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Further results on χ2 for matched samples

Abstract The problem of χ2 tests of a linear hypothesis H0 for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c −1) functional restrictions: F 1(p) = 0, ... , F C−1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ2 and the modified minimum χ2 (=χ*2) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).