Agresti, Categorical Data Analysis, sec 3.3.5-3.3.7, has a brief discussion of
partioning chi-square with a bunch of references including several by Goodman.
This describes some ways of breaking down the total chi-square into components
that are asymptotically chi^2(1) as sample size increases in a table of fixed
size. As Don Burrill points out, the approximate statistic based on
a single cell does not have this property unless we allow the number of
cells to get large.
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