1st Edition

Introduction to Multivariate Analysis

By Chris Chatfield, A. Collins Copyright 1980
    248 Pages
    by Chapman & Hall

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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.

    Part One: Introduction
    1 Introduction
    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
    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
    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
    Part Two: Finding New Underlying Variables
    4. Principal component analysis
    4.1 Introduction
    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
    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
    6.2 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 T²-distribution
    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
    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
    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
    Part Four: Multidimensional Scaling and Cluster Analysis
    10. Multidimensional scaling
    10.1 Introduction
    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
    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
    Answers to exercises
    Name Index
    Subject Index

    "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."