BOOK

## Applied Multivariate Statistical Analysis: Pearson New International Edition

**Richard A. Johnson** |
** Dean W. Wichern**

(2013)

### Additional Information

#### Book Details

### Abstract

**For courses in Multivariate Statistics, Marketing Research, Intermediate Business Statistics, Statistics in Education, and graduate-level courses in Experimental Design and Statistics.**

Appropriate for experimental scientists in a variety of disciplines, this market-leading text offers a readable introduction to the statistical analysis of multivariate observations. Its primary goal is to impart the knowledge necessary to make proper interpretations and select appropriate techniques for analyzing multivariate data. Ideal for a junior/senior or graduate level course that explores the statistical methods for describing and analyzing multivariate data, the text assumes two or more statistics courses as a prerequisite.

### Table of Contents

Section Title | Page | Action | Price |
---|---|---|---|

Cover | Cover | ||

Table of Contents | i | ||

Chapter 1: Aspects of Multivariate Analysis | 1 | ||

1.1 Introduction | 1 | ||

1.2 Applications of Multivariate Techniques | 3 | ||

1.3 The Organization of Data | 5 | ||

Arrays | 5 | ||

Descriptive Statistics | 6 | ||

Graphical Techniques | 11 | ||

1.4 Data Displays and Pictorial Representations | 19 | ||

Linking Multiple Two-Dimensional Scatter Plots | 20 | ||

Graphs of Growth Curves | 24 | ||

Stars | 26 | ||

Chernoff Faces | 27 | ||

1.5 Distance | 30 | ||

1.6 Final Comments | 37 | ||

Exercises | 37 | ||

References | 47 | ||

Chapter 2: Sample Geometry and Random Sampling | 49 | ||

2.1 Introduction | 49 | ||

2.2 The Geometry of the Sample | 49 | ||

2.3 Random Samples and the Expected Values of the Sample Mean and Covariance Matrix | 57 | ||

2.4 Generalized Variance | 61 | ||

Generalized Variance Determined by and its Geometrical Interpretation | 72 | ||

Another Generalization of Variance | 75 | ||

2.5 Sample Mean, Covariance, and Correlation as Matrix Operations | 75 | ||

2.6 Sample Values of Linear Combinations of Variables | 78 | ||

Exercises | 82 | ||

References | 86 | ||

Chapter 3: Matrix Algebra and Random Vectors | 87 | ||

3.1 Introduction | 87 | ||

3.2 Some Basics of Matrix and Vector Algebra | 87 | ||

Vectors | 87 | ||

3.3 Positive Definite Matrices | 98 | ||

3.4 A Square-Root Matrix | 103 | ||

3.5 Random Vectors and Matrices | 104 | ||

3.6 Mean Vectors and Covariance Matrices | 106 | ||

Partitioning the Covariance Matrix | 111 | ||

The Mean Vector and Covariance Matrix for Linear Combinations of Random Variables | 113 | ||

Partitioning the Sample Mean Vector and Covariance Matrix | 115 | ||

3.7 Matrix Inequalities and Maximization | 116 | ||

Supplement 3A: Vectors and Matrices: Basic Concepts | 120 | ||

Vectors | 120 | ||

Matrices | 125 | ||

Exercises | 141 | ||

References | 148 | ||

Chapter 4: The Multivariate Normal Distribution | 149 | ||

4.1 Introduction | 149 | ||

4.2 The Multivariate Normal Density and its Properties | 149 | ||

Additional Properties of the Multivariate Normal Distribution | 156 | ||

4.3 Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation | 168 | ||

The Multivariate Normal Likelihood | 168 | ||

Maximum Likelihood Estimation of μ and Σ | 170 | ||

Sufficient Statistics | 173 | ||

4.4 The Sampling Distribution of X and S | 173 | ||

Properties of the Wishart Distribution | 174 | ||

4.5 Large-Sample Behavior of X and S | 175 | ||

4.6 Assessing the Assumption of Normality | 177 | ||

Evaluating the Normality of the Univariate Marginal Distributions | 177 | ||

Evaluating Bivariate Normality | 182 | ||

4.7 Detecting Outliers and Cleaning Data | 187 | ||

Steps for Detecting Outliers | 189 | ||

4.8 Transformations to Near Normality | 192 | ||

Transforming Multivariate Observations | 195 | ||

Exercises | 200 | ||

References | 208 | ||

Chapter 5: Inferences About a Mean Vector | 210 | ||

5.1 Introduction | 210 | ||

5.2 The Plausibility of μ0 as a Value for a Normal Population Mean | 210 | ||

5.3 Hotelling’s T2 and Likelihood Ratio Tests | 216 | ||

General Likelihood Ratio Method | 219 | ||

5.4 Confidence Regions and Simultaneous Comparisons of Component Means | 220 | ||

Simultaneous Confidence Statements | 223 | ||

A Comparison of Simultaneous Confidence Intervals with One-at-a-Time Intervals | 229 | ||

The Bonferroni Method of Multiple Comparisons | 232 | ||

5.5 Large Sample Inferences about a Population Mean Vector | 234 | ||

5.6 Multivariate Quality Control Charts | 239 | ||

Charts for Monitoring a Sample of Individual Multivariate Observations for Stability | 241 | ||

Control Regions for Future Individual Observations | 247 | ||

Control Ellipse for Future Observations | 248 | ||

T2-Chart for Future Observations | 248 | ||

Control Charts Based on Subsample Means | 249 | ||

Control Regions for Future Subsample Observations | 251 | ||

5.7 Inferences about Mean Vectors When Some Observations are Missing | 251 | ||

5.8 Difficulties Due to Time Dependence in Multivariate Observations | 256 | ||

Supplement 5A: Simultaneous Confidence\rIntervals and Ellipses as Shadows of the p-Dimensional Ellipsoids | 258 | ||

Exercises | 261 | ||

References | 272 | ||

Chapter 6: Comparisons of Several Multivariate Means | 273 | ||

6.1 Introduction | 273 | ||

6.2 Paired Comparisons and a Repeated Measures Design | 273 | ||

Paired Comparisons | 273 | ||

A Repeated Measures Design for Comparing Treatments | 279 | ||

6.3 Comparing Mean Vectors from Two Populations | 284 | ||

Assumptions Concerning the Structure of the Data | 284 | ||

Further Assumptions When n1 and n2 are Small | 285 | ||

Simultaneous Confidence Intervals | 288 | ||

The Two-Sample Situation When Σ1 ≠ Σ2 | 291 | ||

An Approximation to the Distribution of T2 for Normal Populations When Sample Sizes are Not Large | 294 | ||

6.4 Comparing Several Multivariate Population Means (One-Way Manova) | 296 | ||

Assumptions about the Structure of the Data for One-Way Manova | 296 | ||

A Summary of Univariate Anova | 297 | ||

Multivariate Analysis of Variance (Manova) | 301 | ||

6.5 Simultaneous Confidence Intervals for Treatment Effects | 308 | ||

6.6 Testing for Equality of Covariance Matrices | 310 | ||

6.7 Two-Way Multivariate Analysis of Variance | 312 | ||

Univariate Two-Way Fixed-Effects Model with Interaction | 312 | ||

Multivariate Two-Way Fixed-Effects Model with Interaction | 315 | ||

6.8 Profile Analysis | 323 | ||

6.9 Repeated Measures Designs and Growth Curves | 328 | ||

6.10 Perspectives and a Strategy for Analyzing Multivariate Models | 332 | ||

Exercises | 337 | ||

References | 358 | ||

Chapter 7: Multivariate Linear Regression Models | 360 | ||

7.1 Introduction | 360 | ||

7.2 The Classical Linear Regression Model | 360 | ||

7.3 Least Squares Estimation | 364 | ||

Sum-of-Squares Decomposition | 366 | ||

Geometry of Least Squares | 367 | ||

7.4 Inferences About the Regression Model | 370 | ||

Inferences Concerning the Regression Parameters | 370 | ||

Likelihood Ratio Tests for the Regression Parameters | 374 | ||

7.5 Inferences from the Estimated Regression Function | 378 | ||

Estimating the Regression Function at Z0 | 378 | ||

Forecasting a New Observation at Z0 | 379 | ||

7.6 Model Checking and Other Aspects of Regression | 381 | ||

Does the Model Fit? | 381 | ||

Leverage and Influence | 384 | ||

Additional Problems in Linear Regression | 384 | ||

7.7 Multivariate Multiple Regression | 387 | ||

Other Multivariate Test Statistics | 398 | ||

Predictions from Multivariate Multiple Regressions | 399 | ||

7.8 The Concept of Linear Regression | 401 | ||

7.9 Comparing the Two Formulations of the Regression Model | 410 | ||

Mean Corrected Form of the Regression Model | 410 | ||

Relating the Formulations | 412 | ||

7.10 Multiple Regression Models with Time Dependent Errors | 413 | ||

Supplement 7A: The Distribution of the Likelihood Ratio for the Multivariate Multiple Regression Model | 418 | ||

Exercises | 420 | ||

References | 428 | ||

Chapter 8: Principal Components | 430 | ||

8.1 Introduction | 430 | ||

8.2 Population Principal Components | 430 | ||

Principal Components for Covariance Matrices with Special Structures | 439 | ||

8.3 Summarizing Sample Variation by Principal Components | 441 | ||

The Number of Principal Components | 444 | ||

Interpretation of the Sample Principal Components | 448 | ||

Standardizing the Sample Principal Components | 449 | ||

8.4 Graphing the Principal Components | 454 | ||

8.5 Large Sample Inferences | 456 | ||

Large Sample Properties of λi and ei | 456 | ||

Testing for the Equal Correlation Structure | 457 | ||

8.6 Monitoring Quality with Principal Components | 459 | ||

Checking a Given Set of Measurements for Stability | 459 | ||

Controlling Future Values | 463 | ||

Supplement 8A: The Geometry of the Sample\rPrincipal Component Approximation | 466 | ||

The p-Dimensional Geometrical Interpretation | 468 | ||

The n-Dimensional Geometrical Interpretation | 469 | ||

Exercises | 470 | ||

References | 480 | ||

Chapter 9: Factor Analysis and Inference for Structured Covariance Matrices | 481 | ||

9.1 Introduction | 481 | ||

9.2 The Orthogonal Factor Model | 482 | ||

9.3 Methods of Estimation | 488 | ||

The Principal Component (and Principal Factor) Method | 488 | ||

A Modified Approach—the Principal Factor Solution | 494 | ||

The Maximum Likelihood Method | 495 | ||

A Large Sample Test for the Number of Common Factors | 501 | ||

9.4 Factor Rotation | 504 | ||

Oblique Rotations | 512 | ||

9.5 Factor Scores | 513 | ||

The Weighted Least Squares Method | 514 | ||

The Regression Method | 516 | ||

9.6 Perspectives and a Strategy for Factor Analysis | 519 | ||

Supplement 9A: Some Computational Details for Maximum Likelihood Estimation | 527 | ||

Recommended Computational Scheme | 528 | ||

Maximum Likelihood Estimators of p = LzLz + ψz | 529 | ||

Exercises | 530 | ||

References | 538 | ||

Chapter 10: Canonical Correlation Analysis | 539 | ||

10.1 Introduction | 539 | ||

10.2 Canonical Variates and Canonical Correlations | 539 | ||

10.3 Interpreting the Population Canonical Variables | 545 | ||

Identifying the Canonical Variables | 545 | ||

Canonical Correlations as Generalizations of Other Correlation Coefficients | 547 | ||

The First r Canonical Variables as a Summary of Variability | 548 | ||

A Geometrical Interpretation of the Population Canonical Correlation Analysis | 549 | ||

10.4 The Sample Canonical Variates and Sample Canonical Correlations | 550 | ||

10.5 Additional Sample Descriptive Measures | 558 | ||

Matrices of Errors of Approximations | 558 | ||

Proportions of Explained Sample Variance | 561 | ||

10.6 Large Sample Inferences | 563 | ||

Exercises | 567 | ||

References | 574 | ||

Chapter 11: Discrimination and Classification | 575 | ||

11.1 Introduction | 575 | ||

11.2 Separation and Classification for Two Populations | 576 | ||

11.3 Classification with Two Multivariate Normal Populations | 584 | ||

Classification of Normal Populations When Σ1 = Σ2 = Σ | 584 | ||

Scaling | 589 | ||

Fisher’s Approach to Classification with Two Populations | 590 | ||

Is Classification a Good Idea? | 592 | ||

Classification of Normal Populations When Σ1 ≠ Σ2 | 593 | ||

11.4 Evaluating Classification Functions | 596 | ||

11.5 Classification with Several Populations | 606 | ||

The Minimum Expected Cost of Misclassification Method | 606 | ||

Classification with Normal Populations | 609 | ||

11.6 Fisher’s Method for Discriminating among Several Populations | 621 | ||

Using Fisher’s Discriminants to Classify Objects | 628 | ||

11.7 Logistic Regression and Classification | 634 | ||

Introduction | 634 | ||

The Logit Model | 634 | ||

Logistic Regression Analysis | 636 | ||

Classification | 638 | ||

Logistic Regression with Binomial Responses | 640 | ||

11.8 Final Comments | 644 | ||

Including Qualitative Variables | 644 | ||

Classification Trees | 644 | ||

Neural Networks | 647 | ||

Selection of Variables | 648 | ||

Testing for Group Differences | 648 | ||

Graphics | 649 | ||

Practical Considerations Regarding Multivariate Normality | 649 | ||

Exercises | 650 | ||

References | 669 | ||

Chapter 12: Clustering, Distance Methods and Ordination | 671 | ||

12.1 Introduction | 671 | ||

12.2 Similarity Measures | 673 | ||

Distances and Similarity Coefficients for Pairs of Items | 673 | ||

Similarities and Association Measures for Pairs of Variables | 677 | ||

Concluding Comments on Similarity | 678 | ||

12.3 Hierarchical Clustering Methods | 680 | ||

Single Linkage | 682 | ||

Complete Linkage | 685 | ||

Average Linkage | 690 | ||

Ward’s Hierarchical Clustering Method | 692 | ||

Final Comments—Hierarchical Procedures | 695 | ||

12.4 Nonhierarchical Clustering Methods | 696 | ||

K-means Method | 696 | ||

Final Comments—Nonhierarchical Procedures | 701 | ||

12.5 Clustering Based on Statistical Models | 703 | ||

12.6 Multidimensional Scaling | 706 | ||

12.7 Correspondence Analysis | 716 | ||

Algebraic Development of Correspondence Analysis | 718 | ||

12.8 Biplots for Viewing Sampling Units and Variables | 726 | ||

Constructing Biplots | 727 | ||

12.9 Procrustes Analysis: A Method for Comparing Configurations | 732 | ||

Constructing the Procrustes Measure of Agreement | 733 | ||

Supplement 12A: Data Mining | 740 | ||

Introduction | 740 | ||

The Data Mining Process | 741 | ||

Model Assessment | 742 | ||

Exercises | 747 | ||

References | 755 | ||

Selected Additional References for Model Based Clustering | 756 | ||

Appendix | 757 | ||

Table 1: Standard Normal Probabilities | 758 | ||

Table 2: Student’s T-Distribution Percentage Points | 759 | ||

Table 3: X2 Distribution Percentage Points | 760 | ||

Table 4: F-Distribution Percentage Points (α = 10) | 761 | ||

Table 5: F-Distribution Percentage Points (α = .05) | 762 | ||

Table 6: F-Distribution Percentage Points (α = .01) | 763 | ||

Index | 765 |