Skip to main content icon/video/no-internet
icon backReturn to Entries

Categorical Data Analysis

Categorical data analysis is a field of statistical analysis devoted to the analysis of dependent variables that are categorical in nature. Development of analytic techniques for inference utilizing categorical random variables began around 1900 when Karl Pearson introduced the chi-square statistic (χ2). From this first introduction of tests of two-way contingency tables, the field has developed to include not only analyses of contingency tables but also more sophisticated analytic techniques such as the generalized linear mixed model. This entry defines categorical variables, outlines the most frequently utilized probability distributions for categorical variables, describes the most commonly used statistical analyses in the field of categorical data analysis, and discusses estimation methods for parameter estimates.

Categorical Variables

Categorical variables are a class of random variables whose outcomes fall into discrete categories as opposed to a continuous range of numbers. Discrete categorical variables can be categorized based on their level of measurement, either nominal or ordinal. Nominal categorical variables contain categories of responses that have an arbitrary ordering. That is, variables measured on this scale cannot be ranked or ordered based on their observed outcomes. The categories are simply placeholders for the outcomes. As an example, gender is measured on a nominal scale, as the following two outcomes, male and female, cannot be ordered in a meaningful way. Ordinal categorical variables, in contrast, contain categories of responses that have a natural ordering to them. The observed outcomes can be ranked or ordered based on this natural ordering, which provides meaning to the categories. As an example, age categories are measured using an ordinal scale, as the following two age categories, 20–29 and 30–39, have a meaningful order to them. The second category, 30–39, represents subjects who are older than those in the first category, 20–29.

Probability Distributions

The use of inferential statistics requires an assumption of the distributional properties of the variables of interest. The distributional assumption of the categorical dependent variable provides the theoretical distribution of responses in the population, which is the basis for the statistical analysis being performed. For categorical data, the four most common distributions utilized in inferential statistics are the binomial distribution, the multinomial distribution, the hypergeometric distribution, and the Poisson distribution.

Binomial Distribution

The binomial distribution for random variable X calculates the probability of observing the count, Y, of the number of successes in a fixed number of trials of a Bernoulli experiment. A Bernoulli experiment is a random event in which there are two outcomes that have a fixed probability of occurring. In a binomial distribution, one of those outcomes is deemed a “success.” In a total of n trails, these successes are counted and the outcome X is the frequency of occurrence of a successful outcome. For example, if the Bernoulli experiment was flipping a coin, the outcome X could be the number of heads that occur in n = 5 trials (note that X ranges from 0 to n).

Multinomial Distribution

The multinomial distribution calculates the probability of observing the counts of each category when multiple outcomes are possible. The multinomial distribution is different from the binomial distribution in that there are three or more outcomes possible in each random experiment. Rather than utilizing the probability of a success, the probability of each outcome is calculated and creates a distribution of the counts of each outcome category. This is a multivariate distribution of all of the outcomes, where each individual outcome falls into a binomial distribution (that category vs. not in that category). The probabilities are calculated based on the occurrence of each category. For example, when asked to select a new drink, the options are “Drink A,” “Drink B,” and “Drink C.” The multinomial distribution will look at the probability of the frequencies of the three outcomes simultaneously in a sample. One example would be the probability of observing the counts of 4, 3, and 3, respectively, in a sample of 10 subjects.

...

  • Loading...
locked icon

Sign in to access this content

Get a 30 day FREE TRIAL

  • Watch videos from a variety of sources bringing classroom topics to life
  • Read modern, diverse business cases
  • Explore hundreds of books and reference titles

Read next

More like this

Sage Recommends

We found other relevant content for you on other Sage platforms.

close

Have you created a personal profile? Login or create a profile so that you can save clips, playlists and searches

close
close
close Signed In to profile You are signed in as:
close
Logged in Institution Institution