Multinomial test

In statistics, the multinomial test is the likelihood-ratio test of the null hypothesis that the parameters of a multinomial distribution equal specified values. It is used for categorical data; see Read and Cressie.

We begin with a sample of $N$ items each of which has been observed to fall into one of $k$ cells. We can define $\mathbf {x} =(x_{1},x_{2},\dots ,x_{k})$ as the observed numbers of items in each cell. Hence $\textstyle \sum _{i=1}^{k}x_{i}=N$.

Next, we define a vector of parameters $H_{0}:\mathbf {\pi } =(\pi _{1},\pi _{2},\dots ,\pi _{k})$, where $\textstyle \sum _{i=1}^{k}\pi _{i}=1$. These are the parameter values under the null hypothesis.

The exact probability of the observed configuration $\mathbf {x}$ under the null hypothesis is given by

$\Pr(\mathbf {x_{0}} )=n!\prod _{i=1}^{k}{\frac {\pi _{i}^{x_{i}}}{x_{i}!}}.$.

Under the alternative hypothesis, each value $\pi _{i}$ is replaced by its maximum likelihood estimate $p_{i}=x_{i}/N$ and the exact probability of the observed configuration $\mathbf {x}$ under the alternative hypothesis is given by

$\Pr(\mathbf {x_{A}} )=n!\prod _{i=1}^{k}{\frac {p_{i}^{x_{i}}}{x_{i}!}}.$

The natural logarithm of the ratio between these two probabilities multiplied by -2 is then the log likelihood ratio statistic

$-2\ln(LR)=\textstyle -2\sum _{i=1}^{k}x_{i}\ln(\pi _{i}/p_{i})$.

As N increases, the distribution of $-2\ln(LR)$ converges to that of chisquare with k-1 degrees of freedom when the null hypothesis is true.