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.

# Statistical Hypothesis Testing on Correlation Matrices

### Statistical Hypothesis Testing on Correlation Matrices

In this chapter, we review the literature devoted to statistical tests on correlation matrices and provide detailed recommendations and examples of a variety of hypothesis tests applied to correlation matrices. Hypothesis tests and confidence intervals for individual correlation coefficients are well-known; for example, a nil-null hypothesis test (i.e., where the null-hypothesized value is 0) on the associated regression coefficient for predicting Y from X (or vice versa) is equivalent to testing the null hypothesis that ρ = 0, or an asymmetric confidence interval can be constructed around r using the Fisher z transformation. Testing correlation coefficients calculated from two independent samples is also straightforward and is covered in other sources. However, hypothesis tests involving correlation coefficients within the ...