Association measures
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.
 Keywords
 multivariate
Usage
Phi(x, y = NULL, ...)
ContCoef(x, y = NULL, correct = FALSE, ...)
CramerV(x, y = NULL, conf.level = NA, method = c("ncchisq", "ncchisqadj", "fisher", "fisheradj"), ...)
YuleQ(x, y = NULL, ...)
YuleY(x, y = NULL, ...)
TschuprowT(x, y = NULL, ...)
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 how to compute 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. This argument only applies for
ContCoef
and indicates, whether the Sakoda's adjusted Pearson's C should be returned. Default isFALSE
.  ...
 further arguments are passed to the function
table
, allowing i.e. to setuseNA
.
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.
Value

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, 579652
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), 777780.
Smithson, M.J. (2003) Confidence Intervals, Quantitative Applications in the Social Sciences Series, No. 140. Thousand Oaks, CA: Sage. pp. 3941
See Also
Examples
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)