Applied Multivariate Analysis

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Springer Science & Business Media, Jun 21, 2007 - Mathematics - 695 pages
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Univariate statistical analysis is concerned with techniques for the analysis of a single random variable. This book is about applied multivariate analysis. It was written to p- vide students and researchers with an introduction to statistical techniques for the ana- sis of continuous quantitative measurements on several random variables simultaneously. While quantitative measurements may be obtained from any population, the material in this text is primarily concerned with techniques useful for the analysis of continuous obser- tions from multivariate normal populations with linear structure. While several multivariate methods are extensions of univariate procedures, a unique feature of multivariate data an- ysis techniques is their ability to control experimental error at an exact nominal level and to provide information on the covariance structure of the data. These features tend to enhance statistical inference, making multivariate data analysis superior to univariate analysis. While in a previous edition of my textbook on multivariate analysis, I tried to precede a multivariate method with a corresponding univariate procedure when applicable, I have not taken this approach here. Instead, it is assumed that the reader has taken basic courses in multiple linear regression, analysis of variance, and experimental design. While students may be familiar with vector spaces and matrices, important results essential to multivariate analysis are reviewed in Chapter 2. I have avoided the use of calculus in this text.
 

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Contents

62 Random Coefficient Regression Models
352
b Estimating the Parameters
353
c Hypothesis Testing
355
63 Univariate General Linear Mixed Models
357
b Covariance Structures and Model Fit
359
c Model Checking
361
d Balanced Variance Component Experimental Design Models
366
e Multilevel Hierarchical Models
367

d Orthogonal Spaces
17
e Vector Inequalities Vector Norms and Statistical Distance
21
24 Basic Matrix Operations
25
a Equality Addition and Multiplication of Matrices
26
b Matrix Transposition
28
c Some Special Matrices
29
d Trace and the Euclidean Matrix Norm
30
e Kronecker and Hadamard Products
32
f Direct Sums
35
25 Rank Inverse and Determinant
41
b Generalized Inverses
47
c Determinants
50
26 Systems of Equations Transformations and Quadratic Forms
55
b Linear Transformations
61
c Projection Transformations
63
d Eigenvalues and Eigenvectors
67
e Matrix Norms
71
f Quadratic Forms and Extrema
72
g Generalized Projectors
73
27 Limits and Asymptotics
76
Multivariate Distributions and the Linear Model
79
33 The Multivariate Normal MVN Distribution
84
a Properties of the Multivariate Normal Distribution
86
b Estimating ft and S
88
c The Matrix Normal Distribution
90
34 The ChiSquare and Wishart Distributions
93
b The Wishart Distribution
96
35 Other Multivariate Distributions
99
c The Beta Distribution
101
d Multivariate t F and χ2 Distributions
104
36 The General Linear Model
106
a Regression ANOVA and ANCOVA Models
107
b Multivariate Regression MANOVA and MANCOVA Models
110
c The Seemingly Unrelated Regression SUR Model
114
d The General MANOVA Model GMANOVA
115
37 Evaluating Normality
118
38 Tests of Covariance Matrices
133
c Testing for a Specific Covariance Matrix
137
d Testing for Compound Symmetry
138
e Tests of Sphericity
139
f Tests of Independence
143
g Tests for Linear Structure
145
39 Tests of Location
149
b TwoSample Case
156
c TwoSample Case Nonnormality
160
e Profile Analysis Two Groups
165
f Profile Analysis₁ ₂
175
310 Univariate Profile Analysis
181
a Univariate OneGroup Profile Analysis
182
Multivariate Regression Models
185
42 Multivariate Regression
186
b Multivariate Regression Estimation and Testing Hypotheses
187
c Multivariate Influence Measures
193
d Measures of Association Variable Selection and LackofFit Tests
197
e Simultaneous Confidence Sets for a New Observation ynew and the Elements ofB
204
Mean Squared Error of Prediction in Multivariate Regression
206
43 Multivariate Regression Example
212
44 OneWay MANOVA and MANCOVA
218
b OneWay MANCOVA
225
c Simultaneous Test Procedures STPfor OneWay MANOVA MANCOVA
230
45 OneWay MANOVAMANCOVA Examples
234
b MANCOVA Example 452
239
46 MANOVAMANCOVA with Unequal i or Nonnormal Data
245
47 OneWay MANOVA with Unequal i Example
246
b Additive TwoWay MAN OVA
252
c TwoWay MANCOVA
256
49 TwoWay MANOVAMANCOVA Example
257
b TwoWay MANCOVA Example 492
261
410 Nonorthogonal TwoWay MANOVA Designs
264
a Nonorthogonal TwoWay MANOVA Designs with andWithout Empty Cells and Interaction
265
b Additive TwoWay MANOVA Designs With Empty Cells
268
412 Higher Ordered Fixed Effect Nested and Other Designs
273
413 Complex Design Examples
276
b Latin Square Design Example 4132
279
414 Repeated Measurement Designs
282
b Extended Linear Hypotheses
286
415 Repeated Measurements and Extended Linear Hypotheses Example
294
b Extended Linear Hypotheses Example 4152
298
416 Robustness and Power Analysis for MR Models
301
417 Power CalculationsPowersas
304
418 Testing for Mean Differences with Unequal Covariance Matrices
307
Seemingly Unrelated Regression Models
310
52 The SUR Model
312
b Prediction
314
53 Seeming Unrelated Regression Example
316
54 The CGMANOVA Model
318
55 CGMANOVA Example
319
56 The GMANOVA Model
320
b Estimation and Hypothesis Testing
321
c Test of Fit
324
e GMANOVA vs SUR
326
57 GMANOVA Example
327
a One Group Design Example 571
328
b Two Group Design Example 572
330
58 Tests of Nonadditivity
333
59 Testing for Nonadditivity Example
335
511 Sum of Profile Designs
338
512 The Multivariate SUR MSUR Model
339
513 Sum of Profile Example
341
514 Testing Model Specification in SUR Models
344
515 Miscellanea
348
Multivariate Random and Mixed Models
351
f Prediction
368
64 Mixed Model Examples
369
a Random Coefficient Regression Example 641
371
b Generalized Randomized Block Design Example 642
376
c Repeated Measurements Example 643
380
d HLM Model Example 644
381
65 Mixed Multivariate Models
385
a Model Specification
386
b Hypothesis Testing
388
c Evaluating Expected Mean Square
391
d Estimating the Mean
392
66 Balanced Mixed Multivariate Models Examples
394
a Twoway Mixed MANOVA
395
67 Double Multivariate Model DMM
400
68 Double Multivariate Model Examples
403
a Double Multivariate MAN OVA Example 681
404
b SplitPlot Design Example 682
407
69 Multivariate Hierarchical Linear Models
415
610 Tests of Means with Unequal Covariance Matrices
417
Discriminant and Classification Analysis
418
72 Two Group Discrimination and Classification
420
a Fishers Linear Discriminant Function
421
b Testing Discriminant Function Coefficients
422
c Classification Rules
424
d Evaluating Classification Rules
427
73 Two Group Discriminant Analysis Example
429
b Brain Size Example 732
432
74 Multiple Group Discrimination and Classification
434
b Testing Discriminant Functions for Significance
435
c Variable Selection
437
d Classification Rules
438
e Logistic Discrimination and Other Topics
439
75 Multiple Group Discriminant Analysis Example
440
Principal Component Canonical Correlation and Exploratory Factor Analysis
445
a Population Model for PCA
446
b Number of Components and Component Structure
449
c Principal Components with Covariates
453
d Sample PCA
455
e Plotting Components
458
83 Principal Component Analysis Examples
460
b Semantic Differential Ratings Example 832
461
c Performance Assessment Program Example 833
465
84 Statistical Tests in Principal Component Analysis
468
b Tests Using a Correlation Matrix
472
85 Regression on Principal Components
474
a GMANOVA Model
475
86 Multivariate Regression on Principal Components Example
476
87 Canonical Correlation Analysis
477
b Sample CCA
482
c Tests of Significance
483
d Association and Redundancy
485
e Partial Part and Bipartial Canonical Correlation
487
f Predictive Validity in Multivariate Regression using CCA
490
g Variable Selection and Generalized Constrained CCA
491
a Rohwer CCA Example 881
492
b Partial and Part CCA Example 882
494
89 Exploratory Factor Analysis
496
a Population Model for EFA
497
b Estimating Model Parameters
502
c Determining Model Fit
506
d Factor Rotation
507
e Estimating Factor Scores
509
f Additional Comments
510
810 Exploratory Factor Analysis Examples
511
b Di Vesta and Walls Example 8102
512
Cluster Analysis and Multidimensional Scaling
515
92 Proximity Measures
516
b Similarity Measures
519
c Clustering Variables
522
a Agglomerative Hierarchical Clustering Methods
523
b Nonhierarchical Clustering Methods
530
c Number of Clusters
531
d Additional Comments
533
a Protein Consumption Example 941
534
b Nonhierarchical Method Example 942
536
c Teacher Perception Example 943
538
d Cedar Project Example 944
541
a Classical Metric Scaling
542
b Nonmetric Scaling
544
c Additional Comments
547
96 Multidimensional Scaling Examples
548
a Classical Metric Scaling Example 961
549
b Teacher Perception Example 962
550
c Nation Example 963
553
Structural Equation Models
556
102 Path Diagrams Basic Notation and the General Approach
558
103 Confirmatory Factor Analysis
567
104 Confirmatory Factor Analysis Examples
575
b Performance Assessment 5Factor Model Example 1042
578
105 Path Analysis
580
106 Path Analysis Examples
586
b Nonrecursive Model Example 1062
590
107 Structural Equations with Manifest and Latent Variables
594
108 Structural Equations with Manifest and Latent Variables Example
595
109 Longitudinal Analysis with Latent Variables
600
1010 Exogeniety in Structural Equation Models
604
Appendix A
609
References
625
Author Index
667
Subject Index
675
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About the author (2007)

"This book is more than an up-to-date textbook on multivariate analysis. It could enable SAS users to take full and informed advantage of the many options offered in the SAS procedures. For non-SAS users, the clear statement of the models should enable them to fit and interpret them with other software."

ISI Short Book Reviews, Vol. 23/2, August 2003