Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / O / O4 / c / Solution

The generalized likelihood ratio compares the constrained maximum from part (b) with the unrestricted maximum from part (a). Taking minus twice its logarithm givesThe unrestricted $3\times 3$ model has $9-1=8$ free parameters; the independence model has $(3-1)+(3-1)=4$. Thus the asymptotic null distribution is ${\chi }_4^2$, with four degrees of freedom. The supplied value givesTherefore reject independence at the 5% level, but not at the 1% level. There is evidence of an association between the two classifications. The fitted expected counts are all about nine or larger, so the usual chi-squared approximation has no very small expected cells here.