Proof: Multinomial test
Theorem: Let $y = [y_1, \ldots, y_k]$ be the number of observations in $k$ categories resulting from $n$ independent trials with unknown category probabilities $p = [p_1, \ldots, p_k]$, such that $y$ follows a multinomial distribution:
\[\label{eq:Mult} y \sim \mathrm{Mult}(n,p) \; .\]Then, the null hypothesis
\[\label{eq:mult-test-h0} H_0: \; p = p_0 = [p_{01}, \ldots, p_{0k}]\]is rejected at significance level $\alpha$, if
\[\label{eq:mult-test-rej} \mathrm{Pr}_\mathrm{sig} = \sum_{x: \; \mathrm{Pr}_0(x) \leq \mathrm{Pr}_0(y)} \mathrm{Pr}_0(x) < \alpha\]where $\mathrm{Pr}_0(x)$ is the probability of observing the numbers of occurences $x = [x_1, \ldots, x_k]$ under the null hypothesis:
\[\label{eq:mult-test-prob} \mathrm{Pr}_0(x) = n! \prod_{j=1}^k \frac{p_{0j}^{x_j}}{x_j!} \; .\]Proof: The alternative hypothesis relative to $H_0$ is
\[\label{eq:bin-test-h1} H_1: \; p_j \neq p_{0j} \quad \text{for at least one} \quad j = 1, \ldots, k \; .\]We can use $y$ as a test statistic. Its sampling distribution is given by \eqref{eq:Mult}. The probability mass function (PMF) of the test statistic under the null hypothesis is thus equal to the probability mass function of the multionomial distribution with category probabilities $p_0$:
\[\label{eq:y-pmf} \mathrm{Pr}(y = x \vert H_0) = \mathrm{Mult}(x; n, p_0) = {n \choose {x_1, \ldots, x_k}} \, \prod_{j=1}^k {p_j}^{x_j} \; .\]The multinomial coefficient in this equation is equal to
\[\label{eq:mult-coeff} {n \choose {k_1, \ldots, k_m}} = \frac{n!}{k_1! \cdot \ldots \cdot k_m!} \; ,\]such that the probability of observing the counts $y$, given $H_0$, is
\[\label{eq:Pr0-y} \mathrm{Pr}(y \vert H_0) = n! \prod_{j=1}^k \frac{p_{0i}^{y_j}}{y_j!} \; .\]The probability of observing any other set of counts $x$, given $H_0$, is
\[\label{eq:Pr0-x} \mathrm{Pr}(x \vert H_0) = n! \prod_{j=1}^k \frac{p_{0i}^{x_j}}{x_j!} \; .\]The p-value is the probability of observing a value of the test statistic that is as extreme or more extreme then the actually observed test statistic. Any set of counts $x$ might be considered as extreme or more extreme than the actually observed counts $y$, if the former is equally probable or less probably than the latter according to the PMF:
\[\label{eq:mult-test-cond} \mathrm{Pr}_0(x) \leq \mathrm{Pr}_0(y) \; .\]Thus, the p-value for the data in \eqref{eq:Mult} is equal to
\[\label{eq:mult-test-p} p = \sum_{x: \; \mathrm{Pr}_0(x) \leq \mathrm{Pr}_0(y)} \mathrm{Pr}_0(x)\]and the null hypothesis in \eqref{eq:mult-test-h0} is rejected, if
\[\label{eq:mult-test-rej-qed} p < \alpha \; .\]- Wikipedia (2023): "Multinomial test"; in: Wikipedia, the free encyclopedia, retrieved on 2023-12-23; URL: https://en.wikipedia.org/wiki/Multinomial_test.
Metadata: ID: P430 | shortcut: mult-test | author: JoramSoch | date: 2023-12-23, 21:41.