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.
items each of which has been observed to fall into one of
as the observed numbers of items in each cell.
Next, defining a vector of parameters
These are the parameter values under the null hypothesis.
The exact probability of the observed configuration
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 where the sum ranges over all outcomes as likely as, or less likely than, that observed.
In practice this becomes computationally onerous 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 can be defined under which each value
is replaced by its maximum likelihood estimate
The exact probability of the observed configuration
under the alternative hypothesis is given by The natural logarithm of the likelihood ratio,
is then the statistic for the likelihood ratio test (The factor
is chosen to make the statistic asymptotically chi-squared distributed, for convenient comparison to a familiar statistic commonly used for the same application.)
If the null hypothesis is true, then as
However it has long been known (e.g. Lawley[2]) that for finite sample sizes, the moments of
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 are a function of
Williams[3] showed that the first moment can be matched as far as
if the test statistic is divided by a factor given by In the special case where the null hypothesis is that all the values
(i.e. it stipulates a uniform distribution), this simplifies to Subsequently, Smith et al.[4] derived a dividing factor which matches the first moment as far as
For the case of equal values of
is the expected number of cases in category
This statistic also converges to a chi-squared distribution with
degrees of freedom when the null hypothesis is true but does so from below, as it were, rather than from above as
does, so may be preferable to the uncorrected version of