tradtonal teachng group usng lectures; a CD-ROM based learnng approach; and a Web-based approach. Before takng the tranng the students took a profcency test n basc math and computer sklls. Followng the tranng they took an end-of-tranng exam. For ths research The sample sze s The dependent varable s END-of-TRAINING The ndependent varables are PROFICIENCY and tranng Type (represented by dummy varables) 6
5 We need to create dummy varables for our Teachng Methods Decson: let the tradtonal method be the reference category so we can compare the alternatve methods to the tradtonal The dummy varables look lke ths Here s the Data 7 Let s look at some descrptve statstcs END-of-TRAINING: Mean s, larger than the medan The Stem-and-Leaf Plot does not show any major problems The Independent Varables The mean PROFICIENCY s 4.67, less spread The Tranng s equally spread across the three methods: the means End-of-Tranng Quantles.% 99.5% 97.5% 9.% 75.% 5.% 5.%.%.5%.5%.% maxmum quartle medan quartle mnmum are Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N Sum Wgt Sum Varance Skewness Kurtoss CV N Mssng Stem and Leaf Stem Leaf represents 4 Count End-of-Tranng Profcency CDROM Web Mean Standard Medan Mode Standard Devaton Sample Varance Coef of Varaton Kurtoss Skewness Range Mnmum Maxmum Sum 46 8 Count More background nformaton If I look at End-of-Tranng by the Tranng Method I can see that the Tradtonal Method has the lowest average score, followed by Web and then CD-Rom The Smple ANOVA of End-of- Tranng by Tranng Method has an R of.7; about 7.% of the varablty n the tranng test s explaned by the tranng method The assumpton of equal varances s not perfect, but not all together unreasonable In fact, an F-test for equal varances could not be rejected Note the means of each method End-of-Tranng End-of-Tranng By Method Oneway Anova Tradtonal Web CD-ROM Method Rsquare Adj Rsquare Root Method Level Tradtonal Web CD-ROM 7 9 Number Means for Oneway Anova Mean Std Std uses a pooled estmate of error varance Lower 95% * Upper 95% The Regresson of the same results The Regresson results are the same as ANOVA Except we get coeffcents for each dummy varable Whch estmate the dfference n means from the reference group The CDROM method has an average score that s 7.9 hgher than the tradtonal method The Web method has an average score that s.6 hgher than the tradtonal method Whole Adj Root 7 9 Parameter s CDROM Web Std *.7*.84* If I do a test of dfferences across all methods and adjust for Expermentwse error, there s a sgnfcant dfference between Tradtonal and the other two methods, but not between CDROM and Web
6 We should also look at the Correlatons End-of-Tranng Profcency CDROM Web End-of-Tranng. Profcency.59. CDROM Web PROFICIENCY s moderately correlated wth the End-of- Tranng score (.59) The CDROM method s postvely correlated wth End-of- Tranng (.48) ndcatng that those who receved ths tranng method had, on average, hgher scores The correlaton of End-of-Tranng wth Web method was also postve, but much lower (.6) Snce subject were assgned to each method randomly, there s lttle to no relatonshp between the teachng method dummy varables and the Profcency score A closer look at End-of-Tranng and Profcency We noted the postve correlaton between End-of-Tranng score and the Profcency score Those who tested as more profcent n math and computer sklls tended to score hgher at the end of the tranng Here s one way to thnk of Profcency: It s not the focus of the research, but Profcency s a good control varable to help assess the effectveness of the tranng methods Regresson Results R s moderately hgh,.786 The F-test s sgnfcant at p <.. We can safely conclude that at least of the coeffcents are sgnfcantly dfferent from zero When we examne the ndvdual t- tests for the varables n the model The coeffcent for Profcency s postve and sgnfcant at p<. The coeffcent for Web s.89 and sgnfcant: The CDROM coeffcent s.77 and s sgnfcant and postve Lookng at the standardzed coeffcents, CDROM was most mportant followed by Profcency and Web. Whole Actual by Predcted Plot End-of- Tranng Actual End-of-Tranng Predcted P<. RSq=.79 RMSE=9.649 Adj Root Lack Of Ft Lack Of Ft Pure Total Parameter s Profcency Web CDROM Std Max RSq.985 Std Beta Compare the two models Table : Comparson of Two s of PRICE on Apartment Attrbutes.5*** -86.7*** Profcency.6*** Web.6***.89*** CDROM 7.9***.77*** R * p <.; ** p<.5; ***p<. R ncreased from.7 to.786 The coeffcents for Web and CDROM ncreased after controllng for Profcency The coeffcent for the CD-Rom method s.77, ndcatng ths tranng method resulted n an average score that s ponts hgher than the tradtonal method, holdng constant the student s profcency The coeffcent for the Web method s.89, ndcatng ths tranng method resulted n an average score that s ponts hgher than the tradtonal method, holdng constant the student s profcency 4
7 Resdual Results - RANDOM!!! Standardzed Resdual Plot Studentzed Resd End-of-Tranng Predcted End-of-Tranng 5 Analyss Conclusons It was mportant to control for the students profcency n the analyss The CD-Rom teachng method produced a result that was on average ponts hgher than the tradtonal method The Web-based teachng method produced a result that was lower, but stll on average ponts hgher than the tradtonal method Both alternatve methods showed an mprovement over the Tradtonal Method. The fnal decson on whch method to use should factor n other thngs. The company may compare costs or other factors (students perceptons of ease of use, convenence, how the nstructor feels about the methods) to determne whch alternatve method s preferred. 6 Summary Multple regresson s a powerful tool It allows us to make nferences about the effect of an ndependent varable on a dependent varable whle controllng for other factors n the model statstcal control There s so much more to learn n regresson More complex tests Non-lnear relatonshps Addng an element of tme to the model Fxes for data problems 7
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