Equivalence of Certain Chi-Squared Test Statistics

In likelihood analysis of categorized data, it is well known that within a restricted class of log-linear models the likelihood kernels for multinomial and product multinomial sampling distributions are identical. In practical terms the estimation procedure for one is appropriate for the other. There does not appear to be a widespread realization that a similar result holds for a wide class of the Grizzle, Starmer, and Koch (1969) weighted least squares techniques. In this report such a fundamental relationship is explicitly presented and illustrated through two analyses of Bartlett's (1935) data.