41. The hypothesis of symmetry in a square two-way contingency table arises when one of the responses AI " ' " Au is observed for each of N subjects on two occasions (e.g. before and after some intervention). If nij is the number of subjects whose responses on the two occasions are (A ;, A), the joint distribution of the n;j is given by (59) with a =

b. The hypothesis H of symmetry states that Pi} = Pji for all i , j, that is, that the intervention has not changed the probabilities. Determine the likelihood-ratio statistic for testing H, and specify its asymptotic distribution. [Bowker (1948).]