Statistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models

Transcription

1 Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1

2 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables Coeffcent of Determnaton Categorcal Explanatory Varables Transformaton of Varables Model Buldng 1999 Prentce-Hall, Inc. Chap. 14-2

3 Populaton Y-ntercept The Multple Regresson Model Relatonshp between 1 dependent & 2 or more ndependent varables s a lnear functon Populaton slopes Random Error Y X X p X p Ŷ b b X b X b p X p e Dependent (Response) varable for sample Independent (Explanatory) varables for sample model 1999 Prentce-Hall, Inc. Chap. 14-3

4 Sample Multple Regresson Model Y Y b0 b1 X1 b2x 2 b p X p e e X 2 X 1 Ŷ b b X b X b p X p 1999 Prentce-Hall, Inc. Chap. 14-4

5 Multple Regresson Model: Example Develop a model for estmatng heatng ol used for a sngle famly home n the month of January based on average temperature and amount of nsulaton n nches. O l (G a l) T e m p ( 0 F) In su la to n Prentce-Hall, Inc. Chap. 14-5

6 Sample Regresson Model: Example Ŷ b b X b X Excel Output b p X C o e ffc e n ts I n te r c e p t X V a r a b l e X V a r a b l e p Ŷ X X 1 2 For each degree ncrease n temperature, the average amount of For each ncrease n one nch heatng ol used s decreased by of nsulaton, the use of heatng gallons, holdng nsulaton ol s decreased by constant. gallons, holdng temperature constant Prentce-Hall, Inc. Chap. 14-6

7 Usng The Model to Make Predctons Estmate the average amount of heatng ol used for a home f the average temperature s 30 0 and the nsulaton s 6 nches. Ŷ X X The estmated heatng ol used s gallons 1999 Prentce-Hall, Inc. Chap. 14-7

8 Coeffcent of Multple Determnaton Excel Output R eg resso n S tatstcs M u lt p le R R S q u a re A d ju s t e d R S q u a re S t a n d a rd E rro r O b s e rva t o n s 15 2 r Y, 12 Adjusted r 2 SSR SST reflects the number of explanatory varables and sample sze s smaller than r Prentce-Hall, Inc. Chap. 14-8

9 Resdual Plots Resduals Vs Y May need to transform Y varable Resduals Vs X 1 May need to transform X 1 varable Resduals Vs X 2 May need to transform X 2 varable Resduals Vs Tme May have autocorrelaton 1999 Prentce-Hall, Inc. Chap. 14-9

10 Resduals Resdual Plots: Example 60 Tem perature R esdual P lot Excel Output Insulaton R esdual P lot No Dscernable Pattern Prentce-Hall, Inc. Chap

11 Testng for Overall Sgnfcance Shows f there s a lnear relatonshp between all of the X varables together and Y Use F test Statstc Hypotheses: H 0 : 1 = 2 = = p = 0 (No lnear relatonshp) H 1 : At least one 0 ( At least one ndependent varable affects Y) 1999 Prentce-Hall, Inc. Chap

12 Test for Overall Sgnfcance Excel Output: Example A N O V A df SS MS F S gnfcance F R e g re sso n E -0 9 R e sd u a l T o ta l p = 2, the number of explanatory varables n - 1 p value MRS MSE = F Test Statstc 1999 Prentce-Hall, Inc. Chap

13 Test for Overall Sgnfcance Example Soluton H 0 : 1 = 2 = = p = 0 H 1 : At least one I 0 a =.05 df = 2 and 12 Crtcal Value(s): a = F Test Statstc: F (Excel Output) Decson: Reject at a = 0.05 Concluson: There s evdence that At least one ndependent varable affects Y 1999 Prentce-Hall, Inc. Chap

14 Test for Sgnfcance: Indvdual Varables Shows f there s a lnear relatonshp between the varable X and Y Use t test Statstc Hypotheses: H 0 : = 0 (No lnear relatonshp) H 1 : 0 (Lnear relatonshp between X and Y) 1999 Prentce-Hall, Inc. Chap

15 t Test Statstc Excel Output: Example C o e ffc e n ts S ta n d a rd E rro r t Test Statstc for X 1 (Temperature) t S ta t I n te r c e p t X V a r a b l e X V a r a b l e t Test Statstc for X 2 (Insulaton) 1999 Prentce-Hall, Inc. Chap

16 t Test : Example Soluton Does temperature have a sgnfcant effect on monthly consumpton of heatng ol? Test at a = H 0 : 1 = 0 H 1 : 1 0 df = 12 Crtcal Value(s): Reject H 0 Reject H Z Test Statstc: t Test Statstc = Decson: Reject H 0 at a = 0.05 Concluson: There s evdence of a sgnfcant effect of temperature on ol consumpton Prentce-Hall, Inc. Chap

17 Confdence Interval Estmate For The Slope Provde the 95% confdence nterval for the populaton slope 1 (the effect of temperature on ol consumpton). b t n S 1 p1 b 1 Coeffcents Lower 95% Upper 95% Inte rce pt X V a ra ble X V a ra ble The average consumpton of ol s reduced by between 4.7 gallons to 6.17 gallons per each ncrease of 1 0 F Prentce-Hall, Inc. Chap

18 Testng Portons of Model Contrbuton of One X to Model (holdng all others constant) Denote by SSR(X all varables except ) 2 r Y 1.2 = Coeffcent of partal determnaton of X 1 wth Y holdng X 2 constant Evaluate Separate Models Useful n Selectng Independent Varables 1999 Prentce-Hall, Inc. Chap

19 Testng Portons of Model: SSR Contrbuton of X 1 gven X 2 has been ncluded: SSR(X 1 X 2 ) = SSR(X 1 and X 2 ) - SSR(X 2 ) From ANOVA secton of regresson for Ŷ b0 b1 X1 b2x 2 From ANOVA secton of regresson for Ŷ b0 b2x Prentce-Hall, Inc. Chap

20 Partal F Test For Contrbuton of X Hypotheses: H 0 : Varable X does not sgnfcantly mprove the model gven all others ncluded H 1 : Varable X sgnfcantly mproves the model gven all others ncluded Test Statstc: F = Wth df = 1 and (n - p -1) SSR( X all others ) MSE 1999 Prentce-Hall, Inc. Chap

21 Coeffcent of Partal Determnaton 2 SSR( X1 X 2 ) Y 1 2 SST SSR( X1 and X 2 ) SSR( X1 X 2 r. ) From ANOVA secton of regresson for Ŷ b0 b1 X1 b2x 2 From ANOVA secton of regresson for Ŷ b0 b2x Prentce-Hall, Inc. Chap

22 Testng Portons of Model: Example Test at the a =.05 level to determne f the varable of average temperature sgnfcantly mproves the model gven that nsulaton s ncluded Prentce-Hall, Inc. Chap

23 Testng Portons of Model: Example H 0 : X 1 does not mprove model (X 2 ncluded) H 1 : X 1 does mprove model A N O V A F SSR( a =.05, df = 1 and 12 Crtcal Value = 4.75 (For X 1 and X 2 ) A N O V A (For X 2 ) SS SS X 1 MSE X 2 ) MS R e g re sso n R e sd u a l T o ta l R e g r e s s o n R e s d u a l T o t a l , , , 717 Concluson: Reject H 0. X 1 does mprove model = Prentce-Hall, Inc. Chap

24 Curvlnear Regresson Model Relatonshp between 1 response varable and 2 or more explanatory varable s a polynomal functon Useful when scatter dagram ndcates non-lnear relatonshp Curvlnear model: Y 0 X 1 The second explanatory varable s the square of the 1st Prentce-Hall, Inc. Chap X 2 2 1

25 Curvlnear Regresson Model Curvlnear models may be consdered when scatter dagram takes on the followng shapes: Y Y Y Y X 1 X 1 2 > 0 2 > 0 2 < 0 2 < 0 X 1 X 1 2 = the coeffcent of the quadratc term 1999 Prentce-Hall, Inc. Chap

26 Testng for Sgnfcance: Curvlnear Model Testng for Overall Relatonshp Smlar to test for lnear model F test statstc = MSR MSE Testng the Curvlnear Effect Compare curvlnear model 2 Y X X 0 1 wth the lnear model Y X Prentce-Hall, Inc. Chap

27 Dummy-Varable Models Categorcal Varable Involved (dummy varable) wth 2 Levels: yes or no, on or off, male or female, Coded 0 or 1 Intercepts Dfferent Assumes Equal Slopes Regresson Model has Same Form: Y X X X p p 1999 Prentce-Hall, Inc. Chap

28 Dummy-Varable Models Assumpton Gven: Ŷ b0 b1x 1 b2x 2 Y = Assessed Value of House X 1 = Square footage of House X 2 = Desrablty of Neghborhood = Desrable (X 2 = 1) Ŷ b0 b1x 1 b2( 1) ( b0 b2 ) b1x 1 Undesrable (X 2 = 0) Ŷ b0 b1x 1 b2( 0) b0 b1x 1 0 f undesrable 1 f desrable Same slopes 1999 Prentce-Hall, Inc. Chap

29 Dummy-Varable Models Assumpton Y (Assessed Value) Intercepts dfferent b 0 + b 2 b 0 Same slopes X 1 (Square footage) 1999 Prentce-Hall, Inc. Chap

30 Evaluatng Presence of Interacton Hypothesze Interacton Between Pars of Independent Varables Contans 2-way Product Terms Y 0 1X1 2X 2 3X1 X 2 Hypotheses: H 0 : 3 = 0 (No nteracton between X 1 and X 2 H 1 : 3 0 (X 1 nteracts wth X 2 ) 1999 Prentce-Hall, Inc. Chap

31 Usng Transformatons For Non-lnear Models that Volate Lnear Regresson Assumptons Determne Type of Transformaton From Scatter Dagram Requres Data Transformaton Ether or Both Independent and Dependent Varables May be Transformed 1999 Prentce-Hall, Inc. Chap

32 Square Root Transformaton Y X X Y 1 > 0 1 < 0 X 1 Smlarly for X 2 Transforms one of above model to one that appears lnear. Often used to overcome heteroscedastcty Prentce-Hall, Inc. Chap

33 Logarthmc Transformaton Y ln( X ) ln( X 2 ) Y 1 > 0 Smlarly for X 2 1 < 0 X 1 Transformed from an orgnal multplcatve model 1999 Prentce-Hall, Inc. Chap

34 Exponental Transformaton Orgnal Model Y e 0 1X 1 2X 2 Y 1 > 0 Smlarly for X 2 1 < 0 X 1 Transformed nto: lny 0 1X1 2X 2 ln Prentce-Hall, Inc. Chap

35 Collnearty Hgh Correlaton Between Explanatory Varables Coeffcents Measure Combned Effect No New Informaton Provded Leads to Unstable Coeffcents Dependng on the explanatory varables VIF Used to Measure Collnearty VIF 1 j 1 R 2 j, 1999 Prentce-Hall, Inc. Chap j R = Coeffcent of Multple Determnaton of X j wth all the others

36 Model Buldng Goal s to Develop Model wth Fewest Explanatory Varables Easer to nterpret Lower probablty of collnearty Stepwse Regresson Procedure Provde lmted evaluaton of alternatve models Best-Subset Approach Uses the C p Statstc Selects model wth small C p near p Prentce-Hall, Inc. Chap

37 Model Buldng Flowchart Choose X 1,X 2, X k Run Regresson to fnd VIFs Any VIF>5? No Run Subsets Regresson to Obtan best models n terms of C p Remove Varable wth Hghest VIF Yes Yes More than One? No Remove ths X Do Complete Analyss Add Curvlnear Term and/or Transform Varables as Indcated Perform Predctons 1999 Prentce-Hall, Inc. Chap

38 Chapter Summary Presented The Multple Regresson Model Consdered Contrbuton of Indvdual Independent Varables Dscussed Coeffcent of Determnaton Addressed Categorcal Explanatory Varables Consdered Transformaton of Varables Dscussed Model Buldng 1999 Prentce-Hall, Inc. Chap