Ebook Applied multivariate statistical analysis (5th edition) Part 1

(BQ) Part 1 book Applied multivariate statistical analysis has contents: Aspects of multivariate analysis, matrix algebra and random vectors; sample geometry and random sampling; sample geometry and random sampling; sample geometry and random sampling; comparisons of several multivariate means.

Applied Multivariate Statistical Analysis FIFTH E DITION Applied Multivariate Statistical Analysis RICHARD A JOH N SON University of Wisconsin-Madison D EAN W WIC H E RN Texas A&M University Prentice Hall PRENTICE HALL, Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Johnson, Richard Arnold Applied multivariate statistical analysis/Richard A Johnson. 5th ed p cm Includes bibliographical references and index ISBN 0-13-092553-5 Multivariate analysis I Wichern, Dean W II Title QA278 J 63 2002 519.5'35 dc21 2001036199 Quincy McDonald Editor-in-Chief: Sally Yagan Acquisitions Editor: David W Riccardi Kathleen Schiaparelli Senior Managing Editor: Linda Mihatov Behrens Assistant Managing Editor: Bayani DeLeon Production Editor: Steven S Pawlowski Manufacturing Buyer: Alan Fischer Manufacturing Manager: Trudy Pisciotti Marketing Manager: Angela Battle Editorial Assistant/Supplements Editor: Joanne Wendelken Managing Editor, Audio/Video Assets: Grace Hazeldine Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Dlustrator: Marita Froimson Vice President/Director Production and Manufacturing: Executive Managing Editor: ã â 2002, 1998, 1992, 1988, 1982 by Prentice-Hall, Inc Upper Saddle River, NJ 07458 All rights reserved No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Printed in the United States of America 10 ISBN 0-13-092553-5 Pearson Education LTD., London Pearson Education Australia PTY, Limited, Sydney Pearson Education Singapore, Pte Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Education de Mexico, S.A de C.V Pearson Education-Japan, Tokyo Pearson Education Malaysia, Pte Ltd To the memory of my mother and my father R A J To Dorothy, Michael, and Andrew D W W Contents PREFACE ASPECTS OF MULTIVARIATE ANALYSIS 1.1 1.3 1.5 XV Introduction Applications of Multivariate Techniques The Organization of Data Arrays, Descriptive Statistics, Graphical Techniques, 11 Data Displays and Pictorial Representations Linking Multiple Two-Dimensional Scatter Plots, 20 Graphs of Growth Curves, 24 Stars, 25 Chernoff Faces, 28 Distance 30 Final Comments Exercises 38 References 48 19 38 MATRIX ALGEBRA AND RANDOM VECTORS 2.1 2.2 2.3 2.4 2.5 2.6 2.7 50 Introduction 50 Some Basics of Matrix and Vector Algebra 50 Vectors, 50 Matrices, 55 Positive Definite Matrices 61 A Square-Root Matrix 66 Random Vectors and Matrices 67 Mean Vectors and Covariance Matrices 68 Partitioning the Covariance Matrix, 74 The Mean Vecto r and Covariance Matrix for Linear Combinations of Random Variables, 76 Partitioning the Sample Mean Vector and Covariance Matrix, 78 Matrix Inequalities and Maximization 79 vii viii Contents Supplement 2A: Vectors and Matrices: Basic Concepts Vectors, 84 Matrices, 89 84 Exercises 104 References 1 3.1 3.2 3.3 3.4 3.5 112 SAMPLE GEOMETRY AND RANDOM SAMPLING Introduction 112 The Geometry of the Sample 112 Random Samples and the Expected Values of the Sample Mean and Covariance Matrix 120 Generalized Variance 124 Situations in which the Generalized Sample Variance Is Zero, 130 Generalized Variance Determined by I R I and Its Geometrical Interpretation, 136 Another Generalization of Variance, 138 Sample Mean, Covariance, and Correlation As Matrix Operations 139 Sample Values of Linear Combinations of Variables Exercises 145 References 148 141 THE MULTIVARIATE NORMAL DISTRIBUTION 4.1 4.2 4.3 4.4 4.5 4.6 Introduction 149 The Multivariate Normal Density and Its Properties 149 149 Additional Properties of the Multivariate Normal Distribution, 156 Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation 168 The Multivariate Normal Likelihood, 168 Maximum Likelihood Estimation of JL and I, 170 Sufficient Statistics, 173 The Sampling Distribution of X and S 173 Properties of the Wishart D istribution, 174 Large-Sample Behavior of X and S 175 Assessing the Assumption of Normality 177 Evaluating the Normality of the Univariate Marginal D istributions, 178 Evaluating Bivariate Normality, 183 4.7 Detecting Outliers and Cleaning Data 4.8 Transformations To Near Normality Steps for Detecting Outliers, 190 189 194 Transforming Multivariate Observations, 198 Exercises 202 References 209 Contents INFERENCES ABOUT A MEAN VECTOR 5.1 5.2 5.3 5.5 5.6 5.7 5.8 21 Introduction 210 The Plausibility of Ito as a Value for a Normal Population Mean 210 Hotelling's T2 and Likelihood Ratio Tests 216 General Likelihood Ratio Method, 219 Confidence Regions and Simultaneous Comparisons of Component Means 220 Simultaneous Confidence Statements, 223 A Comparison of Simultaneous Confidence Intervals with One-at-a- Time Intervals, 229 The Bonferroni Method of Multiple Comparisons, 232 Large Sample Inferences about a Population Mean Vector 234 Multivariate Quality Control Charts 239 Charts for Monitoring a Sample of Individual Multivariate Observations for Stability, 241 Contro l Regions for Future Individual Observations, 247 Control Ellipse for Future Observations, 248 T -Chart for Future Observations, 248 Control Charts Based on Subsample Means, 249 Control Regions for Future Subsample Observations, 251 Inferences about Mean Vectors when Some Observations Are Missing 252 Difficulties Due to Time Dependence in Multivariate Observations 256 Supplement SA: Simultaneous Confidence Intervals and Ellipses as Shadows of the p-Dimensional Ellipsoids 258 Exercises 260 References 270 COMPARISONS OF SEVERAL MULTIVARIATE MEANS 6.1 6.2 6.3 6.4 ix 272 Introduction 272 Paired Comparisons and a Repeated Measures Design 272 Paired Comparisons, 272 A Repeated Measures Design for Comparing Treatments, 278 Comparing Mean Vectors from Two Populations Assumptions Concerning the Structure of the Data, 283 Further Assumptions when n1 and n2 Are Small, 284 Simultaneous Confidence Intervals, 287 The Two-Sample Situation when �1 i= �2, 290 283 Comparing Several Multivariate Population Means ( One-Way Manova) 293 Assumptions about the Structure of the Data for One-way MAN OVA, 293 A Summary of Univariate AN OVA, 293 Multivariate Analysis of Variance (MAN OVA), 298 Chapter Exercises 339 TABLE 6.9 CARAPACE M EAS U R E M E NTS (I N M I LLI M ETE RS) FOR PAI NTED TU RTLES Femal e Mal e Length Width Height Length Width Height 10310398 818486 423838 939496 747880 373535 105109 8688 4244 102101 8485 3839 123123 9592 5046 103104 8183 3739 133133 10299 5151 107106 8382 3938 133134 100102 5148 112113 8988 4040 136138 10298 4951 114116 8690 4340 138141 10599 5153 117117 9091 4141 147149 108107 5557 120119 9389 4140 153155 107115 6356 120121 9395 4442 155158 117115 6062 127125 9693 4545 159162 118124 6361 128131 9595 4645 177 132 67 135 106 47 FiCompar nd simeulwitaneous conf i d ence i n t e r v al s f o r t h e component mean di f e r e nces t h t h e Bonf e r o ni i n t e r v al s ions of the obsIn therevfatYouirsitophasnsmay e ofwiasshtutdyo ofconsithe dcosert oflogaritransthpormiticngtrmiansformat l k f r o m f a r m s t o dai r y plonants, a sfuurelvey, was rtaekenpair,ofandfirms engaged i n mi l k t r a ns p or t a t i o n Cos t dat a artrueckspre sented in Table 6.10 on page 340capifortal, all36measgasuorleindeonanda per-m23iledibasesiesl, r dif ehresencesis ofiequal n the mean cosot rvects is orresj.ectSeted in Par.01.t a, find the linear IfTescombitht efonhypot cos t vect at i o n of mean component s mos t r e s p ons i b l e f o r t h e r e j e ct i o n Cons t r u ct 99% s i m ul t a neous conf i d ence i n t e r v al s f o r t h e pai r s of mean com­ ponents Which costs, if any, appear to be quite dif erent? ( x1 ) ( x2 ) ( x3 ) ( xi ) ( x2 ) ( x3 ) (c) Hin t: 6.19 X1 = (a) (b) (c) X2 = X3 = n1 = n2 = a = 340 Chapter Com parisons of Severa l M u ltivariate Means (d) Comment on t h e val i d i t y of t h e as s u mpt i o ns us e d i n your anal y s i s Not e i n parfiedtiasculmular tthivatarobsiateeoutrvatliieornss 9(Sandee Exer21 fcoirsegas5.2o2linande tr[u2cks] ) Repeat have beenPartiadenti­ these observations deleted Comment on the results Gasoline trucks Diesel trucks 16.7.4149 12.2.4730 11.3.2923 7.8.5420 12.5.2163 17.9.1151 9.4.2942 5.1.7358 7.9.7758 10.10.2186 14.3.7322 11.5.2939 11.14.2205 5.5.7058 10.9.8678 12.9.6709 12.4.7172 29.11.2008 13.13.3520 10.14.2978 10.9.4650 11.6.4357 8.9.8959 19.14.0530 29.12.6181 15.7.0691 10.3.2238 9.9.7150 2.5.0946 20.13.6848 7.9.9501 5.3.6803 8.9.1133 11.9.7671 17.11.7856 35.17.0180 10.11.2151 5.6.0175 10.7.6171 9.8.0539 10.13.2145 20.17.4656 12.10.2174 14.2.2569 14.6.0399 15.8.2990 12.6.2920 16.19.0389 10.8.8188 2.6.7005 12.12.2143 11.9.5944 16.5.7679 22.14.7676 12.8.3541 14.7.7023 11.12.0681 10.10.4873 21.17.5625 28.10.4676 26.12.9156 17.8.4244 16.7.8189 11.7.8183 12.13.2128 21.19.4240 16.14.7903 13.10.7378 17.14.5589 12.03 9.22 23.09 10.8.3928 4.5.4196 17.4.2006 9.12.7720 11.8.6539 6.5.8539 8.13.9.274029 11.2.7.291256 4.6.6.279231 8.15.2816 11.9.4825 13.8.0176 9.12.4189 4.9.6178 11.9.4949 17.32 6.86 4.44 with TABLE xl M I LK TRANSPORTATION-COST DATA x2 x3 Source: Data courtesy of M Keaton xl x2 x3 Chapter 6.20 ( x1 ) ( x2 ) The thookail lengtbilhesdinkimitesl arimeetgievrens in Tablande wi6.1n1.g lSiengtmilhasr imeasn milurimementeterss for fefomalr 45e hook-Plobtiltheedmalkiteeshookwere bgiilvened kiintTable datea5.as12.a scatter diagram, and (visually) check fTesor toutfolrieequalrs (Nitoty ofe, imean n partivectculaorr, sobsfoerrtvhateipopul on 31 awitiothns of mal284.e )and female hookb i l e d ki t e s Set I f i s r e j e ct e d, f i n d t h e l i n ­ tearbefo elocombi irme inconduct atneatanyioninoutmosg ltihteirrssestfpeoonssundt iAlbinletParefronrtattahievfoelrreyjt,ehyoucte imalonmayofe hook-want(bYilouteodmayikintteerwantdatpreat 284 f o r obs e r v at i o n 31 as a mi s p r i n t and conduct t h e t e s t wi t h 184 vatfDetoriotehrnims31iobsnfeoetrhrtvehate95%imalon.confeDoeshook-idenceitbimakel eredgikioanytne fdatodir fa eisretncereateinandd?)thi95%s casseimhowultaobsneouser­ confAre imaldencee orinfteemalrvalesbiforrdtshgeneral e component s of ly larger? (lengtTaihl ) l(eWngtinhg) le(ngtTaihl ) l(eWngtinhg) le(ngtTaihl ) l(eWngtinhg) 186206180 278277308 195183185 285276282 284176185 287277281 184177 290273 202177 254308 191177 267295 177176 267284 170177 268260 199197 299310 200191 281287 186177 272274 190180 273278 193212 271302 192178 281266 189194 280290 181195 254297 204191 276290 191186 287286 187190 281284 178177 265275 187186 288275 ' Usbondsing Moody sBaabond(torp-atminedigs, suammplquales iofty20) corAapor(maitdedlbonds e-highwerquale isteyl)ecorctepd.orForate and 20 each ofXthe corcurrersepntondiratniog (compani e s , t h e r a t i o s a meas u r e of s h or t t e r m l i q ui d i t y ) XX3 ldebtong tteor-mequiinttyerreasttiora(tae meas(a measureuofre fofinanciinteraelsrtiscoverk or laege)verage) X4 rate of return on equity (a measure of profitability) male (a) x1 (b) a = x1 H0 : p - IL = = H0 • = x1 (c) IL l - IL IL l - IL • (d) TABLE 6.1 xl MALE HOOK-B I LLED KITE DATA x2 xl Source: Data courtesy of S Temple 6.21 341 Exercises = = = = x2 xl x2 = 342 Chapter Com parisons of Severa l M u ltivariate Means were recorded The summar y s t a t i s t i c s ar e as f o l o ws : n1 ==45920, x1 2==54[2.2-.87,02612.600,-.2.44347, 14.830] , and -.202654 27.-.456589 - 053089 - 216702 -.n22==4420, -x2.2==67 [2.404,.1027.155,6.8.54524, 12.840], -.094489 16.-.403289 - 400200 19.-.074419 -.070219 19.-.400044 - 002494 61.-.085494 and 81 12 01 83 - == - 008312 21.-.9.439498894 - 400270494 34.9 033540488 81 - Does pool i n g appear r e as o nabl e her e ? Comment on t h e pool i n g pr o cedur e iArn tehtishecasfine.ancial characteristics of firms with Aa bonds dif erent from those wioftmean h Baavectbondsors? SetUsing==th.e05.pooled covariance matrix, test for the equality Calfor creuljeactteinthgeH0:linearILl -combiIL2 ==natiionnsParoft mean component s mos t r e s p ons i b l e b.in a company's ability to satisfy its Bond r a t i n g compani e s ar e i n t e r e s t e d outmorsetaofndithneg fdebtoregoioblngigfatiniancions aasl rtahtieyos matmiguhtrebe Doesusefulitinappear as i f one or helping to clas ify aResearbondcherass in"hitergeh"steord i"medi u m" qual i t y ? Expl a i n n as s e s i n g pul m onar y f u nc t i o n i n non pat h ol o gi c al pop­ ulwerateiocolns lasectkeded satubjdefectinsitteo irnutneronvalsa andtreadmithe gasl untcontil exhaus t i o n Sampl e s of a i r e nt s anal y zed The r e s u l t s consvarumptiablioens fweror 25e males and 25 females are given in Tablon 4emeas6.12uonrespageof oxygen 343 The XX21 ==== rreessttiinngg volvoluumeme 0022 ((LmL/kgjmi /min) n) XX43 ====maximaximmumum volvoluumeme 0022 ((mL/miLjkg/mi n ) n) Look== 0f5.or Igender di f e r e nces by t e s t i n g f o r equal i t y of gr o up means Us e s t mo f you r e j e ct == f i n d t h e l i n ear combi n at i o n H0: IL IL l rConsesponstruictblet.he 95% simultaneous confidence intervals for each JLl i == 1, 2, 3, Compare with the corresponding Bonfer oni intervals Aa bond companies: Baa bond companies: S pooled (a) (b) a (c) (d) 6.22 (a) a (b) i 0, - JL I ' TABLE xl w � w Res(Lt/miingn0) 0.0.3349 0.0.4381 0.0.3336 0.0.4438 0.0.2321 0.0.3524 0.0.4301 0.0.3424 0.0.5306 0.0.0.444208 0.0.5550 0.0.4340 OXYG E N-CO N S U M PTI O N DATA Males Maxi m um Res t i n g 2 (mL/kg/mi n ) ( L /mi n ) 3.5.7081 2.3.3878 5.3.9153 4.3.6103 4.4.5.705771 4.3.3.931519 3.6.4.376591 3.2.3.855209 7.5.8379 3.3.4077 4.4.9957 4.3.4536 6.4.8608 3.3.3816 6.5.4939 3.3.2109 6.6.0300 4.3.0806 6.6.0454 5.3.0805 5.4.5275 4.5.0203 4.58 2.82 Xz Source: Data courtesy of S Rokicki x3 x4 Maxi(mL/kg/mi mum 0n)2 30.43.8875 44.46.0501 48.48.47.750025 48.48.9826 48.50.5368 51.55.3154 56.58.4679 42.51.49.279509 63.46.2303 55.58.0808 57.50.4356 32.48 - - - xl Res(Lt/miingn0) 0.0.2298 0.0.3301 0.0.0.221518 0.0.2396 0.0.3371 0.0.0.233539 0.0.2188 0.0.2424 0.0.3340 0.0.2371 0.0.0.336576 - - Females mum0n)2 Res(mL/kg/mi ting 0n2 ) Maxi(Lm/mumin)02 Maxi(mL/kg/mi 5.3.0945 2.1.9531 33.35.8825 4.5.8978 2.1.9301 36.37.4870 4.1.5774 2.2.4329 38.39.3109 4.5.2686 2.1.9128 39.39.2941 7.6.2322 2.1.7251 28.42.4971 4.5.2100 2.2.7106 37.31.8100 4.5.4606 2.3.0506 38.51.3800 2.4.0801 2.2.4508 36.37.7608 6.4.5659 3.1.0855 38.46.9156 5.4.5.771723 2.2.1.459783 40.43.30.466009 5.11.0156 2.2.3023 39.39.3446 5.5.2373 2.2.4258 34.35.0876 - - - x3 Xz - - - - - - - - x4 - - - 344 Chapter Com pa risons of Several M u ltivariate Means Thethusdattheya idon Tablnotere6.pr12ewersenteacolrandom lected sfaromplm gre.aComment duate-studenton tvolhe posuntesierblse, andim­ pl i c at i o ns of t h i s i n f o r m at i o n Cons t r u ct a onew ay MANOVA us i n g t h e wi d t h meas u r e ment s f r o m t h e i r i s Tablin mean e 11.5component Constructs 95%for thsiemtulwtoaneous confes fiodrenceeachinpaiterrvalofspopul for diaf-ftdatieornseances.inComment r e s p ons on t h e val i d i t y of t h e as s u mpt i o n t h at Resthe ienartecrherbresedihaveng ofsuaggesresidteentd thpopulat a cahtiangeon wiitnh simkulmil gsizraentoverpopultimatieonsis eviFourdencemea-of sruiordement1 is 4000s were madeperioofd 2malis 3300e Egyptiaandn skperullsifoodr3thisre1850e differeThent timdate aperarieosdshown: pein Table 6.13 see the skull data on the CD-ROM The measured variables are MaxBreath BasHeight BasLength NasHeight PerTimioed 138131 4948 11 131125 8992 132132 131119 5044 11 9996 143137 10089 136138 5456 11 130136 10893 4848 11 139125 10299 5151 11 134134 134131 138134 124133 10197 4848 22 138148 134129 4551 22 98104 126135 124 4552 22 9598 136 132133 145 5448 22 100 130 102 131133 134125 9694 5046 22 132133 130131 91100 5250 33 138130 137127 9499 51 33 45 136134 133 4952 33 91 123 95 136133 137131 54 33 10196 49 133133 138138 10091 5546 33 (c) 6.23 I1 6.24 B.C , B.C , ( TABLE (x i ) ) B.C EGYPTIAN SKULL DATA (x2 ) Source: Data courtesy of J Jackson (x3 ) == (x4) I == I X1 == maxibasibmreumgmatbriecadtheihghtof ofskulsklul(ml (m)mm) = bas i a l v eol a r l e ngt h of s k ul l ( m m) X4 = nasal height of skull (mm) Cons t r u ct a onew ay MAN OVA of t h e Egypt i a n s k ul l dat a Us e a = Con­ struct 95%s sdiimf ulertamong aneousthconfe populidenceatioinsntreerprvalessetnto edetd byertmheintehrwhiee tcimh emeanperi­ component odsCons Artruectthaeone-usuwalayMANOVA as s u mpt i o ns r e al i s t i c f o r t h es e dat a ? Expl a i n MANOVA of t h e c r u deo i l dat a l i s t e d i n Tabl e 11 Con­ scomponent truct 95%s sdiimf ulertaamong neous thconfe populidenceatioinsnt.er(vYalous mayto detwantermtioneconswhiidcerh tmean fMANOVA ormations asofsutmpthe dationsa.)to make them more closely conform to the usraunsal­ Awoulprodjerectactwastodesanielgnedectrtiocalintvesime-tigoatf-eushowe priconscingusmercheme.s in GrTheeencosBay,t ofWielescctonsriciitny, dseakfohourr somes Hourcustolymerconss wasumptseiot nat(ieingkihtlotwatimest-hthoure coss) wast of mea­elec­ tdursruirceiitndygondurpeakainhotgperofsfi-uopmmer day i n Jul y and compared, f o r bot h t h e t e s t gr o up and tthhee contexperroilmgrentoup,al rwiattehs basbegan.elineTheconsruesmptponsioens,measured on a similar day before l o g( cur r e nt cons u mpt i o n) l o g( b as e l i n e cons u mpt i o n) fprorotduced he hourthseendifol onwig 9ng summar 11 y s(taatpeakisticshour) , and ( a peak hour ) : = 28, = [ 153, -.231, -.322, -.339] = 58, = [ 151, 80, 256, 257] and 380455 732255 222833 213299 = 228 233 592 239 232 199 239 479 Perin elfoectrmricaalprconsofileuanalmptyiosin?s DoesWhattiismte-heofnat-usuerpre oficinthgissediemf etroence,makeif aany?dif eCom­ rence ment ( U s e a s i g ni f i c ance l e vel of a = f o r any s t a t i s t i c al t e s t s ) Asbandsparandt ofwithveesstuwerdyeofaslkoedve tando resmarpondriatgeo tihnesExampl e 2, a s a mpl e of hus ­ e ques t i o ns : What i s t h e l e vel of pas s i o nat e l o ve you f e el f o r your par t n er ? What is the level of passionate love that your partner feels for you? Chapter Exercises x2 x3 6.25 6.26 A.M Test group: Control group: nl n2 A.M P.M., P.M x1 x2 S pooled Source: Data courtesy of Statistical Laboratory, University of Wisconsin 6.27 345 346 Chapter Com parisons of Severa l M u ltivariate Means None at all What i s t h e l e vel of c o mpani o nat e l o ve t h at you f e el f o r your par t n er ? WhatTheis trheespleonsveleofs wercompani o nat e l o ve t h at your par t n er f e el s f o r you? e recorded on the fol owing 5-point scale Very little Some A great deal Tremendous amount Thi5-poirtynhust-scbalandse resandpons30ewitovesQuesgavetiothne 1,resXponseas i5-n Tablpoinet-6.sc1al4,ewherrespeonsX1 e Husband rating wife Wife rating husband 25 53 45 45 44 45 55 55 44 53 54 45 44 45 55 55 33 33 54 55 43 43 54 54 34 44 54 54 43 43 55 45 44 54 53 53 43 44 54 44 45 54 54 54 45 55 55 55 44 43 45 45 44 44 54 54 43 34 54 55 43 44 54 54 45 55 54 45 45 55 54 54 54 45 44 44 35 43 44 44 44 44 44 44 54 53 44 44 35 43 55 55 23 45 55 55 53 35 43 43 44 43 54 54 43 43 45 45 43 44 45 54 44 44 53 53 44 44 55 54 == TABLE xl x2 = SPOUSE DATA x3 x4 Source: Data courtesy of E Hatfield xl x2 x3 x4 a to Chapter Exercises Ques t i o n 2, x == a 5p oi n t s c al e r e s p ons e t o Ques t i o n 3, and == a 5p oi n t ­ x scalPle orest tphonse meane to vectQuesortisofnor4.husbands and wives as sample profiles ITess thtefohusr parbandal erlaprtinogfiwilesfewiprthofile==par.0a5.l eIlfttohtehpre wioffileersaappear ting hustboandbe parproaflileel?, tpresotffiloers coiarencicoidentncidprentof,ilteesstatfotrhelesvelameprloefvelilesofwistihgnifi==cance 05 WhatFinallyconcl, if thue­ sTwoisopn(ecis)ecans ofbebitidrngawnfliefsro(gmenusthis analysis? are so similar morphological­ lsyu,chthatasfsoerxmanyratiosyearof emers theyginwerg fleietshandoughtbittiongbehabithetssawerme.e fBioundologitocalexidistf eDorencesthe d ininparthet tiwnoTablspecie 6.es15 on pageand348 and on theTesCD-t fRorOMthe itequalnaxonomi dicatitye anyofc datthdiefatwelirsoetnceepopul a t i o n mean vect o r s us i n g == I f t h e hypot h es e s ofcombiequalnatmean vect o r s i s r e j e ct e d, det e r m i n e t h e mean component s ( o r l i n ear yourUsingusteheofiodatnsnoraofmonmean albone-theorcomponent ymimetnerahlodsconts) fmosoerntthtiesrneeTablspdatonseai.b1.l8e, fionrvesrejteigctatinegequalH0 • iJusty tbe­ify tweenTestthuse idoming n==ant.0and5 nondominant bones Cons t r u ct 95% s i m ul t a neous conf i d ence i n t e r v al s f o r t h e mean di f e r e nces Cons t r u ct t h e Bonf e r o ni 95% s i m ul t a neous i n t e r v al s , and compar e t h es e wi t h t h e i n t e r v al s i n Par t b Tablin Table 6.e11.6 8on, 1pageyear349aftecrontthaeiinr sparthetibonecipatimionniern aanl contexperentims, fentor tahleprfiorgrst 24am.subjCom­ects pareTesthteusdatinagfro==m bot.05.h tables to determine whether there has been bone los Cons t r u ct 95% s i m ul t a neous conf i d ence i n t e r v al s f o r t h e mean di f e r e nces Cons t r u ct t h e Bonf e r o ni 95% s i m ul t a neous i n t e r v al s , and compar e t h es e wi t h t h e i n t e r v al s i n Par t b Peanut ted Steavartesi.eItniesanwiefth­ froerstptecto develstaro seeoanverp iiammlprporvarovedtiaablnteplcrsa ontThepsi,ncrdatparopatsscfiofoernttonehisetsstorwoututo-hfiernactelnoyUnircompar exper i m ent ar e gi v en igeogr n Tablaephi6.c1al7 onlocatpageions349.(1, 2)Thrand,ee varin tiehtiisescas(5e,,6,thande thr8)eewervareiagrbloewns reatprwiestehnttiwnog yitherlede andvariathbleestwaroeimportant grade-grain characteristics were measured The XX21 ==== YiSoundeld (pmatlotuweire kerghtn) els (weight in grams-maximum of250 grams) X == Seed s i z e ( w ei g ht , i n gr a ms , of 0 s e eds ) TherPere werforme tawtowro-eplfaicctatoironsMANOVA of the experusiinmgentthe data in Table 6.17 Test for a locat== i0o5.n effect, a variety effect, and a location-variety interaction Use (a) (b) a a 6.28 Leptoconops) L carteri L torrens? a 6.29 (a) a (b) (c) 6.30 ( a) a (b) (c) 6.31 347 (a) a Chapter 348 TABLE Com parisons of Severa l M u ltivariate Means B ITING-FLY DATA xl x2 ( lWiengtngh) ( wiWidnthg ) L torrens L carteri 8587 9492 9691 929190 87 106105 103100 109104 95104 90104 9410386 82103 101103 10099 100 99110 99103 10110395 1059999 4138 4443 4344 4243 4138 4746 4441 4445 4044 4046 404819 4143 454343 4441 454442 4346 4747 4350 47 ( Thlpalrpd ) (Thipalrpd) (Fourpalpth) ( Lengtantennalh of ) ( Lengtantennalh of) x3 x4 lengt31 h 3236 3235 3636 3636 35 3834 3435 3636 3534 3737 3738 354239 4044 4042 43 3841 3538 384036 403937 wi13dth 1154 1714 1216 1147 11 1514 1514 1315 1415 1214 1114 1412 114155 1815 16 1417 1614 1514 1154 1614 Source: Data courtesy of William Atchley Xs x6 lengt25 h segment9 12 2227 138 2826 109 2426 99 2623 99 24 2631 1010 2324 1010 2730 1110 2329 99 2230 109 2531 96 253332 1099 2529 119 3131 1110 34 10 3336 99 3132 1010 3137 118 3223 1111 3334 127 x7 segment8 139 90 99 99 10 1011 1010 1010 1010 1010 7109 89 911 1010 10 910 1010 811 1110 117 13 Chapter TA BLE 6 bject Sunumber 12 435 67 89 1011 1213 1415 1617 1819 212220 2324 Exercises 349 M I N ERAL CO NTENT I N B O N ES (AFTER YEAR) Domiradinuants 082757 887573 864011 894786 997791 882551 797012 790556 793265 887943 697349 477663 Radi1.051us 888017 689813 786534 892306 982526 773065 882675 772764 798214 593706 960037 743 FactVarioerty2 55 55 66 66 88 88 Source: Data courtesy of Everett Smith TABLE FactLocatoiro1n 11 22 11 22 11 22 Domi n ant humer2.268us 1.1.795318 1.1.664368 1.1.389651 1.1.1.799423133 1.2.365209 1.1.484670 1.1.784247 1.2.912390 1.2.214264 1.2.513073 1.1.404241 PEAN UT DATA xYielld 195.194.33 189.180.74 203.195.09 202.197.76 193.187.50 201.200.50 Source: Data courtesy of Yolanda Lopez Domiulnnaant 680269 775655161 770853 684487 866954 667092 782346 665693 587783 850240 587004 585 Humer u s Ul n a 2.1.724610 698964 1.1.744356 769838 1.1.637861 561915 1.1.688615 778715 1.2.710676 765689 1.1.695180 752626 1.1.480920 757380 1.1.587960 750629 1.1.994197 774085 1.1.292899 762769 1.2.315930 747998 1.1.246511 664034 x x3 SdMat153.K1 er SeedSi51.4 ze 167.139.75 55.53.75 121.156.81 44.49.48 166.166.01 45.60.48 161.164.58 54.57.81 165.166.18 58.65.06 173.8 67.2 50 Chapter Com pa risons of Severa l M u ltivariate Means a Analpear ytozebethesarteissfidieuald?s DifrosmcusPars t Do the usual MANOVA assumptions ap­ Useffeinctgsthareereaddisulttsivine?ParIft nota, can, doeswe concl utdeerathctationthefe lfoecatct siohnowandjorup fovarr soimeety t h e i n vartwo-iafblacteso,rbutANOVAs not for others? Check by running three separate univariate UsLartwoingfgerolronumber cateachionchars2,corcaactnreewesrpisondtconclic?toDiubetdesctustehrsatyiyoureoneld andvaransiwgreterayde-gr i, susbetinagtien95%r charthanBonfatctheereotirsthoicernis sIn oneimulexpertaneousimentintienrvolvalvsinfogrrpaiemotrs eofsvarensiientg,ietsh e spectral reflectance of three sgrpoeciwiensgofse1-asyoearn -Theold sseeedledliinngsgs wasweremeasgrownurewid atthvartwoiodiusf wavel e ngt h s dur i n g t h e e r e nt l e vel s of nut r i e nt : tsheeedloptingsimusaleldevelwer, coded and a s u bopt i m al l e vel , coded - The s p eci e s of (LP) Twoperofcentthesvarpeectsiiatrblkaalerssepmeasfrluectceau(ncerSeS),datwerJapanes ewaveleengtlarhch560(JL)nm, and(grelen)odgepole pine per c ent s p ect r a l r e f l e ct a nce at wavel e ngt h 720 nm ( n ear i n f r a r e d) Thetrientcellelvelmeansare (asCM)fol foowsr Jul Thesian daye aver235agesfor eachare bascombied onnatfoiourn ofreplspiecicateiosnsand nu­ Nut r i e nt Speci e s 720CM 560CM 10.13.4351 25.38.6933 JL 7.10.7408 24.25.2155 LP 17.10.7408 29.41.4205 LPJL Treating thetoceltesltmeans aseciiensdiefvfiedctualandobsaenutrvatriieontnsef, perffect.orUsmea two-.0w5ay MANOVA f o r a s p Cons t r u ct a t w ow ay AN OVA f o r t h e 560CM obs e r v at i o ns and anot h er t w o­ waythe MAN ANOVAOVAforretshuelt720CM obs e r v at i o ns Ar e t h es e r e s u l t s cons i s t e nt wi t h s i n Par t a? I f not , can you expl a i n any di f e r e nces ? Refvariearblteos Exercise 6.32 The data in Table 6.18 are measurements on the perperccentent ssppectectrraall rreeflfleectctaancence atat wavel waveleengtngthh 720560 nmnm ((gnreareen)infrared) fofor1-thyrearee -soplecid seesedl(siitnkgsa stparkenuce at[SS]th,rJapanes erelantrcthim[JesL],(Jandulialnodgepol e pi[1n],eJul[LiP]an) e e di f e day 150 daywere235all gr[2]o,wnandwiJulthiathnedayopti320mal[l3e]vel) durofinnutg trhieentgr.owing season The seedlings (b) (c) (d) 6.32 +, X1 X2 == == ss ss + + + (a) a (b) 6.33 X1 X2 == == == Chapter (a) Exercises Perspecifoersmefafetctw,o-aftaimcteorefMAN OVA us i n g t h e dat a i n Tabl e Tes t f o r a f e c t and s p eci e s t i m e i n t e r a ct i o n Us e Dodatayou? Dithsicnuks thwietush urealfeMAN OVAto a rasessiudmptual analionsyaresis, sandatisftiheed posfor stihbeilitthyesofe r e nce correlated observations over time Speci e s Ti m e Repl i c at i o n 720nm 560nm 11 21 9.8.7343 19.19.5145 11 43 9.8.2371 19.16.2347 22 21 10.10.2123 25.25.3020 22 43 10.10.4622 27.26.2128 33 21 15.16.2225 38.36.8679 17.12.2774 40.67.7540 33 43 12.11.0037 33.32.3073 JLJL 11 21 12.12.4128 31.33.3331 JLJL 11 43 15.14.2381 40.40.0408 JLJL 22 21 9.14.6359 40.33.1905 JLJL 22 43 38.44.7741 77.78.5174 JLJL 33 21 36.37.2671 71.45.4003 JLJL 33 43 8.7.7943 23.20.2877 LPLP 11 21 8.7.3876 22.21.7186 LPLP 11 43 8.6.4759 26.22.3723 LPLP 22 21 8.7.5344 26.24.6877 LPLP 22 43 14.13.5041 44.37.4943 LPLP 33 21 13.12.7373 37.60.9873 LPLP 33 34 a = (b) 351 TABLE SPECTRAL RE FLECTANCE DATA ss ss ss ss ss ss ss ss ss ss ss ss Source: Data courtesy of Mairtin Mac Siurtain 352 Chapter Compar isons of Severa l M u ltivariate Means ForDoesestienrtseraracte iparon tsihcowularuply ifnotreonerestevard iinabltheebutintenotractfioornthofe otspheciere?s andChecktime rCanunniyoung athuniinkvarofianotate thwero-fmetactohrodANofOVAanalyfzionrgeachthesofe datthea tworoaredisfponserentes pertralimreentflectaladesnceignumber n thatswoul? d allow for a potential time trend in the spec­ Ref(a) PlerottotExampl e ver s u s t i m e and t h os e of h e pr o f i l e s , t h e component s of ver s u s timte,thonattlhineearsamegrogrwtaph.h is adequat Commente Takeon the compar.01 ison Tes RefmumerlitkoelExampl e but t r e at al l 31 s u bj e ct s as a s i n gl e gr o up The ihood estimate of the 1) f3 is p whereTheisesthtiemsaatmpled covar e covariancesianceofmattherixmaxi mum likelihood estimators are Cov - - - 1) 2) Fit a quadratic growth curve to this single group and comment on the fit (c) (d) 6.34 ( ) e x­ x2 x1 (b) 6.35 by a == m a xi­ (q == + X ( B' S - B ) - B ' S - x S -._ (n "' ( Pn ) == (n p + (n q) (n p + q )n ( B' S- B ) - REFERENCES Anderson, T W An Introduction to Multivariate Statistical Analysis (2d ed ) New York: John Wiley, 1984 Bacon-Shone, J , and W K Fung "A New Graphical Method for Detecting Single and Multiple Outliers in Univariate and Multivariate Data." 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New York: McGraw-Hill, 1976 14 Pearson, E S , and H Hartley, eds Biometrika Tables for Statisticians vol II Cam­ bridge, England: Cambridge University Press, 1972 15 Potthoff, R F and S N Roy "A generalized multivariate analysis of variance model use­ ful especially for growth curve problems." Biometrika, 51 (1 964) , 313-326 16 Scheffe , H The Analysis of Variance New York: John Wiley, 1959 17 Tiku, M L , and N Balakrishnan "Testing the Equality of Variance-Covariance Matri­ ces the Robust Way." Communications in Statistics-Theory and Methods, 14, no 12 (1985), 3033-3051 Tiku, M L , and M Singh "Robust Statistics for Testing Mean Vectors o f Multivariate Distributions." Communications in Statistics-Theory and Methods, 11, no ( 982) , 985-1 001 19 Timm, N H Multivariate Analysis with Applications in Education and Psychology Mon­ terey, California: Brooks/Cole, 1975 20 Wilks, S S "Certain Generalizations in the Analysis of Variance." Biometrika, 24 (1932) , 471-494 ... 59 61 54 10 0 68 68 59 68 77 43 14 5 82 95 95 10 2 93 10 4 85 95 10 9 Wt Lngth Lngth Lngth Lngth 14 1 14 0 14 5 14 6 15 0 14 2 13 9 15 7 16 8 16 2 15 9 15 8 14 0 17 1 16 8 17 4 17 2 17 6 16 8 17 8 17 6 83 17 0 17 7 17 1 17 5... 64.5 67 11 3.5 14 2.0 12 4.0 12 5.0 12 9.5 12 3.0 14 0.0 97.0 16 2.0 12 6.5 13 6.0 11 6.0 13 5.0 14 15 16 17 18 19 20 21 22 23 24 25 10 .067 10 .0 91 10.888 7. 610 7.733 12 . 015 10 .049 14 9 9 .15 8 12 .13 2 6.978... 13 1 50 12 6.70 11 5 .10 13 0.80 12 4.60 18 . 31 14.20 12 0.30 11 5.70 11 7. 51 109. 81 109 .10 11 5 .10 11 8. 31 12.60 1 6.20 1 8.00 13 1 00 12 5.70 70.42 72.47 78.20 74.89 71 21 78.39 69.02 73 .10 79.28 76.48