Chapter 13: Multiple Regression
|
|
- Dylan Roberts
- 6 years ago
- Views:
Transcription
1 Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to estmate the populaton parameter β 0, β 1, and β usng the followng regresson equaton: 1
2 Usng Data Analyss - Regresson for the frst 3 columns of Potato.xls: The ft equaton allows us now to predct estmated values for the dfferent expermental condtons. For example ph 4 and pressure 15 leads to: ^ Y Our textbook now refers to Mntab for nformaton about a confdence nterval estmate of the average as well as a predcton nterval estmate for a future ndvdual value. The coeffcent of Multple Determnaton (R ) SSR SST R Y Ths essentally means that 41.58% of the varaton can be explaned by the effect of ph and pressure on the sold content. The remanng 58.4% s due to random scatterng of the data. Sometmes an adjusted R s suggested, whch takes the number of data ponts (n) and the number of explanatory varables (k ) nto account. R adj 1 n ( 1 R ) 1 ( ) Y.1 n k
3 13. Resdual Analyss By plottng the dfference between the actual data pont value and the predcted value (resdual), one can check the data for possble trends. The resduals are plotted aganst the varous parameters, X 1 and X, as well as aganst the data pont values, Y, or the tme Testng of Sgnfcance of the Multple Regresson Model Essentally testng: H 0 : β 1 β 0 (s there a slope wth respect to any of the parameters?) SST n ( Y Y ) 1 SSR n ( Y Y ) 1 ˆ SSE n ( Y Yˆ ) Confdence Interval Estmate for each Slope The standard error for each regresson coeffcent s provded n the PhStat analyss. Testng for the sgnfcance of each (slope) regresson coeffcent s: or the Confdence Interval Estmate for the slope s: b j ± tn k 1 s b j 3
4 13.5 Testng Portons of the Multple Regresson Model Whch parameters are really mportant?? One runs the data analyss frst wth all parameters ncluded and then a second tme wth all parameters except the one beng tested. The SS-regresson then allows us to defne the dfference between the two stuatons as beng the contrbuton of that specfc (excluded) parameter. The followng example demonstrates ths for the ph-parameter n potato.xls Analyss ncludng ph and Lower Pressure: Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 54 ANOVA df SS MS F Sgnfcance F Regresson E-06 Resdual Total Coeffcents Standard Error t Stat P-value Intercept PH E-06 Lower Pressure Result wth ph beng excluded: Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 54 ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Intercept E-15 Lower Pressure The sgnfcance of ph s now determned through the F-rato of MSR(X ph ) / MSR(error) ( ) / (58.41 / 51) 6.31 The F-cut-off s 4.04 for 1 and 51 degrees of freedom at the α 5% level. Thus, ph s sgnfcantly mprovng the regresson ft. 4
5 Coeffcent of Partal Determnaton Smlar to the complete analyss of R as descrbed n 13.1, one can also analyze how much varaton n the data can be explaned by a specfc parameter as follows: In our current example: R R Y1. Y ( ph ) ( lower pressure) For a fxed lower pressure 34.03% of the varaton can be explaned by the varaton n ph. And alternatvely 1.7% of the varaton can be explaned by the varaton of lower pressure. 5
6 13.6 The Quadratc Curvlnear Regresson Model Essentally the testng procedure and nterpretaton of the results s the same as for the lnear regresson analyss. However, the second parameter now uses the square value of the frst parameter. In a very smlar way the regresson models can be expanded to cubc or any other hgher order relatons. The predcted Y-values are: Y ˆ b + b X + b 0 The sgnfcance of the Quadratc Curvlnear Model s gven by the F-rato between the mean square regresson and the mean square error. F MSR / MSE As well as the Coeffcent of Multple Determnaton : R Y.1 SSR / SST Estmaton ntervals for ndvdual regresson coeffcents are descrbed usng the t-test wth t b / s b s b beng the standard error of the correspondng parameter n the Excel data analyss output. Detals about the calculaton of the standard error of a parameter are outsde the scope of ths class, but the systematc calculaton s descrbed on page 417 n Appled Statstcs and Probablty for Engneers, 3 rd edton by Montgomery and Runger. The t-dstrbuton s to be used for n {# of ft-parameters} for the correspondng degrees of freedom Dummy-Varable Models Categorcal varables can be substtuted wth dummy varables. For example, operator A s assgned a value of 0 and operator B s assgned a value of 1 (or wet 0 and dry 1). However, we should use ths only n cases where the dummy varable has only two values. We can the further evaluate f the slope of the lnear regresson for our other parameters s affected by the dummy varable by addng an nteracton factor. 1 1 X 1 6
7 13.8 Usng Transformatons n Regresson Models Smlarly to what was already descrbed n secton 13.6, we can use all other knds of transformatons and regresson models. E.g.: Square root transformaton Y β + β X + β X + ε Multplcatve model β β Y β0 X1 1 X ε transforms to lny ln β0 + β1 ln X1 + β ln X + ln ε Exponental model β X X Y e 0 + β1 1 β + ε transforms to lny β0 + β1 X1 + β X + ln ε In short, we can use any possble mathematcal relaton as long as we can descrbe the model as a combnaton of lnear factors Collnearty Collnearty descrbes a stuaton, where factors are hghly correlated. It then becomes dffcult to separate the ndvdual effects from the cross-correlated effects and the effectveness of a specfc model can hghly fluctuate dependng on whch parameters are beng ncluded. One way to evaluate collnearty s to use the varance nflatonary factor: 1 VIFj wth R 1 R j beng the coeffcent of determnaton when usng j all other X-varables except X j. Values close to 1 ndcate uncorrelated varables, whereas values of 5 or hgher are consdered sgnfcant. In other words, a large VIF value ndcates that two (or more) varables are actually closely related. The two varables are not ndependent of each other but they are rather two measurements of the same effect. For example, measurng the bouncng heght of a ball and correlatng that to the speed of a fallng ball 5 cm before httng the ground (varable 1) and the heght of where the ball was released ntally (varable ) are hghly correlated varables. There s no need to nclude both varables n a model. 7
8 13.10 Model Buldng We lke to acheve a model that ncludes the fewest number of varables. 8
9 Frst we elmnate hghly correlated varables. We then could use varous combnatons of varables and see f all of them show sgnfcance above a certan threshold n our regresson model. In a more systematc way, we can use all possble combnaton of varables. The followng fgure shows the result for the potato-processng data. We can now evaluate the adjusted R values of the dfferent models. The larger the R value s, the more of the data scatterng s beng explaned by the model. An alternatve approach use the so-called C p statstc for evaluaton f a certan model should be consdered. Here models where C p s less than the number of varables beng used n the model + 1 are beng consdered for further evaluaton. Thereafter we agan evaluate each model n detal and see f all varables show a sgnfcant effect (P-values below a gven cut-off value). Nevertheless, choosng an approprate model s a hghly subjectve process! 9
10 13.11 Ptfalls n Multple Regresson The regresson coeffcent for one partcular varable s nterpreted for the case that all other varables are held constant. We need resdual plots for each ndependent varable. Interacton plots are needed for each parameter when usng dummy varables. VIF evaluaton s needed to decde on parameters to be ncluded n a model. Examne several alternatve subsets for models. Sample sze 10 x larger than the # of varables n the model. Evaluate a suffcently wde range for each varable. Stablty over tme s a key necessty to use a fttng model for predctve purposes. 10
Statistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationChapter 15 Student Lecture Notes 15-1
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
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationChapter 15 - Multiple Regression
Chapter - Multple Regresson Chapter - Multple Regresson Multple Regresson Model The equaton that descrbes how the dependent varable y s related to the ndependent varables x, x,... x p and an error term
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationLecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management
Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MTH352/MH3510 Regression Analysis
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 014-015 MTH35/MH3510 Regresson Analyss December 014 TIME ALLOWED: HOURS INSTRUCTIONS TO CANDIDATES 1. Ths examnaton paper contans FOUR (4) questons
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationY = β 0 + β 1 X 1 + β 2 X β k X k + ε
Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationStatistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationCorrelation and Regression
Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationPubH 7405: REGRESSION ANALYSIS. SLR: INFERENCES, Part II
PubH 7405: REGRESSION ANALSIS SLR: INFERENCES, Part II We cover te topc of nference n two sessons; te frst sesson focused on nferences concernng te slope and te ntercept; ts s a contnuaton on estmatng
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More information[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationIntroduction to Regression
Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationANOVA. The Observations y ij
ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2
More informationTopic 23 - Randomized Complete Block Designs (RCBD)
Topc 3 ANOVA (III) 3-1 Topc 3 - Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn,
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationSTAT 511 FINAL EXAM NAME Spring 2001
STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte
More informationwhere I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).
11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationRegression. The Simple Linear Regression Model
Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng the Estmated Regresson Equaton for Estmaton and Predcton Resdual Analss: Valdatng
More information18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
More information28. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More informationECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston
More information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationIntroduction to Analysis of Variance (ANOVA) Part 1
Introducton to Analss of Varance (ANOVA) Part 1 Sngle factor The logc of Analss of Varance Is the varance explaned b the model >> than the resdual varance In regresson models Varance explaned b regresson
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationDiagnostics in Poisson Regression. Models - Residual Analysis
Dagnostcs n Posson Regresson Models - Resdual Analyss 1 Outlne Dagnostcs n Posson Regresson Models - Resdual Analyss Example 3: Recall of Stressful Events contnued 2 Resdual Analyss Resduals represent
More informationa. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
More informationChapter 14 Simple Linear Regression Page 1. Introduction to regression analysis 14-2
Chapter 4 Smple Lnear Regresson Page. Introducton to regresson analyss 4- The Regresson Equaton. Lnear Functons 4-4 3. Estmaton and nterpretaton of model parameters 4-6 4. Inference on the model parameters
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationChapter 6. Supplemental Text Material
Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.
More informationReduced slides. Introduction to Analysis of Variance (ANOVA) Part 1. Single factor
Reduced sldes Introducton to Analss of Varance (ANOVA) Part 1 Sngle factor 1 The logc of Analss of Varance Is the varance explaned b the model >> than the resdual varance In regresson models Varance explaned
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationIntroduction to Dummy Variable Regressors. 1. An Example of Dummy Variable Regressors
ECONOMICS 5* -- Introducton to Dummy Varable Regressors ECON 5* -- Introducton to NOTE Introducton to Dummy Varable Regressors. An Example of Dummy Varable Regressors A model of North Amercan car prces
More informationChapter 5 Multilevel Models
Chapter 5 Multlevel Models 5.1 Cross-sectonal multlevel models 5.1.1 Two-level models 5.1.2 Multple level models 5.1.3 Multple level modelng n other felds 5.2 Longtudnal multlevel models 5.2.1 Two-level
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan
More informationBiostatistics 360 F&t Tests and Intervals in Regression 1
Bostatstcs 360 F&t Tests and Intervals n Regresson ORIGIN Model: Y = X + Corrected Sums of Squares: X X bar where: s the y ntercept of the regresson lne (translaton) s the slope of the regresson lne (scalng
More informationTopic 7: Analysis of Variance
Topc 7: Analyss of Varance Outlne Parttonng sums of squares Breakdown the degrees of freedom Expected mean squares (EMS) F test ANOVA table General lnear test Pearson Correlaton / R 2 Analyss of Varance
More informationBiostatistics. Chapter 11 Simple Linear Correlation and Regression. Jing Li
Bostatstcs Chapter 11 Smple Lnear Correlaton and Regresson Jng L jng.l@sjtu.edu.cn http://cbb.sjtu.edu.cn/~jngl/courses/2018fall/b372/ Dept of Bonformatcs & Bostatstcs, SJTU Recall eat chocolate Cell 175,
More informationREGRESSION ANALYSIS II- MULTICOLLINEARITY
REGRESSION ANALYSIS II- MULTICOLLINEARITY QUESTION 1 Departments of Open Unversty of Cyprus A and B consst of na = 35 and nb = 30 students respectvely. The students of department A acheved an average test
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationLinear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
More informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationPBAF 528 Week Theory Is the variable s place in the equation certain and theoretically sound? Most important! 2. T-test
PBAF 528 Week 6 How do we choose our model? How do you decde whch ndependent varables? If you want to read more about ths, try Studenmund, A.H. Usng Econometrcs Chapter 7. (ether 3 rd or 4 th Edtons) 1.
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationTopic- 11 The Analysis of Variance
Topc- 11 The Analyss of Varance Expermental Desgn The samplng plan or expermental desgn determnes the way that a sample s selected. In an observatonal study, the expermenter observes data that already
More informationβ0 + β1xi. You are interested in estimating the unknown parameters β
Ordnary Least Squares (OLS): Smple Lnear Regresson (SLR) Analytcs The SLR Setup Sample Statstcs Ordnary Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals) wth OLS
More informationTests of Single Linear Coefficient Restrictions: t-tests and F-tests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Tests of Sngle Lnear Coeffcent Restrctons: t-tests and -tests Basc Rules Tests of a sngle lnear coeffcent restrcton can be performed usng ether a two-taled t-test
More informationRegression Analysis. Regression Analysis
Regresson Analyss Smple Regresson Multvarate Regresson Stepwse Regresson Replcaton and Predcton Error 1 Regresson Analyss In general, we "ft" a model by mnmzng a metrc that represents the error. n mn (y
More informationFREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,
FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationSystematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal
9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd
More informationPHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University
PHYS 45 Sprng semester 7 Lecture : Dealng wth Expermental Uncertantes Ron Refenberger Brck anotechnology Center Purdue Unversty Lecture Introductory Comments Expermental errors (really expermental uncertantes)
More informationOutline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Lecture k r Factorial Designs with Replication
EEC 66/75 Modelng & Performance Evaluaton of Computer Systems Lecture 3 Department of Electrcal and Computer Engneerng Cleveland State Unversty wenbng@eee.org (based on Dr. Ra Jan s lecture notes) Outlne
More informationexperimenteel en correlationeel onderzoek
expermenteel en correlatoneel onderzoek lecture 6: one-way analyss of varance Leary. Introducton to Behavoral Research Methods. pages 246 271 (chapters 10 and 11): conceptual statstcs Moore, McCabe, and
More information( )( ) [ ] [ ] ( ) 1 = [ ] = ( ) 1. H = X X X X is called the hat matrix ( it puts the hats on the Y s) and is of order n n H = X X X X.
( ) ( ) where ( ) 1 ˆ β = X X X X β + ε = β + Aε A = X X 1 X [ ] E ˆ β β AE ε β so ˆ = + = β s unbased ( )( ) [ ] ˆ Cov β = E ˆ β β ˆ β β = E Aεε A AE ε ε A Aσ IA = σ AA = σ X X = [ ] = ( ) 1 Ftted values
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur Two-way classfcaton wth nteractons
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More information