In statistics, a **categorical variable** (also called **qualitative variable**) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property.^{ [1] } In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a **level**. The probability distribution associated with a random categorical variable is called a categorical distribution.

- Examples of categorical variables
- Notation
- Number of possible values
- Categorical variables and regression
- Dummy coding
- Effects coding
- Contrast coding
- Nonsense coding
- Embeddings
- Interactions
- See also
- References
- Further reading

**Categorical data** is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cross tabulations, or from observations of quantitative data grouped within given intervals. Often, purely categorical data are summarised in the form of a contingency table. However, particularly when considering data analysis, it is common to use the term "categorical data" to apply to data sets that, while containing some categorical variables, may also contain non-categorical variables.

A categorical variable that can take on exactly two values is termed a * binary variable * or a **dichotomous variable**; an important special case is the Bernoulli variable. Categorical variables with more than two possible values are called **polytomous variables**; categorical variables are often assumed to be polytomous unless otherwise specified. Discretization is treating continuous data as if it were categorical. Dichotomization is treating continuous data or polytomous variables as if they were binary variables. Regression analysis often treats category membership with one or more quantitative dummy variables.

Examples of values that might be represented in a categorical variable:

- The roll of a six-sided die: possible outcomes are 1,2,3,4,5, or 6.
- Demographic information of a population: gender, disease status.
- The blood type of a person: A, B, AB or O.
- The political party that a voter might vote for, e. g.
*Green Party*,*Christian Democrat*,*Social Democrat*, etc. - The type of a rock: igneous, sedimentary or metamorphic.
- The identity of a particular word (e.g., in a language model): One of
*V*possible choices, for a vocabulary of size*V*.

For ease in statistical processing, categorical variables may be assigned numeric indices, e.g. 1 through *K* for a *K*-way categorical variable (i.e. a variable that can express exactly *K* possible values). In general, however, the numbers are arbitrary, and have no significance beyond simply providing a convenient label for a particular value. In other words, the values in a categorical variable exist on a nominal scale: they each represent a logically separate concept, cannot necessarily be meaningfully ordered, and cannot be otherwise manipulated as numbers could be. Instead, valid operations are equivalence, set membership, and other set-related operations.

As a result, the central tendency of a set of categorical variables is given by its mode; neither the mean nor the median can be defined. As an example, given a set of people, we can consider the set of categorical variables corresponding to their last names. We can consider operations such as equivalence (whether two people have the same last name), set membership (whether a person has a name in a given list), counting (how many people have a given last name), or finding the mode (which name occurs most often). However, we cannot meaningfully compute the "sum" of Smith + Johnson, or ask whether Smith is "less than" or "greater than" Johnson. As a result, we cannot meaningfully ask what the "average name" (the mean) or the "middle-most name" (the median) is in a set of names.

Note that this ignores the concept of alphabetical order, which is a property that is not inherent in the names themselves, but in the way we construct the labels. For example, if we write the names in Cyrillic and consider the Cyrillic ordering of letters, we might get a different result of evaluating "Smith < Johnson" than if we write the names in the standard Latin alphabet; and if we write the names in Chinese characters, we cannot meaningfully evaluate "Smith < Johnson" at all, because no consistent ordering is defined for such characters. However, if we do consider the names as written, e.g., in the Latin alphabet, and define an ordering corresponding to standard alphabetical order, then we have effectively converted them into ordinal variables defined on an ordinal scale.

Categorical random variables are normally described statistically by a categorical distribution, which allows an arbitrary *K*-way categorical variable to be expressed with separate probabilities specified for each of the *K* possible outcomes. Such multiple-category categorical variables are often analyzed using a multinomial distribution, which counts the frequency of each possible combination of numbers of occurrences of the various categories. Regression analysis on categorical outcomes is accomplished through multinomial logistic regression, multinomial probit or a related type of discrete choice model.

Categorical variables that have only two possible outcomes (e.g., "yes" vs. "no" or "success" vs. "failure") are known as *binary variables* (or *Bernoulli variables*). Because of their importance, these variables are often considered a separate category, with a separate distribution (the Bernoulli distribution) and separate regression models (logistic regression, probit regression, etc.). As a result, the term "categorical variable" is often reserved for cases with 3 or more outcomes, sometimes termed a *multi-way* variable in opposition to a binary variable.

It is also possible to consider categorical variables where the number of categories is not fixed in advance. As an example, for a categorical variable describing a particular word, we might not know in advance the size of the vocabulary, and we would like to allow for the possibility of encountering words that we haven't already seen. Standard statistical models, such as those involving the categorical distribution and multinomial logistic regression, assume that the number of categories is known in advance, and changing the number of categories on the fly is tricky. In such cases, more advanced techniques must be used. An example is the Dirichlet process, which falls in the realm of nonparametric statistics. In such a case, it is logically assumed that an infinite number of categories exist, but at any one time most of them (in fact, all but a finite number) have never been seen. All formulas are phrased in terms of the number of categories actually seen so far rather than the (infinite) total number of potential categories in existence, and methods are created for incremental updating of statistical distributions, including adding "new" categories.

Categorical variables represent a qualitative method of scoring data (i.e. represents categories or group membership). These can be included as independent variables in a regression analysis or as dependent variables in logistic regression or probit regression, but must be converted to quantitative data in order to be able to analyze the data. One does so through the use of coding systems. Analyses are conducted such that only *g* -1 (*g* being the number of groups) are coded. This minimizes redundancy while still representing the complete data set as no additional information would be gained from coding the total *g* groups: for example, when coding gender (where *g* = 2: male and female), if we only code females everyone left over would necessarily be males. In general, the group that one does not code for is the group of least interest.^{ [2] }

There are three main coding systems typically used in the analysis of categorical variables in regression: dummy coding, effects coding, and contrast coding. The regression equation takes the form of * Y = bX + a*, where

Dummy coding is used when there is a control or comparison group in mind. One is therefore analyzing the data of one group in relation to the comparison group: *a* represents the mean of the control group and *b* is the difference between the mean of the experimental group and the mean of the control group. It is suggested that three criteria be met for specifying a suitable control group: the group should be a well-established group (e.g. should not be an “other” category), there should be a logical reason for selecting this group as a comparison (e.g. the group is anticipated to score highest on the dependent variable), and finally, the group's sample size should be substantive and not small compared to the other groups.^{ [3] }

In dummy coding, the reference group is assigned a value of 0 for each code variable, the group of interest for comparison to the reference group is assigned a value of 1 for its specified code variable, while all other groups are assigned 0 for that particular code variable.^{ [2] }

The *b* values should be interpreted such that the experimental group is being compared against the control group. Therefore, yielding a negative b value would entail the experimental group have scored less than the control group on the dependent variable. To illustrate this, suppose that we are measuring optimism among several nationalities and we have decided that French people would serve as a useful control. If we are comparing them against Italians, and we observe a negative *b* value, this would suggest Italians obtain lower optimism scores on average.

The following table is an example of dummy coding with *French* as the control group and C1, C2, and C3 respectively being the codes for *Italian*, *German*, and *Other* (neither French nor Italian nor German):

Nationality | C1 | C2 | C3 |

French | 0 | 0 | 0 |

Italian | 1 | 0 | 0 |

German | 0 | 1 | 0 |

Other | 0 | 0 | 1 |

In the effects coding system, data are analyzed through comparing one group to all other groups. Unlike dummy coding, there is no control group. Rather, the comparison is being made at the mean of all groups combined (*a* is now the grand mean). Therefore, one is not looking for data in relation to another group but rather, one is seeking data in relation to the grand mean.^{ [2] }

Effects coding can either be weighted or unweighted. Weighted effects coding is simply calculating a weighted grand mean, thus taking into account the sample size in each variable. This is most appropriate in situations where the sample is representative of the population in question. Unweighted effects coding is most appropriate in situations where differences in sample size are the result of incidental factors. The interpretation of *b* is different for each: in unweighted effects coding *b* is the difference between the mean of the experimental group and the grand mean, whereas in the weighted situation it is the mean of the experimental group minus the weighted grand mean.^{ [2] }

In effects coding, we code the group of interest with a 1, just as we would for dummy coding. The principal difference is that we code −1 for the group we are least interested in. Since we continue to use a *g* - 1 coding scheme, it is in fact the −1 coded group that will not produce data, hence the fact that we are least interested in that group. A code of 0 is assigned to all other groups.

The *b* values should be interpreted such that the experimental group is being compared against the mean of all groups combined (or weighted grand mean in the case of weighted effects coding). Therefore, yielding a negative *b* value would entail the coded group as having scored less than the mean of all groups on the dependent variable. Using our previous example of optimism scores among nationalities, if the group of interest is Italians, observing a negative *b* value suggest they obtain a lower optimism score.

The following table is an example of effects coding with *Other* as the group of least interest.

Nationality | C1 | C2 | C3 |

French | 0 | 0 | 1 |

Italian | 1 | 0 | 0 |

German | 0 | 1 | 0 |

Other | −1 | −1 | −1 |

The contrast coding system allows a researcher to directly ask specific questions. Rather than having the coding system dictate the comparison being made (i.e., against a control group as in dummy coding, or against all groups as in effects coding) one can design a unique comparison catering to one's specific research question. This tailored hypothesis is generally based on previous theory and/or research. The hypotheses proposed are generally as follows: first, there is the central hypothesis which postulates a large difference between two sets of groups; the second hypothesis suggests that within each set, the differences among the groups are small. Through its a priori focused hypotheses, contrast coding may yield an increase in power of the statistical test when compared with the less directed previous coding systems.^{ [2] }

Certain differences emerge when we compare our a priori coefficients between ANOVA and regression. Unlike when used in ANOVA, where it is at the researcher's discretion whether they choose coefficient values that are either orthogonal or non-orthogonal, in regression, it is essential that the coefficient values assigned in contrast coding be orthogonal. Furthermore, in regression, coefficient values must be either in fractional or decimal form. They cannot take on interval values.

The construction of contrast codes is restricted by three rules:

- The sum of the contrast coefficients per each code variable must equal zero.
- The difference between the sum of the positive coefficients and the sum of the negative coefficients should equal 1.
- Coded variables should be orthogonal.
^{ [2] }

Violating rule 2 produces accurate *R*^{2} and *F* values, indicating that we would reach the same conclusions about whether or not there is a significant difference; however, we can no longer interpret the *b* values as a mean difference.

To illustrate the construction of contrast codes consider the following table. Coefficients were chosen to illustrate our a priori hypotheses: Hypothesis 1: French and Italian persons will score higher on optimism than Germans (French = +0.33, Italian = +0.33, German = −0.66). This is illustrated through assigning the same coefficient to the French and Italian categories and a different one to the Germans. The signs assigned indicate the direction of the relationship (hence giving Germans a negative sign is indicative of their lower hypothesized optimism scores). Hypothesis 2: French and Italians are expected to differ on their optimism scores (French = +0.50, Italian = −0.50, German = 0). Here, assigning a zero value to Germans demonstrates their non-inclusion in the analysis of this hypothesis. Again, the signs assigned are indicative of the proposed relationship.

Nationality | C1 | C2 |

French | +0.33 | +0.50 |

Italian | +0.33 | −0.50 |

German | −0.66 | 0 |

Nonsense coding occurs when one uses arbitrary values in place of the designated “0”s “1”s and “-1”s seen in the previous coding systems. Although it produces correct mean values for the variables, the use of nonsense coding is not recommended as it will lead to uninterpretable statistical results.^{ [2] }

*Embeddings* are codings of categorical values into high-dimensional real-valued (sometimes complex-valued) vector spaces, usually in such a way that ‘similar’ values are assigned ‘similar’ vectors, or with respect to some other kind of criterion making the vectors useful for the respective application. A common special case are word embeddings, where the possible values of the categorical variable are the words in a language and words with similar meanings are to be assigned similar vectors.

An interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive. Interactions may arise with categorical variables in two ways: either categorical by categorical variable interactions, or categorical by continuous variable interactions.

This type of interaction arises when we have two categorical variables. In order to probe this type of interaction, one would code using the system that addresses the researcher's hypothesis most appropriately. The product of the codes yields the interaction. One may then calculate the *b* value and determine whether the interaction is significant.^{ [2] }

Simple slopes analysis is a common post hoc test used in regression which is similar to the simple effects analysis in ANOVA, used to analyze interactions. In this test, we are examining the simple slopes of one independent variable at specific values of the other independent variable. Such a test is not limited to use with continuous variables, but may also be employed when the independent variable is categorical. We cannot simply choose values to probe the interaction as we would in the continuous variable case because of the nominal nature of the data (i.e., in the continuous case, one could analyze the data at high, moderate, and low levels assigning 1 standard deviation above the mean, at the mean, and at one standard deviation below the mean respectively). In our categorical case we would use a simple regression equation for each group to investigate the simple slopes. It is common practice to standardize or center variables to make the data more interpretable in simple slopes analysis; however, categorical variables should never be standardized or centered. This test can be used with all coding systems.^{ [2] }

**Analysis of variance** (**ANOVA**) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the *t*-test beyond two means.

In statistics, the **logistic model** is used to model the probability of a certain class or event existing such as pass/fail, win/lose, alive/dead or healthy/sick. This can be extended to model several classes of events such as determining whether an image contains a cat, dog, lion, etc. Each object being detected in the image would be assigned a probability between 0 and 1, with a sum of one.

In statistics and econometrics, particularly in regression analysis, a **dummy variable** is one that takes only the value 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome. They can be thought of as numeric stand-ins for qualitative facts in a regression model, sorting data into mutually exclusive categories.

An ** F-test** is any statistical test in which the test statistic has an

**Analysis of covariance** (**ANCOVA**) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

In statistics, an **interaction** may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.

**Binary data** is data whose unit can take on only two possible states, traditionally labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra.

In statistics, **multicollinearity** is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. In this situation, the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. Multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set; it only affects calculations regarding individual predictors. That is, a multivariate regression model with collinear predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others.

In statistics, **classification** is the problem of identifying which of a set of categories (sub-populations) an observation belongs to. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagnosis to a given patient based on observed characteristics of the patient.

In statistics, **multinomial logistic regression** is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables.

**Multilevel models** are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

In statistics, particularly in analysis of variance and linear regression, a **contrast** is a linear combination of variables whose coefficients add up to zero, allowing comparison of different treatments.

**Omnibus tests** are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

**Choice modelling** attempts to model the decision process of an individual or segment via revealed preferences or stated preferences made in a particular context or contexts. Typically, it attempts to use discrete choices in order to infer positions of the items on some relevant latent scale. Indeed many alternative models exist in econometrics, marketing, sociometrics and other fields, including utility maximization, optimization applied to consumer theory, and a plethora of other identification strategies which may be more or less accurate depending on the data, sample, hypothesis and the particular decision being modelled. In addition, choice modelling is regarded as the most suitable method for estimating consumers' willingness to pay for quality improvements in multiple dimensions.

In statistics and regression analysis, **moderation** occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the **moderator variable** or simply the **moderator**. The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical or quantitative variable that affects the direction and/or strength of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.

**Bivariate analysis** is one of the simplest forms of quantitative (statistical) analysis. It involves the analysis of two variables, for the purpose of determining the empirical relationship between them.

In statistics and in machine learning, a **linear predictor function** is a linear function of a set of coefficients and explanatory variables, whose value is used to predict the outcome of a dependent variable. This sort of function usually comes in linear regression, where the coefficients are called regression coefficients. However, they also occur in various types of linear classifiers, as well as in various other models, such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights".

**Log-linear analysis** is a technique used in statistics to examine the relationship between more than two categorical variables. The technique is used for both hypothesis testing and model building. In both these uses, models are tested to find the most parsimonious model that best accounts for the variance in the observed frequencies.

In statistics, **linear regression** is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called *simple linear regression*; for more than one, the process is called **multiple linear regression**. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

- ↑ Yates, Daniel S.; Moore, David S.; Starnes, Daren S. (2003).
*The Practice of Statistics*(2nd ed.). New York: Freeman. ISBN 978-0-7167-4773-4. Archived from the original on 2005-02-09. Retrieved 2014-09-28. - 1 2 3 4 5 6 7 8 9 10 Cohen, J.; Cohen, P.; West, S. G.; Aiken, L. S. (2003).
*Applied multiple regression/correlation analysis for the behavioural sciences (3rd ed.)*. New York, NY: Routledge. - ↑ Hardy, Melissa (1993).
*Regression with dummy variables*. Newbury Park, CA: Sage.

- Andersen, Erling B. 1980.
*Discrete Statistical Models with Social Science Applications*. North Holland, 1980. - Bishop, Y. M. M.; Fienberg, S. E.; Holland, P. W. (1975).
*Discrete Multivariate Analysis: Theory and Practice*. MIT Press. ISBN 978-0-262-02113-5. MR 0381130. - Christensen, Ronald (1997).
*Log-linear models and logistic regression*. Springer Texts in Statistics (Second ed.). New York: Springer-Verlag. pp. xvi+483. ISBN 0-387-98247-7. MR 1633357. - Friendly, Michael.
*Visualizing categorical data.*SAS Institute, 2000. - Lauritzen, Steffen L. (2002) [1979].
*Lectures on Contingency Tables*(PDF) (updated electronic version of the (University of Aalborg) 3rd (1989) ed.). - NIST/SEMATEK (2008)
*Handbook of Statistical Methods*

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