Testing for Symmetry.

Consider a multinomial sample arranged in an I ×I table. In square tables with similar categories for the two factors, it is sometimes of interest to test H0 : pij = pji for all i and j.

(a) Give a procedure for testing this hypothesis based on testing equality of probabilities (homogeneity of proportions) in a 2 × I(I − 1)/2 table. If you think of the I × I table as a matrix, the rows indicate whether a cell is above or below the diagonal. The columns are corresponding off diagonal pairs. Illustrate the test for a 4 × 4 table.

(b) Give a justification for the procedure in terms of a (conditional)

sampling model.

(c) The data in Table 2.9 were given by Fienberg (1980), Yule (1900), and earlier by Galton. They report the relative heights of 205 married couples.

TABLE 2.9. Heights of Married Couples Wife Husband Tall Medium Short Tall 18 28 14 Medium 20 51 28 Short 12 25 9 Test for symmetry and do any other appropriate analysis for these data.
Do the data display symmetry?