Outline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Lecture k r Factorial Designs with Replication
|
|
- Candice Palmer
- 5 years ago
- Views:
Transcription
1 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 Revew of lecture k-p Fractonal Factoral Desgns One factor experment 3 4 k r Factoral Desgns wth Replcaton k r Factoral Desgns wth Replcaton r replcatons of k experments k r observatons Allows estmaton of expermental errors Model: y q 0 + q A x A + q B x B + q AB x A x B +e e expermental error Computaton of effects: use sgn table Estmaton of errors: e y yˆ y q + q x + q x + q x x 0 A A B B AB A B Allocaton of Varaton: SST ( y y ) rq + rq + rq + e.. A B AB, SST SSA + SSB + SSAB + SSE Effects are random varables Errors ~ N(0, σ e) y ~ N( y.., σ e) q 0 s normal wth varance /( r ) σ e
2 Confdence Intervals for Effects 5 Multplcatve Models for r Experments 6 Varance of errors: Estmated varance of q 0 : Smlarly, s q A s q B SSE s e ee MSE ( r ) s ( r ) Δ s /( r q ) 0 e sq se r Confdence ntervals (CI) for the effects: q t [ r ] s α / ; ( ) q CI does not nclude a zero > sgnfcant AB Addtve model. Not vald f effects do not add E.g., executon tme of workloads Executon tme y v w The two effects multply. Logarthm > addtve model: log( y ) log( v ) + log( w ) Correct model: where, y' log( y ) Takng an antlog of addtve effects q. to get the multplcatve effects u. y' q+ qx + qx + q xx + e 0 A A B B AB A B k-p Fractonal Factoral Desgns Desgn Large number of factors Large number of experments Full factoral desgn too expensve Use a fractonal factoral desgn Study 7 factors wth only experments! k-p desgn allows analyzng k factors wth only k-p experments k- desgn requres only half as many experments k- desgn requres only one quarter of the experments
3 7-4 Desgn Full factoral desgn s easy to analyze due to orthogonalty of sgn vectors Fractonal factoral desgns also use orthogonal vectors The sum of each column s zero: x 0, th varable, th experment The sum of the products of any two columns s zero x xl 0 l The sum of the squares of each column s 7-4 x Desgn Model: y q + q x + q x + q x + q x + q x + q x + q x 0 A A B B Effects can be computed usng nner products q q C C y + y y + y y + y y + y y x y y + y + y y y + y + y y x A A B B D D E E F F G G Desgn Preparng Sgn Table for k-p Desgn Factors A through G explans 37.6%, 4.74%, 43.40%, 6.75%, 0%,.06%, and 0.03% of varaton, respectvely > Use only factors C and A for further expermentaton Prepare a sgn table for a full factoral desgn wth k-p factors Mark the frst column I Mark the next k-p columns wth the k-p factors Of the ( k-p k + p ) columns on the rght, choose p columns and mark them wth the p factors whch were not chosen n step 3
4 Example: 7-4 Desgn 3 Example: 4- Desgn 4 Start wth 3 desgn Step : k7, p4 (3 factors), sgn table> Step : mark st column I Mark next 3 columns (.e., A, B, C) wth 3 factors (.e., A, B, C) Mark AB, AC, BC, ABC wth factors D, E, F, G Start wth 3 desgn Step : Step : Step 3: Step 4: choose the rghtmost column and mark t D > can study effects q A, q B, q C, and q D along wth nteractons q AB, q AC, and q BC Confoundng: 4- Desgn Confoundng: only the combned nfluence of two or more effects can be computed q q q + q D q A D ABC q q D ABC y + y y3 + y4 y5 + y6 y7 + y yxa y + y + y3 y4 + y5 y6 y7 + y yxd y + y + y3 y4 + y5 y6 y7 + y yxaxbxc ABC y x x x A B C y + y + y3 y4 + y5 y6 y + y Effects of D and ABC are confounded. Not a problem f q ABC s neglgble 7 5 Confoundng: 4- Desgn Confoundng representaton: D ABC Other confoundngs: y + y y3 + y4 y5 + y qa qbcd yxa > A BCD A BCD, B ACD, C ABC, AB CD, AC BD, BC AD, ABC D, and I ABCD y + y I ABCD > confoundng of ABCD wth the mean
5 7 Other Fractonal Factoral Desgns Algebra of Confoundng A fractonal factoral desgn s not unque p dfferent fractonal factoral desgns possble Confoundng: IABD, ABD, BAD, CABCD, DAB, ACBCD, BCACD, ABCCD Not as good as the prevous desgn Gven ust one confoundng, t s possble to lst all other confoundngs Rules: I s treated as unty Any term wth a power of s erased 9 0 Algebra of Confoundng Algebra of Confoundng I ABCD Multplyng both sdes by A : A A BCD BCD Multplyng both sdes by B, C, D,and AB : B AB CD ACD C ABC D ABD D ABCD ABC AB A B CD CD and so on. generator polynomal: I ABCD For the second desgn : I ABC In a k-p desgn, p effects are confounded together In the desgn: DAB, EAC, FBC, GABC > IABD, IACE, IBCF, IABCG >IABDACEBCFABCG Usng products of all subsets: IABDACEBCFABCGBCDEACDFCDG ABEFBEGAFGDEFADEGBDFG CEFGABCDEFG Other confoundngs: ABDCEABCFBCGABCDECDFACDGBEF ABEGFGADEFDEGABDFGACEFGBCDEFG 5
6 Desgn Resoluton Desgn Resoluton Order of an effect number of factors ncluded n t Order of ABCD4, order of I0 Order of a confoundng sum of the order of two terms E.g., ABCDE s of order 5 Example : 4 I ABCD R IV Resoluton IV IV A BCD, B ACD, C ABD, AB CD, AC BD, BC AD, ABC D,and I ABCD Resoluton of a desgn mnmum of orders of confoundngs Notaton: Resoluton III k p R III III Example : I ABD R III desgn Desgn Resoluton 3 Case Study: Latex vs. troff 4 Example 3: I ABD ACE BCF ABCG BCDE ACDF CDG ABEF BEG AFG DEF ADEG BDFG CEFG ABCDEFG Ths s a resoluton-iii desgn A desgn of hgher resoluton s consdered a better desgn Desgn: 6- wth IBCDEF 6
7 Case Study: Latex vs. troff Conclusons Over 90% of the varaton s due to: bytes, program, and equatons and a second order nteracton Text fle szes were sgnfcantly dfferent makng ts effect more than that of the programs Hgh percentage of varaton explaned by the program equaton nteracton > Choce of the text formattng program depends upon the number of equatons n the text. troff not as good for equatons 5 Case Study: Latex vs. troff Conclusons (contnued) Low program bytes nteracton > Changng the fle sze affects both programs n a smlar manner In next phase, reduce range of fle szes. Alternately, ncrease the number of levels of fle szes 6 Case Study: Scheduler Desgn 7 Case Study: Scheduler Desgn Three classes of obs: word processng, data processng, and background data processng Desgn: 5- wth IABCDE Measured throughputs T W : for word processng T I : for nteractve data processng T B : for batch data processng 7
8 Case Study: Scheduler Desgn 9 Case Study: Scheduler Desgn 30 Effects and varaton explaned Conclusons For work processng throughput (T W ): A (preempton), B (tme slce), and AB are mportant For nteractve obs: E (farness), A (preempton), BE, and B (tme slce) For background obs: A (preempton), AB, B (tme slce), E (farness) May use dfferent polces for dfferent classes of workloads Factors C (queue assgnment) or any of ts nteracton do not have any sgnfcant mpact on the throughput Case Study: Scheduler Desgn 3 One Factor Experments 3 Conclusons (contnued): Factor D (requeung) s not effectve Preempton (A) mpacts all workloads sgnfcantly Tme slde (B) mpacts less than preempton Farness (E) s mportant for nteractve obs and slghtly mportant for background obs Used to compare alternatves of a sngle categorcal varable For example, several processors, several cachng schemes Model: y μ + α + e r number of replcatons y th response wth th alternatve μ mean response α effect of alternatve e error term α 0
9 Computaton of Effects r a a r a y arμ+ r α + e arμ r a μ y y ar r r y y ( μ+ α + e ). r r r r r e 0 r μ+ α + μ+ α + α y μ y y Example: Code Sze Comparson Entres n a row are unrelated. Otherwse, need a two factor analyss 34 Analyss of Code Sze Comparson Data 35 Interpretaton 36 Average processor requres 7.7 bytes of storage The effects of the processors R, V, and Z are - 3.3, -4.5, and 37.7, respectvely. That s R requres 3.3 bytes less than an average processor V requres 4.5 bytes less than an average processor Z requres 37.7 bytes more than an average processor 9
10 Estmatng Expermental Errors 37 Example 3 Estmated response for th alternatve Error: ˆ y μ + α e y yˆ Sum of squared errors (SSE): SSE r a e SSE ( 30.4) ( 54.4) (76.6) Allocaton of Varaton 39 Example 40 y μ + α + e + μα + μe + α e,,,, y μ+ α + e y μ + α + e + cross product terms SSY SS0+ SSA + SSE r a r a a μ μ α α SS0 ar SSA r Total varaton of y (SST): SST ( y,..,, y ) y ary SSY SS0 SSA + SSE.. SSY SS0 arμ 3 5 (7.7) 5.7 SSA r α 5[( 3.3) + ( 4.5) + (37.6) ] 099. SST SSY SS SSE SST SSA Percent varaton explaned by processors / % 9.6% of varaton n code sze s due to expermental errors (programmer dfferences). Is 0.4% statstcally sgnfcant? 0
11 Analyss of Varance (ANOVA) 4 F-Test 4 Importance sgnfcance Important > explans a hgh percent of varaton Sgnfcance > hgh contrbuton to the varaton compared to that by errors Degree of freedom number of ndependent values requred to compute SSY SS0 + SSA + SSE ar + ( a ) + a( r ) Note that the degrees of freedom also add up Purpose: to check f SSA s sgnfcantly greater than SSE Errors are normally dstrbuted > SSE and SSA have ch-square dstrbutons The rato (SSA/ν A )/ (SSE/ν e ) has an F dstrbuton. Where ν A a degree of freedom for SSA ν e a(r ) degree of freedom for SSE Computed rato > F[ α ; ν A, ν e] > SSA s sgnfcantly hgher than SSE SSA/ν A s called mean square of A or (MSA). Smlarly, MSE SSE/ ν e ANOVA Table for One Factor Experment 43 Example: Code Sze Comparson 44 Computed F-value < F from table > the varaton n the code szes s mostly due to expermental errors and not because of any sgnfcant dfference among the processors
12 Vsual Dagnostc Tests 45 Example 46 Assumptons: Factors effects are addtve Errors are addtve Errors are ndependent of factor levels Errors are normally dstrbuted Errors have the same varance for all factor levels Vsual tests: Normal quantle-quantle plot: Lnear > Normalty Resduals versus predcted response: No trend > ndependence Scale of errors << scale of response > gnore vsble trends Horzontal and vertcal scales smlar Resduals are not small Varaton due to factors s small compared to the unexplaned varaton No vsble trend n the spread Example 47 Confdence Intervals for Effects 4 S-shape > shorter tals than normal Estmates are random varables For the confdence ntervals, use t values at r(a-) degree of freedom Mean response: ŷ μ + α Contrasts h α : e.g., to compare alternatves, α α
13 Example: Code Sze Comparson Error varance: Se 763. Std dev of errors (var. of errors).7 Std dev of μ s / ar.7 / 5.9 e Std dev of α s / {( a ) /( ar)}.7 / ( /5) 3.4 e For 90% confdence, t [0.95;].7 49 Example: Code Sze Comparson 90% confdence ntervals: μ 97.7 (.7)(.9) (46.9,.5) α 3.3 (.7)(3.4) ( 7.0, 44.4) α 4.5 (.7)(3.4) (.,33.) α 37.6 (.7)(3.4) ( 0.0,95.4) 3 The code sze on an average processor s sgnfcantly dfferent from zero Processor effects are not sgnfcant 50 Example: Code Sze Comparson 5 Example: Code Sze Comparson 5 Usng h, h -, h 3 0, ( h 0): Mean α α y y se.7 Std dev of α α 56. (/5) ( h / ar) 90% CI for α α. (.7)(56.) (.7,.) CI ncludes zero > one sn t superor to other Smlarly, 90% CI for α α 3 ( ) (.7)(56.) ( 50.9, 4.9) 90% CI for α α 3 ( ) (.7)(56.) ( 6.,37.7) Any one processor s not superor to another 3
14 Unequal Sample Szes 53 Parameter Estmaton 54 Model: y By defnton: μ + α + e a r α 0 Here, r s the number of observatons at th level (alternatve) Total number of observatons: a N r Analyss of Varance 55 Example: Code Sze Comparson 56 All means are obtaned by dvdng by the number of observatons added The column effects are.5, 3.75, and -.9 4
15 Analyss of Varance 57 Example: Code Sze Comparson Sums of Squares: SSY y SS N μ SSA 5α + 4α + 3α SSE ( 30.40) ( 54.40) ( 9.33) SST SSY SS Degrees of freedom: SSY SS0 + SSA + SSE N + ( a ) + N a ANOVA Table: Code Sze Comparson 59 Dervaton of Standard Devaton 60 Concluson: varaton due to processors s nsgnfcant as compared to that due to modelng errors Consder the effect of processor Z: Snce, α y y ( y3 + y3 + y33) ( y + y + + y3 + y4 + y3 + y3 + y33) 3 ( y3 + y3 + y33) ( y + y + + y3 + y4) 4 5
16 Dervaton of Standard Devaton 6 Error n α 3 errors n terms on the rght hand sze: eα ( e e3 + e33) ( e + e + + e3 + e4 ) 4 e 's are normally dstrbuted > α3 s normal wth sα 3s e + s e 4 6
Dr. 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 informationChapter 13: Multiple Regression
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
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 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 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 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 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 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 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 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 informationF statistic = s2 1 s 2 ( F for Fisher )
Stat 4 ANOVA Analyss of Varance /6/04 Comparng Two varances: F dstrbuton Typcal Data Sets One way analyss of varance : example Notaton for one way ANOVA Comparng Two varances: F dstrbuton We saw that the
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 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 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 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 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 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 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 informationStatistics 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 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 informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
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 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 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 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 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 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 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 informationFractional Factorial Designs
k-p Fractional Factorial Designs Fractional Factorial Designs If we have 7 factors, a 7 factorial design will require 8 experiments How much information can we obtain from fewer experiments, e.g. 7-4 =
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 informationCS 147: Computer Systems Performance Analysis
CS 147: Computer Systems Performance Analysis Fractional Factorial Designs CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview Overview Overview Example Preparing
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 informationChapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout
Serk Sagtov, Chalmers and GU, February 0, 018 Chapter 1. Analyss of varance Chapter 11: I = samples ndependent samples pared samples Chapter 1: I 3 samples of equal sze one-way layout two-way layout 1
More informationStatistical Evaluation of WATFLOOD
tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
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 informationTwo-factor model. Statistical Models. Least Squares estimation in LM two-factor model. Rats
tatstcal Models Lecture nalyss of Varance wo-factor model Overall mean Man effect of factor at level Man effect of factor at level Y µ + α + β + γ + ε Eε f (, ( l, Cov( ε, ε ) lmr f (, nteracton effect
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 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 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 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 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 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 informationT E C O L O T E R E S E A R C H, I N C.
T E C O L O T E R E S E A R C H, I N C. B rdg n g En g neern g a nd Econo mcs S nce 1973 THE MINIMUM-UNBIASED-PERCENTAGE ERROR (MUPE) METHOD IN CER DEVELOPMENT Thrd Jont Annual ISPA/SCEA Internatonal Conference
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 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 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 information1-FACTOR ANOVA (MOTIVATION) [DEVORE 10.1]
1-FACTOR ANOVA (MOTIVATION) [DEVORE 10.1] Hgh varance between groups Low varance wthn groups s 2 between/s 2 wthn 1 Factor A clearly has a sgnfcant effect!! Low varance between groups Hgh varance wthn
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 information2 k, 2 k r and 2 k-p Factorial Designs
2 k, 2 k r and 2 k-p Factorial Designs 1 Types of Experimental Designs! Full Factorial Design: " Uses all possible combinations of all levels of all factors. n=3*2*2=12 Too costly! 2 Types of Experimental
More informationChapter 13 Analysis of Variance and Experimental Design
Chapter 3 Analyss of Varance and Expermental Desgn Learnng Obectves. Understand how the analyss of varance procedure can be used to determne f the means of more than two populatons are equal.. Know the
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 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 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 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 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 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 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 informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
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 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 informationAssignment 5. Simulation for Logistics. Monti, N.E. Yunita, T.
Assgnment 5 Smulaton for Logstcs Mont, N.E. Yunta, T. November 26, 2007 1. Smulaton Desgn The frst objectve of ths assgnment s to derve a 90% two-sded Confdence Interval (CI) for the average watng tme
More informationANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected.
ANSWERS CHAPTER 9 THINK IT OVER thnk t over TIO 9.: χ 2 k = ( f e ) = 0 e Breakng the equaton down: the test statstc for the ch-squared dstrbuton s equal to the sum over all categores of the expected frequency
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 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 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 informationTHE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for
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 information17 Nested and Higher Order Designs
54 17 Nested and Hgher Order Desgns 17.1 Two-Way Analyss of Varance Consder an experment n whch the treatments are combnatons of two or more nfluences on the response. The ndvdual nfluences wll be called
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 informationModeling and Simulation NETW 707
Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must
More informationBETWEEN-PARTICIPANTS EXPERIMENTAL DESIGNS
1 BETWEEN-PARTICIPANTS EXPERIMENTAL DESIGNS I. Sngle-factor desgns: the model s: y j = µ + α + ε j = µ + ε j where: y j jth observaton n the sample from the th populaton ( = 1,..., I; j = 1,..., n ) µ
More informationOutline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3.
Outlne 3. Multple Regresson Analyss: Estmaton I. Motvaton II. Mechancs and Interpretaton of OLS Read Wooldrdge (013), Chapter 3. III. Expected Values of the OLS IV. Varances of the OLS V. The Gauss Markov
More informationTurbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH
Turbulence classfcaton of load data by the frequency and severty of wnd gusts Introducton Oscar Moñux, DEWI GmbH Kevn Blebler, DEWI GmbH Durng the wnd turbne developng process, one of the most mportant
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 informationBasically, if you have a dummy dependent variable you will be estimating a probability.
ECON 497: Lecture Notes 13 Page 1 of 1 Metropoltan State Unversty ECON 497: Research and Forecastng Lecture Notes 13 Dummy Dependent Varable Technques Studenmund Chapter 13 Bascally, f you have a dummy
More informationReminder: Nested models. Lecture 9: Interactions, Quadratic terms and Splines. Effect Modification. Model 1
Lecture 9: Interactons, Quadratc terms and Splnes An Manchakul amancha@jhsph.edu 3 Aprl 7 Remnder: Nested models Parent model contans one set of varables Extended model adds one or more new varables 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 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 informationLINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables
LINEAR REGRESSION ANALYSIS MODULE VIII Lecture - 7 Indcator Varables Dr. Shalabh Department of Maematcs and Statstcs Indan Insttute of Technology Kanpur Indcator varables versus quanttatve explanatory
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
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 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 informationLecture 2: Prelude to the big shrink
Lecture 2: Prelude to the bg shrnk Last tme A slght detour wth vsualzaton tools (hey, t was the frst day... why not start out wth somethng pretty to look at?) Then, we consdered a smple 120a-style regresson
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 information17 - LINEAR REGRESSION II
Topc 7 Lnear Regresson II 7- Topc 7 - LINEAR REGRESSION II Testng and Estmaton Inferences about β Recall that we estmate Yˆ ˆ β + ˆ βx. 0 μ Y X x β0 + βx usng To estmate σ σ squared error Y X x ε s ε we
More informationBose (1942) showed b t r 1 is a necessary condition. PROOF (Murty 1961): Assume t is a multiple of k, i.e. t nk, where n is an integer.
Resolvable BIBD: An ncomplete bloc desgn n whch each treatment appears r tmes s resolvable f the blocs can be dvded nto r groups such that each group s a complete replcaton of the treatments (.e. each
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More information# c i. INFERENCE FOR CONTRASTS (Chapter 4) It's unbiased: Recall: A contrast is a linear combination of effects with coefficients summing to zero:
1 INFERENCE FOR CONTRASTS (Chapter 4 Recall: A contrast s a lnear combnaton of effects wth coeffcents summng to zero: " where " = 0. Specfc types of contrasts of nterest nclude: Dfferences n effects Dfferences
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 informationMD. LUTFOR RAHMAN 1 AND KALIPADA SEN 2 Abstract
ISSN 058-71 Bangladesh J. Agrl. Res. 34(3) : 395-401, September 009 PROBLEMS OF USUAL EIGHTED ANALYSIS OF VARIANCE (ANOVA) IN RANDOMIZED BLOCK DESIGN (RBD) ITH MORE THAN ONE OBSERVATIONS PER CELL HEN ERROR
More informationA Robust Method for Calculating the Correlation Coefficient
A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal
More informationCHAPTER 8. Exercise Solutions
CHAPTER 8 Exercse Solutons 77 Chapter 8, Exercse Solutons, Prncples of Econometrcs, 3e 78 EXERCISE 8. When = N N N ( x x) ( x x) ( x x) = = = N = = = N N N ( x ) ( ) ( ) ( x x ) x x x x x = = = = Chapter
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 informationOne Factor Experiments
One Factor Experiments Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu These slides are available on-line at: http://www.cse.wustl.edu/~jain/cse567-06/ 20-1 Overview!
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
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 informationUncertainty as the Overlap of Alternate Conditional Distributions
Uncertanty as the Overlap of Alternate Condtonal Dstrbutons Olena Babak and Clayton V. Deutsch Centre for Computatonal Geostatstcs Department of Cvl & Envronmental Engneerng Unversty of Alberta An mportant
More information