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In statistics, the multinomial test is the 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 {\displaystyle ~N~} items each of which has been observed to fall into one of k {\displaystyle k} categories. It is possible to define x = {\displaystyle ~\mathbf {x} =~} as the observed numbers of items in each cell. Hence ∑ i = 1 k x i = N . {\displaystyle ~\sum _{i=1}^{k}x_{i}=N~.}
Next, defining a vector of parameters H 0 : π = , {\displaystyle ~H_{0}:{\boldsymbol {\pi }}=~,} where: ∑ i = 1 k π i = 1 . {\displaystyle ~\sum _{i=1}^{k}\pi _{i}=1~.} These are the parameter values under the null hypothesis.
The exact probability of the observed configuration x {\displaystyle ~\mathbf {x} ~} under the null hypothesis is given by