contingency table

Dictionary: contingency table

A statistical table that shows the observed frequencies of data elements classified according to two variables, with the rows indicating one variable and the columns indicating the other variable.

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Statistics Dictionary: contingency table

Variant: cross-classification; cross-tabulation

A table displaying the frequencies for each combination of two or more variables. The variables are either categorical variables or numerical variables for which the possible outcomes have been arranged in groups. The term was first used by Karl Pearson in 1904. Each location in a table is called a cell, and the corresponding frequency is the cell frequency.

Suppose A and B are two categorical variables having J and K categories, respectively. There are therefore JK possible category combinations. The table described would be called a J×K table. One simple model for such a table is the independence model (see also chi-squared test). Multidimensional contingency tables summarize information from more than two categorical variables. A three-variable table might be called a J×K×L table. Models used include logit models and, most commonly, log-linear models.

Marketing Dictionary: contingency table

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Method used in statistical analysis to define how one set of variables is controlled by variations in another set. For example, the effect of price on demand can be expressed by a contingency table that shows the number of units sold at various price levels.

Business Dictionary: Contingency Table

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Table presenting sample observations classified by two or more characteristics, such as R and C, into as many classes. The division of homeowners in a condominium project by sex (R = male or female) and by age groups (C = 20 to 30, 31 to 40, and 41 and above) would be representative of a typical contingency table.

Encyclopedia of Public Health: Contingency Table

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A contingency table is a display of data in columns and rows, arranged to facilitate the discovery of any relationship that may exist between different sets of data. The simplest type of contingency table displays two sets of data, one each in the columns and rows. The simplest of all is a fourfold, or 2 × 2, table. More complex contingency tables can be constructed with a further subclassification of data in the columns or rows, or in both columns and rows. Many varieties of data exist that can be arranged in this sort of table.

Some of the common variables that contingency tables show are: dichotomous (either-or); nominal (i.e. unordered, qualitative, classes, races); and ordinal (i.e. arranged along a scale that may or may not be continuous from zero to infinity, have defined upper and lower limits, or a defined mathematical relationship). Sometimes a relationship between columns and rows is intuitively obvious merely from inspection, and may show that the values in the columns and those in the rows vary either directly or inversely—that is, as the numbers increase across the rows, they also increase down the columns, or vice versa. At other times there may be no obvious relationship, but one may be revealed by appropriate statistical tests for association or correlation among the variables displayed in the table.

— JOHN M. LAST

Wikipedia: Contingency table

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It has been suggested that this article or section be merged with Cross tabulation . (Discuss)

In statistics, contingency tables are used to record and analyse the relationship between two or more variables, most usually categorical variables.

Suppose that we have two variables, sex (male or female) and handedness (right- or left-handed). We observe the values of both variables in a random sample of 100 people. Then a contingency table can be used to express the relationship between these two variables, as follows:

	right-handed	left-handed	TOTAL
male	43	9	52
female	44	4	48
TOTAL	87	13	100

The figures in the right-hand column and the bottom row are called marginal totals and the figure in the bottom right-hand corner is the grand total.

The table allows us 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. However the two proportions are not identical, and the statistical significance of the difference between them can be tested with a Pearson's chi-square test, a G-test or Fisher's exact test or Barnard's test, provided the entries in the table represent a random sample from the population contemplated in the null hypothesis. If the proportions of individuals in the different columns varies between rows (and, therefore, vice versa) we say that the table shows contingency between the two variables. If there is no contingency, we say that the two variables are independent.

The example above is for the simplest kind of contingency table, in which each variable has only two levels; this is called a 2 x 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 hard to represent on paper. The relationship between ordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, though this is less often done since the distributions of ordinal variables can be summarised efficiently by the median.

The degree of association between the two variables can be assessed by a number of coefficients: the simplest is the phi coefficient defined by

$\phi=\sqrt{\frac{\chi^2}{N}}$

where χ² is derived from the Pearson test, and N is the grand total number of observations. φ varies from 0 (corresponding to no association between the variables) to 1 (complete association). This coefficient can only be used for 2 x 2 tables. Alternatives include the tetrachoric correlation coefficient (also only useful for 2 × 2 tables), the contingency coefficient C and Cramér's V. C suffers from the disadvantage that it does not reach a maximum of 1 with complete association in asymmetrical tables (those where the numbers of row and columns are not equal). The tetrachoric correlation coefficient the Pearson product-moment correlation coefficient between hypothetical row and column variables with Normal distributions, that would reproduce the observed contingency table if they were divided into two categories in the appropriate proportions. It should not be confused with the Pearson product-moment correlation coefficient computed by assigning values 0 and 1 to the cells. In tables with more than two levels for each variable an analogous quantity is called the polychoric correlation coefficient.

The formulae for the other coefficients are:

$C=\sqrt{\frac{\chi^2}{N+\chi^2}}$

$V=\sqrt{\frac{\chi^2}{N(k-1)}}$

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

C can be adjusted so it reaches a maximum of 1 when there is complete association in a table of any number of rows and columns by dividing it by $\sqrt{\frac{k-1}{k}}$ .

The term contingency table was first used by Karl Pearson in "On the Theory of Contingency and its Relation to Association and Normal Correlation" in Drapers' Company Research Memoirs (1904) Biometric Series I.[1]

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
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