# Cramer's V, Pearson's Contingency Coefficient and Phi Coefficient Yule's Q and Y, Tschuprow's T Calculate Cramer's V, Pearson's contingency coefficient and phi, Yule's Q and Y and Tschuprow's T of `x`, if `x` is a table. If both, `x` and `y` are given, then the according table will be built first. ```r Phi(x, y = NULL, ...) ContCoef(x, y = NULL, correct = FALSE, ...) CramerV(x, y = NULL, conf.level = NA, method = c("ncchisq", "ncchisqadj", "fisher", "fisheradj"), correct = FALSE, ...) YuleQ(x, y = NULL, ...) YuleY(x, y = NULL, ...) TschuprowT(x, y = NULL, correct = FALSE, ...) ``` ## Arguments - `x`: can be a numeric vector, a matrix or a table. - `y`: NULL (default) or a vector with compatible dimensions to `x`. If y is provided, `table(x, y, ...)` is calculated. - `conf.level`: confidence level of the interval. This is only implemented for Cramer's V. If set to `NA` (which is the default) no confidence interval will be calculated. See examples for calculating bootstrap intervals. - `method`: string defining the method to calculate confidence intervals for Cramer's V. One out of `"ncchisq"` (using noncentral chisquare), `"ncchisqadj"`, `"fisher"` (using fisher z transformation), `"fisheradj"` (using fisher z transformation and bias correction). Default is `"ncchisq"`. - `correct`: logical. Applying to `ContCoef` this indicates, whether the Sakoda's adjusted Pearson's C should be returned. For `CramerV()` and `TschuprowT()` it defines, whether a bias correction should be applied or not. Default is `FALSE`. - ``...``: further arguments are passed to the function `table`, allowing i.e. to set `useNA`. ## Details For x either a matrix or two vectors `x` and `y` are expected. In latter case `table(x, y, ...)` is calculated. The function handles `NAs` the same way the `table` function does, so tables are by default calculated with `NAs` omitted. A provided matrix is interpreted as a contingency table, which seems in the case of frequency data the natural interpretation (this is e.g. also what `chisq.test` expects). Use the function `PairApply` (pairwise apply) if the measure should be calculated pairwise for all columns. This allows matrices of association measures to be calculated the same way `cor` does. `NAs` are by default omitted pairwise, which corresponds to the `pairwise.complete` option of `cor`. Use `complete.cases`, if only the complete cases of a `data.frame` are to be used. (see examples) The maximum value for Phi is $\sqrt(min(r, c) - 1)$. The contingency coefficient goes from 0 to $\sqrt(\frac{min(r, c) - 1}{min(r, c)})$. For the corrected contingency coefficient and for Cramer's V the range is 0 to 1. A Cramer's V in the range of [0, 0.3] is considered as weak, [0.3,0.7] as medium and > 0.7 as strong. The minimum value for all is 0 under statistical independence. ## Returns a single numeric value if no confidence intervals are requested, and otherwise a numeric vector with 3 elements for the estimate, the lower and the upper confidence interval ## References Yule, G. Uday (1912) On the methods of measuring association between two attributes. **Journal of the Royal Statistical Society, LXXV**, 579-652 Tschuprow, A. A. (1939) **Principles of the Mathematical Theory of Correlation**, translated by M. Kantorowitsch. W. Hodge & Co. Cramer, H. (1946) **Mathematical Methods of Statistics**. Princeton University Press Agresti, Alan (1996) **Introduction to categorical data analysis**. NY: John Wiley and Sons Sakoda, J.M. (1977) Measures of Association for Multivariate Contingency Tables, **Proceedings of the Social Statistics Section of the American Statistical Association** (Part III), 777-780. Smithson, M.J. (2003) **Confidence Intervals, Quantitative Applications in the Social Sciences Series**, No. 140. Thousand Oaks, CA: Sage. pp. 39-41 Bergsma, W. (2013) A bias-correction for Cramer's V and Tschuprow's T **Journal of the Korean Statistical Society** 42(3) DOI: 10.1016/j.jkss.2012.10.002 ## Author(s) Andri Signorell , Michael Smithson (confidence intervals for Cramer V) ## See Also `table`, `PlotCorr`, `PairApply`, `Assocs` ## Examples ```r tab <- table(d.pizza$driver, d.pizza$wine_delivered) Phi(tab) ContCoef(tab) CramerV(tab) TschuprowT(tab) # just x and y CramerV(d.pizza$driver, d.pizza$wine_delivered) # data.frame PairApply(d.pizza[,c("driver","operator","area")], CramerV, symmetric = TRUE) # useNA is passed to table PairApply(d.pizza[,c("driver","operator","area")], CramerV, useNA="ifany", symmetric = TRUE) d.frm <- d.pizza[,c("driver","operator","area")] PairApply(d.frm[complete.cases(d.frm),], CramerV, symmetric = TRUE) m <- as.table(matrix(c(2,4,1,7), nrow=2)) YuleQ(m) YuleY(m) # Bootstrap confidence intervals for Cramer's V # http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf, p. 1821 tab <- as.table(rbind( c(26,26,23,18, 9), c( 6, 7, 9,14,23))) d.frm <- Untable(tab) n <- 1000 idx <- matrix(sample(nrow(d.frm), size=nrow(d.frm) * n, replace=TRUE), ncol=n, byrow=FALSE) v <- apply(idx, 2, function(x) CramerV(d.frm[x,1], d.frm[x,2])) quantile(v, probs=c(0.025,0.975)) # compare this to the analytical ones CramerV(tab, conf.level=0.95) ```

CramerV function