Understanding Correlation Matrices Alexandria R Hadd Joseph Lee Rodgers

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6. Understanding
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7. Quantitative Applications in the Social Sciences
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10. Understanding
Correlation Matrices
Alexandria Hadd
Spelman College
Joseph Lee Rodgers
Vanderbilt University
Quantitative Applications in the Social Sciences, Volume 186

11. Copyright © 2021 by SAGE Publications, Inc.
All rights reserved. Except as permitted by U.S.
copyright law, no part of this work may be reproduced
or distributed in any form or by any means, or stored in
a database or retrieval system, without permission in
writing from the publisher.
All third party trademarks referenced or depicted herein
are included solely for the purpose of illustration and are
the property of their respective owners. Reference to
these trademarks in no way indicates any relationship
with, or endorsement by, the trademark owner.
Printed in the United States of America
Library of Congress Cataloging-in-Publication Data
Names: Hadd, Alexandria, author. | Rodgers, Joseph Lee,
1953- author.
Title: Understanding correlation matrices / Alexandria
Hadd, Spelman College, Joseph Lee Rodgers,
Vanderbilt University.
Description: First Edition. | Thousand Oaks : SAGE
Publications, Inc, 2020. | Includes bibliographical
references.
Identifiers: LCCN 2020031187 | ISBN 9781544341095
(paperback) | ISBN 9781544341071 (epub) | ISBN
9781544341088 (epub) | ISBN 9781544341101 (ebook)
Subjects: LCSH: Correlation (Statistics) | Matrices.
Classification: LCC HA31.3 .H333 2020 | DDC
519.5/37—dc23
LC record available at https://lccn.loc.gov/2020031187
This book is printed on acid-free paper.
20 21 22 23 24 10 9 8 7 6 5 4 3 2 1
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12. TABLE OF CONTENTS
Series Editors Introduction xi
Preface xiii
Acknowledgments xv
About the Authors xvii
Chapter 1: Introduction 1
The Correlation Coefficient: A Conceptual Introduction 2
The Covariance 3
The Correlation Coefficient and Linear Algebra:
Brief Histories 5
Examples of Correlation Matrices 8
Summary 15
Chapter 2: The Mathematics of Correlation Matrices 17
Requirements of Correlation Matrices 18
Eigenvalues of a Correlation Matrix 20
Pseudo-Correlation Matrices and Positive Definite Matrices 21
Smoothing Techniques 23
Restriction of Correlation Ranges in the Matrix 25
The Inverse of a Correlation Matrix 25
The Determinant of a Correlation Matrix 26
Examples 27
Racial Composition of NBA and Sponsor Cities 27
Girls’ Intelligence Across Development 27
Summary 28
Chapter 3: Statistical Hypothesis Testing on Correlation Matrices 29
Hypotheses About Correlations in a Single Correlation Matrix 30
Testing Equality of Two Correlations in a Correlation
Matrix (No Variable in Common) 30
Testing Equality of Two Correlations in a Correlation
Matrix (Variable in Common) 32

13. Testing Equality to a Specified Population Correlation
Matrix 33
Hypotheses About Two or More Correlation Matrices 37
Testing Equality of Two Correlation Matrices From
Independent Groups 37
Testing Equality of Several Correlation Matrices 40
Testing Equality of Several Correlations From
Independent Samples 42
Testing for Linear Trend of Eigenvalues 43
Summary 45
Chapter 4: Methods for Correlation/Covariance
Matrices as the Input Data 47
Factor Analysis 48
Summary 48
Example 50
Resources for Software and Additional Readings 51
Structural Equation Modeling 52
Summary 52
Examples 54
Resources for Software and Additional Readings 57
Meta-Analysis of Correlation Matrices 58
Summary 58
Recent MASEM Examples 59
Resources for Software and Additional Readings 59
Summary 60
Chapter 5: Graphing Correlation Matrices 63
Graphing Correlations 65
Graphing Correlation Matrices 69
The Scatterplot Matrix 70
The Scatterplot Matrix, Enhanced 70
Corrgrams Using the corrplot Package in R 74
Heat Maps 75
Parallel Coordinate Plots 78
Eigenvector Plots 81
Summary 84

14. Chapter 6: The Geometry of Correlation Matrices 85
What Is Correlation Space? 85
The 3 × 3 Correlation Space 87
Properties of Correlation Space: The Shape and Size 90
Convexity of the Space 90
Number of Vertices and Edges 90
Volume Relative to Space of Pseudo-Correlations 91
Uses of Correlation Space 92
Similarity of Correlation Matrices 92
Generating Random Correlation Matrices 94
Fungible Correlation Matrices 94
Defining Correlation Spaces Using Angles 95
Example Using 3 × 3 and 4 × 4 Correlation Space 96
Summary 99
Chapter 7: Conclusion 101
References 105
Index 113

16. xi
SERIES EDITOR’S INTRODUCTION
In 1988, Joseph Rodgers and Alan Nicewander published an article in The
American Statistician that described 13 different ways to look at the cor-
relation coefficient. This classic article is one of my favorites. It holds up
one of the workhorses of modern statistics and examines it from multiple
perspectives, twisting and turning it to develop new understandings and
insights. In Understanding Correlation Matrices, Alexandria Hadd and
Joseph Rodgers team up to provide a parallel treatment for the correlation
matrix.
Correlation matrices (along with their unstandardized counterparts,
covariance matrices) underlie much of the statistical machinery in common
use today. Multiple regression models, confirmatory and exploratory factor
analysis, principal components analysis, and structural equation models
often start with a correlation (or covariance) matrix. The correlation matrix
is so common, we hardly ever give it a second look unless, of course, there
are problems that prevent us from estimating the model we have specified.
This book shines a light on the correlation matrix, examining it from
­ multiple perspectives: mathematical, statistical, and geometric.
A correlation matrix is more than a matrix filled with correlation coeffi-
cients. The value of one coefficient in the matrix puts constraints on the
values of the others. This is a major theme of the book. As Hadd and Rodgers
explain, a “true” correlation matrix must be symmetric, with diagonal ele-
ments equal to one, off-diagonals containing correlation coefficients
bounded by −1 and 1, and nonnegative eigenvalues. If the first three condi-
tions are met, but not the fourth, the matrix is a pseudo-correlation matrix.
This can happen, for example, when the correlation coefficients are esti-
mated based on different samples. Hadd and Rodgers explore these and
other characteristics of correlation matrices in chapters devoted to statisti-
cal hypothesis testing on correlation matrices, methods for graphing
­ correlation matrices, and the geometry of correlation space.
Although some of the topics are advanced, the book is written to be
accessible to readers with no background in linear algebra. The key points
are illustrated with a wide range of lively examples, including correlations
between intelligence measured at different ages through adolescence;

17. xii   
correlations between public health expenditures, health life expectancy,
adult mortality, and other country characteristics; correlations between
well-being and state-level vital statistics; correlations between the racial
composition of cities and professional sports teams; and correlations
between childbearing intentions and childbearing outcomes over the
­ reproductive life course.
One of the reasons that I so thoroughly enjoy methods and statistics is
that topics can be understood on many levels. The deeper you dig, the more
you learn. This book is a great illustration: Even sophisticated readers will
find a new appreciation for an old friend, the correlation matrix. I invite you
to join Ali and Joe, and dig deep.
Barbara Entwisle
—Series Editor

18. xiii
PREFACE
This book was written to present, for the first time, a description of the cor-
relation matrix and its many facets. Both introductory and advanced
courses in statistics and quantitative methods focus carefully on the correla-
tion coefficient, and some of those also discuss the correlation matrix as it
pertains to advanced modeling methods. However, there is no comprehen-
sive, dedicated, and unified treatment of the correlation matrix. The
assumption might be that if students understand the correlation coefficient,
they’ll understand the correlation matrix. In fact, this assumption would be
entirely incorrect. The goal of this book is to present the correlation matrix
and many of its valuable details—and to do so assuming the least amount
of mathematical background possible.
Teachers, students, and researchers in many disciplines will find this
book valuable. The first disciplinary arena is obvious—that would be sta-
tistics, in both its theoretical and applied forms. But all disciplines that rely
on quantitative analysis to support their research mission will find an intro-
duction to the correlation matrix to be a valuable contribution to their
scholarly repertoire. These include psychology, sociology, education
­ economics, political science, business, communications, social work,
anthropology, medical sciences, and biology, and there are many others. If
a discipline includes an introductory course in statistics within its introduc-
tory ­curriculum, then this book will facilitate that effort.
The first author researched the correlation matrix for her master’s thesis
and made important breakthroughs (several of which are presented in
Chapter 6); the seeds of this book were sown during that work in graduate
school. The second author has been working on the theory and application
of the correlation coefficient for his whole career; much of that treatment
has contributed to statistical pedagogy. We continue a commitment to
strong pedagogy within the current monograph. Why should we pay atten-
tion to the correlation coefficient, and how does it transition to the
­correlation matrix?
An argument can be made that the “invention” of the correlation coeffi-
cient by Karl Pearson was the most important development in the field of
statistics. The correlation measures the relationship between two variables,

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