In statistics, a **contingency table** (also known as a **cross tabulation** or **crosstab**) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term *contingency table* was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation",^{ [1] } part of the * Drapers' Company Research Memoirs Biometric Series I* published in 1904.

- Example
- Standard contents of a contingency table
- Measures of association
- Odds ratio
- Phi coefficient
- Cramér's V and the contingency coefficient C
- Tetrachoric correlation coefficient
- Lambda coefficient
- Uncertainty coefficient
- Others
- See also
- References
- Further reading
- External links

A crucial problem of multivariate statistics is finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can use information theory concepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies.

A pivot table is a way to create contingency tables using spreadsheet software.

Suppose there are two variables, sex (male or female) and handedness (right- or left-handed). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male right-handed and left-handed, female right-handed and left-handed. Such a contingency table is shown below.

Handed- ness Sex | Right-handed | Left-handed | Total |
---|---|---|---|

Male | 43 | 9 | 52 |

Female | 44 | 4 | 48 |

Total | 87 | 13 | 100 |

The numbers of the males, females, and right- and left-handed individuals are called marginal totals. The grand total (the total number of individuals represented in the contingency table) is the number in the bottom right corner.

The table allows users to see at a glance that the proportion of men who are right-handed is about the same as the proportion of women who are right-handed although the proportions are not identical. The strength of the association can be measured by the odds ratio, and the population odds ratio estimated by the sample odds ratio. The significance of the difference between the two proportions can be assessed with a variety of statistical tests including Pearson's chi-squared test, the *G*-test, Fisher's exact test, Boschloo's test, and Barnard's test, provided the entries in the table represent individuals randomly sampled from the population about which conclusions are to be drawn. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), it is said that there is a *contingency* between the two variables. In other words, the two variables are *not* independent. If there is no contingency, it is said that the two variables are *independent*.

The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 × 2 contingency table. In principle, any number of rows and columns may be used. There may also be more than two variables, but higher order contingency tables are difficult to represent visually. The relation between ordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare. For more on the use of a contingency table for the relation between two ordinal variables, see Goodman and Kruskal's gamma.

- Multiple columns (historically, they were designed to use up all the white space of a printed page). Where each row refers to a specific sub-group in the population (in this case men or women), the columns are sometimes referred to as
*banner points*or*cuts*(and the rows are sometimes referred to as*stubs*). - Significance tests. Typically, either
*column comparisons*, which test for differences between columns and display these results using letters, or,*cell comparisons*, which use color or arrows to identify a cell in a table that stands out in some way. *Nets*or*netts*which are sub-totals.- One or more of: percentages, row percentages, column percentages, indexes or averages.
- Unweighted sample sizes (counts).

The degree of association between the two variables can be assessed by a number of coefficients. The following subsections describe a few of them. For a more complete discussion of their uses, see the main articles linked under each subsection heading.

The simplest measure of association for a 2 × 2 contingency table is the odds ratio. Given two events, A and B, the odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the odds ratio is 1; if the odds ratio is greater than 1, the events are positively associated; if the odds ratio is less than 1, the events are negatively associated.

The odds ratio has a simple expression in terms of probabilities; given the joint probability distribution:

the odds ratio is:

A simple measure, applicable only to the case of 2 × 2 contingency tables, is the phi coefficient (φ) defined by

where χ^{2} is computed as in Pearson's chi-squared test, and *N* is the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or −1 (complete association or complete inverse association), provided it is based on frequency data represented in 2 × 2 tables. Then its sign equals the sign of the product of the main diagonal elements of the table minus the product of the off–diagonal elements. φ takes on the minimum value −1.0 or the maximum value of +1.0 if and only if every marginal proportion is equal to 0.5 (and two diagonal cells are empty).^{ [2] }

Two alternatives are the *contingency coefficient**C*, and Cramér's V.

The formulae for the *C* and *V* coefficients are:

- and

*k* being the number of rows or the number of columns, whichever is less.

*C* suffers from the disadvantage that it does not reach a maximum of 1.0, notably the highest it can reach in a 2 × 2 table is 0.707 . It can reach values closer to 1.0 in contingency tables with more categories; for example, it can reach a maximum of 0.870 in a 4 × 4 table. It should, therefore, not be used to compare associations in different tables if they have different numbers of categories.^{ [3] }

*C* can be adjusted so it reaches a maximum of 1.0 when there is complete association in a table of any number of rows and columns by dividing *C* by where *k* is the number of rows or columns, when the table is square ^{[ citation needed ]}, or by where *r* is the number of rows and *c* is the number of columns.^{ [4] }

Another choice is the tetrachoric correlation coefficient but it is only applicable to 2 × 2 tables. Polychoric correlation is an extension of the tetrachoric correlation to tables involving variables with more than two levels.

Tetrachoric correlation assumes that the variable underlying each dichotomous measure is normally distributed.^{ [5] } The coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."^{ [6] }

The tetrachoric correlation coefficient should not be confused with the Pearson correlation coefficient computed by assigning, say, values 0.0 and 1.0 to represent the two levels of each variable (which is mathematically equivalent to the φ coefficient).

The lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level. Values range from 0.0 (no association) to 1.0 (the maximum possible association).

Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.

The uncertainty coefficient, or Theil's U, is another measure for variables at the nominal level. Its values range from −1.0 (100% negative association, or perfect inversion) to +1.0 (100% positive association, or perfect agreement). A value of 0.0 indicates the absence of association.

Also, the uncertainty coefficient is conditional and an asymmetrical measure of association, which can be expressed as

- .

This asymmetrical property can lead to insights not as evident in symmetrical measures of association.^{ [7] }

- Gamma test: No adjustment for either table size or ties.

- Kendall's tau: Adjustment for ties.

- Confusion matrix
- Pivot table, in spreadsheet software, cross-tabulates sampling data with counts (contingency table) and/or sums.
- TPL Tables is a tool for generating and printing crosstabs.
- The iterative proportional fitting procedure essentially manipulates contingency tables to match altered joint distributions or marginal sums.
- The multivariate statistics in special multivariate discrete probability distributions. Some procedures used in this context can be used in dealing with contingency tables.
- OLAP cube, a modern multidimensional computing form of contingency tables
- Panel data, multidimensional data over time

In probability theory and statistics, the **chi-square distribution** with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the **central chi-square distribution**, a special case of the more general noncentral chi-square distribution.

In statistics, **correlation ** or **dependence ** is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense **correlation** is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

In statistics, the **Pearson correlation coefficient**, also referred to as **Pearson's r**, the

**Pearson's chi-squared test** is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900. In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to *Pearson χ-squared* test or statistic are used.

In statistics, **Spearman's rank correlation coefficient** or **Spearman's ρ**, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function.

In statistics, **canonical-correlation analysis** (**CCA**), also called **canonical variates analysis**, is a way of inferring information from cross-covariance matrices. If we have two vectors *X* = (*X*_{1}, ..., *X*_{n}) and *Y* = (*Y*_{1}, ..., *Y*_{m}) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of *X* and *Y* which have maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Jordan in 1875.

In statistics, an **effect size** is a number measuring the strength of the relationship between two variables in a statistical population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter of a hypothetical statistical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In probability theory and statistics, the **noncentral chi-square distribution** is a noncentral generalization of the chi-square distribution. It often arises in the power analysis of statistical tests in which the null distribution is a chi-square distribution; important examples of such tests are the likelihood-ratio tests.

In probability theory and statistics, **partial correlation** measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. If we are interested in finding to what extent there is a numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another, confounding, variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.

The **Matthews correlation coefficient** (MCC) or **phi coefficient** is used in machine learning as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975. The MCC is defined identically to Pearson's phi coefficient, introduced by Karl Pearson, also known as the Yule phi coefficient from its introduction by Udny Yule in 1912. Despite these antecedents which predate Matthews's use by several decades, the term MCC is widely used in the field of bioinformatics and machine learning.

**Exact statistics**, such as that described in exact test, is a branch of statistics that was developed to provide more accurate results pertaining to statistical testing and interval estimation by eliminating procedures based on asymptotic and approximate statistical methods. The main characteristic of exact methods is that statistical tests and confidence intervals are based on exact probability statements that are valid for any sample size. Exact statistical methods help avoid some of the unreasonable assumptions of traditional statistical methods, such as the assumption of equal variances in classical ANOVA. They also allow exact inference on variance components of mixed models.

In statistics, the **phi coefficient** is a measure of association for two binary variables. Introduced by Karl Pearson, this measure is similar to the Pearson correlation coefficient in its interpretation. In fact, a Pearson correlation coefficient estimated for two binary variables will return the phi coefficient. The phi coefficient is related to the chi-squared statistic for a 2×2 contingency table

A **correlation coefficient** is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.

In statistics, the **Cochran–Mantel–Haenszel test** (CMH) is a test used in the analysis of stratified or matched categorical data. It allows an investigator to test the association between a binary predictor or treatment and a binary outcome such as case or control status while taking into account the stratification. Unlike the McNemar test which can only handle pairs, the CMH test handles arbitrary strata size. It is named after William G. Cochran, Nathan Mantel and William Haenszel. Extensions of this test to a categorical response and/or to several groups are commonly called Cochran–Mantel–Haenszel statistics. It is often used in observational studies where random assignment of subjects to different treatments cannot be controlled, but confounding covariates can be measured.

In statistics, **Cramér's V** is a measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It is based on Pearson's chi-squared statistic and was published by Harald Cramér in 1946.

In statistics, **Tschuprow's T** is a measure of association between two nominal variables, giving a value between 0 and 1 (inclusive). It is closely related to Cramér's V, coinciding with it for square contingency tables. It was published by Alexander Tschuprow in 1939.

The **evaluation of binary classifiers** compares two methods of assigning a binary attribute, one of which is usually a standard method and the other is being investigated. There are many metrics that can be used to measure the performance of a classifier or predictor; different fields have different preferences for specific metrics due to different goals. For example, in medicine sensitivity and specificity are often used, while in computer science precision and recall are preferred. An important distinction is between metrics that are independent on the prevalence, and metrics that depend on the prevalence – both types are useful, but they have very different properties.

In statistics, **Yule's Y**, also known as the

**Ordinal data** is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories is not known. These data exist on an **ordinal scale**, one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a ranking. It also differs from interval and ratio scales by not having category widths that represent equal increments of the underlying attribute.

- ↑ Karl Pearson, F.R.S. (1904).
*Mathematical contributions to the theory of evolution*. Dulau and Co. - ↑ Ferguson, G. A. (1966).
*Statistical analysis in psychology and education*. New York: McGraw–Hill. - ↑ Smith, S. C., & Albaum, G. S. (2004)
*Fundamentals of marketing research*. Sage: Thousand Oaks, CA. p. 631 - ↑ Blaikie, N. (2003)
*Analyzing Quantitative Data*. Sage: Thousand Oaks, CA. p. 100 - ↑ Ferguson.
^{[ full citation needed ]} - ↑ Ferguson, 1966, p. 244
- ↑ https://towardsdatascience.com/the-search-for-categorical-correlation-a1cf7f1888c9

- 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. - Lauritzen, Steffen L. (1979).
*Lectures on Contingency Tables (Aalborg University)*(PDF) (4th edition (first electronic edition), 2002 ed.). - Gokhale, D. V.; Kullback, Solomon (1978).
*The Information in Contingency Tables*. Marcel Dekker. ISBN 0-824-76698-9.

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