Past exam of the statistics course of the University of Oxford / 2015 / 2 / b

The count vector has a multinomial distribution, with likelihood functionMaximize its log-likelihood using a Lagrange multiplier for the sum constraint. The equations $n_i/{\pi }_i=\lambda$ and ${\sum }_i{\pi }_i=1$ give $\lambda =n$ and ${\hat{\pi }}_i=n_i/n$. Since all counts are positive and the log-likelihood is strictly concave on the interior of the probability simplex, this is the global maximum-likelihood estimate.

Under the uniform null hypothesis, each category has expected countThe common multinomial coefficient cancels from the likelihood ratio:Taking minus twice its logarithm gives exactlyThere are $k-1$ free category probabilities, because their sum is one, and the null fixes all of them. Thus its regular asymptotic chi-squared distribution has $k-1$ statistical degrees of freedom.

For four categories the number of statistical degrees of freedom is $4-1=3$. The supplied 5% critical value is therefore $q_3=7.82$, and $7.3<7.82$. HenceThe approximate p-value is $\mathrm{Pr}({\chi }_3^2\ge 7.3)\approx 0.063$. This conclusion assumes the large-sample multinomial approximation is adequate, in particular sufficiently large expected category counts. It means there is insufficient evidence against the uniform null hypothesis, rather than establishing that the probabilities are equal.