Cramer's V

Measures the strength of association between two categorical variables. These can be provided as two data vectors x and y, or as a contingency table (see Association).

Usage

cramerV(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  method = c("ncchisq", "ncchisqadj", "fisher", "fisheradj"),
  correct = FALSE,
  ...
)

Arguments

x: either a contingency table, a two-column object (matrix, data.frame or list), or a vector of observations (together with y)
y: optional second vector. If x is not a vector, y must be NULL.
conf.level: confidence level of the interval. If set to NA (the default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().
method: character string specifying the confidence interval method: "ncchisq" (default, using the noncentral chi-squared distribution), "ncchisqadj", "fisher" (using fisher z transformation), or "fisheradj" (using the Fisher z transformation and bias correction)
correct: logical; whether to apply the bias correction of Bergsma (2013); defaults to FALSE
...: further arguments, passed on to normalizeToConfusion() and table() for building the table - useNA is the usual one.

Value

if conf.level = NA, a numeric scalar containing Cramer's V; otherwise a named numeric vector with elements:

est: point estimate of Cramer's V.
lci: lower confidence interval bound.
uci: upper confidence interval bound.

Details

Cramer's V ranges from 0 to 1, with 0 indicating statistical independence.

correct = TRUE applies Bergsma's bias correction to the point estimate and to the interval: both are put through the same transformation of the chi-squared statistic, so estimate and bounds always live on the same scale. Formerly the two noncentral chi-squared methods returned an uncorrected interval around a corrected estimate.

All four methods are analytical; there is no bootstrap here, and none is needed - unlike contCoef, where no generally accepted analytical interval exists.

For a one-sided interval the bound is computed at the adjusted level \(2\gamma - 1\) and the open side is closed at the boundary of the attainable range, 0 or 1, rather than at an infinity V can never reach. sides = "left" names the side carrying the finite bound and is the analogue of alternative = "greater" in a test.

The two Fisher methods rest on \(\mathrm{Var}(\mathrm{atanh}(V)) \approx 1/(n-3)\). Where that approximation has nothing to say - three or fewer observations, or a perfect association, where the transformation is infinite - the bounds are NA and a warning names the reason. The estimate itself is still returned.

Note

Based on code by Michael Smithson (confidence intervals), adapted to conform to package standards.

References

Cramer, H. (1946) Mathematical Methods of Statistics. Princeton University Press

Agresti, Alan (1996) Introduction to categorical data analysis. NY: John Wiley and Sons

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

Examples


tab <- table(Pizza$driver, Pizza$wine_delivered)
cramerV(tab)
#> [1] 0.1328222

# just x and y
cramerV(Pizza$driver, Pizza$wine_delivered)
#> [1] 0.1328222

# data.frame
bedrock::pairApply(Pizza[, c("driver", "operator", "area")], cramerV,
                   symmetric = TRUE)
#>             driver   operator       area
#> driver   1.0000000 0.23585686 0.65018461
#> operator 0.2358569 1.00000000 0.08670047
#> area     0.6501846 0.08670047 1.00000000

# useNA is passed on to table()
bedrock::pairApply(Pizza[, c("driver", "operator", "area")], cramerV,
                   useNA = "ifany", symmetric = TRUE)
#>             driver   operator       area
#> driver   1.0000000 0.20253639 0.53066544
#> operator 0.2025364 1.00000000 0.07847762
#> area     0.5306654 0.07847762 1.00000000

d.frm <- Pizza[, c("driver", "operator", "area")]
bedrock::pairApply(d.frm[complete.cases(d.frm), ], cramerV, symmetric = TRUE)
#>             driver  operator      area
#> driver   1.0000000 0.2345141 0.6504665
#> operator 0.2345141 1.0000000 0.0869935
#> area     0.6504665 0.0869935 1.0000000

# one-sided: "left" carries the finite lower bound, the upper one opens
# to the maximum V can attain
cramerV(tab, conf.level = 0.95, sides = "left")
#>        est        lci        uci 
#> 0.13282221 0.06256252 1.00000000 


# 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 <- bedrock::untable(tab)

set.seed(1)
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))
#>      2.5%     97.5% 
#> 0.2771562 0.5591357 

# compare this to the analytical ones
cramerV(tab, conf.level = 0.95)
#>       est       lci       uci 
#> 0.4064888 0.2211575 0.5410535