Categorical Data and the Multinomial Experiment

The Multinomial Distribution

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For $n$ independent trials, each of which leads to a success for exactly one of $k$ categories and with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

The binomial distribution is the probability distribution of the number of successes for one of just two categories in $n$ independent Bernoulli trials, with the same probability of success on each trial. In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number $k$ of possible outcomes, with probabilities $p_1, \cdots , p_k$ (so that $p_i \geq 0$ for $i = 1, \cdots, k$ and the sum is $1$), and there are $n$ independent trials. Then if the random variables X_i indicate the number of times outcome number $i$ is observed over the $n$ trials, the vector $X = (X_1, \cdots , X_k)$ follows a multinomial distribution with parameters $n$ and $p$, where $p = (p_1, \cdots , p_k)$.

The Multinomial Experiment

In statistics, the multinomial experiment is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values. It is used for categorical data. It is really an extension of the binomial experiment, where there were only two categories: success or failure. One example of a multinomial experiment is asking which of six candidates a voter preferred in an election.

Properties for the Multinomial Experiment

• The experiment consists of $n$ identical trials.
• There are $k$ possible outcomes for each trial. These outcomes are sometimes called classes, categories, or cells.
• The probabilities of the $k$ outcomes, denoted by $p_1$, $p_2$, $\cdots$, $p_k$, remain the same from trial to trial, and they sum to one.
• The trials are independent.
• The random variables of interest are the cell counts $n_1$, $n_2$, $\cdots$, $n_k$, which refer to the number of observations that fall into each of the $k$ categories.