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Cramér's V (φc) |
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In statistics, Cramér's V (sometimes referred to as Cramér's phi or Cramers C and denoted as φc) is a popular[citation needed] 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 (ordered, numerical, etc)
Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1×k table (e.g: 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 the two variables are equal to each 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]
Calculation[]
Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (k is the smaller of the number of rows r or columns c).
The formula for the φc coefficient is:
where:
- is the phi coefficient.
- is derived from Pearson's chi-squared test
- is the grand total of observations and
- being the number of rows or the number of columns, whichever is less.
The p-value for the significance of φc is the same one that is calculated using the Pearson's chi-squared test [citation needed].
The formula for the variance of φc is known.[3]
Unlike the contingency tablethe value of V is relatively independent of the number of columns or rows.
See also[]
Other measures of correlation for nominal data:
- The phi coefficient
- Tschuprow's T
- The uncertainty coefficient
- The Lambda coefficient
Other related articles:
References[]
- ↑ Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, p282. ISBN 0-691-08004-6
- ↑ 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)
- Cramér, H. (1999). Mathematical Methods of Statistics, Princeton University Press
External links[]
- A Measure of Association for Nonparametric Statistics (Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386)
- Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient
Statistics | |
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Descriptive statistics |
Mean (Arithmetic, Geometric) - Median - Mode - Power - Variance - Standard deviation |
Inferential statistics |
Hypothesis testing - Significance - Null hypothesis/Alternate hypothesis - Error - Z-test - Student's t-test - Maximum likelihood - Standard score/Z score - P-value - Analysis of variance |
Survival analysis |
Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models |
Probability distributions | |
Correlation |
Confounding variable - Pearson product-moment correlation coefficient - Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient) |
Regression analysis |
Linear regression - Nonlinear regression - Logistic regression |
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