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Categorical variable
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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.

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. Ordinal variables have a meaningful ordering, while nominal variables have no meaningful ordering.

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 categorical variables

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

  • 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.

Notation

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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.

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.

Number of possible values

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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 have not 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 and regression

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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 b is the slope and gives the weight empirically assigned to an explanator, X is the explanatory variable, and a is the Y-intercept, and these values take on different meanings based on the coding system used. The choice of coding system does not affect the F or R2 statistics. However, one chooses a coding system based on the comparison of interest since the interpretation of b values will vary.[2]

Dummy coding

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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

Effects coding

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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

Contrast coding

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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:

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

Violating rule 2 produces accurate R2 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

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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

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Embeddings are codings of categorical values into low-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.

Interactions

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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.

Categorical by categorical variable interactions

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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]

Categorical by continuous variable interactions

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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]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Categorical variable

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A categorical variable, also known as a qualitative variable, is a type of variable in that represents data through distinct categories or labels without an inherent numerical order or meaningful arithmetic operations between the categories. These variables are used to classify observations into groups based on shared characteristics, such as attributes or types, and are fundamental in descriptive and inferential for analyzing non-numeric data patterns. Unlike quantitative variables, which involve measurable numerical values with consistent intervals, categorical variables focus on grouping rather than magnitude, enabling analyses like frequency distributions and associations between groups. Categorical variables are broadly classified into two subtypes: nominal and ordinal. Nominal variables consist of categories with no implied order or , where the labels serve solely for identification, such as (male, female) or (blue, brown, green). Ordinal variables, in contrast, maintain a clear or sequence among categories, though the intervals between ranks are not necessarily equal, as seen in (high school, bachelor's degree, ) or (low, medium, high). This distinction is crucial because it influences the choice of statistical tests and visualizations, with nominal data often analyzed via chi-square tests and allowing for measures like medians. Common examples of categorical variables include race, , age groups (e.g., under 18, 18-35, over 35), and favorite flavors, which can be binned from continuous data for targeted analysis. In practice, these variables are visualized and summarized using tools like bar graphs, pie charts, and contingency tables to display frequencies or proportions, facilitating insights into relationships, such as the distribution of by hair color in a sample. For instance, in a of 20 individuals, a two-way table might reveal that 50% of redheads have eyes, highlighting categorical associations without assuming numerical differences. The role of categorical variables extends to various fields, including social sciences, , and , where they form the basis for modeling predictors like treatment types in clinical trials or user preferences in surveys. Proper handling, such as encoding for nominal variables in regression models, ensures accurate , as misclassifying them as quantitative can lead to invalid conclusions. Overall, understanding categorical variables is essential for robust , as they capture qualitative diversity that quantitative measures alone cannot address.

Definition and Types

Definition

In statistics, a variable refers to any characteristic, number, or that can be measured or counted and that varies across or units of . A categorical variable, also known as a qualitative variable, is a specific type of variable that assigns each to one of a limited, usually fixed, number of discrete categories or labels, where the categories lack inherent numerical meaning or a natural order. These categories represent distinct groups based on qualitative properties rather than measurable , enabling the classification of data into non-overlapping groupings. A defining feature of categorical variables is that their categories must be mutually exclusive, ensuring that each belongs to exactly one category without overlap, and exhaustive, meaning the set of categories encompasses all possible outcomes for the variable. This structure facilitates the analysis of associations and distributions within datasets, distinguishing categorical variables from numerical ones, which support arithmetic operations and possess intrinsic ordering. The origins of categorical variables trace back to early 20th-century statistical developments, particularly Karl Pearson's foundational work on contingency tables in 1900, which introduced methods for examining relationships between such variables through chi-squared tests. This innovation built on prior probabilistic ideas but formalized the treatment of categorical data as a core component of .

Nominal Variables

Nominal variables represent a fundamental subtype of categorical variables, characterized by categories that lack any intrinsic order, ranking, or numerical progression. These variables serve to classify observations into distinct groups based solely on qualitative differences, such as or , where one category cannot be considered inherently greater or lesser than another. Unlike other forms of categorical data, nominal variables treat all categories as equals, with no implied hierarchy or magnitude. A key characteristic of nominal variables is the equality among their categories, which precludes the application of arithmetic operations like addition or subtraction across values. This equality makes them particularly suitable for statistical tests that assess associations or between groups, such as the chi-square test of , which evaluates whether observed frequencies in a deviate significantly from expected values under a of no relationship. For instance, in analyzing survey data on preferred beverage types (e.g., , , soda), a chi-square test can determine if preferences differ by demographic group without assuming any ordering. In the typology of measurement scales proposed by S.S. Stevens, nominal measurement occupies the lowest level, serving primarily as a or naming system without quantitative implications. Stevens defined nominal scales as those permitting only the determination of equality or inequality between entities, with permissible statistics limited to mode, chi-square measures, and contingency coefficients. This foundational framework underscores that nominal data cannot support more advanced operations, such as or , distinguishing it from higher scales like ordinal or interval. The implications for of nominal variables are significant, as traditional measures of like the or are inapplicable due to the absence of numerical ordering or spacing. Instead, descriptive focuses on frequencies—the of occurrences within each category—and the mode, which identifies the most frequent category. For example, in a of blood types (A, B, AB, O), one would report the percentage distribution and highlight the most common type, rather than averaging the categories. This approach ensures that interpretations remain aligned with the qualitative nature of the data, avoiding misleading quantitative summaries.

Ordinal Variables

Ordinal variables represent a subtype of categorical variables characterized by categories that have a natural, meaningful order, but with intervals between successive categories that are not necessarily equal or quantifiable. This ordering allows for the of observations, such as classifying severity levels in assessments or degrees in surveys, without implying that the difference between adjacent categories is uniform across the scale. For instance, a intensity scale might order responses as "none," "mild," "moderate," "severe," and "extreme," where each step indicates increasing intensity, yet the psychological or physiological gap between "mild" and "moderate" may differ from that between "severe" and "extreme." In S. S. Stevens' foundational typology of measurement scales, ordinal variables occupy the second level, following nominal scales, emphasizing the ability to determine relative position or rank while prohibiting operations that assume equal spacing, such as calculating arithmetic means without qualification. A classic example is the , originally developed for attitude measurement, which typically features five or seven ordered response options from "strongly disagree" to "strongly agree," capturing subjective intensity without assuming equidistant intervals. Unlike nominal variables, which treat categories as unordered and interchangeable, ordinal variables enable directional comparisons, such as identifying whether one response is "higher" than another. Key characteristics of ordinal variables include their suitability for ranking-based analyses, where the focus is on order rather than magnitude of differences, making them ideal for non-parametric statistical tests that avoid assumptions of normality or equal intervals. The Wilcoxon rank-sum , for example, ranks all observations from two independent groups and compares the sum of ranks to assess differences in , providing a robust method for in comparative studies. This approach preserves the ordinal nature by treating categories as ranks, circumventing issues with unequal spacing that could invalidate parametric alternatives. For descriptive analysis, medians and modes serve as appropriate central tendency measures for ordinal variables, with the median indicating the middle value in an ordered and the mode highlighting the most common category; these avoid the pitfalls of assuming interval properties. Means, however, require caution as they imply equal distances between categories, potentially leading to misleading interpretations unless specific assumptions hold, such as the presence of five or more categories with roughly symmetric response thresholds. Under such conditions, may be approximated as interval for parametric methods, though this should be justified empirically to maintain validity.

Examples

Everyday Examples

Categorical variables appear frequently in daily life, where they classify observations into distinct groups using labels rather than numerical values that imply magnitude or order. For instance, serves as a classic example of a nominal categorical variable, categorizing individuals into groups such as , , , or without any inherent ranking or numerical computation between the categories. These labels simply assign qualitative distinctions to describe characteristics, allowing for grouping and comparison based on frequencies rather than arithmetic operations. Another relatable example is education level, which represents an ordinal categorical variable by ordering categories like elementary school, high school, , or , where the sequence implies progression but the differences between levels are not quantifiable numerically. Here, the variable assigns hierarchical labels to reflect relative standing without enabling direct mathematical calculations, such as addition or averaging across levels. Binary categorical variables, a special case with exactly two categories, often arise in preferences or simple choices, such as yes/no responses to questions like "Do you prefer over ?" These are frequently represented as 0 and 1 for convenience in data handling, but the core function remains labeling mutually exclusive options without numerical meaning. Nominal and ordinal types, as defined earlier, encompass these everyday applications by providing structured ways to categorize non-numeric attributes in observations.

Domain-Specific Examples

In , serves as a classic nominal categorical variable, classifying individuals into mutually exclusive groups such as A, B, AB, or O based on the . This variable is crucial for informing transfusion decisions and investigating disease associations; for instance, contingency tables have been used to analyze links between blood types and infection risks, like higher susceptibility in type A individuals compared to type O. With four categories, it exemplifies multi-category complexity, requiring methods that account for multiple levels to detect subtle associations without assuming order. In , tumor stage represents an ordinal categorical variable, categorizing cancer progression into ordered levels such as stage I (localized), II (regional spread), III (advanced regional), and IV (metastatic). This staging informs treatment planning and ; contingency tables help evaluate associations between stages and outcomes, such as survival rates post-therapy, by cross-tabulating stage groups with response categories to guide designs. The multi-level nature (often four or more stages) adds complexity, as analyses must respect the inherent ordering while handling uneven category distributions across patient cohorts. Social sciences frequently employ political affiliation as a nominal categorical variable, grouping respondents into categories like Democrat, Republican, Independent, or other parties without implied . It aids in studying voter behavior and policy preferences; contingency tables reveal associations, such as between affiliation and support for , enabling researchers to quantify partisan divides in surveys. Multi-category setups, with three or more affiliations, highlight analytical challenges like sparse cells in tables, necessitating robust tests for . In , product categories function as a nominal categorical variable, segmenting items into groups such as , apparel, groceries, or books for and targeting purposes. These inform sales strategies and customer segmentation; contingency tables cross-tabulate categories with purchase behaviors to identify patterns, like higher sales among certain demographics, supporting targeted campaigns. With numerous categories (often exceeding five in retail datasets), this variable underscores the intricacies of multi-category , where high dimensionality can complicate association detection without aggregation.

Notation and Properties

Standard Notation

In statistical literature, categorical variables serving as predictors are commonly denoted by an uppercase letter such as XX, with categories distinguished by subscripts to indicate specific levels, for instance XjX_j for the jj-th category among KK possible values. For binary cases, this simplifies to X=0X = 0 or X=1X = 1, or equivalently X1X_1 and X2X_2. To represent membership in a particular category, the indicator function I(X=k)I(X = k) is frequently used, where it equals 1 if the variable XX takes the value corresponding to category kk and 0 otherwise; this notation facilitates modeling and computation in analyses involving multiple categories. In software environments for data analysis, categorical variables employ specialized notations for efficient storage and manipulation. In the R programming language, they are implemented as factors, which internally map category labels to integer codes while preserving the categorical structure. Similarly, in Python's pandas library, the 'category' dtype designates such variables, optimizing memory usage for datasets with repeated category labels.

Number of Possible Values

A categorical variable consists of a fixed, of categories, conventionally denoted by kk levels where k≥2k \geq 2. This structure distinguishes it from continuous variables, as the possible values are discrete and exhaustive within the defined set, enabling straightforward enumeration in . The binary case, where k=2k=2, represents the simplest form of a categorical variable, often termed dichotomous, with outcomes such as yes/no or success/failure. This configuration minimizes analytical demands, as it aligns directly with binary logistic models or simple proportions without requiring additional partitioning. For multicategory variables, where k>2k > 2, the analysis grows in complexity due to the need to account for multiple distinctions among levels, often necessitating techniques like contingency tables or multinomial models to capture inter-category relationships. A key implication arises in hypothesis testing and regression, where the for the variable equal k−1k-1, reflecting the redundancy in representing all levels independently. This adjustment ensures unbiased estimation while preventing overparameterization in models.

Finiteness and Exhaustiveness

Categorical variables are defined by a of discrete categories, in contrast to continuous variables that allow for an infinite range of values within intervals. This finiteness ensures that the possible outcomes are limited and countable, facilitating discrete probability modeling and avoiding the complexities associated with uncountable spaces. For instance, a variable representing might include only a handful of options such as , , , and , rather than any conceivable shade along a . A key structural requirement for categorical variables is exhaustiveness, where the categories are mutually exclusive—each belongs to exactly one category—and collectively complete, encompassing all possible values that the variable can take in the population or sample. This property prevents overlap and omission, ensuring that the variable fully partitions the outcome space. In statistical analyses, such as contingency tables, this completeness allows marginal probabilities to sum to unity across categories. Violations of finiteness or exhaustiveness can occur when categories are incomplete, such as in surveys where respondents provide responses outside predefined options, leading to unclassified data. To address this, practitioners often introduce an "other" category to capture residual cases and restore exhaustiveness without discarding information. Alternatively, for missing or uncategorized entries, imputation strategies like multiple imputation by chained equations (MICE) can estimate values based on observed patterns, preserving the variable's discrete nature while minimizing bias. Theoretically, finiteness and exhaustiveness underpin the validity of probability distributions for categorical variables, particularly the , which models counts across a fixed number of categories with probabilities summing to one. This framework supports inference in models like for multicategory outcomes, ensuring parameters are identifiable and estimates are consistent. Without these properties, the assumption of a closed outcome space would fail, complicating likelihood-based analyses.

Descriptive Analysis

Visualization Techniques

Visualization techniques for categorical variables enable the graphical representation of data distributions, proportions, and relationships, facilitating exploratory analysis and effective communication without relying on numerical computations. Bar charts are a primary method for displaying the frequencies or counts of categories, where each bar's height corresponds to the number of observations in a given category, making it suitable for both nominal and ordinal variables. For instance, in a dataset of preferred fruits, a bar chart can clearly show the count for each fruit type, allowing quick identification of the most common preferences. Pie charts represent proportions of categories as slices of a circle, where the angle of each slice reflects the relative frequency, offering an intuitive view for simple datasets with few categories. However, pie charts can distort perceptions of differences between slices, especially when categories have similar proportions or when more than a handful of categories are present, leading experts to recommend them only for emphasizing parts of a whole in limited cases. For exploring associations between two or more categorical variables, mosaic plots extend the concept of stacked bar charts by dividing a into tiles whose areas represent joint frequencies or proportions, visually highlighting deviations from . This technique is particularly useful for contingency tables, as the tile widths and heights proportionally encode marginal distributions while shading can indicate residuals for . Best practices in these visualizations include clearly labeling categories and axes to ensure interpretability, using distinct colors for differentiation without relying on color alone for those with visual impairments, and avoiding three-dimensional effects that can introduce perspective distortions and mislead viewers. Software tools like in support these methods through functions such as geom_bar() for bar charts and geom_mosaic() via extensions for mosaic plots, while in Python offers similar capabilities with plt.bar() for categorical bars and extensions like statsmodels for mosaic displays. These graphical approaches reveal underlying patterns, such as imbalances in category distributions or unexpected associations, in a non-numerical manner that enhances accessibility for diverse audiences and supports initial .

Summary Measures

Summary measures for categorical variables provide numerical summaries of their distributions and associations without relying on graphical representations. For in nominal , the mode is the appropriate measure, defined as the category with the highest . This captures the most common value, as arithmetic means are inapplicable due to the lack of numerical ordering. To describe the overall distribution, counts indicate the absolute number of occurrences for each category, while percentages express these as proportions of the total sample size. These measures are often presented in contingency tables, offering a tabular overview of category prevalences. For ordinal categorical variables, which possess a natural ordering, the serves as a measure by identifying the category at the 50th when data are ranked. Associations between two categorical variables are commonly assessed using of independence, which evaluates whether observed frequencies differ significantly from expected frequencies under the of no association. The test statistic is calculated as χ2=∑(O−E)2E,\chi^2 = \sum \frac{(O - E)^2}{E}, where OO denotes observed frequencies and EE expected frequencies across all cells of the . Introduced by in 1900, this statistic follows a under the , enabling p-value computation for significance testing. A key limitation of summary measures for nominal categorical variables is the absence of a standard variance metric, as categories lack quantifiable distances or intervals for dispersion calculation. Such measures are thus restricted to counts, proportions, and modes, complementing visualization techniques for a fuller descriptive .

Encoding Techniques

Dummy Coding

Dummy coding is a fundamental technique for encoding categorical variables into numerical form suitable for statistical modeling, particularly in . It involves creating binary indicator variables, each taking values of 0 or 1, to represent the presence or absence of specific categories. For a categorical variable with kk levels, exactly k−1k-1 dummy variables are generated, omitting one category as the or baseline to avoid redundancy. The construction of dummy variables follows a straightforward rule: for each non-reference category jj (where j=1,2,…,k−1j = 1, 2, \dots, k-1), the dummy variable DjD_j is set to 1 if the observation falls into category jj, and 0 otherwise. The reference category is implicitly represented when all dummy variables are 0. This omission is crucial to prevent the dummy variable trap, a form of perfect that would arise if all kk dummies were included alongside a model intercept, as the dummies would sum to a constant. A primary advantage of dummy coding lies in its interpretability, especially in models. The coefficient for each dummy variable quantifies the average difference in the outcome variable between that category and the reference category, controlling for other predictors. This direct comparison facilitates clear insights into category-specific effects. As an illustration, consider a binary gender variable with categories "" and "." One dummy variable DmaleD_{\text{male}} can be defined such that Dmale=1D_{\text{male}} = 1 for s and 0 for s, treating as the . In a regression model, the on DmaleD_{\text{male}} would estimate the additional effect on the response associated with being compared to .

Effects Coding

Effects coding is a scheme for encoding categorical predictors in statistical models, such as linear regression, by assigning values that allow to represent deviations from the grand mean of the response variable across all categories. For a categorical variable with kk levels, this method employs k−1k-1 binary indicator variables, where each variable corresponds to one non- level. The coding assigns +1 to observations in the corresponding level, 0 to levels neither corresponding nor the , and -1 to the level, ensuring the design matrix columns sum to zero in balanced designs. In the regression model, the intercept β0\beta_0 estimates the overall yˉ\bar{y} of the dependent variable, while each βj\beta_j for the jj-th effects-coded variable estimates the deviation of the for level jj from the grand , given by βj=yˉj−yˉ\beta_j = \bar{y}_j - \bar{y}. This interpretation holds under ordinary estimation with balanced data, where the sample sizes per category are equal. For illustration, consider a categorical variable with four levels (A, B, C, D), treating D as reference; the coding for three variables is:
LevelVariable 1Variable 2Variable 3
A100
B010
C001
D-1-1-1
This setup yields coefficients where β1=yˉA−yˉ\beta_1 = \bar{y}_A - \bar{y}, β2=yˉB−yˉ\beta_2 = \bar{y}_B - \bar{y}, and β3=yˉC−yˉ\beta_3 = \bar{y}_C - \bar{y}. The primary advantages of effects coding include the property that the coefficients sum to zero (∑βj=0\sum \beta_j = 0), facilitating tests of overall effects and maintaining interpretability of main effects independent of other factors in multifactor designs. It promotes among predictors, leading to equal standard errors and higher statistical power in balanced experiments compared to non-orthogonal schemes. Relative to dummy coding, effects coding avoids designating a specific reference category, instead centering all interpretations around the grand mean for a more symmetric view of categorical effects.

Contrast Coding

Contrast coding assigns specific numerical weights to the levels of a categorical variable in regression or ANOVA models to test targeted hypotheses about differences between group means, rather than estimating all parameters separately. These weights are chosen such that they sum to zero across levels, ensuring and allowing the model's intercept to represent the grand mean of the response variable. This approach is particularly useful for planned comparisons, where researchers specify contrasts in advance to increase statistical power and focus on theoretically relevant differences. One common type is the treatment versus control contrast, which compares treatment levels to a designated control or reference level, often using weights adjusted for hypothesis testing. For instance, in a design with one control and multiple treatments, a single overall contrast can test the average treatment effect against the control by assigning -1 to the control and +1/n to each treatment level (where n is the number of treatment levels), enabling a test of whether the mean of the treatments differs from the control. Individual comparisons of each treatment to the control use separate contrast variables. Another type, the Helmert contrast, compares the mean of each level to the mean of all subsequent levels, facilitating sequential hypothesis tests such as whether the first level differs from the average of the rest. This is defined for k levels with weights that partition the comparisons orthogonally, such as for three levels: first contrast (1, -0.5, -0.5), second (0, 1, -1). Polynomial contrasts, suitable for ordinal categorical variables, model trends like linear or quadratic effects across ordered levels by assigning weights derived from orthogonal polynomials, such as for a linear trend in four levels: (-3/√10, -1/√10, 1/√10, 3/√10), normalized for unit variance. A straightforward example for a two-group categorical variable (e.g., control and treatment) uses weights of -0.5 and +0.5, respectively. In a Y=β0+β1X+[ϵ](/page/Epsilon)Y = \beta_0 + \beta_1 X + [\epsilon](/page/Epsilon), where XX is the contrast-coded predictor, the intercept β0\beta_0 estimates the grand mean, and β1\beta_1 estimates the signed difference between group means (full difference if groups are balanced). This setup directly tests the H0:μ1−μ2=[0](/page/0)H_0: \mu_1 - \mu_2 = [0](/page/0) via the t-statistic on β1\beta_1. Effects coding, which compares each level to the grand mean using weights that sum to zero (e.g., +1 and -1 for two groups, scaled), serves as a special case of contrast coding for omnibus mean comparisons. The primary advantages of contrast coding include its efficiency in parameter estimation, as it uses k-1 orthogonal predictors for k levels, reducing and compared to unadjusted dummy coding while enabling precise hypothesis tests. It enhances power for a priori contrasts by concentrating variance on specific comparisons, minimizing Type II errors in experimental designs. Additionally, for , polynomial contrasts reveal underlying trends without assuming arbitrary group differences, supporting interpretable inferences in fields like and social sciences.

Advanced Representations

Nonsense Coding

Nonsense coding refers to a method of representing categorical variables in statistical models by assigning arbitrary or randomly selected numerical values to each category, without any intent to impose meaningful structure or order. This approach contrasts with structured schemes like dummy or effects coding, as the chosen values bear no relation to the categories' substantive differences. According to O'Grady and Medoff (1988), nonsense coding uses any non-redundant set of coefficients to indicate category membership, but its parameters are only interpretable under limited conditions, often leading to misleading conclusions about category effects. The purpose of nonsense coding is primarily pedagogical: it demonstrates that the overall fit of a regression model, such as the multiple RR or the accuracy of predicted values, remains invariant across different coding schemes for categorical predictors, including arbitrary ones. Gardner (n.d.) illustrates this in the context of a 2x2 design, where nonsense coding yields the same R2R^2 value (e.g., 0.346) and cell mean estimates as standard codings, but alters the numerical values and significance tests of the regression coefficients. This highlights how overparameterized models can achieve good predictive performance even with non-informative representations, underscoring the distinction between statistical fit and substantive insight. A concrete example involves a three-level categorical variable, such as treatment groups A, B, and C, coded arbitrarily as 3 for A, 7 for B, and 1 for C. In a multiple , the resulting coefficients for these codes would reflect linear combinations of category effects but lack any direct, meaningful interpretation—unlike dummy coding, where coefficients represent deviations from a reference category. The key lesson is that the choice of coding profoundly influences the ability to draw valid inferences about categorical effects, even if predictive utility is preserved.

Embeddings

Embeddings represent categorical variables as low-dimensional dense vectors learned directly from data, enabling machine learning models to infer latent relationships and similarities among categories without relying on predefined structures. This approach treats categories as entities to be mapped into a continuous Euclidean space, where proximity reflects functional or semantic similarity, as demonstrated in entity embedding techniques for function approximation problems. For instance, in text processing, word embeddings like those from Word2Vec model words as categorical tokens, capturing contextual analogies such as "king" - "man" + "woman" ≈ "queen" through vector arithmetic. These embeddings are typically learned end-to-end within architectures, starting with categorical inputs converted to indices or encodings, which are then projected via a trainable layer into a fixed-size of lower dimensionality than the number of categories. The learning process optimizes the vectors based on the overall model objective, such as minimizing prediction error in supervised tasks, allowing the embeddings to adaptively encode category interactions with other features. This contrasts with sparse traditional codings by producing compact, dense representations that generalize better across datasets. A key advantage of embeddings is their ability to quantify category similarities using metrics like cosine distance, where vectors for related categories (e.g., "dog" and "puppy" in an animal classification task) cluster closely, facilitating downstream tasks like clustering or nearest-neighbor search. They are especially valuable for high-cardinality variables, where the explosion of unique categories would render one-hot encodings computationally prohibitive, reducing parameter count while preserving expressive power. In applications, embeddings have transformed by enabling efficient handling of vocabulary as categorical variables, powering tasks from to since the introduction of efficient training methods in the . In recommendation systems, they represent user preferences or item attributes as categories, improving personalization by learning latent factors that capture user-item affinities, as extended from entity embedding principles to large-scale . This development, building on foundational neural language models, has become a standard in pipelines for categorical data since Mikolov et al.'s 2013 work.

Regression Applications

Incorporating Categorical Predictors

To incorporate categorical predictors into a linear regression model, the categories are first encoded into a set of binary indicator variables (dummies) or contrast variables, with one category typically omitted as the reference to avoid perfect multicollinearity. These encoded variables then replace the original categorical predictor in the model specification, allowing the regression to estimate category-specific effects alongside other predictors. The resulting model takes the form Y=β0+∑j=1k−1βjDj+ϵ,Y = \beta_0 + \sum_{j=1}^{k-1} \beta_j D_j + \epsilon, where YY is the response variable, β0\beta_0 is the intercept (representing the mean of YY for the reference category when all other predictors are zero), DjD_j are the indicator variables for the k−1k-1 non-reference categories (each Dj=1D_j = 1 if the observation belongs to category jj, and 0 otherwise), βj\beta_j are the coefficients for those categories, and ϵ\epsilon is the error term. This approach, originally formalized for handling qualitative factors in econometric models, enables the linear regression framework to accommodate non-numeric predictors without altering the core estimation procedure. The coefficients βj\beta_j in this model are interpreted as the adjusted difference in the of YY between category jj and the category, holding all other predictors constant; for example, a positive βj\beta_j indicates that category jj is associated with a higher response than the . This interpretation depends on the chosen encoding scheme, such as dummy coding where βj\beta_j directly measures the deviation from the , but remains consistent across valid encodings like contrasts as long as the is clearly defined. In practice, the intercept β0\beta_0 provides the baseline prediction for the group, while the βj\beta_j terms quantify incremental effects. Key assumptions for this incorporation include in the parameters (the effects of the categorical predictors enter the model additively through the linear predictor) and no among the encoded variables, which is ensured by excluding one category as the to prevent linear dependence. Violation of the no- assumption would lead to unstable estimates, but the category omission resolves this for categorical predictors alone; interactions or correlated covariates may introduce additional issues requiring separate diagnostics. These assumptions align with the standard framework, ensuring unbiased and efficient estimation under ordinary least squares. In software implementations, categorical predictors are integrated seamlessly into linear models via functions like R's lm(), which automatically applies treatment contrasts (dummy coding) to factor variables upon model fitting, or generalized linear model (GLM) frameworks that extend this to non-normal responses while maintaining the same encoding process. For instance, specifying a factor variable in lm(Y ~ categorical_factor + other_predictors, data = dataset) generates the necessary dummies internally, with coefficients output relative to the first level as reference unless contrasts are customized. This built-in handling simplifies analysis in tools like R or SAS, reducing manual preprocessing while supporting extensions to GLMs for broader applicability.

Interactions

In , interactions involving categorical variables arise when the effect of a categorical predictor on the response variable depends on the value of another predictor, necessitating the inclusion of product terms to model this dependency accurately. For instance, the influence of a categorical factor such as treatment type on an outcome may vary across levels of another variable, like dosage, requiring terms that capture these conditional effects. This approach ensures that the model reflects real-world complexities where categorical effects are not uniform. The rationale for incorporating such interactions stems from the observation that assuming additive effects alone can lead to biased interpretations of main effects, particularly when categorical predictors moderate relationships with other variables. By including interaction terms, researchers can account for non-additive influences, enhancing the model's explanatory power and validity, as supported by theoretical expansions like that justify product terms for smooth functions. In multiple regression, the general form extends the standard by adding cross-products of encoded categorical variables and other predictors; for a categorical variable encoded as dummy indicators DjD_j and another predictor ZkZ_k, the term βjkDjZk\beta_{jk} D_j Z_k is included to represent varying slopes or intercepts across categories. Detection of these interactions typically involves statistical tests and graphical methods to assess their significance before inclusion. Analysis of variance (ANOVA) can test the overall significance of interaction terms through their p-values in the model output, indicating whether the combined effects deviate from additivity. Alternatively, added variable plots, which partial out main effects to visualize the relationship between residuals and the interaction term, or residual plots against the product of predictors, help identify non-random patterns suggestive of interactions. Encoding techniques, such as dummy coding, are briefly referenced here to form these product terms appropriately.

Categorical-Categorical Interactions

In regression models, interactions between two categorical variables capture how the effect of one categorical predictor on the outcome varies across levels of the other categorical predictor. This allows for modeling non-additive relationships, where the combined influence of the categories differs from the sum of their individual main effects. Such interactions are particularly useful in scenarios like designs or observational studies involving multiple grouping factors. To incorporate these interactions, categorical variables are first encoded using dummy variables. For a categorical predictor with k levels, k-1 dummy variables are created, typically with one level as the reference (coded 0) and others as 1 or -1 depending on the scheme (e.g., dummy or effect coding). The interaction terms are then formed by taking the cross-products of these dummy variables from each predictor. For two categorical variables with k and m levels, this results in (k-1)(m-1) interaction terms, which fully parameterize the deviations from additivity across all non-reference combinations. This approach ensures while spanning the space of possible cell-specific effects in the . The coefficients of these interaction terms represent the conditional effects or deviations from the main effects. Specifically, the for a particular cross-product indicates the additional change in the outcome when both corresponding categories are active, relative to the levels, holding other factors constant. This yields stratified estimates: for instance, the effect of one variable's levels can be interpreted separately within each level of the other variable, revealing how associations differ across subgroups. In effect coding, these coefficients further reflect deviations from the grand mean, facilitating comparisons to overall averages. A classic example occurs in analyzing treatment effects moderated by gender, akin to two-way ANOVA models. Consider a study regressing patient recovery scores on treatment (placebo vs. drug, encoded as a single dummy) and (female as reference). The interaction term (treatment × male) captures whether the drug's benefit differs for males versus females. If the interaction is positive and significant, it suggests the treatment elevates recovery more for males (e.g., +5 points beyond the ) than females, allowing tailored inferences like stratified ratios or means. To reduce the complexity of the full set of interaction terms, especially with many levels, analysts can parameterize the model using cell means or predefined contrasts. Cell means coding directly estimates the mean outcome for each combination of categories, equivalent to a with k × m parameters (one per cell, with overparameterization resolved via constraints). Alternatively, simplified contrasts—such as planned comparisons (e.g., treatment vs. control within each )—collapse multiple terms into fewer interpretable ones, focusing on specific hypotheses while maintaining model parsimony. These methods, often implemented via software options like LSMEANS, aid in post-hoc testing and visualization without fitting the exhaustive cross-product set.

Categorical-Continuous Interactions

Categorical-continuous interactions in regression models allow the effect of a continuous predictor on the outcome to vary across levels of a categorical predictor, enabling the modeling of heterogeneous slopes. This is achieved by creating interaction terms that multiply a dummy-coded representation of the categorical variable with the continuous variable. For a categorical variable with kk categories, k−1k-1 dummy variables DjD_j (where j=1,…,k−1j = 1, \dots, k-1) are used, and the interaction terms are formed as Dj×XD_j \times X, where XX is the continuous predictor; the corresponding coefficients βj\beta_j in the model Y=β0+β1X+∑γjDj+∑βj(DjX)+ϵY = \beta_0 + \beta_1 X + \sum \gamma_j D_j + \sum \beta_j (D_j X) + \epsilon capture the differences in slopes relative to the reference category. The interpretation of these interactions reveals separate regression lines for each category of the categorical variable, differing in both intercepts (from the main effect of the dummies) and slopes (from the interaction terms). For the reference category, the slope is simply β1\beta_1; for category jj, it becomes β1+βj\beta_1 + \beta_j, indicating how the continuous variable's influence adjusts per group. This approach visualizes as parallel or non-parallel lines in scatterplots stratified by category, highlighting moderation effects where the continuous predictor's impact is not uniform. A common example involves examining how the effect of age (continuous) on (outcome) differs by (categorical). In such a model, the interaction terms test whether the age-income is steeper for one , say males, compared to females as the , reflecting potential labor market disparities. This setup, analyzed in behavioral applications, underscores how demographic categories can moderate age-related trajectories. To assess the significance of these interactions, an is employed on the set of interaction coefficients collectively, evaluating whether the slopes differ significantly across categories beyond what main effects alone explain; a significant F-statistic (e.g., with (k−1,n−p−1)(k-1, n - p - 1), where pp is the number of predictors) supports including the interaction in the model. Follow-up t-tests on individual βj\beta_j can probe specific category differences if the overall test is significant.

References

  1. https://stats.oarc.ucla.edu/other/mult-pkg/whatstat/what-is-the-difference-between-categorical-ordinal-and-interval-variables/
  2. https://online.stat.psu.edu/stat200/lesson/1/1.1/1.1.1
  3. http://www.stat.yale.edu/Courses/1997-98/101/catdat.htm
  4. https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/variables
  5. https://support.minitab.com/en-us/minitab/help-and-how-to/statistical-modeling/regression/supporting-topics/basics/what-are-categorical-discrete-and-continuous-variables/
  6. https://statisticsbyjim.com/glossary/categorical-variables/
  7. https://www.sciencedirect.com/science/article/abs/pii/S0378375807001073
  8. https://www.uvm.edu/~statdhtx/methods8/Supplements/Chi-Square-Folder/ChiSquareTests.pdf
  9. https://pmc.ncbi.nlm.nih.gov/articles/PMC3900058/
  10. https://www.science.org/doi/10.1126/science.103.2684.677
  11. https://soc.utah.edu/sociology3112/central-tendency-variability.php
  12. https://online.stat.psu.edu/stat504/book/export/html/630
  13. https://pmc.ncbi.nlm.nih.gov/articles/PMC8217997/
  14. https://pages.gseis.ucla.edu/faculty/richardson/Courses/stevens1946.pdf
  15. https://pmc.ncbi.nlm.nih.gov/articles/PMC3886444/
  16. https://pmc.ncbi.nlm.nih.gov/articles/PMC4754273/
  17. https://psychmodels.ucdavis.edu/sites/g/files/dgvnsk12156/files/inline-files/rhemtullabrosseauliardsavalei_pm.pdf
  18. https://www.scribbr.com/methodology/types-of-variables/
  19. https://stats.libretexts.org/Courses/Adler_University/Graduate-Level_Statistics_in_Psychology/03%3A_Variable_Types/3.03%3A_Categorical_Variables-_Variables_That_Vary_by_Type
  20. https://pmc.ncbi.nlm.nih.gov/articles/PMC5506151/
  21. https://journals.asm.org/doi/10.1128/spectrum.01108-24
  22. https://www.bmj.com/about-bmj/resources-readers/publications/statistics-square-one/1-data-display-and-summary
  23. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson2/section2.html
  24. https://pmc.ncbi.nlm.nih.gov/articles/PMC4419298/
  25. https://www.researchgate.net/figure/Types-of-variables-in-biomedical-research_tbl1_276284455
  26. https://methods.sagepub.com/ency/edvol/the-sage-encyclopedia-of-social-science-research-methods/chpt/nominal-variable
  27. https://www.povertyactionlab.org/evaluation/understanding-connection-between-party-affiliation-partisanship-and-political-beliefs
  28. https://socialsci.libretexts.org/Courses/Orange_Coast_College/SOC_200%253A_Introduction_to_Sociology_Research_Methods_%28Ridnor%29/04%253A_Conceptualization_Operationalization_and_Measurement/4.04%253A_Levels_of_Measurment
  29. https://www.appinio.com/en/blog/market-research/nominal-data
  30. https://www.appinio.com/en/blog/market-research/cross-tabulation-analysis
  31. https://www.insightsforprofessionals.com/marketing/marketing-technology/examples-of-nominal-data-in-business
  32. https://www.stat.cmu.edu/~hseltman/309/Book/chapter16.pdf
  33. https://scpoecon.github.io/ScPoEconometrics/categorical-vars.html
  34. https://www.statlect.com/fundamentals-of-probability/indicator-functions
  35. https://r4ds.had.co.nz/factors.html
  36. https://pandas.pydata.org/docs/user_guide/categorical.html
  37. https://mregresion.files.wordpress.com/2012/08/agresti-introduction-to-categorical-data.pdf
  38. https://pmc.ncbi.nlm.nih.gov/articles/PMC6707495/
  39. https://stats.oarc.ucla.edu/spss/faq/coding-systems-for-categorical-variables-in-regression-analysis-2/
  40. http://galton.uchicago.edu/~burbank/stat224/lectures/05chapter.html
  41. https://lin-yu.me/books/AGRESTI_categorical_data_analysis_2nd.pdf
  42. https://www.aptech.com/blog/introduction-to-categorical-variables/
  43. https://pressbooks.rampages.us/sociology-research-methods/chapter/7-2-deciding-on-the-correct-level-of-measurement/
  44. https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-018-0653-0
  45. https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.pdf
  46. https://openintro-ims.netlify.app/explore-categorical
  47. https://pmc.ncbi.nlm.nih.gov/articles/PMC8556474/
  48. https://www.jmp.com/en/statistics-knowledge-portal/exploratory-data-analysis/mosaic-plot
  49. https://guides.library.ucla.edu/c.php?g=180624&p=9224430
  50. https://r-graph-gallery.com/ggplot2-package.html
  51. https://matplotlib.org/stable/gallery/lines_bars_and_markers/categorical_variables.html
  52. https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php
  53. https://thomaselove.github.io/2019-431-book/summarizing-categorical-variables.html
  54. https://www.scribbr.com/statistics/ordinal-data/
  55. https://www.scribbr.com/statistics/chi-square-tests/
  56. https://www.stat.purdue.edu/docs/research/tech-reports/1983/tr83-53.pdf
  57. https://statisticsbyjim.com/basics/nominal-ordinal-interval-ratio-scales/
  58. https://stats.oarc.ucla.edu/spss/faq/coding-systems-for-categorical-variables-in-regression-analysis/
  59. https://www.statlect.com/fundamentals-of-statistics/dummy-variable
  60. https://www.mathworks.com/help/stats/dummy-indicator-variables.html
  61. https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faqwhat-is-effect-coding/
  62. https://sites.globalhealth.duke.edu/rdac/wp-content/uploads/sites/27/2020/08/Core-Guide_Dummy-and-Effect-Coding_16-03-20.pdf
  63. https://www.taylorfrancis.com/books/mono/10.4324/9780203774441/applied-multiple-regression-correlation-analysis-behavioral-sciences-jacob-cohen-patricia-cohen-stephen-west-leona-aiken
  64. https://stats.oarc.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/
  65. https://www.diva-portal.org/smash/get/diva2:325460/fulltext01.pdf
  66. https://publish.uwo.ca/~gardner/DataAnalysisDotCalm/Manuscripts/2X2AnalysisOfVarianceAndMultipleRegression.pdf
  67. https://arxiv.org/abs/1604.06737
  68. https://arxiv.org/abs/1301.3781
  69. https://online.stat.psu.edu/stat462/node/86/
  70. https://stats.oarc.ucla.edu/spss/seminars/introduction-to-regression-with-spss/introreg-lesson3/
  71. https://stats.oarc.ucla.edu/r/modules/coding-for-categorical-variables-in-regression-models/
  72. https://statisticsbyjim.com/regression/interaction-effects/
  73. https://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/19/lecture-19.pdf
  74. https://pmc.ncbi.nlm.nih.gov/articles/PMC11805792/
  75. https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-how-do-i-interpret-the-coefficients-of-an-effect-coded-variable-involved-in-an-interaction-in-a-regression-model/
  76. https://pages.stat.wisc.edu/~ane/st572/notes/lec08.pdf
  77. https://www.theanalysisfactor.com/interaction-dummy-variables-in-linear-regression/
  78. https://online.stat.psu.edu/stat501/book/export/html/938
  79. https://www.lesahoffman.com/SPLH861/861_Lecture3_Interactions_Categorical.pdf
  80. https://stats.oarc.ucla.edu/stata/webbooks/reg/chapter6/regression-with-statachapter-6-more-on-interactions-of-categorical-variablesdraft-version/
  81. https://www.routledge.com/Applied-Multiple-RegressionCorrelation-Analysis-for-the-Behavioral-Sciences/Cohen-Cohen-West-Aiken/p/book/9780805822236
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