Abstract
This chapter introduces permutation methods for measures of correlation and regression, the best-known of which is Pearson’s product-moment correlation coefficient. Included in this chapter are six example analyses illustrating computation of exact permutation probability values for correlation and regression, calculation of measures of effect size for measures of correlation and regression, the effects of extreme values on conventional (ordinary least squares) and permutation (least absolute deviation) correlation and regression, exact and Monte Carlo permutation procedures for measures of correlation and regression, application of permutation methods to correlation and regression with rank-score data, and analysis of multiple correlation and regression. Included in this chapter are permutation versions of ordinary least squares correlation and regression, least absolute deviation correlation and regression, Spearman’s rank-order correlation coefficient, Kendall’s rank-order correlation coefficient, Spearman’s footrule measure of correlation, and a permutation-based alternative for the conventional measures of effect size for correlation and regression: Pearson’s r 2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that whereas a permutation approach eschews estimated population parameters and degrees of freedom, the summations are divided by N, not N − 1. Thus \(S_{x}^{2}\) and \(S_{y}^{2}\) denote the sample variances, not the estimated population variances.
- 2.
In this section, a caret(∧) over a symbol such as \(\hat {\alpha }\) or \(\hat {\beta }\) indicates an OLS regression model predicted value of a corresponding population parameter, while a tilde (∼) over a symbol such as \(\tilde {\alpha }\) or \(\tilde {\beta }\) indicates a LAD regression model predicted value of a corresponding population parameter.
- 3.
One degree of freedom is lost due to the sample estimate (\(\hat {\alpha }_{yx}\)) of the population intercept and one degree of freedom is lost due to the sample estimate (\(\hat {\beta }_{yx}\)) of the population slope.
- 4.
For simplification and clarity the formulæ and examples are limited to untied rank-score data.
- 5.
Note that in Eq. (10.3) N is a constant, so only the sum-of-squared differences need be calculated for each arrangement of the observed data.
References
Barrodale, I., Roberts, F.D.K.: A improved algorithm for discrete ℓ 1 linear approximation. J. Numer. Anal. 10, 839–848 (1973)
Barrodale, I., Roberts, F.D.K.: Solution of an overdetermined system of equations in the ℓ 1 norm. Commun. ACM 17, 319–320 (1974)
Berry, K.J., Mielke, P.W.: Least sum of absolute deviations regression: distance, leverage, and influence. Percept. Motor Skill. 86, 1063–1070 (1998)
Berry, K.J., Mielke, P.W.: A Monte Carlo investigation of the Fisher Z transformation for normal and nonnormal distributions. Psychol. Rep. 87, 1101–1114 (2000)
Hotelling, H., Pabst, M.R.: Rank correlation and tests of significance involving no assumption of normality. Ann. Math. Stat. 7, 29–43 (1936)
Johnston, J.E., Berry, K.J., Mielke, P.W.: Permutation tests: precision in estimating probability values. Percept. Motor Skill. 105, 915–920 (2007)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Berry, K.J., Johnston, J.E., Mielke, P.W. (2019). Correlation and Regression. In: A Primer of Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-030-20933-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-20933-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20932-2
Online ISBN: 978-3-030-20933-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)