Multivariate Analysis

532 Pages·24.35 MB·english

by  Mardia K.V.

MULTIVARIATE
ANALYSIS
K. V. MARDIA
Department of Statistics.
University of Leeds. Leeds. U.K.
1. T. KENT
Department of Statistics.
University of Leeds, Leeds. U.K.
1. M. BIBBY
Faculty of Mathematics,
The Open University.
Milton Keynes, U.K.
This is a volume in
PROBABILITY AND MATHEMATICAL STATISTICS
A Series of Monographs and Textbooks
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Copyright © 1979 Academic Press LImited To Neela Mardia
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British Librar), Calaloguing in Publica/lOll Dolo
Mardi.-, KantiJal Varichand
Multivariale analysIS. - (Pruoal.ilily IUUJ
matbematical statistics).
I. Multivariate anaJysis
I. Title II. Kent. J T
111. Bibby, John IV. Series
519.5'3 QA27S 79-40922
ISBN ~12-471250-9
ISBN 0-12-471252-5 Pbk
Printed in Great Britain by TJ Press (Potdstow) LId., Pacl~ow. Corn" ,.11
PREFACE
"Everything is Tetaled with every OIher thing. and this
relation involves the! emergence of a relational quality. The
qllalities cannot be known a prior;, though a good number of
them can be deduced from certain fundamental characteris Multivariate Analysis deals with observations on more than one variable
tiCS." where there i. some inherent interdependence between the: variable.
- JSUlB philosophy With several texts already available in this area, one may very well
enquire of tbe authors as to the need for yet another book. Most of the
available books fall into two categories, either theoretical Or data analytic.
The present book not only combines the two approaches but it also
The! lama Pliliosophy o( Non·Absohmsm by S. Mookcrjec:. q ..... emphasizes modern developments. The choice of material for the book
Mahalanobis (1957) has been guided by the need to give suitable mailer for the. beginner as
well as illustrating some deeper aspects of the sUbject for the research
worker. Practical examples are kept to the forefront and. wherever
feasible. each technique is motivated by such an example.
The book is aimed at final year under!,'Taduates and postgraduate
students in Mathematic /Statistics with sections suitahle lor practitioners
and research workers. The book assumes a basic knowledge of
Mathematical Statistics at undergraduate level. An eLementary course on
Linear Algebra is also assumed. In particular. we assume an exposure to
Matrix Algebra to the level required to read Appendix A.
Broadly speaking, Chapters 1-6 and Chapter 12 can be de cribed as
containing direct extensions of univariate ideas and techniques. The
remaining chapters concentrate on specifically multivariate problems
which bave no meaningful analogues in the univariate case. Chaptcr 1 is
primarily concerned with giving exploratory analyses for multivariate data
and briefly introduces some 01 the important techniques, tools, and
diagrammatic representations. Chapter 2 introduces various distributions
together with some fundamental results, whereas Chapler 3 concentrates
exclusively on normal distribution theory. Chapters 4-6 deal with prob
lems in inference. Chapter 7 gives an over-view 01 Econometrics, whilst
Principal Component Analysis, Factor Analysis, Canonical Correlation
Analysis. and Discriminant Analysis are discussed from both theoretical
and practical points of view in Chapters 8-11. Chapter 12 is on Mul
tivariate Analysis of Variance, which can be better understood in terms of
PREFACE viii ix PREFACE
the techniques of previous chapters. The later chapters look into the asterisks. Mathematically orientated students may like to proceed to
presently developing techniques of Ouster Analysis, Multidimensional Chapter 2, omitting the data analytic ideas of Chapter 1.
'Scaling, and Directional Data. Various new methods of presentation are utilized in Ihe book. For
Each chapter concludes with a set of exercises. Solving these will nOI instance the data matrix is emphasized throughout, a density-free ap
only enable the reader to understand the material better but will also proach is given for normal theory_ the union intersection principle is used
serve to complement the chapter itself. In general, the questions have in testing as well as the likelihood ratio principle, and graphical methods
in-built answers, but. where desirable, hints for the solution of theoretical are used in explanation. In view of the computer packages generally
problems are provided. Some of the numerical exercises are designed to available. most 01 the numerical work is laken for granted and therelore,
be run On a computer, but as the main aim is On interpretation, the except for a few particular cases, emphasis is not placed on numerical
answers are provided. We found NAG roulines and GLIM most useful, calculations, The style of preselltation is generally kept descriptive excepl
but nowadays any computer centre will have some suitable statistics and where rigour is found to be necessary for theoretical reslllts, whicb are
matrix algebra routines. then put in the form of theorems. If any details of the proof of a theorem
There are three appendices, A. S, and C, which respectively provide a are felt tedious but simple, they are then relegated to the exercises.
sufficient background of matrix algebra, a summary of univariate statis Several important topics not usually found in multivariate texts are
tics, and some tables of critical values. The aim of Appendix A on Matrix discussed in detail. Examples of such material include the complete
Algebra is not only to provide a summary of results, but also to give chapters on Econometrics, Cluster Analysis, Multidimensional Scaling,
sufficient guidance to master these for students having little previous and Directional Data. Further material is also included in parts of other
knowledge. Equations from Appendix A are referred to as (A.x.x) to chapters: methods of graphical presentation, measures of multivariate
distinguiSh them from (I.x.x), etc. Appendix A also includes a summary skewness and kurtosis, the singular multinonnal distribution, various
of results in n-dimensional geometry which are used liberally in the book. non-normal distributions and families of distributions, a density-free
Appendix B gives a summary of important univariate distributions. approach to normal distribution theory, Bayesian and robust estimators, a
The reference list is by no means exhaustive. Only directly relevant recent solution to the Fisher-Beluens problem. a test of multinorrnality, a
articles are quoted and lor a fuller bibliography we refer the reader to non-parametric lest, discarding of variables in regression, principal com
Anderson. Das Gupta, and Styan (1972) and Subrahmaniam and Subrahma ponent analysis and discrimination analysis, correspondence analysis,
niam (l1J73). The relerence list also serves as an author index. A suhject allometry, the jack-knifing method in discrimination, canonkal analysis of
index is provided. qualitative and quantitative variables, and a test of dimensionality in
The material in the book can be used in several different ways. For MANOV A. It is hoped thaI coverage 01 these developments will be
example, a one-semester elementary cOurse of 40 lectures could cover the helpful for studeots as well as research workers.
following topics. Appendix A; Chapter 1 (Sections 1.1-1.7); Chapter 2 Tbere are various other topics which have oot been tOllched upon
(Sections 2.1-2.5); Chapter 3 (Sections 3.4.1.3.5,3.6.1. assuming results partly because of lack of space as well as our own preferences. such as
from previous sections. Definitions 3.7.1, 3.7.2); Chapter 4 (Section Control Theory, Multivariate Time Series, Latent Variable Models, Path
4.2.2); Chapler 5 (Sections 5_1, 5.2.1a, 5.2.lb, 5.2.2a. 5.2.2b, 5.3.2b. Analysis, Growth Curves, Portfolio Analysis, and various Multivariate
5.5); Chapter 8 (Sections 8.1, 8.2.1. 8.2.2, 8.2.5, 8.2.6, 8.4.3, 8.7); Designs.
Chapter 9 (Sections 9.1-9.3, 9.4 (without details), 9.5, 9.6, 9.8); Chapler In addition to various research papers, we have been inlluenced by
JO (Sections 10.1, 10.2); Chapler 11 (Sections 11.1, 11.2.1-11.2.3, particular texts in this area, especially Anderson (195&), Kendall (1975),
11.3.1. 11.6.1). Further material which can be introduced is Chapter 12 Kshirsagar (1972). Morrison (1976). Press (1972), and Rao (1973). All
(Sections 12.1-12.3, 12.6); Chapter 13 (Sections 13.1, 13.3.1); Chapter these are recommended to the reader.
14 (Sections 14_1. 14.2). This material has been covered in 40 lectures The authors would be most grateful to readers who draw their attention
spread over two terms in different British universities. Alternatively. a to any errors o~ obscurities in the book, or suggest other improvements.
one-semester course with more emphasis on foundation rather than
applications could be based on Appendix A and Chapters 1-5. Two January 1979 Kanti Mardia
semester courses could indude all the chapters, excluding Chaplers 7 and John Kent
15 On Econometrics and Directional Data. as well a, the ,,,clions with John Bibby
ACKNOWLEDGEMENTS CONTENTS
First of all we wish to express our gratitude to pioneers in this field. In Preface vji
particular. w~ should mention M. S. Bartlett, R. A. Fisher, H. Hotelling,
D. G. Kendall. M. G. Kendall, P. C. Mahalanobis. C. R. Rao, S. . Roy, Acknowledgements X
W. S. Torgerson, and S. S. Wilks.
Chapter t-lntroduction
We are grateful to authors and editors who have generously granted us 1.1 Objec1S and variables 1
permission to reproduce figures and tables. 1.2 Some multivariate problems and techniques 2
We are also grateful to many colleagues for their valuable help and 1.3 The data matru 8
1.4 Summary statistics 9
comments, in panicular Martin Beale, Cltristopber Bingbam. Lesley
1.5 Linear combinations 13
Butler. Richard Cormack, David Cox. Ian Curry, Peter Fisk. Allan
\.6 Ge<>metricai ideas 16
Gordon, John Gower, Peler Harris. Chunni Khatri, Conrad Le<er, E.ric 1.7 Graphical representations l7
Okell, Ross Renner. David Salmond. Cyril Smith. and Peter Zemroch. * 1.8 Measures of multivariate skewness and kurtosis 20
We are also indebted to Joyce Snell for making various comments on an Exercises and complements . ... .. . 22
earlier draft of the book which have led to considerable improvement.
Chapter 2-Basic Properties of Random Vedors 26
We should also express our gratitude to Rob Edwards for his help in
2.1 Cumulative distribution functions and probability density func-
various facets of the book. for calculations, for proof· reading, for diag tioos ., ... . . . 26
rams, etc. 2.2 Population moments 28
Some of the questions are laken from examination papers in British 2.3 Characteristic [unctions 33
2.4 Transformations 35
universities and we are grateful to various unnamed colleagues. Since tbe
2,) The multinormal distribution 36
original sources of questions are difficult to trace, we apologize to any 2,6 Some multivariate generalizations of univariate distributions 43
colleague who recognizes a question of his own. 2.7 Families of distributions 45
The authors would like to thank their wives, Pavan Mardia, Susan 2.8 Random samples . . 49
Kent, and Zorina Bibby. 2.9 Limit theo'rems 51
Exercises and complements 53
Finally our thanks go to Barbara Forsyth and Margaret Richardson for
typing a difficult manuscript with great skill.
Chapter 3- onnal Distribution Theory S9
3.1 Characterization and properties 59
KVM
3.2 Linear forms . . . .. " . 62
JTK 3.3 Transformations of normal data matrices 64
JMB 3.4 The Wishart distribution ,.... .. 66
XII xiii CONTENTS
3.5 The Hotelling T2distribution .... . 73 8.7 Discarding of variables ....... . 242
8.8 Principal component anaJysis in regression 244
3.6 Mahalanobis distance . .. .... . 76
Exerc.ises and complements 246
3.7 Statistics based on the Wishart distribution 80
3.8 Other distributions related to the multinormal 8S
Chapter 9-Factor Anal}D 2SS
Exercises and complements 86
9.1 Introduction . . . . 255
9.2 The factor model . . 256
Chapter 4---Eotimation 96
9.3 Principal factor analysis 261
4. t Likelihood and sufficiency 96
9.4 Maximum likelihood factor analysis 263
4.2 Maximum likelihood estimation 102
9.5 Goodness of fit test 267
4.3 Other techniques and concepts 109
9.6 Rotation of factors . . . . . . . 268
Exercises and complements 113
9.7 Factorscores 273
9.8 Relationships between factor analysis and principal component
Chapter 5-Hypothesis Testing 120
analysis • . . . . . . . . . . 275
5.1 Introduction ..... 120
9.9 Analysis of covariance structures . , 276
5.2 The techniques introduced 123
Exercises and complements ...• . . 276
5.3 The techniques further iUustrated 131
'5.4 The Behrens-FISher problem 142 Chapter IO-Canonical Correlation Analysis 281
5.5 Simultaneous confidence intervals 144 10.1 Introduction . . . ....... 281
5.6 Multivariate hYJX>thesis testing: some general points 147 10.2 Mathematical development . . . . 282
'5.7 Non-normal data . . . . . . _ . . . . . . . . 148 10.3 Qualitative data and dummy variables 290
5.8 A non-parametric test for the bivariate two-sample problem 149 10.4 Qualitative and quanta live data 293
Exercises and complements ..... . lSI Exercises and complements 295
Chapter &--Multivariale Regression Analysis 157 Chapter lI-Discriminllnt AnalY8is 300
6.1 Introduction . . . . . . . . . 157 11.1 Introduction ..... . 300
6.2 Maximum likelihood estimation 158 11.2 Discrimination when the populations are known 301
6.3 The general linear hypothesis . • 161 11.3 Discrimination under estimation 309
6.4 Design matrices of degenerate rank 164 11.4 Is discrimination worthwbile? . . . 318
6.5 Multiple correlation 167 ] 1.5 Fisber's linear discriminant function 318
6.6 Least squares estimatiOn 171 J 1.6 Probabilities of misclassificalion 320
6.7 Discarding of variables 175 11.7 Discarding of variables . . . . . . 322
Exercises and complements . . 180 t 1.8 When does correlation improve discrimination? 324
Exercises and complements ...... . 325
Chapter 7-Eronometri.. ........ .... . 185
7.1 Introduction . . . . . . . . . .. .... . 185 Chapter 12-Multivariate Anal}. .i s of Variance . . . . . 333
7.2 Instrumental variables and two-slage least squares 186 12.1 Introduction . . .. .... ... .... . 333
7.3 Simultaneous equation systems 191 12.2 Formulation of multivariate one-way classification 333
7.4 Single-equation estimators 199 12.3 The likelihood ratio principle . . . . . . . . 334
7.5 System estima tors 203 12.4 Testing fixed contrasts . . . . . . . . . . . 337
7.6 Comparison of estimators 208 12.5 Canonical variables and a test of dimensionality 338
Exercises and complements 208 12.6 The union intersection approach 348
12.7 Two-way classification 350
Chapter 8-Principal Component Analysis . . . . . . . 213 Exercises and complements 356
8.1 Introduction . . . . . . . . . . . . . . . . . 213
8.2 Definition and properties of principal components 214 Chapter 13-Clnster Analysis 360
8.3 Sampling properties of principal components . 229 13.1 Introduction . . . . 360
8.4 Testing bypotheses about principal components 233 13.2 A probabilistic formulation 361
8.5 Correspondenoe analysis ... . .. . . . 237 13.3 Hierarchical methods 369
8.6 Allometry-the measurement of size and sbape 239 13.4 Distances and simiJanl'ics 375
CONTENTS )(IV xv CONTENTS
13.5 Other methods and comparative approach Table C.2 Upper percemage points of the t.. distribution . . . 491
Exercises and complements . . . . . . TableC.3 pper percentage points of the F",.v) distribution 492
Table C.4 Upper percentage points 00 of 8(p. 1)1. '-'2)' the largest
Chapter 14-Multidimensional Scalin~ 394 eigenvalueofIB- O(W+B)I=Oforp=2. 494
14.1 Introduction ...... . 394
14.2 ClassicaJ solution .. .. . 397 References . 497
14.3 Duality between principal coordinate analysis and principal corn-
ponent analysis . . . . . . _. . .. ..... . . . -lO-l List of Main otations and Abbreviations . 508
14.4 Oplimal properties of the classical solution and goodness of 6t 406
J 4.5 Seriation ......... . .. .. . 409 Subject Index 510
14.6 Non-metric methods . . . . . . . . . . . 413
14.7 Goodness of fit measure; Procrustes rotation .....• 416 Author Index 519
.] 4.8 Multi-sample problem and canonical variates 4J9
Exercises and complements ......,... . 420
Chapter IS-Directional Data 424
15.1 Introduction 424
15.2 Descriptive measures 426
15.3 Basic distributions . . 421\
15.4 Distribution theory 435
]5.5 Maximum lik.elihood estimators for the Von Mise$-Fi~her distribu-
tion . . . . . . . . . . . . . . 437
15.6 Test of uniformity; the Rayleigh test 439
15.7 Some other problems 441
Exercises and complements 446
Appendix A-Matrix Algebra 452
A.l Introduction 452
A.2 Matrix operations 455
A.3 Further particular matrices and types of matrix 460
A.4 Vector spaces. rank. and linear equations 462
A.5 Linear translormations . . . . . 465
A.6 Eigenvalues and eigenvectors . . 466
A.7 Quadratic lorms and definiteness 474
• A.8 Generalized inverse . . . . . . 476
A.9 Matrix differentiation and maximjzation probJems 478
A.IO Geometrical idea. . . . . . . . . . 481
Appendix B-Univariate Statistics 486
B.1 Introduction .. . .. 486
B.2 Normal distribution 486
B.3 Chi·squared distribution. 487
B.4 F and beta variables 487
B.S I distribution .. ... 488
Appendix C-Tables .. . .. • .. .. . ....... 4119
X:
Table C. 1 Upper percentage pOints of the distribution 4<)()
1
Introduction
1.1 Objects and V~bles
I
Multivariate analysis deals with data containing observations on two or
more variables each measured on a set of objects. For example. we may
have the set of examination marks achieved by certain students, or the
cork deposit in various directions of a set of trees. or flower measure
ments for different species of iris (see Tables 1.2.1, 1.4.1, and 1.2.2,
respectively). Each of these data has a set of "variables" (the examination
marks, trunk thicknesses, and flower measurements) and a set of "ob
jects" (the students. trees, and flowers). In general, if there are" objects.
0", .. . 0", and p variables, Xl •. ... Xp• the data contains lip pieces of
information. These may be conveniently arranged using an (II x p) "data
matrix". in which each row corresponds to an object and each column
corresponds to a variable. For instance. three variables on five "objects"
(students) are shown as a (5 x 3) data matrix in Table 1.1.1.
Table. 1.1.1 Data matrix with five students as object..~ where Xl =age in years
at entry to university. x, = marks Oul of 100 in an examination al the end of the
first year, and X.3 = sex
Variables
Objects x, x, X3't
1 18.45 70 1
2 18.41 6S 0
3 18.39 71 0
4 18.70 72 0
5 18".34 94 1
t 1 indicates male~ 0 indicutes female.
MULTIVARIATE ANALYSIS 2 3 INTRODUCTION
Note that the variables need not all be of the same type: in Tahlc 1.1. I. Table 1.2. J Marks in open-book and closed-book examination out of J 00 t
Xl is a "'continuousU variable, X2 is a discrete variable, and ,t"l i.o,,; a
dichotomous variable. Note also that attribute, characteristic. description. Mechanics (C) Vectors (C) Algebra (0) Anatysis (0) Statistics (0)
item, measurement, and response are synonyms for "variable", wherca~
77 82 67 67 81
individual, observation, plot, reading, and unit caD be used in place of
63 78 80 70 81
"object",
75 73 71 66 81
55 72 63 70 68
63 63 65 70 63
1.2 Some Multivariate Problems and Techniques 53 tit 72 64 73
51 67 65 65 68
59 70 68 62 56
We may now illustrate various categories of multivariate technique. 62 60 58 62 70
64 72 60 62 45
52 64 60 63 54
1.2.1 Generalizations of univariate techniques 55 67 59 62 44
50 50 64 55 63
Most univariate questions are capable of at least one multivariate
65 63 58 56 37
generalization. For instance, using Table 1.2.1 we may ask, as an exam 31 55 60 57 73
ple. "What is the appropriate underlying parent distribution of examina 60 64 56 54 40
of
tion marks on various papers of a set students?" "What are the 44 69 53 53 53
42 69 61 55 45
summary statistics?" "Are the differences between average marks on
62 46 6J 57 45
different papers significant?", etc. These problems are direct generaliza
31 49 62 63 62
tions of univariate problems and their motivation is easy to grasp. See for 44 6t 52 62 46
example Chapters 2-6 and 12. 49 4t 6t 49 64
12 58 6J 63 67
49 53 49 62 47
1.2.2 Dependence and regession 54 49 56 47 53
54 53 46 S9 44
Referring to Table 1.2.1. we may enquire as to the degree of dependence 44 56 55 61 ~
between performance on different papers taken by the same students. It 18 44 50 57 81
may be useful. for counselling or other purposes, to have some idea of 46 52 65 50 35
32 45 49 57 64
how final degree marks ("dependent" variables) are affected by previous
30 69 50 52 45
examination results or by other variables such as age or sex ("explana
46 49 53 59 37
tory" variables). This presents the so-called regression problem. which is 40 27 54 61 61
examined in Chapter 6. 31 42 48 54 68
30 59 51 45 51
56 40 56 54 35
1.2.3 linear combinations 46 56 57 49 32
45 42 55 56 40
Given examination marks on different topics (as in Table 1.2.1), the
42 60 54 49 33
question arises of how to combine or average these marks in a suitable 40 63 53 54 25
way. A straightforward method would use the simple arithmetic mean. 23 55 59 53 44
but this procedure may not always be suitable. For instance, if the marks 4 48 49 51 37
41 63 49 46 34
on some papers vary more than others, we may wish to weight them
46 52 53 41 40
differently. This leads us to search for a linear combination (weighted
sum) which is "optimal'" in some sense. If all the examination papers rail to indicates open-book. C indk:atcs closed book.
MULTIVARIATE ANALYSIS 4 5 INTRODUcnON
Table 1.2.1 Conrinued in one group then principal compollent aMlysis and factor analysis are
two tech.niques which can help to answer such questions (see Chapters 8
Mechanics (C) Vectors (C) Algebra (0) Analysis (0) Statistics (O)
and 9). In some situations tbe papers may rail into more tban one
46 61 46 38 41 group--for instance. in Table 1.2.1 some examinations were "open
40 57 51 52 31 book" wh.i1e others were "closed-book". In sucb situations we may wish
49 49 45 48 39 to investigate the use of linear combinations within each. group separately.
22 58 53 56 41 This leads to the method known as canonical corTelatiOtt analysis, which is
35 60 47 54 33
discussed in Cbapter 10.
48 56 49 42 32
The idea of taking linear combinations is an important one in mul
31 57 50 54 34
17 53 57 43 51 tivariate analysis. and we shall return to it in Section 1.5.
49 57 47 39 26
59 50 47 15 46
37 56 49 28 45 1.2.4 Assignment and dissection
40 43 4R 21 61
Table 1.2.2 gives three (50 x 4) data matrices. In each malTix the "ob
35 35 41 51 50
38 44 54 47 24 jects" are 50 irises of species Iris setosa, Iris versicolollr, and Iris virginica,
43 43 38 34 49 respectively. The "variables" are
39 46 46 32 43
62 44 36 22 42 x, =sepaJ lengtb, X2 = sepal width.
4384 3482 5410 4447 3239 X3 = petal length, X4 = petal width.
I~ 51 40 56 30 If a new iris of unknown species has measurements x,=5.1, x.,=3.2,
35 36 46 48 29
x,=2.7. x4=0.7 we may ask to which species it belongs. This presents
59 53 37 22 19
41 41 43 30 33 the problem of discriminallt analysis, which is discussed in Chapter 11.
31 52 37 n ~ Howe er. if we were presented with the 150 observations of Table 1.2.2
17 51 52 35 31 in an unclassified manner (say, before the three species were established)
34 30 50 47 36 then the aim could have been to dissect the population ioto homogeneous
46 40 47 29 17
groups. This problem is handled by cluster analysis (see Chapter 13).
10 46 36 47 39
46 37 45 15 30
30 34 43 46 18
1.2.5 Building configurations
13 51 50 25 31
49 50 38 23 9 In some cases the data consists not of an (n X p) data matrix, but of
18 32 31 ~ ~ 1n(n-1) "distances" between aU pairs o( points. To get an intuitive feel
8 42 4~ 26 40 for tbe structure of such data. a configuration can be constructed of n
23 38 36 48 15
30 24 43 33 25 points in a Euclidean space of low dimension (e.g. p = 2 or 3). Hopefully
3 9 5L 47 ~ the distances between the 11 points of the configuration will closely match
7 51 43 17 22 the original distances. The problems of building and interpreting such
15 40 43 23 18
configurations are studied in Chapter 14, on Inllitidimensional scaling.
15 38 39 28 17
5 30 44 36 18
12 30 32 35 21 1.2.6 Directional data
5 26 15 20 W
o 40 21 <) 14 Tbere are various problems which arise when the variables are
directional-that is, the multivariate observations are constrained to lie
on a hypersphere. For a discus.ion of these problems see Chapter 15.