Multinomial test

Based on the definition provided by Read and Cressie:

We begin with a system of n items observed to occur in k cells. We can define $\mathbf {x} =(x_{1},x_{2},...,x_{k})$ as the observed numbers of items in each cell.

Next, we define a hypothetical distribution $H_{0}:\pi _{0}=(\pi _{01},\pi _{02},...,\pi _{0k})$, where $\textstyle \sum _{i=1}^{k}\pi _{0i}=1$. This is the expected distribution under the null hypothesis.

The exact probability of observing the observed configuration x is given by

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

To test the significance of whether an observed outcome x* came from a population distributed according to $H_{0}$, you must first compute the probability of all possible outcomes x, then compute the cumulative probability of observing x* plus all other outcomes that are equal or less probable than that of observing x*. This is the maximum likelihood test. We can reject H₀ if the cumulative probability is less than or equal to our chosen significance level.