Cramér's V
In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φc) 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 and was published by Harald Cramér in 1946.[1]
Usage and interpretation[]
φc is the intercorrelation of two discrete variables[2] and may be used with variables having two or more levels. φc 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 φc may be used with nominal data types or higher (notably, ordered or numerical).
Cramér's V may also be applied to goodness of fit 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.[citation needed]
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.
φc2 is the mean square canonical correlation between the variables.[citation needed]
In the case of a 2 × 2 contingency table Cramér's V is equal to the Phi coefficient.
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.[citation needed]
V may be viewed as the association between two variables as a percentage of their maximum possible variation. V2 is the mean square canonical correlation between the variables.[citation needed]
Calculation[]
Let a sample of size n of the simultaneously distributed variables and for be given by the frequencies
- number of times the values were observed.
The chi-squared statistic then is:
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:
where:
- is the phi coefficient.
- is derived from Pearson's chi-squared test
- is the grand total of observations and
- being the number of columns.
- being the number of rows.
The p-value for the significance of V is the same one that is calculated using the Pearson's chi-squared test.[citation needed]
The formula for the variance of V=φc is known.[3]
In R, the function cramerV()
from the package rcompanion
[4] calculates V using the chisq.test function from the stats package. In contrast to the function cramersV()
from the lsr
[5] 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[6]
where
and
Then estimates the same population quantity as Cramér's V but with typically much smaller mean squared error. The rationale for the correction is that under independence, .[7]
See also[]
Other measures of correlation for nominal data:
- The phi coefficient
- Tschuprow's T
- The uncertainty coefficient
- The Lambda coefficient
- The Rand index
- Davies–Bouldin index
- Dunn index
- Jaccard index
- Fowlkes–Mallows index
Other related articles:
- Contingency table
- Effect size
- Cluster analysis § External evaluation
References[]
- ^ Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, page 282 (Chapter 21. The two-dimensional case). ISBN 0-691-08004-6 (table of content Archived 2016-08-16 at the Wayback Machine)
- ^ Sheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press.
- ^ Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32. (pages 15–16)
- ^ "Rcompanion: Functions to Support Extension Education Program Evaluation". 2019-01-03.
- ^ "Lsr: Companion to "Learning Statistics with R"". 2015-03-02.
- ^ Bergsma, Wicher (2013). "A bias correction for Cramér's V and Tschuprow's T". Journal of the Korean Statistical Society. 42 (3): 323–328. doi:10.1016/j.jkss.2012.10.002.
- ^ Bartlett, Maurice S. (1937). "Properties of Sufficiency and Statistical Tests". Proceedings of the Royal Society of London. Series A. 160 (901): 268–282. doi:10.1098/rspa.1937.0109. JSTOR 96803.
External links[]
- A Measure of Association for Nonparametric Statistics (Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386)
- Nominal Association: Phi and Cramer's Vl[dead link] from the homepage of Pat Dattalo.
- Statistical ratios
- Summary statistics for contingency tables
- Covariance and correlation