hoeffd {Hmisc} | R Documentation |
Matrix of Hoeffding's D Statistics
Description
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.
Usage
hoeffd(x, y)
## S3 method for class 'hoeffd'
print(x, ...)
Arguments
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 |
Details
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).
Value
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.
Author(s)
Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com
References
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.
See Also
Examples
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)