HoeffD {DescTools}R Documentation

Matrix of Hoeffding's D Statistics


Computes a matrix of Hoeffding's (1948) D statistics for all possible pairs of columns of a matrix. D is a measure of the distance between F(x,y) and G(x)H(y), where F(x,y) is the joint CDF of X and Y, and G and H are marginal CDFs. Missing values are deleted in pairs rather than deleting all rows of x having any missing variables. The D statistic is robust against a wide variety of alternatives to independence, such as non-monotonic relationships. The larger the value of D, the more dependent are X and Y (for many types of dependencies). D used here is 30 times Hoeffding's original D, and ranges from -0.5 to 1.0 if there are no ties in the data. print.HoeffD prints the information derived by HoeffD. The higher the value of D, the more dependent are x and y.


HoeffD(x, y)
## S3 method for class 'HoeffD'
print(x, ...)



a numeric matrix with at least 5 rows and at least 2 columns (if y is absent), or an object created by HoeffD


a numeric vector or matrix which will be concatenated to x




Uses midranks in case of ties, as described by Hollander and Wolfe. P-values are approximated by linear interpolation on the table in Hollander and Wolfe, which uses the asymptotically equivalent Blum-Kiefer-Rosenblatt statistic. For P<.0001 or >0.5, P values are computed using a well-fitting linear regression function in log P vs. the test statistic. Ranks (but not bivariate ranks) are computed using efficient algorithms (see reference 3).


a list with elements D, the matrix of D statistics, n the matrix of number of observations used in analyzing each pair of variables, and P, the asymptotic P-values. Pairs with fewer than 5 non-missing values have the D statistic set to NA. The diagonals of n are the number of non-NAs for the single variable corresponding to that row and column.


Frank Harrell <f.harrell@vanderbilt.edu>
Department of Biostatistics
Vanderbilt University


Hoeffding W. (1948) A non-parametric test of independence. Ann Math Stat 19:546–57.

Hollander M., Wolfe D.A. (1973) Nonparametric Statistical Methods, pp. 228–235, 423. New York: Wiley.

Press W.H., Flannery B.P., Teukolsky S.A., Vetterling, W.T. (1988) Numerical Recipes in C Cambridge: Cambridge University Press.

See Also

rcorr, varclus


x <- c(-2, -1, 0, 1, 2)
y <- c(4,   1, 0, 1, 4)
z <- c(1,   2, 3, 4, NA)
q <- c(1,   2, 3, 4, 5)

HoeffD(cbind(x, y, z, q))

# Hoeffding's test can detect even one-to-many dependency
x <- seq(-10, 10, length=200)
y <- x * sign(runif(200, -1, 1))
plot(x, y)

HoeffD(x, y)

[Package DescTools version 0.99.51 Index]