Understanding Correlation Matrices: (Quantitative Applications in the Social Sciences)

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.

Table of Contents:
Series Editors Introduction Preface Acknowledgments 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 Summary 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 Examples Summary 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 Summary Chapter 4: Methods for Correlation/Covariance Matrices as the Input Data Factor Analysis Structural Equation Modeling Meta-Analysis of Correlation Matrices Summary Chapter 5: Graphing Correlation Matrices Graphing Correlations Graphing Correlation Matrices Summary 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 Summary Chapter 7: Conclusion References Index

About the Author :
Alexandria Ree Hadd is an Assistant Professor of Psychology at Spelman College in Atlanta, where she teaches courses on statistics and research methods to undergraduate students. She earned her Masters and Ph.D. in Quantitative Psychology at Vanderbilt University and her B.S. in Psychology and Mathematics from Oglethorpe University. Her Masters thesis – titled “Correlation Matrices in Cosine Space” -- was specifically on the properties of correlation matrices. She also researched correlations in her dissertation, which was titled “A Comparison of Confidence Interval Techniques for Dependent Correlations.” At Vanderbilt, she taught introductory statistics and was a teaching assistant for a number of graduate statistics/methods courses. In addition to correlation matrices, her research interests include applying modeling techniques to developmental, educational, and environmental psychology questions. In her spare time, her hobbies include hiking, analog collaging, attending art and music shows, and raising worms (who are both pets and dedicated composting team members). Joseph Lee Rodgers is Lois Autrey Betts Chair of Psychology and Human Development at Vanderbilt University in Nashville.  He moved to Vanderbilt in 2012 from the University of Oklahoma, where he worked from 1981 until 2012, and where he holds the title George Lynn Cross Emeritus Professor of Psychology.  Joe earned his Ph.D. in Quantitative Psychology from the L. L. Thurstone Psychometric Laboratory at the University of North Carolina, Chapel Hill, in 1981 (and also minored in Biostatistics at UNC).  He has held short-term teaching/research positions at Ohio State, University of Hawaii, UNC, Duke, University of Southern Denmark, and Penn.  He has published six books and over 175 papers and chapters on statistics/quantitative methods, demography, behavior genetics, and developmental and social psychology.  His best-known paper, “Thirteen Ways to Look at the Correlation Coefficient,” was published in American Statistician in 1988.  Joe is married to Jacci Rodgers, an academic accountant (and currently an associate dean of Peabody College at Vanderbilt), and they have two adult daughters; Rachel works for an international development company in DC, and Naomi is a Ph.D. student in Geology at USC in Los Angeles.  Joe’s hobbies include playing tennis and golf, reading, and music.

Review :
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. 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.