Applied multivariate statistical analysis

Cover......Page 1
Table of Contents......Page 4
 1.1 Introduction......Page 6
 1.2 Applications of Multivariate Techniques......Page 8
  Arrays......Page 10
  Descriptive Statistics......Page 11
  Graphical Techniques......Page 16
 1.4 Data Displays and Pictorial Representations......Page 24
  Linking Multiple Two-Dimensional Scatter Plots......Page 25
  Graphs of Growth Curves......Page 29
  Stars......Page 31
  Chernoff Faces......Page 32
 1.5 Distance......Page 35
 Exercises......Page 42
 References......Page 52
 2.2 The Geometry of the Sample......Page 54
 2.3 Random Samples and the Expected Values of the Sample Mean and Covariance Matrix......Page 62
 2.4 Generalized Variance......Page 66
  Generalized Variance Determined by and its Geometrical Interpretation......Page 77
 2.5 Sample Mean, Covariance, and Correlation as Matrix Operations......Page 80
 2.6 Sample Values of Linear Combinations of Variables......Page 83
 Exercises......Page 87
 References......Page 91
  Vectors......Page 92
 3.3 Positive Definite Matrices......Page 103
 3.4 A Square-Root Matrix......Page 108
 3.5 Random Vectors and Matrices......Page 109
 3.6 Mean Vectors and Covariance Matrices......Page 111
  Partitioning the Covariance Matrix......Page 116
  The Mean Vector and Covariance Matrix for Linear Combinations of Random Variables......Page 118
  Partitioning the Sample Mean Vector and Covariance Matrix......Page 120
 3.7 Matrix Inequalities and Maximization......Page 121
  Vectors......Page 125
  Matrices......Page 130
 Exercises......Page 146
 References......Page 153
 4.2 The Multivariate Normal Density and its Properties......Page 154
  Additional Properties of the Multivariate Normal Distribution......Page 161
  The Multivariate Normal Likelihood......Page 173
  Maximum Likelihood Estimation of μ and Σ......Page 175
 4.4 The Sampling Distribution of X and S......Page 178
  Properties of the Wishart Distribution......Page 179
 4.5 Large-Sample Behavior of X and S......Page 180
  Evaluating the Normality of the Univariate Marginal Distributions......Page 182
  Evaluating Bivariate Normality......Page 187
 4.7 Detecting Outliers and Cleaning Data......Page 192
  Steps for Detecting Outliers......Page 194
 4.8 Transformations to Near Normality......Page 197
  Transforming Multivariate Observations......Page 200
 Exercises......Page 205
 References......Page 213
 5.2 The Plausibility of μ0 as a Value for a Normal Population Mean......Page 215
 5.3 Hotelling’s T2 and Likelihood Ratio Tests......Page 221
  General Likelihood Ratio Method......Page 224
 5.4 Confidence Regions and Simultaneous Comparisons of Component Means......Page 225
  Simultaneous Confidence Statements......Page 228
  A Comparison of Simultaneous Confidence Intervals with One-at-a-Time Intervals......Page 234
  The Bonferroni Method of Multiple Comparisons......Page 237
 5.5 Large Sample Inferences about a Population Mean Vector......Page 239
 5.6 Multivariate Quality Control Charts......Page 244
  Charts for Monitoring a Sample of Individual Multivariate Observations for Stability......Page 246
  Control Regions for Future Individual Observations......Page 252
  T2-Chart for Future Observations......Page 253
  Control Charts Based on Subsample Means......Page 254
 5.7 Inferences about Mean Vectors When Some Observations are Missing......Page 256
 5.8 Difficulties Due to Time Dependence in Multivariate Observations......Page 261
 Supplement 5A: Simultaneous ConfidenceIntervals and Ellipses as Shadows of the p-Dimensional Ellipsoids......Page 263
 Exercises......Page 266
 References......Page 277
  Paired Comparisons......Page 278
  A Repeated Measures Design for Comparing Treatments......Page 284
  Assumptions Concerning the Structure of the Data......Page 289
  Further Assumptions When n1 and n2 are Small......Page 290
  Simultaneous Confidence Intervals......Page 293
  The Two-Sample Situation When Σ1 ≠ Σ2......Page 296
  An Approximation to the Distribution of T2 for Normal Populations When Sample Sizes are Not Large......Page 299
  Assumptions about the Structure of the Data for One-Way Manova......Page 301
  A Summary of Univariate Anova......Page 302
  Multivariate Analysis of Variance (Manova)......Page 306
 6.5 Simultaneous Confidence Intervals for Treatment Effects......Page 313
 6.6 Testing for Equality of Covariance Matrices......Page 315
  Univariate Two-Way Fixed-Effects Model with Interaction......Page 317
  Multivariate Two-Way Fixed-Effects Model with Interaction......Page 320
 6.8 Profile Analysis......Page 328
 6.9 Repeated Measures Designs and Growth Curves......Page 333
 6.10 Perspectives and a Strategy for Analyzing Multivariate Models......Page 337
 Exercises......Page 342
 References......Page 363
 7.2 The Classical Linear Regression Model......Page 365
 7.3 Least Squares Estimation......Page 369
  Sum-of-Squares Decomposition......Page 371
  Geometry of Least Squares......Page 372
  Inferences Concerning the Regression Parameters......Page 375
  Likelihood Ratio Tests for the Regression Parameters......Page 379
  Estimating the Regression Function at Z0......Page 383
  Forecasting a New Observation at Z0......Page 384
  Does the Model Fit?......Page 386
  Additional Problems in Linear Regression......Page 389
 7.7 Multivariate Multiple Regression......Page 392
  Other Multivariate Test Statistics......Page 403
  Predictions from Multivariate Multiple Regressions......Page 404
 7.8 The Concept of Linear Regression......Page 406
  Mean Corrected Form of the Regression Model......Page 415
  Relating the Formulations......Page 417
 7.10 Multiple Regression Models with Time Dependent Errors......Page 418
 Supplement 7A: The Distribution of the Likelihood Ratio for the Multivariate Multiple Regression Model......Page 423
 Exercises......Page 425
 References......Page 433
 8.2 Population Principal Components......Page 435
  Principal Components for Covariance Matrices with Special Structures......Page 444
 8.3 Summarizing Sample Variation by Principal Components......Page 446
  The Number of Principal Components......Page 449
  Interpretation of the Sample Principal Components......Page 453
  Standardizing the Sample Principal Components......Page 454
 8.4 Graphing the Principal Components......Page 459
  Large Sample Properties of λi and ei......Page 461
  Testing for the Equal Correlation Structure......Page 462
  Checking a Given Set of Measurements for Stability......Page 464
  Controlling Future Values......Page 468
 Supplement 8A: The Geometry of the SamplePrincipal Component Approximation......Page 471
  The p-Dimensional Geometrical Interpretation......Page 473
  The n-Dimensional Geometrical Interpretation......Page 474
 Exercises......Page 475
 References......Page 485
 9.1 Introduction......Page 486
 9.2 The Orthogonal Factor Model......Page 487
  The Principal Component (and Principal Factor) Method......Page 493
  A Modified Approach—the Principal Factor Solution......Page 499
  The Maximum Likelihood Method......Page 500
  A Large Sample Test for the Number of Common Factors......Page 506
 9.4 Factor Rotation......Page 509
  Oblique Rotations......Page 517
 9.5 Factor Scores......Page 518
  The Weighted Least Squares Method......Page 519
  The Regression Method......Page 521
 9.6 Perspectives and a Strategy for Factor Analysis......Page 524
 Supplement 9A: Some Computational Details for Maximum Likelihood Estimation......Page 532
  Recommended Computational Scheme......Page 533
  Maximum Likelihood Estimators of p = LzLz + ψz......Page 534
 Exercises......Page 535
 References......Page 543
 10.2 Canonical Variates and Canonical Correlations......Page 544
  Identifying the Canonical Variables......Page 550
  Canonical Correlations as Generalizations of Other Correlation Coefficients......Page 552
  The First r Canonical Variables as a Summary of Variability......Page 553
  A Geometrical Interpretation of the Population Canonical Correlation Analysis......Page 554
 10.4 The Sample Canonical Variates and Sample Canonical Correlations......Page 555
  Matrices of Errors of Approximations......Page 563
  Proportions of Explained Sample Variance......Page 566
 10.6 Large Sample Inferences......Page 568
 Exercises......Page 572
 References......Page 579
 11.1 Introduction......Page 580
 11.2 Separation and Classification for Two Populations......Page 581
  Classification of Normal Populations When Σ1 = Σ2 = Σ......Page 589
  Scaling......Page 594
  Fisher’s Approach to Classification with Two Populations......Page 595
  Is Classification a Good Idea?......Page 597
  Classification of Normal Populations When Σ1 ≠ Σ2......Page 598
 11.4 Evaluating Classification Functions......Page 601
  The Minimum Expected Cost of Misclassification Method......Page 611
  Classification with Normal Populations......Page 614
 11.6 Fisher’s Method for Discriminating among Several Populations......Page 626
  Using Fisher’s Discriminants to Classify Objects......Page 633
  The Logit Model......Page 639
  Logistic Regression Analysis......Page 641
  Classification......Page 643
  Logistic Regression with Binomial Responses......Page 645
  Classification Trees......Page 649
  Neural Networks......Page 652
  Testing for Group Differences......Page 653
  Practical Considerations Regarding Multivariate Normality......Page 654
 Exercises......Page 655
 References......Page 674
 12.1 Introduction......Page 676
  Distances and Similarity Coefficients for Pairs of Items......Page 678
  Similarities and Association Measures for Pairs of Variables......Page 682
  Concluding Comments on Similarity......Page 683
 12.3 Hierarchical Clustering Methods......Page 685
  Single Linkage......Page 687
  Complete Linkage......Page 690
  Average Linkage......Page 695
  Ward’s Hierarchical Clustering Method......Page 697
  Final Comments—Hierarchical Procedures......Page 700
  K-means Method......Page 701
  Final Comments—Nonhierarchical Procedures......Page 706
 12.5 Clustering Based on Statistical Models......Page 708
 12.6 Multidimensional Scaling......Page 711
 12.7 Correspondence Analysis......Page 721
  Algebraic Development of Correspondence Analysis......Page 723
 12.8 Biplots for Viewing Sampling Units and Variables......Page 731
  Constructing Biplots......Page 732
 12.9 Procrustes Analysis: A Method for Comparing Configurations......Page 737
  Constructing the Procrustes Measure of Agreement......Page 738
  Introduction......Page 745
  The Data Mining Process......Page 746
  Model Assessment......Page 747
 Exercises......Page 752
 References......Page 760
 Selected Additional References for Model Based Clustering......Page 761
Appendix......Page 762
 Table 1: Standard Normal Probabilities......Page 763
 Table 2: Student’s T-Distribution Percentage Points......Page 764
 Table 3: X2 Distribution Percentage Points......Page 765
 Table 4: F-Distribution Percentage Points (α = 10)......Page 766
 Table 5: F-Distribution Percentage Points (α = .05)......Page 767
 Table 6: F-Distribution Percentage Points (α = .01)......Page 768
Index......Page 770