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AbstractLet (Y, Xl,…, XK) be a random vector distributed according to a multivariate normal distribution where Xl,…, XK are considered as predictor variables and y is the predictand. Let ri, and Ri denote the population and sample correlation coefficients, respectively, between Y and Xi. The population correlation coefficient ri is a measure of the predictive power of Xi. The author has derived the joint distribution of Rl,…, RK and its asymptotic property. The given result is useful in the problem of selecting the most important predictor variable corresponding to the largest absolute value of ri.

Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open‐source software package JASP.

Abstract  In interpreting the product moment correlation coefficient we can take advantage of the fact that functions of it exist which admit interpretation as probabilities of error in certain situations.

Non-zero correlation coefficients have non-normal distributions, affecting both means and standard deviations. Previous research suggests that z transformation may effectively correct mean bias for N's less than 30. In this study, simulations with small (20 and 30) and large (50 and 100) N's found that mean bias adjustments for larger N's are seldom needed. However, z transformations improved confidence intervals even for N = 100. The improvement was not in the estimated standard errors so much as in the asymmetrical CI's estimates based upon the z transformation. The resulting observed probabilities were generally accurate to within 1 point in the first non-zero digit. These issues are an order of magnitude less important for accuracy than design issues influencing the accuracy of the results, such as reliability, restriction of range, and N. DOI:10.2458/azu_jmmss_v1i2_gorsuch