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 also computes the mean and maximum
absolute values of the difference between the joint empirical CDF and
the product of the marginal empirical CDFs.
hoeffd(x, y) ## S3 method for class 'hoeffd' print(x, ...)
x |
a numeric matrix with at least 5 rows and at least 2 columns (if
|
y |
a numeric vector or matrix which will be concatenated to |
... |
ignored |
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
Department of Biostatistics
Vanderbilt University
fh@fharrell.com
Hoeffding W. (1948): A non-parametric test of independence. Ann Math Stat 19:546–57.
Hollander M. and Wolfe D.A. (1973). Nonparametric Statistical Methods, pp. 228–235, 423. New York: Wiley.
Press WH, Flannery BP, Teukolsky SA, Vetterling, WT (1988): Numerical Recipes in C. Cambridge: Cambridge University Press.
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 set.seed(1) x <- seq(-10,10,length=200) y <- x*sign(runif(200,-1,1)) plot(x,y) hoeffd(x,y)
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