CORRELATION BETWEEN TWO VECTOR VARIABLES
Abstract
H. Ruben (1966) has suggested a simple approximate normalization for the correlation coefficient in normal samples, by representing it as the ratio of a linear combination of a standard normal variable and a chi variable to an independent chi variable and then using Fisher's approximation to a chi variable. This result is extended in this paper to a matrix, which in a sense is the correlation coefficient between two vector variables x and y. The result is then used to obtain large sample null and non-null (but in the linear case) distributions of the Hotelling-Lawley criterion and the Pillai criterion in multivariate analysis. Williams (1955) and Bartlett (1951) have derived some exact tests for the goodness of fit of a single hypothetical function to bring out adequately the entire relationship between two vectors x and y, by factorizing Wilks' lambda suitably. These factors are known as 'direction' and 'collinearity' factors, as they refer to the direction and collinearity aspects of the null hypothesis. In this paper, the other two criteria viz. the Hotelling-Lawley and Pillai criteria are partitioned into direction and collinearity parts and large sample tests corresponding to them are derived for testing the goodness of fit of an assigned function.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 04, 1969
- Accession Number
- AD0684428
Entities
People
- A. M. Kshirsagar
Organizations
- Southern Methodist University