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## contingency table

contingency table Contingency tables, often referred to as cross-classifications or cross-tabulations, are tables of counts which describe and analyse the relationship between two or more variables in a data-set. They contain row variables across the horizontal axis and column variables down the vertical. Cell entries give the number of cases (persons, households, or other unit of analysis) that occur in each cell. The cells themselves are formed by combining one category from each of the row and column variables. Marginal totals (or marginals) give the total number of cases found in each category of the variables—in other words they are the row and column totals. Normally, cell entries are expressed as either row or column percentages (depending on the point the analyst wishes to make), with the total numbers of cases shown in the marginals. These elements are shown in tabular form below.

The above is a template for a typical 2 × 2 cross-tabulation. However, contingency tables can take more complex forms, with three or more variables and several categories of each. In such complex presentations, it is often difficult both to determine the nature of the causal relationships to be found among the variables, and then to demonstrate these to the reader. For example, it may be hard to eliminate spurious correlations, or to establish three-way interaction effects, that is, those cases where two variables are associated, but the strength or direction of the association varies across different categories of the variables. For these reasons the analysis of complex contingency tables is nowadays normally undertaken using the mathematical techniques of loglinear analysis. See also MULTIVARIATE ANALYSIS; TABULAR PRESENTATION.

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# CONTINGENCY TABLE

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 inverselythat 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

(see also: Data Sources and Collection Methods; Epidemiology; Statistics for Public Health )

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## contingency table

contingency table In statistical analysis, a frequency distribution of sample data classified by two or more factors, each with two or more classes. A simple example is a medical clinical trial of two treatments in which the number of patients assigned to each treatment is classified according to whether improvement was observed or not. If there is no significant difference between the proportions of patients improving, there is said to be no interaction between the two classifications of the table. The statistical analysis of contingency tables depends on certain assumptions (random assignment to classes, absence of other relevant factors) that make the interpretation controversial, and care must be taken in applying the tests correctly. See also chi-squared distribution.

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## contingency table

contingency table(two-way table) A table of data for two methods of classification of the same individuals (e.g. leaf shape and flower structure or hair colour and eye colour). This type of data can then be analysed statistically for association between these properties using a χ2 test.

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## contingency table

contingency table (two-way table) A table of data for two methods of classification of the same individuals, e.g. leaf shape and flower structure. This type of data can then be analysed statistically for association between these properties using a χ2 test.

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## contingency table

contingency table (two-way table) A table of data for two methods of classification of the same individuals (e.g. hair colour and eye colour). This type of data can then be analysed statistically for association between these properties using a χ2 test.

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