Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / O / 4 / Solution

Up to a parameter-independent multinomial factor, the likelihood function is $L={\prod }_{i,j}{\theta }_{ij}^{n_{ij}}$. A Lagrange multiplier for $\sum {\theta }_{ij}=1$ gives $n_{ij}/{\theta }_{ij}=\lambda$, whence $\lambda =n$ andSince all counts are positive, the log-likelihood function is strictly concave on the positive simplex and tends to minus infinity at its boundary, proving this is the unique maximum.

Under independence of random variables, ${\theta }_{ij}={\varphi }_i{\rho }_j$, so the log-likelihood function separates as ${\sum }_i{n}_{i\cdot }\log {\varphi }_i+{\sum }_j{n}_{\cdot j}\log {\rho }_j$. Applying the same constrained maximization twice gives ${\hat{\varphi }}_i=n_{i\cdot }/n$, ${\hat{\rho }}_j=n_{\cdot j}/n$, andTwice the difference between the unrestricted and null maximized log-likelihood functions isthe required likelihood-ratio test statistic. Under the null hypothesis, its asymptotic chi-squared distribution has $(r-1)(c-1)$ statistical degrees of freedom, the parameter-count difference $(rc-1)-(r+c-2)$. Here that is four. Since $25.192>q_4=18.467$, reject independence at the 0.1 percent significance level. For four statistical degrees of freedom, the asymptotic upper-tail probability is $e^{-25.192/2}(1+25.192/2)\approx 4.60\times 10^{-5}$. This establishes an association in these data; it is not a causal conclusion.