Tschuprow's T

Computes Tschuprow's T, a measure of association between two categorical variables based on the chi-squared statistic.

Usage

tschuprowT(x, y = NULL, correct = FALSE, ...)

Arguments

x: a vector of categorical data (then y must be given) or a two-dimensional contingency table (matrix or table)
y: optional second categorical vector. If provided, a contingency table is constructed from x and y.
correct: logical; if TRUE, applies a bias correction according to Bergsma (2013).
...: additional arguments passed to base::table(). This refers only to the vector interface.

Value

a numeric scalar containing Tschuprow's T

Details

If y is provided, a contingency table is created using table(x, y, ...). Otherwise, x is assumed to already be a two-dimensional contingency table.

Tschuprow's T is defined as: $$ T = \sqrt{ \frac{\chi^2}{n \sqrt{(r - 1)(c - 1)}} } $$ where $\chi^2$ is the chi-squared statistic, $n$ is the total sample size, and $r$ and $c$ are the number of rows and columns of the contingency table.

If correct = TRUE, a bias-corrected version is computed based on Bergsma (2013), which adjusts the estimate especially for small samples. It replaces $\phi^2 = \chi^2/n$ by $\tilde\phi^2 = \max(0, \phi^2 - (r-1)(c-1)/(n-1))$ and the dimensions by $\tilde r = r - (r-1)^2/(n-1)$ and $\tilde c = c - (c-1)^2/(n-1)$.

For a 2x2 table T coincides with Cramer's V and with the absolute value of the phi coefficient; the sign of the association is not reported.

References

Tschuprow, A. A. (1939). Principles of the Mathematical Theory of Correlation. W. Hodge & Co.

Bergsma, W. (2013). A bias-correction for Cramer's V and Tschuprow's T. Journal of the Korean Statistical Society, 42(3), 323–328. https://doi.org/10.1016/j.jkss.2012.10.002

Examples

# Example with vectors
x <- c("A", "A", "B", "B")
y <- c("yes", "no", "yes", "no")
tschuprowT(x, y)
#> [1] 0

# Example with contingency table
tab <- matrix(c(10, 20, 30, 40), nrow = 2)
tschuprowT(tab)               # 0.08908708
#> [1] 0.08908708

# Bias-corrected version: the correction exceeds the estimate here,
# so the corrected value is 0
tschuprowT(tab, correct = TRUE)
#> [1] 0