1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
|
\name{hoeffd}
\alias{hoeffd}
\alias{print.hoeffd}
\title{
Matrix of Hoeffding's D Statistics
}
\description{
Computes a matrix of Hoeffding's (1948) \code{D} statistics for all possible
pairs of columns of a matrix. \code{D}
is a measure of the distance
between \code{F(x,y)} and \code{G(x)H(y)}, where \code{F(x,y)} is the joint CDF of \code{X} and \code{Y},
and \code{G} and \code{H} are marginal CDFs. Missing values are deleted in pairs rather than deleting all rows
of \code{x} having any missing variables.
The \code{D} statistic is robust against a wide
variety of alternatives to independence, such as non-monotonic relationships.
The larger the value of \code{D}, the more dependent are \code{X} and \code{Y} (for many types
of dependencies). \code{D} used here is 30 times Hoeffding's original \code{D}, and
ranges from -0.5 to 1.0 if there are no ties in the data.
\code{print.hoeffd} prints the information derived by \code{hoeffd}. The higher
the value of \code{D}, the more dependent are \code{x} and \code{y}.
}
\synopsis{hoeffd(x, y)}
\usage{
hoeffd(x)
hoeffd(x, y)
\method{print}{hoeffd}(x, \dots)
}
\arguments{
\item{x}{
a numeric matrix with at least 5 rows and at least 2 columns (if
\code{y} is absent), or an object created by \code{hoeffd}
}
\item{y}{
a numeric vector or matrix which will be concatenated to \code{x}
}
\item{\dots}{ignored}
}
\value{
a list with elements \code{D}, the
matrix of D statistics, \code{n} the
matrix of number of observations used in analyzing each pair of variables,
and \code{P}, the asymptotic P-values.
Pairs with fewer than 5 non-missing values have the D statistic set to NA.
The diagonals of \code{n} are the number of non-NAs for the single variable
corresponding to that row and column.
}
\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 \code{P<.0001} or \code{>0.5}, \code{P} values are
computed using a well-fitting linear regression function in \code{log P} vs.
the test statistic.
Ranks (but not bivariate ranks) are computed using efficient
algorithms (see reference 3).
}
\author{
Frank Harrell
\cr
Department of Biostatistics
\cr
Vanderbilt University
\cr
f.harrell@vanderbilt.edu
}
\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.
}
\seealso{
\code{\link{rcorr}}, \code{\link{varclus}}
}
\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)
}
\keyword{nonparametric}
\keyword{htest}
% Converted by Sd2Rd version 1.21.
|
