The usual Pearson chi-square statistic for testing for independence in a two-way contingency table is where the Y ij are the observed frequencies in the table, and the µ ij are the estimated expected...

The usual Pearson chi-square statistic for testing for independence in a two-way contingency table is

where the Y_ij
are the observed frequencies in the table, and the µ_ij
are the estimated expected frequencies under independence. The estimated expected frequencies can be computed from the maximum-likelihood estimates for the loglinear model of independence, or they can be computed directly as µ_ij
= Y_i+Y_+j/n. The likelihood-ratio statistic for testing for independence can also be computed from the estimated expected counts as

Both test statistics have (r – 1) (c – 1) degrees of freedom. The two tests are asymptotically equivalent and usually produce similar results. Applying these formulas to the two-way table for voter turnout and intensity of partisan preference in Table 15.4 (page 435), compute both test statistics, verifying that the direct formula for G²
₀
produces the same result as given in the text. Do the Pearson and likelihood-ratio tests agree?