Introduction to multivariate analysis 0412160404, 9780412160301, 0412160307, 9780412160400
636 70 2MB
English Pages x,246p : ill ( cased ); 24cm [259] Year 1980;2000
Table of contents :
Cover......Page 1
Half Title......Page 2
Title Page......Page 6
Copyright Page......Page 7
Table of Contents......Page 8
Preface......Page 12
PART ONE: INTRODUCTION......Page 14
1: Introduction......Page 16
1.1 Examples......Page 16
1.2 Notation......Page 18
1.3 Review of objectives and different approaches......Page 19
1.4 Some general comments......Page 22
1.5 Review of books on multivariate analysis......Page 23
1.6 Some matrix algebra revision......Page 24
1.7 The general linear model......Page 28
Exercises......Page 30
2: Multivariate distributions......Page 31
2.1 Multivariate, marginal and conditional distributions......Page 31
2.2 Means, variances, covariances and correlations......Page 36
2.3 The multivariate normal distribution......Page 41
2.4 The bivariate normal distribution......Page 42
2.5 Other multivariate distributions......Page 44
2.5.1 Multivariate discrete distributions......Page 44
2.5.2 Multivariate continuous distributions......Page 44
Exercises......Page 45
3: Preliminary data analysis......Page 47
3.1 Processing the data......Page 47
3.1.1 Data editing......Page 49
3.2 Calculating summary statistics......Page 51
3.2.1 Interpreting the sample correlation matrix......Page 53
3.2.2 The rank of R......Page 61
3.3 Plotting the data......Page 62
3.4 The analysis of discrete data......Page 64
Exercises......Page 66
PART TWO: FINDING NEW UNDERLYING VARIABLES......Page 68
4: Principal component analysis......Page 70
4.1 Introduction......Page 70
4.2 Derivation of principal components......Page 71
4.2.1 Principal components from the correlation matrix......Page 74
4.2.2 Estimating the principal components......Page 75
4.3 Further results on PCA......Page 76
4.3.1 Mean-corrected component scores......Page 77
4.3.2 The inverse transformation......Page 77
4.3.3 Zero eigenvalues......Page 77
4.3.4 Small eigenvalues......Page 77
4.3.5 Repeated roots......Page 78
4.3.6 Orthogonality......Page 79
4.3.7 Component loadings/component correlations......Page 79
4.3.8 Off-diagonal structure......Page 80
4.3.9 Uncorrelated variables......Page 81
4.4 The problem of scaling in PCA......Page 81
4.5 Discussion......Page 84
4.5.1 The identification o f important components......Page 85
4.5.2 The use o f components in subsequent analyses......Page 88
4.6 PCA for multivariate normal data......Page 90
4.7 Summary......Page 91
Exercises......Page 92
5: Factor analysis......Page 95
5.1 Introduction......Page 95
5.2 The factor-analysis model......Page 96
5.3 Estimating the factor loadings......Page 99
5.4 Discussion......Page 100
PART THREE: PROCEDURES BASED ON THE MULTIVARIATE NORMAL DISTRIBUTION......Page 104
6: The multivariate normal distribution......Page 106
6.1 Introduction......Page 106
6.2 Definition of the multivariate normal distribution......Page 106
6.3 Properties of the multivariate normal distribution......Page 110
6.4 Linear compounds and linear combinations......Page 114
6.5 Estimation of the parameters of the distribution......Page 116
6.6 The Wishart distribution......Page 117
6.7 The joint distribution of the sample mean vector and the sample covariance matrix......Page 120
6.8 The Hotelling T2-distribution......Page 121
Exercises......Page 124
7: Procedures based on normal distribution theory......Page 127
7.1 Introduction......Page 127
7.2 One-sample procedures......Page 127
7.3 Confidence intervals and further analysis......Page 130
7.4 Tests of structural relations among the components of the mean......Page 135
7.5 Two-sample procedures......Page 137
7.6 Confidence intervals and further analysis......Page 138
7.7 Tests of structural relations among the components of the means......Page 142
7.8 Discriminant analysis......Page 146
Exercises......Page 151
8: The multivariate analysis of variance......Page 153
8.1 Introduction......Page 153
8.2 MANOVA calculations......Page 153
8.3 Testing hypotheses......Page 157
8.3.1 The special case: The univariate procedure......Page 157
8.3.2 The multivariate model for Example 8.1......Page 158
8.3.3 Multivariate test procedures......Page 159
8.3.4 Distributional approximations......Page 161
8.3.5 Applications o f the methodology......Page 162
8.4 Further analysis......Page 164
8.5 The dimensionality of the alternative hypothesis......Page 165
8.6 Canonical variates analysis......Page 166
8.7 Linear functional relationships......Page 172
8.8 Discriminant analysis......Page 173
Exercises......Page 173
9: The multivariate analysis of covariance and related topics......Page 175
9.1 Introduction......Page 175
9.2 Multivariate regression......Page 175
9.2.1 The special case: Univariate multiple regression......Page 176
9.2.2 The general case: Multivariate regression......Page 178
9.3 Canonical correlation......Page 182
9.4 The multivariate analysis of covariance......Page 186
9.4.1 The special case: Univariate analysis of covariance......Page 186
9.4.2 The multivariate case: An example......Page 189
9.4.3 The multivariate case: General results......Page 191
9.5 The test for additional information......Page 193
9.6 A test of an assigned subset of linear compounds......Page 197
Exercises......Page 199
PART FOUR: MULTIDIMENSIONAL SCALING AND CLUSTER ANALYSIS......Page 200
10: Multidimensional scaling......Page 202
10.1 Introduction......Page 202
10.2 Measures of similarity and dissimilarity......Page 203
10.2.1 Similarity coefficients for binary data......Page 207
10.3 Classical scaling......Page 211
10.3.1 The calculation of co-ordinate values from Euclidean distances......Page 211
10.3.2 The relationship between classical scaling and principal component analysis......Page 213
10.3.3 Classical scaling for a dissimilarity matrix......Page 214
10.3.4 Interpretation of the results......Page 215
10.3.5 Some related methods......Page 217
10.4 Ordinal scaling......Page 217
10.4.1 The iterative procedure......Page 218
10.4.2 Interpreting the results......Page 221
10.5 A comparison......Page 222
10.6 Concluding remarks......Page 223
Exercises......Page 223
11: Cluster analysis......Page 225
11.1 Introduction......Page 225
11.1.1 Objectives......Page 227
11.1.2 Clumping, dissection and clustering variables......Page 228
11.1.3 Some other preliminary points......Page 228
11.2 Visual approaches to finding a partition......Page 229
11.3 Hierarchical trees......Page 232
11.4 Single-link clustering......Page 233
11.5 Some other clustering procedures......Page 237
11.5.1 Method or algorithm?......Page 239
11.6 A comparison of procedures......Page 239
11.6.1 Some desirable conditions for hierarchical clustering methods......Page 239
11.6.2 A comparison......Page 240
Exercises......Page 242
References......Page 244
Answers to exercises......Page 248
Name Index......Page 254
Subject Index......Page 256
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