Chapter 15 Student Lecture Notes 15-1

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Chapter 15 Student Lecture Notes 15-1 Basc Busness Statstcs (9 th Edton) Chapter 15 Multple Regresson Model Buldng 004 Prentce-Hall, Inc. Chap 15-1 Chapter Topcs The Quadratc Regresson Model Usng Transformatons n Regresson Models Influence Analyss Collnearty Model Buldng Ptfalls n Multple Regresson and Ethcal Issues 004 Prentce-Hall, Inc. Chap 15- The Quadratc Regresson Model Relatonshp between the Response Varable and the Explanatory Varable s a Quadratc Polynomal Functon Useful When Scatter Dagram Indcates Non- Lnear Relatonshp Quadratc Model : = β0 + β1x1 + βx1 + ε The Second Explanatory Varable s the Square of the Frst Varable 004 Prentce-Hall, Inc. Chap 15-3 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15- Quadratc Regresson Model (contnued) Quadratc model may be consdered when a scatter dagram takes on the followng shapes: β > 0 β > 0 β < 0 β < 0 β = the coeffcent of the quadratc term 004 Prentce-Hall, Inc. Chap 15-4 Testng for Sgnfcance: Quadratc Model Testng for Overall Relatonshp Smlar to test for lnear model M SR F test statstc = M SE Testng the Quadratc Effect Compare quadratc model = β + β X + β X + ε 0 1 1 1 wth the lnear model = β + β X + ε 0 1 1 Hypotheses H0 : β = 0 (No quadratc effect) H : β 0 (Quadratc effect s present) 1 004 Prentce-Hall, Inc. Chap 15-5 Determne f a quadratc model s needed 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. Heatng Ol Example Ol (Gal) Temp ( 0 F) Insulaton 75.30 40 3 363.80 7 3 164.30 40 10 40.80 73 6 94.30 64 6 30.90 34 6 366.70 9 6 300.60 8 10 37.80 3 10 11.40 63 3 31.40 65 10 03.50 41 6 441.10 1 3 33.00 38 3 5.50 58 10 004 Prentce-Hall, Inc. Chap 15-6 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-3 60 Temperature Resdual Plot Heatng Ol Example: Resdual Analyss Possble nonlnear relatonshp (contnued) 40 Resduals 0 0 0 0 40 60 80-0 Insulaton Resdual Plot -40-60 0 4 6 8 10 1 No dscernable pattern 004 Prentce-Hall, Inc. Chap 15-7 Heatng Ol Example: t Test for Quadratc Model Testng the Quadratc Effect Model wth quadratc nsulaton term = β0+ β1x1 + βx + β3x + ε Model wthout quadratc nsulaton term = β + β X + β X + ε 0 1 1 Hypotheses H0 : β 3 = 0 (No quadratc term n nsulaton) H1: β3 0 (Quadratc term s needed n nsulaton) (contnued) 004 Prentce-Hall, Inc. Chap 15-8 H 0 : β 3 = 0 H 1 : β 3 0 df = 11 Crtcal Values: -.010 Example Soluton Is quadratc term n nsulaton needed on monthly consumpton of heatng ol? Test at α = 0.05. Reject H 0 Reject H 0.05.05 0.010 1.6611 Z Test Statstc: b3 β 3 1.8667 0 t = = = 1.6611 S 1.138 004 Prentce-Hall, Inc. Chap 15-9 b3 Decson: Do not reject H 0 at α = 0.05. Concluson: There s not suffcent evdence for the need to nclude quadratc effect of nsulaton on ol consumpton. Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-4 Example Soluton n PHStat PHStat Regresson Multple Regresson Excel spreadsheet for the heatng ol example Mcrosoft Excel Worksheet 004 Prentce-Hall, Inc. Chap 15-10 Usng Transformatons Ether or Both Independent and Dependent Varables May Be Transformed Can Be Based on Theory, Logc or Scatter Dagrams 004 Prentce-Hall, Inc. Chap 15-11 Inherently Non-Lnear Models Non-Lnear Models that Can Be Expressed n Lnear Form Can be estmated by least squares n lnear form Requre Data Transformaton 004 Prentce-Hall, Inc. Chap 15-1 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-5 Transformed Multplcatve Model (Log-Log) 1 Orgnal: 0 X β β = β 1 X ε ( ) = ( β ) + β ( X ) + β ( X ) + ( ε ) Transformed: ln ln ln ln ln 0 1 1 β 1 > 1 0< β1 < 1 1< β1 < 0 β 1 = 1 β 1 < 1 Smlarly for X 004 Prentce-Hall, Inc. Chap 15-13 Square Root Transformaton = β + β X + β X + ε 0 1 1 β 1 > 0 Smlarly for X β 1 < 0 Transforms non-lnear model to one that appears lnear. Often used to overcome heteroscedastcty. 004 Prentce-Hall, Inc. Chap 15-14 Exponental Transformaton (Log-Lnear) Orgnal Model 0 1X1 X e β + β + = β ε β 1 > 0 β 1 < 0 Transformed Into: ln = β0+ β1x1 + βx + ln ε1 004 Prentce-Hall, Inc. Chap 15-15 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-6 Interpretaton of Coeffcents Transformed Exponental Model ( s Transformed nto ln ) The coeffcent of the ndependent varable X k can be approxmately nterpreted as: a 1 unt change n X k leads to an estmated average rate of change of 100( b k ) percentage n 004 Prentce-Hall, Inc. Chap 15-16 Interpretaton of Coeffcents (contnued) Transformed Multplcatve Model The Dependent Varable s transformed to ln The Independent Varable X s transformed to ln X The coeffcent of the ndependent varable X k can be approxmately nterpreted as a 1 percent rate of change n X k leads to an estmated average rate of change of b k percentage n. Therefore, b k s the elastcty of wth respect to a change n. X k 004 Prentce-Hall, Inc. Chap 15-17 Influence Analyss To Determne Observatons that Have Influental Effect on the Ftted Model Potentally Influental Ponts Become Canddates for Removal from the Model Crtera Used are: The hat matrx elements h The studentzed deleted resduals t Cook s dstance statstc D All 3 Crtera are Complementary Only when all 3 crtera provde a consstent result should an observaton be removed 004 Prentce-Hall, Inc. Chap 15-18 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-7 The Hat Matrx Element h h = + n ( X X) n ( X X ) = 1 ( ) 1 h > k+ 1 / n If, X s an Influental Pont X may be consdered a canddate for removal from the model 004 Prentce-Hall, Inc. Chap 15-19 ( k ) The Hat Matrx Element h : Heatng Ol Example n= 15 k = + 1 / n= 0.4 No h > 0.4 No observaton appears to be a canddate for removal from the model Ol (Gal) Temp Insulaton h 75.30 40 3 0.1567 363.80 7 3 0.185 164.30 40 10 0.1757 40.80 73 6 0.467 94.30 64 6 0.1618 30.90 34 6 0.0741 366.70 9 6 0.306 300.60 8 10 0.351 37.80 3 10 0.68 11.40 63 3 0.446 31.40 65 10 0.759 03.50 41 6 0.0676 441.10 1 3 0.174 33.00 38 3 0.1574 5.50 58 10 0.68 004 Prentce-Hall, Inc. Chap 15-0 The Studentzed Deleted Resduals t n k 1 t = e SSE ( 1 h) e e : the resdual for observaton SSE : error sum of squares An observaton s consdered nfluental f t tn k n k s the crtcal value of a two-tal test at 10% level of sgnfcance t > 004 Prentce-Hall, Inc. Chap 15-1 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-8 n k 11 The Studentzed Deleted Resduals t :Example n= 15 k = t = t = 1.7957 t 10 and t 13 are nfluental ponts for potental removal from the model Ol (Gal) Temp Insulaton 75.30 40 3-0.377 363.80 7 3 0.3474 164.30 40 10 0.843 40.80 73 6-0.1871 94.30 64 6 0.0066 30.90 34 6-1.0571 366.70 9 6-1.1776 300.60 8 10-0.8464 t 10 37.80 3 10 0.0341 11.40 63 3-1.8536 31.40 65 10 1.0304 t 13 03.50 41 6-0.6075 441.10 1 3.9674 33.00 38 3 1.1681 5.50 58 10 0.43 004 Prentce-Hall, Inc. Chap 15- t Cook s Dstance Statstc D = e h D kmse ( 1 h ) e = the resdual for observaton MSE = mean square error of the ftted regresson model h = hat matrx element of observaton If D > F k + 1, n k 1, an observaton s consdered nfluental F k + 1, n k 1 s the crtcal value of the F dstrbuton at a 50% level of sgnfcance 004 Prentce-Hall, Inc. Chap 15-3 Cook s Dstance Statstc D : Heatng Ol Example n= 15 k = F = F = 0.835 k+ 1, n k 1 3,1 No D > 0.835 No observaton appears to be a canddate for removal from the model Usng the 3 crtera, there s nsuffcent evdence for the removal of any observaton from the model. Ol (Gal) Temp Insulaton D 75.30 40 3 0.0094 363.80 7 3 0.0098 164.30 40 10 0.0496 40.80 73 6 0.0041 94.30 64 6 0.0001 30.90 34 6 0.095 366.70 9 6 0.134 300.60 8 10 0.138 37.80 3 10 0.0001 11.40 63 3 0.3083 31.40 65 10 0.134 03.50 41 6 0.0094 441.10 1 3 0.4941 33.00 38 3 0.084 5.50 58 10 0.006 004 Prentce-Hall, Inc. Chap 15-4 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-9 Collnearty (Multcollnearty) Hgh Correlaton between Explanatory Varables Coeffcent of Multple Determnaton Measures Combned Effect of the Correlated Explanatory Varables Lttle or No New Informaton Provded Leads to Unstable Coeffcents (Large Standard Error) 004 Prentce-Hall, Inc. Chap 15-5 Large Overlap reflects collnearty between Temp and Insulaton Venn Dagrams and Collnearty Temp Ol Insulaton Large Overlap n varaton of Temp and Insulaton s used n explanng the varaton n Ol but NOT n estmatng and 004 Prentce-Hall, Inc. Chap 15-6 β β 1 Detect Collnearty (Varance Inflatonary Factor) VIF j Used to Measure Collnearty R j = coeffcent of multple 1 VIF determnaton from the j = ( 1 Rj ) regresson of X j on all the other explantory varables If VIF 5, s Hghly Correlated wth the j > X j Other Explanatory Varables 004 Prentce-Hall, Inc. Chap 15-7 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-10 Detect Collnearty n PHStat PHStat Regresson Multple Regresson Check the Varance Inflatonary Factor (VIF) box Excel spreadsheet for the heatng ol example Snce there are only two explanatory varables, only one VIF s reported n the Excel spreadsheet No VIF s > 5 There s no evdence of collnearty Mcrosoft Excel Worksheet 004 Prentce-Hall, Inc. Chap 15-8 Model Buldng Goal s to Develop a Good Model wth the Fewest Explanatory Varables Easer to nterpret Lower probablty of collnearty Stepwse Regresson Procedure Provdes lmted evaluaton of alternatve models Best-Subset Approach Uses the or C p Statstc r adj r adj Selects the model wth the largest or small C p near k+1 004 Prentce-Hall, Inc. Chap 15-9 Model Buldng Flowchart Choose,X, X p 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 es es More than One? No Remove ths X Do Complete Analyss Add Curvlnear Term and/or Transform Varables as Indcated Perform Predctons 004 Prentce-Hall, Inc. Chap 15-30 Statstcs for Managers Usng Mcrosoft Excel, /e

Chapter 15 Student Lecture Notes 15-11 Ptfalls and Ethcal Issues Fal to Understand that the Interpretaton of the Estmated Regresson Coeffcents are Performed Holdng All Other Independent Varables Constant Fal to Evaluate Resdual Plots for Each Independent Varable Fal to Evaluate Interacton Terms 004 Prentce-Hall, Inc. Chap 15-31 Ptfalls and Ethcal Issues (contnued) Fal to Obtan VIF for Each Independent Varable and Remove Varables that Exhbt a Hgh Collnearty wth Other Independent Varables Before Performng Sgnfcance Test on Each Independent Varable Fal to Examne Several Alternatve Models Fal to Use Other Methods When the Assumptons Necessary for Least-Squares Regresson Have Been Serously Volated 004 Prentce-Hall, Inc. Chap 15-3 Chapter Summary Descrbed the Quadratc Regresson Model Dscussed Usng Transformatons n Regresson Models Presented Influence Analyss Descrbed Collnearty Dscussed Model Buldng Addressed Ptfalls n Multple Regresson and Ethcal Issues 004 Prentce-Hall, Inc. Chap 15-33 Statstcs for Managers Usng Mcrosoft Excel, /e

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