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| In [[statistics]], a '''contingency table''' (also referred to as '''[[cross tabulation]]''' or '''cross tab''') is a type of [[table]] in a [[matrix (mathematics)|matrix]] format that displays the (multivariate) [[frequency distribution]] of the variables. The term ''contingency table'' was first used by [[Karl Pearson]] in "On the Theory of Contingency and Its Relation to Association and Normal Correlation",<ref>http://ia600408.us.archive.org/18/items/cu31924003064833/cu31924003064833.pdf</ref> part of the ''[[Drapers' Company]] Research Memoirs Biometric Series I'' published in 1904.
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| A crucial problem of multivariate statistics is finding (direct-)dependence structure underlying the variables contained in high dimensional contingency tables. If some of the [[conditional independence]]s 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.
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| ==Example==
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| Suppose that we have 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 and right-handed, male and left-handed, female and right-handed, and female and left-handed. Such a contingency table is shown below.
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| <center> | |
| {| class="wikitable"
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| |-----
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| ! !! Right-handed !! Left-handed !! Total
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| |-----
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| ! Males
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| | 43 || 9 || 52
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| |-----
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| ! Females
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| | 44 || 4 || 48
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| |-----
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| ! Totals
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| | 87 || 13 || 100
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| |}</center>
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| The numbers of the males, females, and right- and left-handed individuals are called '''[[marginal distribution|marginal totals]]'''. The '''grand total''', i.e., the total number of individuals represented in the contingency table, is the number in the bottom right corner. | |
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| 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 although the proportions are not identical. The [[statistical significance|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|''G''-test]], [[Fisher's exact test]], and [[Barnard's test]], provided the entries in the table represent individuals randomly sampled from the population about which we want to draw a conclusion. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), we say that there is a ''contingency'' between the two variables. In other words, the two variables are ''not'' independent. If there is no contingency, we say that the two variables are ''independent''.
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| 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 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 difficult to represent on paper. The relation between [[ordinal variable]]s, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare.
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| ==Measures of association==
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| {{Main|Phi coefficient}}
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| {{Main|Cramér's V}}
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| The degree of association between the two variables can be assessed by a number of coefficients: the simplest is the [[phi coefficient]] defined by
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| : <math>\phi=\sqrt{\frac{\chi^2}{N}}</math>, | |
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| where χ<sup>2</sup> is derived from [[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). This coefficient can only be calculated for frequency data represented in 2 x 2 tables. φ can reach a minimum value -1.00 and a maximum value of 1.00 ''only'' when every marginal proportion is equal to .50 (and two diagonal cells are empty). Otherwise, the phi coefficient cannot reach those minimal and maximal values.<ref>Ferguson, G. A. (1966). ''Statistical analysis in psychology and education''. New York: McGraw-Hill.</ref>
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| Alternatives include the [[polychoric correlation|tetrachoric correlation coefficient]] (also only applicable to 2 × 2 tables), the ''contingency coefficient'' ''C'', and '''[[Cramér's V]]'''.
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| ''C'' suffers from the disadvantage that it does not reach a maximum of 1 or the minimum of -1; the highest it can reach in a 2 x 2 table is .707; the maximum it can reach in a 4 × 4 table is 0.870. It can reach values closer to 1 in contingency tables with more categories. It should, therefore, not be used to compare associations among tables with different numbers of categories.<ref>Smith, S. C., & Albaum, G. S. (2004) ''Fundamentals of marketing research''. Sage: Thousand Oaks, CA. p. 631</ref> Moreover, it does not apply to asymmetrical tables (those where the numbers of row and columns are not equal).
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| The formulae for the ''C'' and ''V'' coefficients are:
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| : <math>C=\sqrt{\frac{\chi^2}{N+\chi^2}}</math> and
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| : <math>V=\sqrt{\frac{\chi^2}{N(k-1)}}</math>,
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| ''k'' being the number of rows or the number of columns, whichever is less.
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| ''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 ''C'' by <math>\sqrt{\frac{k-1}{k}}</math> (recall that ''C'' only applies to tables in which the number of rows is equal to the number of columns and therefore equal to ''k'').
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| The [[polychoric correlation|tetrachoric correlation coefficient]] assumes that the variable underlying each dichotomous measure is normally distributed.<ref>Ferguson.</ref> The tetrachoric correlation coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."<ref>Ferguson, p. 244</ref> The tetrachoric correlation should not be confused with the [[Pearson product-moment correlation coefficient]] computed by assigning, say, values 0 and 1 to represent the two levels of each variable (which is mathematically equivalent to the phi coefficient). An extension of the tetrachoric correlation to tables involving variables with more than two levels is the [[polychoric correlation]] coefficient.
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| The '''[[Goodman and Kruskall's lambda|Lambda coefficient]]''' is a measure of the strength of association of the cross tabulations when the variables are measured at the [[Level of measurement|nominal level]]. Values range from 0 (no association) to 1 (the theoretical 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.
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| The [[uncertainty coefficient]] is another measure for variables at the nominal level.
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| The values range from -1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
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| *'''[[Gamma test (statistics)|Gamma test]]''': No adjustment for either table size or ties.
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| *'''[[Kendall tau]]''': Adjustment for ties.
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| **'''[[Tau b]]''': For square tables.
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| **'''[[Tau c]]''': For rectangular tables.
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| == See also ==
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| *The [[Pivot table|pivot]] operation in spreadsheet software can be used to generate a contingency table from sampling data.
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| *[[TPL Tables]] is a tool for generating and printing cross tabs.
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| *The [[iterative proportional fitting]] procedure essentially manipulates contingency tables to match altered joint distributions or marginal sums.
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| *The [[multivariate statistics]] in special multivariate discrete probability distributions. Some procedures used in this context can be used in dealing with contingency tables.
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| ==References==
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| {{reflist}}
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| * Andersen, Erling B. 1980. ''Discrete Statistical Models with Social Science Applications''. North Holland, 1980.
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| * {{cite book
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| |title=Discrete Multivariate Analysis: Theory and Practice
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| |last=Bishop |first=Y. M. M.
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| |first2=S. E. |last2=Fienberg |authorlink2=Stephen Fienberg
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| |first3=P. W. |last3=Holland
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| |year=1975
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| |publisher=MIT Press|isbn=978-0-262-02113-5 |mr=381130
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| }}
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| * {{cite book | last=Christensen | first=Ronald | title=Log-linear models and logistic regression | edition=Second | series=Springer Texts in Statistics | publisher=Springer-Verlag | location=New York | year=1997 | pages=xvi+483 | ISBN=0-387-98247-7 | mr=1633357 }}
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| * {{cite book | first=Steffen L. | last=Lauritzen | authorlink=Steffen L. Lauritzen | title=Lectures on Contingency Tables | year=2002 electronic (1979, 1982, 1989) | edition=updated electronic version of the (University of Aalborg) 3rd (1989) | url=http://www.stats.ox.ac.uk/~steffen/papers/cont.pdf }}
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| * {{cite book
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| |title=The Information in Contingency Tables
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| |last=Gokhale |first=D. V.
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| |first2=Solomon |last2=Kullback |authorlink2=Solomon Kullback
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| |year=1978
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| |publisher=Marcel Dekker|isbn=0-824-76698-9
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| }}
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| ==External links==
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| {{commons category|Contingency tables}}
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| * [http://www.physics.csbsju.edu/stats/contingency.html On-line analysis of contingency tables: calculator with examples]
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| * [http://people.revoledu.com/kardi/tutorial/Questionnaire/ContingencyTable.html Interactive cross tabulation, chi-squared independent test & tutorial]
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| * [http://statpages.org/ctab2x2.html Fisher and chi-squared calculator of 2 × 2 contingency table ]
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| *[http://www.andrews.edu/~calkins/math/edrm611/edrm13.htm More Correlation Coefficients]
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| *[http://www2.chass.ncsu.edu/garson/pa765/assocnominal.htm Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient]
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| * [http://www.custominsight.com/articles/crosstab-sample.asp Customer Insight com Cross Tabulation]
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| * [http://www.csupomona.edu/~jlkorey/POWERMUTT/Topics/displaying_categorical_data.html The POWERMUTT Project: IV. DISPLAYING CATEGORICAL DATA]
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| * [http://www.childrensmercy.org/stats/journal/oddsratio.asp StATS: Steves Attempt to Teach Statistics Odds ratio versus relative risk (January 9, 2001)]
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| * [ftp://ftp.cdc.gov/pub/Software/epi_info/EIHAT_WEB/Lesson5AnalysisCreatingStatistics.pdf Epi Info Community Health Assessment Tutorial Lesson 5 Analysis: Creating Statistics]
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| {{Statistics|descriptive}}
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| [[Category:Categorical data]]
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| [[Category:Data analysis]]
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| [[Category:Statistical dependence]]
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