Improved likelihood ratio tests for complete contingency tables

摘要: SUMMARY Lawley (1956) describes how asymptotic likelihood ratio tests can in general be improved by multiplying the -2 log A test statistic a multiplier chosen so that null distribution of modified is better approximated its x2 distribution. This paper applies this technique to hypotheses concerning complete contingency tables. Improved are derived for with closed form maximum estimators. An composite hypothesis against alternative obtained comparing distribution, where maximized likelihoods under two hypotheses. showed such scale factor resulting has same moments as X2 ignoring quantities order n-2, n size sample. When - 2 expressed an explicit function observations most easily found calculating directly expectation far terms n-1. These have been used area multivariate analysis. In recent years considerable progress made developing methods analyzing multidimensional tables using linear models. comprehensive review given Plackett (1 974). these models cell frequencies may regarded independent Poisson variables whose expectations hierarchical set main effect and interaction parameters. The testing goodness fit model takes S = 2EX (X/u), X ,u estimators their expectations. practice (X/,u) replaced zero when 0. Following Nelder & Wedderburn (1972), will termed deviance model. To S1 additional estimated parameters S2, iS S1-S2