Cramér's V (statistics)
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In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Cramér's V (sometimes referred to as Cramér's phi and denoted as φ''c'') is a measure of
association Association may refer to: *Club (organization), an association of two or more people united by a common interest or goal *Trade association, an organization founded and funded by businesses that operate in a specific industry *Voluntary associatio ...
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 Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statist ...
in 1946.


Usage and interpretation

φ''c'' is the intercorrelation of two discrete variablesSheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press. and may be used with variables having two or more levels. φ''c'' is a symmetrical measure: it does not matter which variable we place in the columns and which in the rows. Also, the order of rows/columns doesn't matter, so φ''c'' may be used with nominal data types or higher (notably, ordered or numerical). Cramér's V may also be applied to
goodness of fit The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
chi-squared models when there is a 1 × ''k'' table (in this case ''r'' = 1). In this case ''k'' is taken as the number of optional outcomes and it functions as a measure of tendency towards a single outcome. Cramér's V varies from 0 (corresponding to no association between the variables) to 1 (complete association) and can reach 1 only when each variable is completely determined by the other. It may be viewed as the association between two variables as a percentage of their maximum possible variation. φ''c''2 is the mean square
canonical correlation 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'' ...
between the variables. In the case of a 2 × 2
contingency table 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 i ...
Cramér's V is equal to the absolute value of
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 ...
. Note that as chi-squared values tend to increase with the number of cells, the greater the difference between ''r'' (rows) and ''c'' (columns), the more likely φc will tend to 1 without strong evidence of a meaningful correlation.Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Routledge. https://doi.org/10.4324/9780203771587 p78-81.


Calculation

Let a sample of size ''n'' of the simultaneously distributed variables A and B for i=1,\ldots,r; j=1,\ldots,k be given by the frequencies :n_= number of times the values (A_i,B_j) were observed. The chi-squared statistic then is: :\chi^2=\sum_\frac\;, where n_=\sum_jn_ is the number of times the value A_i is observed and n_=\sum_in_ is the number of times the value B_j is observed. Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the minimum dimension minus 1: :V = \sqrt = \sqrt\;, where: * \varphi is the phi coefficient. * \chi^2 is derived from Pearson's chi-squared test * n is the grand total of observations and * k being the number of columns. * r being the number of rows. The
p-value In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means ...
for the significance of ''V'' is the same one that is calculated using the
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 formula for the variance of ''V''=φ''c'' is known. In R, the function cramerV() from the package rcompanion calculates ''V'' using the chisq.test function from the stats package. In contrast to the function cramersV() from the lsr package, cramerV() also offers an option to correct for bias. It applies the correction described in the following section.


Bias correction

Cramér's V can be a heavily biased estimator of its population counterpart and will tend to overestimate the strength of association. A bias correction, using the above notation, is given by :\tilde V = \sqrt   where : \tilde\varphi^2 = \max\left(0,\varphi^2 - \frac\right)   and : \tilde k = k - \frac   : \tilde r = r - \frac   Then \tilde V estimates the same population quantity as Cramér's V but with typically much smaller
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
. The rationale for the correction is that under independence, E varphi^2\frac.


See also

Other measures of correlation for nominal data: * 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 ...
*
Tschuprow's T 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 ...
* The
uncertainty coefficient In statistics, the uncertainty coefficient, also called proficiency, entropy coefficient or Theil's U, is a measure of nominal association. It was first introduced by Henri Theil and is based on the concept of information entropy. Definition Sup ...
* The Lambda coefficient * The
Rand index The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is financed ...
*
Davies–Bouldin index The Davies–Bouldin index (DBI), introduced by David L. Davies and Donald W. Bouldin in 1979, is a metric for evaluating clustering algorithms. This is an internal evaluation scheme, where the validation of how well the clustering has been d ...
*
Dunn index The Dunn index (DI) (introduced by J. C. Dunn in 1974) is a metric for evaluating clustering algorithms. This is part of a group of validity indices including the Davies–Bouldin index or Silhouette (clustering), Silhouette index, in that it is an ...
*
Jaccard index The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets. It was developed by Grove Karl Gilbert in 1884 as his ratio of verification (v) and now is freque ...
*
Fowlkes–Mallows index The Fowlkes–Mallows index is an external evaluation method that is used to determine the similarity between two clusterings (clusters obtained after a clustering algorithm), and also a metric to measure confusion matrices. This measure of simi ...
Other related articles: *
Contingency table 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 i ...
*
Effect size In statistics, an effect size is a value measuring the strength of the relationship between two variables in a 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 ...
*


References


External links


A Measure of Association for Nonparametric Statistics
(Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386)

from the homepage of Pat Dattalo. {{DEFAULTSORT:Cramer's V Statistical ratios Summary statistics for contingency tables Covariance and correlation