CORRECTED_CORRELS: Corrected Pearson correlation coefficients

In EFA.dimensions: Exploratory Factor Analysis Functions for Assessing Dimensionality

Corrected Pearson correlation coefficients

Description

Corrects Pearson correlation coefficients for attenuation due to unreliability and tests for overall differences between the original and corrected correlation matrices.

Usage

CORRECTED_CORRELS(cormat, Ncases = 100, alphas, 
                  systematic=FALSE, Nperms = 1000, verbose=TRUE)

Arguments

cormat	A square correlation matrix.
Ncases	The number of cases upon which cormat is based. Example: Ncases = 250
alphas	A vector of reliability coefficients (e.g., Cronbach alpha values) for the variables in cormat, in the same order as the variables in cormat. Example: alphas <- c(.81, .95, .64, .49, .36)
systematic	The kind of permutations for the tests for significance differences between the uncorrected and corrected correlation matrices. The options are TRUE (for all possible permutations), or FALSE (the default), in which case random matrix permutations will be conducted.
Nperms	(optional) The number of random matrix permutation for when systematic = FALSE. Example: Nperms = 1000
verbose	(optional) Should detailed results be displayed in console? TRUE (default) or FALSE

Details

This function uses the simple, traditional formula for correcting correlations for attentuation due to unreliability, as described in Schmidt and Hunter (1996), Schmitt (1996), and Trafimow (2016).

Spearman's rho and Kendall's tau correlations between the values in the entered and corrected correlation matrices are provided when there are three or more variables. These are rank correlations that do not assume that similarity increases are linear and which are more robust to outliers than are Pearson correlations. The corresponding permutation significance tests of overall equality between the entered and corrected correlation matrices are "Mantel" tests.

The Steiger (1980) and Jennrich (1970) tests of the overall equality of the entered and corrected correlation matrices are conducted using functions from the psych package (Revelle, 2023).

See the MatrixCorrelation package for additional tests of differences between correlation matrices.

Value

A list with the following elements:

cormat	The entered correlation matrix
alphas	The entered alphas
corxd	The matrix of corrected correlations
resids	The differences between the entered and corrected correlations
data_in_rows	The input data and results in row format
cor_spearman	The Spearman's rho correlation between the entered and corrected correlations
cor_kendall	The Kendall's tau correlation between the entered and corrected correlations
Steiger_test	Results of the Steiger test for equality of the entered and corrected correlation matrices
Jennrich_test	Results of the Jennrich test for equality of the entered and corrected correlation matrices

Author(s)

Brian P. O'Connor

References

Jennrich, R. I. (1970) An asymptotic test for the equality of two correlation matrices. Journal of the American Statistical Association, 65, 904-912.

Revelle, W. (2023). psych: Procedures for Psychological, Psychometric, and Personality Research. R package version 2.3.6, https://CRAN.R-project.org/package=psych

Schmidt, F. L., & Hunter, J. E. (1996). Measurement error in psychological research: Lessons from 26 research scenarios. Psychological Methods, 1(2), 199223.

Schmitt, N. (1996). Uses and abuses of coefficient alpha. Psychological Assessment, 8(4), 350353.

Steiger, J. H. (1980). Testing pattern hypotheses on correlation matrices: Alternative statistics and some empirical results. Multivariate Behavioral Research, 15, 335-352.

Trafimow, D. (2016). The attenuation of correlation coefficients: A statistical literacy issue. Teaching Statistics, 38(1), 2528.

Examples

# data from Schmitt (1996)
cormat <- '
 1.0
 .70  1.0
 .51  .47  1.0  
 .52  .55  .45  1.0
 .32  .35  .31  .28  1.0'
cormat_noms <- c('v1', 'v2', 'v3', 'v4', 'v5')
cormat <- data.matrix( read.table(text=cormat, fill=TRUE, col.names=cormat_noms, 
                                  row.names=cormat_noms ))
cormat[upper.tri(cormat)] <- t(cormat)[upper.tri(cormat)]

alphas <- c(.81, .95, .64, .49, .36)

CORRECTED_CORRELS(cormat=cormat, alphas=alphas, 
                  Nperms = 10000, systematic=FALSE, verbose=TRUE)

EFA.dimensions documentation built on Sept. 14, 2026, 9:08 a.m.