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1.APPLIED MULTIVARIATE METHODS
1.1 An Overview of Multivariate Methods 1
Variable-and Individual-Directed Techniques 2
Creating New Variables 2
Principal Components Analysis 3
Factor Analysis 3
Discriminant Analysis 4
Canonical Discriminant Analysis 5
Logistic Regression 5
Cluster Analysis 5
Multivariate Analysis of Variance 6
Canonical Variates Analysis 7
Canonical Correlation Analysis 7
Where to Find the Preceding Topics 7
1.2 Two Examples 8
Independence of Experimental Units 11
1.3 Types of Variables U
1.4 Data Matrices and Vectors 12
Variable Notation 13
Data Matrix 13
Data Vectors 13
Data Subscripts 14
1.5 The Multivariate Normal Distribution 15
Some Definitions 15
Summarizing Multivariate Distributions 16
Mean Vectors and Variance-Covariance Matrices 16
Correlations and Correlation Matrices 17
The Multivariate Normal Probability Density Function 19
Bivariate Normal Distributions 19
1.6 Statistical Computing 22
Cautions About Computer Usage 22
Missing Values 22
Replacing Missing Values by Zeros 23
Replacing Missing Values by Averages 23
Removing Rows of the Data Matrix 23
Sampling Strategies 24
Data Entry Errors and Data Verification 24
1.7 Multivariate Outliers 25
Locating Outliers 25
Dealing with Outliers 25
Outliers May Be Influential 26
1.8 Multivariate Summary Statistics 26
1.9 Standardized Data and/or Z Scores 27
Exercises 28
2.SAMPLE CORRELATIONS
2.1 Statistical Tests and Confidence Intervals 35
Are the Correlations Large Enough to Be Useful? 36
Confidence Intervals by the Chart Method 36
Confidence Intervals by Fishers Approximation 38
Confidence Intervals by Rubens Approximation 39
Variable Groupings Based on Correlations 40
Relationship to Factor Analysis 46
2.2 Summary 46
Exercises 47
3.MULTIVARIATE DATA PLOTS
3.1 Three-Dimensional Data Plots 55
3.2 Plots of Higher Dimensional Data 59
Chernoff Faces 61
Star Plots and Sun-Ray Plots 63
Andrews Plots 65
Side-by-Side Scatter Plots 66
3.3 Plotting to Check for Multivariate Normality 67
Summary 73
Exercises 73
4.EIGENVALUES AND EIGENVECTORS
4.1 Trace and Determinant 77
Examples 78
4.2 Eigenvalues 78
4.3 Eigenvectors 79
Positive Definite and Positive Semidefinite Matrices 80
4.4 Geometric Descriptions (p = 2) 82
Vectors 82
Bivariate Normal Distributions 83
4.5 Geometric Descriptions (p = 3) 87
Vectors 87
Trivariate Normal Distributions 87
4.6 Geometric Descriptions (p > 3) 90
Summary 91
Exercises 91
5.PRINCIPAL COMPONENTS ANALYSIS
5.1 Reasons for Using Principal Components Analysis 93
Data Screening 93
Clustering 95
Discriminant Analysis 95
Regression 95
5.2 Objectives of Principal Components Analysis 96
5.3 Principal Components Analysis on the Variance-Covariance
Matrix 96
Principal Component Scores 98
Component Loading Vectors 98
5.4 Estimation of Principal Components 99
Estimation of Principal Component Scores 99
5.5 Determining the Number of Principal Components 99
Method 1 100
Method 2 100
5.6 Caveats 107
5.7 PCA on the Correlation Matrix P 109
Principal Component Scores 110
Component Correlation Vectors 110
Sample Correlation Matrix 110
Determining the Number of Principal Components 110
5.8 Testing for Independence of the Original Variables 111
5.9 Structural Relationships 111
5.10 Statistical Computing Packages 112
SASR PRINCOMP Procedure 112
Principal Components Analysis Using Factor Analysis
Programs 118
PCA with SPSSs FACTOR Procedure 124
Summary 142
Exercises 142
6. FACTOR ANALYSIS
6.1 Objectives of Factor Analysis 147
6.2 Caveats 148
6.3 Some History of Factor Analysis 148
6.4 The Factor Analysis Model 150
Assumptions 150
Matrix Form of the Factor Analysis Model 151
Definitions of Factor Analysis Terminology 151
6.5 Factor Analysis Equations 151
Nonuniqueness of the Factors 152
6.6 Solving the Factor Analysis Equations 153
……
7.DISCRIMINANT ANALYSIS
8.LOGISTIC REGRESSION METHODS
9.CLUSTER ANALYSIS
10.MEAN VECTORS AND VARIANCE-COVARIANCE MATRICES
11.MULTIVARIATE ANALYSIS OF VARIANCE
12.PREDICTION MODELS AND MULTIVARIATE REGRESSION
APPENDIX A:MATRIX RESULTS
APPENDIX B:WORK ATTITUDES SURVEY
APPENDIX C:FAMILY CONTROL STUDY
REFERENCES
INDEX
A researcher wanted to compare fathers attitudes with their daughters dating behaviors. When several different variables have been measured on the fathers and several others measured on the daughters, canonical correlation analysis might be used to identify new variables that summarize any relationships that might exist between these two sets of family members.
Canonical correlation analysis is a generalization of multiple correlation in regression problems. It requires that the response variables be divided into two groups. The assignment of variables into these two groups must always be motivated by the nature of the response variables and never by an inspection of the data. For example, a legitimate assignment would be one in which the variables in one group are easy to obtain and inexpensive to measure, while the variables in the other group are hard to obtain or expensive to measure. Another would be measurements on fathers versus measurements on their daughters.
One basic question that canonical correlation analysis is expected to answer is whether the variables in one group can be used to predict the variables in the other group. When they can, then canonical correlation analysis attempts to summarize the relationships between the two sets of variables by creating new variables from each of the two groups of original variables.
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