Goodness of fit of an assigned set of scores for the analysis of association in a contingency table

The problem of association between two attributes in ap×q contingency table can be looked upon as the problem of relationship between two vector variablesx andy. If there is only one true non-zero canonical correlation betweenx andy, the association between the two attributes is of rank 1 and in this case, one set of scores is adequate to describe the association completely; these scores are nothing but the coefficients in the canonical variates corresponding to the true non-zero canonical correlation. Given a set of hypothetical scoresα ₁,α₁,⋯,α_p for the rows, one is interested in testing their goodness of fit. Tests for this are suggested in this paper. For obtaining these tests, a preliminary result about direction and collinearity factors in discriminant analysis, whenS irrelevant variables are eliminated, is needed. This is derived in part one of this paper. This research was sponsored by the Office of Naval Research, Contract No. N00014-68-A-0515, Project No. NR042-260. Reproduction in whole or in part is permitted for any purpose of the United States Government.