In
statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of
table
Table may refer to:
* Table (furniture), a piece of furniture with a flat surface and one or more legs
* Table (landform), a flat area of land
* Table (information), a data arrangement with rows and columns
* Table (database), how the table data ...
in a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
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
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
in "On the Theory of Contingency and Its Relation to Association and Normal Correlation", part of the ''
Drapers' Company
The Worshipful Company of Drapers is one of the 110 livery companies of the City of London. It has the formal name The Master and Wardens and Brethren and Sisters of the Guild or Fraternity of the Blessed Mary the Virgin of the Mystery of Dr ...
Research Memoirs Biometric Series I'' published in 1904.
A crucial problem of
multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
is finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the
conditional independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabil ...
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
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
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
A pivot table is a table of grouped values that aggregates the individual items of a more extensive table (such as from a database, spreadsheet, or business intelligence program) within one or more discrete categories. This summary might include ...
is a way to create contingency tables using spreadsheet software.
Example
Suppose there are two variables, sex (male or female) and
handedness
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjec ...
(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.
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
An odds ratio (OR) is a statistic that quantifies the strength of the association between 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 (du ...
, 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
Pearson's chi-squared test (\chi^2) 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 (e.g ...
, the
''G''-test,
Fisher's exact test
Fisher's exact test is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample sizes are small, it is valid for all sample sizes. It is named after its inventor, Ronald Fisher, a ...
,
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
In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both ...
.
Standard contents of a contingency table
* 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).
Measures of association
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.
Odds ratio
The simplest measure of association for a 2 × 2 contingency table is the
odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between 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 (du ...
. 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:
:
Phi coefficient
A simple measure, applicable only to the case of 2 × 2 contingency tables, is the
phi coefficient
In statistics, the phi coefficient (or mean square contingency coefficient and denoted by φ or rφ) is a measure of association for two binary variables. In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a ...
(φ) defined by
:
where is computed as in
Pearson's chi-squared test
Pearson's chi-squared test (\chi^2) 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 (e.g ...
, 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
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
every marginal proportion is equal to 0.5 (and two diagonal cells are empty).
Cramér's ''V'' and the contingency coefficient ''C''
Two alternatives are the ''contingency coefficient'' ''C'', and
Cramér's V In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φ''c'') 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 an ...
.
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.
''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 , or by