Some symmetric, invariant measures of multivariate association

Abstract

canonical correlation, multivariate analysis, association, redundancy,

On association in regression: the coefficient of determination revisited

“…This measure coincides with γ 1 in Cramer's and Nicewanders review ( [7]: 1979) of multivariate measures of association under the assumption of a joint multivariate Gaussian distribution. Based on the decomposition of covariance various measures have been derived which focus on ratios of (re-scaled) between variances to (re-scaled) total variances.…”

“…An issue we finally address is the classification of measures of multivariate dependence into (symmetric) measures of association and (asymmetric) measures of redundancy. The measures discussed by Cramer and Nicewander ( [7]: 1979) under the assumption of joint Normality are all -like ours -measures of dependence between X and Y . They can be rewritten in information theoretic terms…”

“…Based on the decomposition of covariance various measures have been derived which focus on ratios of (re-scaled) between variances to (re-scaled) total variances. For instance the re-scaled variance ratio (called γ 3 by Cramer and Nicewander ( [7]: 1979)…”

“…Although measures of association and dependence between variables X and Y have been extensively discussed, especially in psychometrics (e.g. [7]: Cramer and Nicewander, 1979; [33]: Särndal, 1974) they have hardly been evaluated taking into account a regression model ( [20]: Kent, 1983). Based on measures of association the determination of Y by X(= x), can be interpreted in two ways: determination of the conditional density p(y|x), capturing the discriminatory power of X(= x) or determination of the (range of) values of Y given X(= x), capturing the explanatory power of X(= x).…”

Measures of Effect Size for Comparative Studies: Applications, Interpretations, and Limitations

“…In addition to eta and omega squared three additional multivariate proportion of variance effect sizes have been recommended (Cramer & Nicewander, 1979;Serlin, 1982) based on different multivariate test criteria (Hotelling-Lawley trace and Bartlet-Pillai trace). These alternative multivariate effect size measures are briefly discussed by Huberty (1994, pp.…”