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.

These are the parameter values under the null hypothesis.

under the null hypothesis is given by

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

increase so it is probably only worth using exact tests for small samples. For larger samples, asymptotic approximations are accurate enough and easier to calculate.

One of these approximations is the likelihood ratio. An alternative hypothesis under the alternative hypothesis is given by

The natural logarithm is then the statistic for the likelihood ratio test

is chosen to make the statistic asymptotically chi-squared distributed, for convenient comparison to a familiar statistic commonly used for the same application.)

converges to that of chi-squared degrees of freedom. However it has long been known (e.g. Lawley are greater than those of chi-squared, thus inflating the probability of type I errors (false positives). The difference between the moments of chi-squared and those of the test statistic Williams if the test statistic is divided by a factor given by

(i.e. it stipulates a uniform distribution), this simplifies to

Subsequently, Smith et al. this factor is

The null hypothesis can also be tested by using Pearson's chi-squared test

for small samples.

References