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Understanding Correlation Matrices

Understanding Correlation Matrices

February 2021 | 136 pages | SAGE Publications, Inc

Correlation matrices (along with their unstandardized counterparts, covariance matrices) underlie the majority the statistical methods that researchers use today. A correlation matrix is more than a matrix filled with correlation coefficients. The value of one correlation in the matrix puts constraints on the values of the others, and the multivariate implications of this statement is a major theme of the volume. Alexandria Hadd and Joseph Lee Rodgers cover many features of correlations matrices including statistical hypothesis tests, their role in factor analysis and structural equation modeling, and graphical approaches. They illustrate the discussion with a wide range of lively examples including correlations between intelligence measured at different ages through adolescence; correlations between country characteristics such as public health expenditures, health life expectancy, and adult mortality; 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. This volume may be used effectively across a number of disciplines in both undergraduate and graduate statistics classrooms, and also in the research laboratory.

Series Editors Introduction
About the Authors
Chapter 1: Introduction
The Correlation Coefficient: A Conceptual Introduction

The Covariance

The Correlation Coefficient and Linear Algebra: Brief Histories

Examples of Correlation Matrices


Chapter 2: The Mathematics of Correlation Matrices
Requirements of Correlation Matrices

Eigenvalues of a Correlation Matrix

Pseudo-Correlation Matrices and Positive Definite Matrices

Smoothing Techniques

Restriction of Correlation Ranges in the Matrix

The Inverse of a Correlation Matrix

The Determinant of a Correlation Matrix



Chapter 3: Statistical Hypothesis Testing on Correlation Matrices
Hypotheses About Correlations in a Single Correlation Matrix

Hypotheses About Two or More Correlation Matrices

Testing for Linear Trend of Eigenvalues


Chapter 4: Methods for Correlation/Covariance Matrices as the Input Data
Factor Analysis

Structural Equation Modeling

Meta-Analysis of Correlation Matrices


Chapter 5: Graphing Correlation Matrices
Graphing Correlations

Graphing Correlation Matrices


Chapter 6: The Geometry of Correlation Matrices
What Is Correlation Space?

The 3 × 3 Correlation Space

Properties of Correlation Space: The Shape and Size

Uses of Correlation Space

Example Using 3 × 3 and 4 × 4 Correlation Space


Chapter 7: Conclusion


Instructor Resource Site
R files for chapters 1 and 3, plus an online appendix which demonstrates the use of the functions, are available on a website for the book at:

This volume provides a useful and interesting discussion about the importance and utility of the correlation matrix as a unified entity, beyond the pairwise correlations themselves. As such it provides readers with useful information about the foundations of several important statistical procedures and models.

William G. Jacoby
Michigan State University

This is an exceptional book that brings together information on a technique that has been around for over a century, the correlation. The authors challenge the reader to see correlations not as individuals but as a community that can be interpreted and acted on as such.

Rick Tivis
Idaho State University