Generalized Multinomial Distribution
- Models counts or probabilities for multiple categorical outcomes across a fixed number of trials.
- Lets you compute probabilities, expected values, variances, and comparisons among categories.
- More flexible than binary models because it supports more than two outcome categories.
Definition
Section titled “Definition”The generalized multinomial distribution is a statistical model used to describe the probabilities of multiple outcomes occurring in a fixed number of trials. It is a generalization of the standard multinomial distribution, which describes the probabilities of multiple outcomes in a sequence of independent and identically distributed trials.
Explanation
Section titled “Explanation”The generalized multinomial distribution models outcomes when each trial results in one of several categories and the total number of trials is fixed. It can be used to calculate probabilities for individual outcomes and for combinations or comparisons of outcomes, and to compute expected values and variances for each category. These statistics support hypothesis testing and inference about the underlying category probabilities. A key advantage is the ability to model more than two outcome categories, making it applicable in settings that require analysis of multiple categories.
Examples
Section titled “Examples”Survey example
Section titled “Survey example”A survey of 100 individuals asks each to choose one of four options: classical, pop, rock, or other. The responses can be modeled as a generalized multinomial distribution with four categories and 100 trials. The distribution can be used to calculate probabilities such as the probability that a given individual chooses classical music, the probability that more individuals choose pop music than rock music, and to compute expected values and variances for the categories to test hypotheses about preferences.
Medical study example
Section titled “Medical study example”A study with 100 individuals randomly assigned to receive either treatment A or treatment B records a binary response where 0 indicates the treatment was not effective and 1 indicates the treatment was effective. The responses can be modeled as a generalized multinomial distribution with two categories and 100 trials. The distribution can be used to calculate probabilities such as the probability that a given individual receives treatment A, the probability that more individuals receive treatment B than treatment A, and to compute expected values and variances for hypothesis testing about treatment effectiveness.
Use cases
Section titled “Use cases”- Survey analysis
- Medical research
- Other situations requiring modeling and analysis of multiple categorical outcomes
Related terms
Section titled “Related terms”- Multinomial distribution
- Binary distribution