Statistical factor analysis and related methods: theory and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Wiley
1994
|
| Schriftenreihe: | Wiley series in probability and mathematical statistics
A Wiley interscience publication |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 737 S. graph. Darst. |
| ISBN: | 0471570826 |
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| 100 | 1 | |a Basilevsky, Alexander |d 1940- |e Verfasser |0 (DE-588)170149021 |4 aut | |
| 245 | 1 | 0 | |a Statistical factor analysis and related methods |b theory and applications |c Alexander Basilevsky |
| 264 | 1 | |a New York, NY [u.a.] |b Wiley |c 1994 | |
| 300 | |a XXIII, 737 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Wiley series in probability and mathematical statistics | |
| 490 | 0 | |a A Wiley interscience publication | |
| 650 | 7 | |a Factoranalyse |2 gtt | |
| 650 | 4 | |a Factor analysis | |
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Datensatz im Suchindex
| _version_ | 1804124448303874048 |
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| adam_text | Contents
1. Preliminaries 1
1.1 Introduction 1
1.2 Rules for Univariate Distributions 5
1.2.1 The Chi Squared Distribution 5
1.2.2 The F Distribution 6
1.2.3 The t Distribution 7
1.3 Estimation 8
1.3.1 Point Estimation: Maximum Likelihood 9
1.3.2 The Likelihood Ratio Criterion 12
1.4 Notions of Multivariate Distributions 15
1.5 Statistics and the Theory of Measurement 19
1.5.1 The Algebraic Theory of Measurement 20
1.5.2 Admissible Transformations and the Classification
of Scales 25
1.5.3 Scale Classification and Meaningful Statistics 28
1.5.4 Units of Measure and Dimensional Analysis for
Ratio Scales 30
1.6 Statistical Entropy 31
1.7 Complex Random Variables 33
Exercises 35
2. Matrixes, Vector Spaces 37
2.1 Introduction 37
2.2 Linear, Quadratic Forms 38
2.3 Multivariate Differentiation 42
2.3.1 Derivative Vectors 42
2.3.2 Derivative Matrices 44
xvii
xviii contents
2.4 Grammian Association Matrices 47
2.4.1 The Inner Product Matrix 49
2.4.2 The Cosine Matrix 50
2.4.3 The Covariance Matrix 51
2.4.4 The Correlation Matrix 52
2.5 Transformation of Coordinates 56
2.5.1 Orthogonal Rotations 57
2.5.2 Oblique Rotations 60
2.6 Latent Roots and Vectors of Grammian Matrices 62
2.7 Rotation of Quadratic Forms 67
2.8 Elements of Multivariate Normal Theory 69
2.8.1 The Multivariate Normal Distribution 70
2.8.2 Sampling from the Multivariate Normal 83
2.9 The Kronecker Product 86
2.10 Simultaneous Decomposition of Two Grammian Matrices 88
2.11 The Complex Multivariate Normal Distribution 90
2.11.1 Complex Matrices, Hermitian Forms 90
2.11.2 The Complex Multivariate Normal 93
Exercises 95
3. The Ordinary Principal Components Model 97
3.1 Introduction 97
3.2 Principal Components in the Population 101
3.3 Isotropic Variation 119
3.4 Principal Components in the Sample 127
3.4.1 Introduction 127
3.4.2 The General Model 128
3.4.3 The Effect of Mean and Variances on PCs 142
3.5 Principal Components and Projections 146
3.6 Principal Components by Least Squares 160
3.7 Nonlinearity in the Variables 162
3.8 Alternative Scaling Criteria 173
3.8.1 Introduction 173
3.8.2 Standardized Regression Loadings 174
3.8.3 Ratio Index Loadings 175
3.8.4 Probability Index Loadings 177
Exercises 178
4. Statistical Testing of the Ordinary Principal Components Model 182
4.1 Introduction 182
CONTENTS XIX
4.2 Testing Covariance and Correlation Matrices 184
4.2.1 Testing for Complete Independence 185
4.2.2 Testing Sphericity 191
4.2.3 Other Tests for Covariance Matrices 194
4.3 Testing Principal Components by Maximum Likelihood 202
4.3.1 Testing Equality of all Latent Roots 202
4.3.2 Testing Subsets of Principal Components 204
4.3.3 Testing Residuals 207
4.3.4 Testing Individual Principal Components 209
4.3.5 Information Criteria of Maximum Likelihood
Estimation of the Number of Components 220
4.4 Other Methods of Choosing Principal Components 223
4.4.1 Estimates Based on Resampling 223
4.4.2 Residual Correlations Test 228
4.4.3 Informal Rules of Thumb 229
4.5 Discarding Redundant Variables 231
4.6 Assessing Normality 234
4.6.1 Assessing for Univariate Normality 234
4.6.2 Testing for Multivariate Normality 235
4.6.3 Retrospective Testing for Multivariate Normality 241
4.7 Robustness, Stability, and Missing Data 242
4.7.1 Robustness 242
4.7.2 Sensitivity of Principal Components 243
4.7.3 Missing Data 246
Exercises 248
5. Extensions of the Ordinary Principal Components Model 250
5.1 Introduction 250
5.2 Principal Components of Singular Matrices 250
5.2.1 Singular Grammian Matrices 251
5.2.2 Rectangular Matrices and Generalized Inverses 252
5.3 Principal Components as Clusters: Linear
Transformations in Exploratory Research 257
5.3.1 Orthogonal Rotations 258
5.3.2 Oblique Rotations 270
5.3.3 Grouping Variables 276
5.4 Alternative Modes for Principal Components 278
5.4.1 Q Mode Analysis 278
5.4.2 Multidimensional Scaling and Principal
Coordinates 282
XX CONTENTS
5.4.3 Three Mode Analysis 286
5.4.4 Joint Plotting of Loadings and Scores 297
5.5 Other Methods for Multivariable and Multigroup
Principal Components 300
5.5.1 The Canonical Correlation Model 300
5.5.2 Modification of Canonical Correlation 308
5.5.3 Canonical Correlation for More than Two Sets
of Variables 310
5.5.4 Multigroup Principal Components 311
5.6 Weighted Principal Components 318
5.7 Principal Components in the Complex Field 321
5.8 Miscellaneous Statistical Applications 322
5.8.1 Further Optimality Properties 322
5.8.2 Screening Data 324
5.8.3 Principal Components of Discrimination and
Classification 326
5.8.4 Mahalanobis Distance and the Multivariate T
Test 328
5.9 Special Types of Continuous Data 330
5.9.1 Proportions and Compositional Data 330
5.9.2 Estimating Components of a Mixture 334
5.9.3 Directional Data 339
Exercises 347
6. Factor Analysis 351
6.1 Introduction 351
6.2 The Unrestricted Random Factor Model in the
Population 353
6.3 Factoring by Principal Components 361
6.3.1 The Homoscedastic Residuals Model 361
6.3.2 Unweighed Least Squares Models 363
6.3.3 The Image Factor Model 365
6.3.4 The Whittle Model 367
6.4 Unrestricted Maximum Likelihood Factor Models 367
6.4.1 The Reciprocal Proportionality Model 367
6.4.2 The Lawley Model 370
6.4.3 The Rao Canonical Correlation Factor Model 379
6.4.4. The Generalized Least Squares Model 381
6.5 Other Weighted Factor Models 382
6.5.1 The Double Heteroscedastic Model 382
6.5.2 Psychometric Models 384
CONTENTS xxi
6.6 Tests of Significance 384
6.6.1 The Chi Squared Test 385
6.6.2 Information Criteria 387
6.6.3 Testing Loading Coefficients 392
6.7 The Fixed Factor Model 394
6.8 Estimating Factor Scores 395
6.8.1 Random Factors: The Regression Estimator 396
6.8.2 Fixed Factors: The Minimum Distance Estimator 398
6.8.3 Interpoint Distance in the Factor Space 400
6.9 Factors Representing Missing Data: The EM
Algorithm 400
6.10 Factor Rotation and Identification 402
6.11 Confirmatory Factor Analysis 414
6.12 Multigroup Factor Analysis 417
6.13 Latent Structure Analysis 418
Exercises 420
7. Factor Analysis of Correlated Observations 432
7.1 Introduction 432
7.2 Time Series as Random Functions 424
7.2.1 Constructing Indices and Indicators 430
7.2.2 Computing Empirical Time Functions 434
7.2.3 Pattern Recognition and Data Compression:
Electrocardiograph Data 437
7.3 Demographic Cohort Data 439
7.4 Spatial Correlation: Geographic Maps 443
7.5 The Karhunen Loeve Spectral Decomposition in the
Time Domain 445
7.5.1 Analysis of the Population: Continuous Space 446
7.5.2 Analysis of a Sample: Discrete Space 454
7.5.3 Order Statistics: Testing Goodness of Fit 461
7.6 Estimating Dimensionality of Stochastic Processes 464
7.6.1 Estimating A Stationary ARMA Process 465
7.6.2 Time Invariant State Space Models 467
7.6.3 Autoregression and Principal Components 469
7.6.4 Kalman Filtering Using Factor Scores 477
7.7 Multiple Time Series in the Frequency Domain 480
7.7.1 Principle Components in the Frequency Domain 481
7.7.2 Factor Analysis in the Frequency Domain 483
7.8 Stochastic Processes in the Space Domain:
Karhunen Loeve Decomposition 486
XXU CONTENTS
7.9 Patterned Matrices 489
7.9.1 Circular Matrices 490
7.9.2 Tridiagonal Matrices 491
7.9.3 Toeplitz Matrices 492
7.9.4 Block Patterned Matrices 495
Exercises 497
8. Ordinal and Nominal Random Data 501
8.1 Introduction 501
8.2 Ordinal Data 501
8.2.1 Ordinal Variables as Intrinsically Continuous:
Factor Scaling 502
8.2.2 Ranks as Order Statistics 508
8.2.3 Ranks as Qualitative Random Variables 512
8.2.4 Conclusions 518
8.3 Nominal Random Variables: Count Data 518
8.3.1 Symmetric Incidence Matrices 519
8.3.2 Asymmetric Incidence Matrices 522
8.3.3 Multivariate Multinominal Data: Dummy Variables 524
8.4 Further Models for Discrete Data 533
8.4.1 Guttman Scaling 534
8.4.2 Maximizing Canonical Correlation 538
8.4.3 Two Way Contingency Tables: Optimal Scoring 540
8.4.4 Extensions and Other Types of Discrete Data 552
8.5 Related Procedures: Dual Scaling and Correspondence
Analysis 561
8.6 Conclusions 564
Exercises 564
9. Other Models for Discrete Data 570
9.1 Introduction 570
9.2 Serially Correlated Discrete Data 570
9.2.1 Seriation 571
9.2.2 Ordination 577
9.2.3 Higher Dimensional Maps 580
9.3 The Nonlinear Horseshoe Effect 583
9.4 Measures of Pairwise Correlation of Dichotomous
Variables 593
9.4.1 Euclidean Measures of Association 594
9.4.2 Non Euclidean Measures of Association 596
contents xxiii
9.5 Mixed Data 597
9.5.1 Point Biserial Correlation 598
9.5.2 Biserial Correlation 599
9.6 Threshold Models 602
9.7 Latent Class Analysis 607
Exercises 621
10. Factor Analysis and Least Squares Regression 624
10.1 Introduction 624
10.2 Least Squares Curve Fitting with Errors in Variables 624
10.2.1 Minimizing Sums of Squares of Errors in
Arbitrary Direction 627
10.2.2 The Maximum Likelihood Model 635
10.2.3 Goodness of Fit Criteria of Orthogonal Norm
Least Squares 639
10.2.4 Testing Significance of Orthogonal Norm Least
Squares 640
10.2.5 Nonlinear Orthogonal Curve Fitting 644
10.3 Least Squares Regression with Multicollinearity 645
10.3.1 Principal Components Regression 647
10.3.2 Comparing Orthogonal Norm and Y Norm
Least Squares Regression 663
10.3.3 Latent Root Regression 665
10.3.4 Quadratic Principal Components Regression 669
10.4 Least Squares Regression with Errors in Variables and
Multicollinearity 671
10.4.1 Factor Analysis Regression: Dependent
Variable Excluded 671
10.4.2 Factor Analysis Regression: Dependent
Variable Included 674
10.5 Factor Analysis of Dependent Variables in MANOVA 676
10.6 Estimating Empirical Functional Relationships 678
10.7 Other Applications 682
10.7.1 Capital Stock Market Data: Arbitrage Pricing 682
10.7.2 Estimating Nonlinear Dimensionality: Sliced
Inverse Regression 684
10.7.3 Factor Analysis and Simultaneous Equations
Models 687
Exercises 687
References 690
Index 733
|
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| isbn | 0471570826 |
| language | English |
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| spelling | Basilevsky, Alexander 1940- Verfasser (DE-588)170149021 aut Statistical factor analysis and related methods theory and applications Alexander Basilevsky New York, NY [u.a.] Wiley 1994 XXIII, 737 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and mathematical statistics A Wiley interscience publication Factoranalyse gtt Factor analysis Faktorenanalyse (DE-588)4016338-6 gnd rswk-swf Faktorenanalyse (DE-588)4016338-6 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006677060&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Statistical factor analysis and related methods theory and applications |
| title_auth | Statistical factor analysis and related methods theory and applications |
| title_exact_search | Statistical factor analysis and related methods theory and applications |
| title_full | Statistical factor analysis and related methods theory and applications Alexander Basilevsky |
| title_fullStr | Statistical factor analysis and related methods theory and applications Alexander Basilevsky |
| title_full_unstemmed | Statistical factor analysis and related methods theory and applications Alexander Basilevsky |
| title_short | Statistical factor analysis and related methods |
| title_sort | statistical factor analysis and related methods theory and applications |
| title_sub | theory and applications |
| topic | Factoranalyse gtt Factor analysis Faktorenanalyse (DE-588)4016338-6 gnd |
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