Multinomial test

Article summary

Multinomial test is the statistical test of the null hypothesis that the parameters of a multinomial distribution equal specified values; it is used for categorical data. Beginning with a sample of ~ N ~ items each of which has been observed to fall into one of k categories. It is possible to define ~ \mathbf{x} = (x_1, x_2, \dots, x_k) ~ as the observed numbers of items in each cell. Hence ~ \sum_{i=1}^k x_{i} = N ~. Next, defining a vector of parameters ~ H_0: \boldsymbol{\pi} = (\pi_{1}, \pi_{2}, \ldots, \pi_{k}) ~, where: ~ \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 : ~ \operatorname\mathbb{P}\left( \mathbf{x}\right )_0 = N! \, \prod_{i=1}^k \frac{\pi_{i}^{x_i}}{x_i!} ~. The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an exact test, this is calculated as : ~ p_\mathcal{[sig]} = \sum_{ \mathbf{y} \, : \; \operatorname{\mathbb{P}}\left( \mathbf{y} \right) \, \le \, \operatorname{\mathbb{P}}\left( \mathbf{x} \right)_0 } \operatorname\mathbb{P}\left( \mathbf{y}\right) ~ where the sum ranges over all outcomes as likely as, or less likely than, that observed.…

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