Chapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout
|
|
- Alicia Helena Sharp
- 6 years ago
- Views:
Transcription
1 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 One-way layout Consder I ndependent IID samples (Y 11,..., Y 1 ),..., (Y I1,..., Y I ) measurng I treatment results. We have one man factor (factor A havng I levels) as the prncple cause of varaton n the data. The goal s to test H 0 : all I treatments have the same effect, vs H 1 : there are systematc dfferences. Data: 70 measurements of chlorphenramne maleate n tablets wth a nomnal dosage of 4 mg. Seven labs made ten measurements each: I = 7, = 10. Normal theory model Lab Mean Normally dstrbuted observatons Y j N(µ, σ ) wth equal varances (compare to the t-tests). In other words, Y j = µ + α + ɛ j, α = 0, ɛ j N(0, σ ), meanng: obs = overall mean + dfferental effect + nose. Sample means as maxmum lkelhood estmates Ȳ. = 1 Y j, Ȳ.. = 1 Y. = 1 Y j, I I j j ˆµ = Ȳ.., ˆµ = Ȳ., ˆα = Ȳ. Ȳ.., ˆα = 0, so that Y j = ˆµ + ˆα + ˆɛ j, where ˆɛ j = Y j Ȳ. are the so-called resduals Decomposton of the total sum of squares: SS T = SS A + SS E. SS T = j (Y j Ȳ..) total sum of squares for the pooled sample wth df T = I 1, SS A = ˆα factor A sum of squares (between-group varaton) wth df A = I 1, SS E = j ˆɛ j error sum of squares (wthn-group varaton) wth df E = I( 1). Mean squares and ther expected values MS A = SS A df A, E(MS A ) = σ + I 1 = SS E df E, E( ) = σ. α, 1
2 One-way F -test Pooled sample varance s p = = s an unbased estmate of σ. Use F = MS A 1 I( 1) (Y j Ȳ.) as test statstc for H 0 : α 1 =... = α I = 0 aganst H 1 : α u α v for some (u, v). Reject H 0 for large values of F, snce E H0 (MS A ) = σ and E H1 (MS A ) = σ + I 1 j α > σ. Null dstrbuton F F n1,n wth degrees of freedom n 1 = I 1 and n = I( 1). If X 1 χ n 1 and X χ n are two ndependent random varables, then X 1/n 1 X /n F n1,n. The normal probablty plot of resduals ˆɛ j supports the normalty assumpton. Nose sze σ s estmated by s p = = One-way Anova table Source df SS MS F P -value Labs Error Total Whch of the ( 7 ) = 1 parwse dfferences are sgnfcant? Usng the 95% CI for a sngle par of ndependent samples (µ u µ v ) we get (Ȳu. Ȳv.) ± t 63 (0.05) sp = (Ȳu. Ȳv.) ± 0.055, 5 where t 63 (0.05) =.00. Ths formula yelds 9 sgnfcant dfferences: Labs Dff The multple comparson problem: the above CI formula s amed at a sngle dfference, and may produce false dscoveres. We need a smultaneous CI formula for all 1 parwse comparsons. Bonferron method Thnk of k ndependent replcatons of a statstcal test. The overall result s postve f we get at least one postve result among these k tests. The overall sgnfcance level α s obtaned, f each sngle test s performed at sgnfcance level α/k: ndeed, assumng the null hypothess s true, the number of postve results s X Bn(k, α k ), and due to ndependence P(X 1 H 0 ) = 1 (1 α k )k α for small values of α. Smultaneuos 100(1 α)% CI formula for ( I ) parwse dfferences (µu µ v ):
3 (Ȳu. Ȳv.) α ± t I( 1) ( I(I 1) ) s p Flexblty of the formula: works for dfferent sample szes as well after replacng by Warnngs: ( I ) parwse Anova comparsons are not ndependent as requred by Bonferron method, Bonferron method gves narrower ntervals compared to the Tukey method. 1 u + 1 v. The Bonferron smultaneuos 95% CI for (α u α v ) (Ȳu. Ȳv.) ± t 63 (.05) sp 4 5 = (Ȳu. Ȳv.) ± 0.086, where t 63 (0.001) = 3.17, detects 3 sgnfcant dfferences between labs (1,4), (1,5), (1,6). Tukey method If I ndependent samples (Y 1,..., Y ) taken from N(µ, σ ) have the same sze, then the sample means Ȳ. N(µ, σ ) are ndependent. Consder the range of dfferences between (Ȳ. µ ): Then we get R(I; ) = max{ȳ1. µ 1,..., ȲI. µ I } mn{ȳ1. µ 1,..., ȲI. µ I }. R(I; ) s p / SR(I, I( 1)), where the so-called studentzed range dstrbuton SR(k, df) has two parameters: the number of samples k, and the number of degrees of freedom used n the varance estmate s p. Tukey s 95% smultaneuos CI = (Ȳu. Ȳv.) ± q I,I( 1) (0.05) sp Usng q 7,60 (0.05) = 4.31 from the SR-dstrbuton table, we fnd four sgnfcant parwse dfferences: (1,4), (1,5), (1,6), (3,4), snce (Ȳu. Ȳv.) ± q 7,63 (0.05) = (Ȳu. Ȳv.) ± Kruskal-Walls test A nonparametrc test, wthout assumng normalty, for H 0 : all observatons are equal n dstrbuton, no treatment effects. Extendng the dea of the rank-sum test, consder the pooled sample of sze N = I. Let R j be the pooled ranks of the sample values Y j, so that j R j = N(N+1) and R.. = N+1 s the mean rank. Kruskal-Walls test statstc K = 1 I N(N+1) =1 ( R. N+1 ) 3
4 Reject H 0 for large K usng the null dstrbuton table. For I = 3, 5 or I 4, 4, use the approxmate null dstrbuton K a χ I 1. In the table below the actual measurements are replaced by ther ranks Wth the observed test statstc K = 8.17 and df = 6, usng χ 5-dstrbuton table we get a P-value Two-way layout Labs Means Suppose the data values are nfluenced by two man factors and a nose: Y jk = µ + α + β j + δ j + ɛ jk, = 1,..., I, j = 1,...,, k = 1,..., K, grand mean + man A-effect +man B-effect + nteracton + nose. Factor A has I levels, factor B has levels, and we have K observatons for each combnaton (, j). Normal theory model Key assumpton: all nose components ɛ jk N(0, σ ) are ndependent and have the same varance. Parameter constrants and numbers of degrees of freedom df A = I 1, because α = 0, df B = 1, because j β j = 0, df AB = I I + 1 = (I 1)( 1), because δ j = 0, j δ j = 0. Maxmum lkelhood estmates: ˆµ = Ȳ..., ˆα = Ȳ.. Ȳ..., ˆβj = Ȳ.j. Ȳ..., ˆδ j = Ȳj. Ȳ... ˆα ˆβ j = Ȳj. Ȳ.. Ȳ.j. + Ȳ..., and the resduals ˆɛ jk = Y j. Ȳjk. Example (ron retenton) Raw data X jk s the percentage of ron retaned n mce. Factor A: I = ron forms, factor B: = 3 dosage levels, K = 18 observatons for each (ron form, dosage level) combnaton. From the graphs we see that the raw data s not normally dstrbuted. However, the transformed data Y jk = ln(x jk ) produce more satsfactory graphs. The sample means and maxmum lkelhood estmates for the transformed data 4
5 ( ) (Ȳj.) = two rows produce two profles: not parallel - possble nteracton, Ȳ... = 1.9, ˆα 1 = 0.14, ˆα = 0.14, ( ) ˆβ 1 = 0.50, ˆβ = 0.08, ˆβ3 = 0.4, (ˆδ j ) = Sums of squares SS T = j k (Y jk Ȳ...) = SS A + SS B + SS AB + SS E, df T = IK 1 SS A = K ˆα, df A = I 1, MS A = SS A df A, E(MS A ) = σ + K I 1 α j β j SS B = IK ˆβ j, df B = 1, MS B = SS B df B, E(MS B ) = σ + IK 1 SS AB = K j δ j, df AB = (I 1)( 1), MS AB = SS AB df AB, E(MS AB ) = σ + SS E = j k (Y jk Ȳj.), df E = I(K 1), = SS E df E, E( ) = σ Three F -tests Pooled sample varance s p = s an unbased estmate of σ. K (I 1)( 1) j δ j Null hypothess No-effect property Test statstcs and null dstrbuton H A : α 1 =... = α I = 0 E(MS A ) = σ F A = MS A F dfa,df E H B : β 1 =... = β = 0 E(MS B ) = σ F B = MS B F dfb,df E H AB : all δ j = 0 E(MS AB ) = σ F AB = MS AB F dfab,df E Reject null hypothess for large values of the respectve test statstc F. Inspect normal probablty plot for the resduals ˆɛ jk. Example (ron retenton) Two-way Anova table for the transformed ron retenton data. Dosage effect was expected from the begnnng. Interacton s not sgnfcant. Source df SS MS F P Iron form Dosage Interacton Error Total Sgnfcant effect due to ron form. Estmated log scale dfference ˆα ˆα 1 = Ȳ.. Ȳ1.. = 0.8 yelds the multplcatve effect of e 0.8 = 1.3 on a lnear scale. 3 Randomzed block desgn Blockng s used to remove the effects of a few of the most mportant nusance varables. Randomzaton s then used to reduce the contamnatng effects of the remanng nusance varables. Block what you can, randomze what you cannot. 5
6 Expermental desgn: randomly assgn I treatments wthn each of blocks. Test the null hypothess of no treatment effects usng the two-way layout Anova. The block effect s antcpated and s not of major nterest. Examples: Block Treatments Observaton A homogeneous plot of land I fertlzers each appled to The yeld on the dvded nto I subplots a randomly chosen subplot subplot (, j) A four-wheel car 4 types of tres tested on the same car tre s lfe-length A ltter of I anmals I dets randomly assgned to I snlngs the weght gan Addtve model If K = 1, then we cannot estmate nteracton. Ths leads to the addtve model wthout nteracton Y j = µ + α + β j + ɛ j. Maxmum lkelhood estmates ˆµ = Ȳ.., ˆα = Ȳ. Ȳ.., ˆβ = Ȳ.j Ȳ.., ˆɛ j = Y j Ȳ.. ˆα ˆβ = Y j Ȳ. Ȳ.j + Ȳ.. Sums of squares SS T = j (Ȳj Ȳ..) = SS A + SS B + SS E, df T = I 1 SS A = ˆα, df A = I 1, MS A = SS A df A SS B = I ˆβ j j, df B = 1 MS B = SS B SS E = df B F A = MS A F B = MS B j ˆɛ j, df E = (I 1)( 1) = SS E df E E( ) = σ Example (tchng) Data: the duraton of the tchng n seconds Y j, wth K = 1 observaton per cell, I = 7 treatments to releve tchng appled to = 10 male volunteers aged F dfa,df E F dfb,df E Subject No Drug Placebo Papaverne Morphne Amnophyllne Pentabarbtal Trpelennamne BG F BS SI BW TS GM SS MU OS Boxplots ndcate volatons of the assumptons of normalty and equal varance. Notce much bgger varance for the placebo group. Two-way Anova table Source df SS MS F P Drugs Subjects Error Total
7 Tukey s method of multple comparson q I,(I 1)( 1) (α) sp 3096 = q 7,54 (0.05) = 75.8 reveals only 10 one sgnfcant dfference: papaverne vs placebo wth = 90. > Fredman test Treatment Mean Nonparametrc test, when ɛ j are non-normal, to test H 0 : no treatment effects. Rankng wthn j-th block: (R 1j,..., R Ij ) = ranks of (Y 1j,..., Y Ij ) so that R 1j R Ij = I(I+1) mplyng 1(R I 1j R Ij ) = I+1 and R.. = I+1. Test statstc Q = 1 I I(I+1) =1 ( R. I+1 ) has an approxmate null dstrbuton Q a χ I 1. Snce Q s a measure of agreement between rankngs, we reject H 0 for large values of Q. Example (tchng) From the values R j and R. below and I+1 = 4, we fnd the Fredman test statstc Q = Usng the ch-square dstrbuton table wth df = 6 we obtan an approxmate P-value to be.14%. We reject the null hypothess of no effect even n the non-parametrc settng. Subject No Drug Placebo Papaverne Morphne Amnophyllne Pentabarbtal Trpelennamne BG F BS SI BW TS GM SS MU OS R , 7
Chapter 12. Analysis of variance
Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout
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 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 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 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 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 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 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 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 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 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 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 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 informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 11 Analysis of Variance - ANOVA. Instructor: Ivo Dinov,
UCLA STAT 3 ntroducton to Statstcal Methods for the Lfe and Health Scences nstructor: vo Dnov, Asst. Prof. of Statstcs and Neurology Chapter Analyss of Varance - ANOVA Teachng Assstants: Fred Phoa, Anwer
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 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 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 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 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 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 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 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 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 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 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 informationThe experimental unit of a study is the object on which measurements are taken.
Contents 4 Analyss of Varance (ANOVA) 1 4.1 Introducton........................................... 1 4.1.1 Termnology...................................... 1 4.1.2 Data...........................................
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 informationBOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS. M. Krishna Reddy, B. Naveen Kumar and Y. Ramu
BOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS M. Krshna Reddy, B. Naveen Kumar and Y. Ramu Department of Statstcs, Osmana Unversty, Hyderabad -500 007, Inda. nanbyrozu@gmal.com, ramu0@gmal.com
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 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 informationUnit 8: Analysis of Variance (ANOVA) Chapter 5, Sec in the Text
Unt 8: Analyss of Varance (ANOVA) Chapter 5, Sec. 13.1-13. n the Text Unt 8 Outlne Analyss of Varance (ANOVA) General format and ANOVA s F-test Assumptons for ANOVA F-test Contrast testng Other post-hoc
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 informationF8: Heteroscedasticity
F8: Heteroscedastcty Feng L Department of Statstcs, Stockholm Unversty What s so-called heteroscedastcty In a lnear regresson model, we assume the error term has a normal dstrbuton wth mean zero and varance
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 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 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 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 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 informationJoint Statistical Meetings - Biopharmaceutical Section
Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve
More informationContents. 2 groups n groups (n > 2) (independent) unpaired. paired t -test. one-way ANOVA ANOVA. (related) paired. two-way ANOVA.
Statstcal ests for Computatonal Intellgence Research and Human Subjectve ests Sldes are downloadable from http://www.desgn.kyushu-u.ac.jp/~takag Hdeyuk AKAGI Kyushu Unversty, Japan http://www.desgn.kyushu-u.ac.jp/~takag/
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 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 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 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 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 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 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 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 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 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 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 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 informationECON 351* -- Note 23: Tests for Coefficient Differences: Examples Introduction. Sample data: A random sample of 534 paid employees.
Model and Data ECON 35* -- NOTE 3 Tests for Coeffcent Dfferences: Examples. Introducton Sample data: A random sample of 534 pad employees. Varable defntons: W hourly wage rate of employee ; lnw the natural
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 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 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 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 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 information+ E 1,1.k + E 2,1.k Again, we need a constraint because our model is over-parameterized. We add the constraint that
TWO WAY ANOVA Next we consder the case when we have two factors, categorzatons, e.g. lab and manufacturer. If there are I levels n the frst factor and J levels n the second factor then we can thnk of ths
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 informationStatistical tables are provided Two Hours UNIVERSITY OF MANCHESTER. Date: Wednesday 4 th June 2008 Time: 1400 to 1600
Statstcal tables are provded Two Hours UNIVERSITY OF MNCHESTER Medcal Statstcs Date: Wednesday 4 th June 008 Tme: 1400 to 1600 MT3807 Electronc calculators may be used provded that they conform to Unversty
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 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 informationDurban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications
Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha ralyaf@ksu.edu.sa, maomar@ksu.edu.sa, aalshha@ksu.edu.sa Department
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 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 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 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 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 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 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 informationI i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic.
Serik Sagitov, Chalmers and GU, February, 08 Solutions chapter Matlab commands: x = data matrix boxplot(x) anova(x) anova(x) Problem.3 Consider one-way ANOVA test statistic For I = and = n, put F = MS
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 information4.3 Poisson Regression
of teratvely reweghted least squares regressons (the IRLS algorthm). We do wthout gvng further detals, but nstead focus on the practcal applcaton. > glm(survval~log(weght)+age, famly="bnomal", data=baby)
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 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 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 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 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 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 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 informationSTAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test
STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test Rebecca Barter April 13, 2015 Let s now imagine a dataset for which our response variable, Y, may be influenced by two factors,
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 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 information4.1. Lecture 4: Fitting distributions: goodness of fit. Goodness of fit: the underlying principle
Lecture 4: Fttng dstrbutons: goodness of ft Goodness of ft Testng goodness of ft Testng normalty An mportant note on testng normalty! L4.1 Goodness of ft measures the extent to whch some emprcal dstrbuton
More informationAnswers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
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 informationUnit 10: Simple Linear Regression and Correlation
Unt 10: Smple Lnear Regresson and Correlaton Statstcs 571: Statstcal Methods Ramón V. León 6/28/2004 Unt 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regresson analyss s a method for studyng the
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 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 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 informationMultiple Contrasts (Simulation)
Chapter 590 Multple Contrasts (Smulaton) Introducton Ths procedure uses smulaton to analyze the power and sgnfcance level of two multple-comparson procedures that perform two-sded hypothess tests of contrasts
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 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 informationProfessor Chris Murray. Midterm Exam
Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam.
More information10-701/ Machine Learning, Fall 2005 Homework 3
10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40
More informationStat 543 Exam 2 Spring 2016
Stat 543 Exam 2 Sprng 2016 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of 11 questons. Do at least 10 of the 11 parts of the man exam.
More informationCHAPTER IV RESEARCH FINDING AND DISCUSSIONS
CHAPTER IV RESEARCH FINDING AND DISCUSSIONS A. Descrpton of Research Fndng. The Implementaton of Learnng Havng ganed the whole needed data, the researcher then dd analyss whch refers to the statstcal data
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More information