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This book provides an introduction to the analysis of multivariate data.It describes multivariate probability distributions, the preliminary analysisof a large -scale set of data, princ iple component and factor analysis,traditional normal theory material, as well as multidimensional scaling andcluster analysis.Introduction to Multivariate Analysis provides a reasonable blend oftheory and practice. Enough theory is given to introduce the concepts andto make the topics mathematically interesting. In addition the authors discussthe use (and misuse) of the techniques in pra ctice and present appropriatereal-life examples from a variety of areas includ ing agricultural research,soc iology and crim inology. The book should be suitable both for researchworkers and as a text for students taking a course on multivariate analysis.

Preface

Part One: Introduction

1 Introduction

Examples

Notation

1.1 Review of Objectives and different approaches

1.2 Some general comments

1.3 Review of books on multivariate analysis

1.4 Some matrix algebra revision

1.5 The general linear model

Exercises**2 Multivariate distributions**2.1 Multivariate, marginal and conditional distributions

2.2 Means, variances, covariances and correlations

2.3 The multivariate normal distribution

2.4 The bivariate normal distribution

2.5 Other multivariate distributions

2.5.1 Multivariate discrete distributions

2.5.2 Multivariate continuous distributions

Exercises

**3. Preliminary data analysis**

3.1 Processing the data

3.1.1 Data editing

3.2 Calculating summary statistics

3.2.1 Interpreting the sample correlation matrix

3.2.2 The rank of R

3.3 Plotting the data

3.4 The analysis of discrete data

Exercises

**Part Two: Finding New Underlying Variables**

4. Principal component analysis

4.1 Introduction

4. Principal component analysis

4.2 Derivation of principal components

4.3.1 Principal components from the correlation matrix

4.2.2 Estimating the principal components

4.3 Further results on PCA

4.3.1 Mean-corrected component scores

4.3.2 The inverse transformation

4.3.3 Zero eigenvalues

4.3.4 Small eigenvalues

4.3.5 Repeated roots

4.3.6 Orthogonality

4.3.7 Component loadings/component correlations

4.3.8 Off-diagonal structure

4.3.9 Uncorrelated variables

4.4 The problem of scaling in PCA

4.5 Discussion

4.5.1 The identification of important components

4.5.2 The use of components in subsequent analyses

4.6 PCA for multivariate normal data

4.7 Summary

Exercises

**5. Factor analysis**

5.1 Introduction

5.2 The factor-analysis model

5.3 Estimating the factor loadings

5.4 Discussion

**Part Three: Procedures Based on the Multivariate Normal Distribution**

6. The multivariate normal distribution

6.1 Introduction

6. The multivariate normal distribution

6.2 Definition of the multivariate normal distribution

6.3 Properties of the multivariate normal distribution

6.4 Linear compounds and linear combinations

6.5 Estimation of the parameters of the distribution

6.6 The Wishart distribution

6.7 The joint distribution of the sample mean vector and the sample covariance matrix

6.8 The Hotelling TÂ²-distribution

Exercises

**7. Procedures based on normal distribution theory**

7.1 Introduction

7.2 One-sample procedures

7.3 Confidence intervals and further analysis

7.4 Tests of structural relations among the components of the mean

7.5 Two-sample procedures

7.6 Confidence intervals and further analysis

7.7 Tests of structural relations among the components of the means

7.8 Discriminant analysis

Exercises

**8. The multivariate analysis of variance**

8.1 Introduction

8.2 MANOVA calculations

8.3 Testing hypotheses

8.3.1 The special case: The univariate procedure

8.3.2 The multivariate model for Example 8.1

8.3.3 Multivariate test procedure

8.3.4 Distributional approximations

8.3.5 Applications of the methodology

8.4 Further analysis

8.5 The dimensionality of the alternative hypothesis

8.6 Canonical variates analysis

8.7 Linear functional relationships

8.8 Discriminant analysis

Exercises

**9. The multivariate analysis of covariance and related topics**

9.1 Introduction

9.2 Multivariate regression

9.2.1 The special case: Univariate multiple regression

9.2.2 The general case: Multivariate regression

9.3 Canonical correlation

9.4 The multivariate analysis of covariance

9.4.1 The special case: Univariate analysis of covariance

9.4.2 The multivariate case: An example

9.4.3 The multivariate case: General results

9.5 The test for additional information

9.6 A test of an assigned subset of linear compounds

Exercises

**Part Four: Multidimensional Scaling and Cluster Analysis**

10. Multidimensional scaling

10.1 Introduction

10. Multidimensional scaling

10.2 Measures of similarity and dissimilarity

10.2.1 Similarity coefficients for binary data

10.3 Classical scaling

10.3.1 The calculation of co-ordinate values from Euclidean distances

10.3.2 The relationship between classical scaling and principal component analysis

10.3.3 Classical scaling for a dissimilarity matrix

10.3.4 Interpretation of the results

10.3.5 Some related methods

10.4 Ordinal scaling

10.4.1 The iterative procedure

10.4.2 Interpreting the results

10.5 A comparison

10.6 Concluding remarks

Exercises

**11 Cluster analysis**

11.1 Introduction

11.1.1 Objectives

11.1.2 Clumping, dissection and clustering variables

11.1.3 Some other preliminary points

11.2 Visual approaches to finding a partition

11.3 Hierarchical trees

11.4 Single-link clustering

11.5 Some other clustering procedures

11.5.1 Method or algorithm?

11.6 A comparison of procedures

11.6.1 Some desirable conditions for hierarchical clustering methods

11.6.2 A comparison

Exercises

**References**

Answers to exercises

Name Index

Subject Index

Answers to exercises

Name Index

Subject Index

### Biography

Chris Chatfield

"The book's simplicity of approach and clear presentation make it a good choice for an undergraduate course or a self-study course in multivariate analysis."

-Journal of the American Statistical Association

"The original aims of the book are fulfilled remarkably well, thus providing a text which is to be welcomed into an area where there has been a recent dearth of introductory material."

-BIAS