Description Usage Arguments Value P-value Ties Big Data References See Also Examples
The function hoeffding.D.test
provides
independence testing for two continuous numeric variables,
that is consistent for absolutely-continuous alternative
bivariate distributions.
It implements the classical D statistic by Hoeffding,
which in terms of CDFs estimates the integral of (Fxy-Fx*Fy)^2 dFxy.
It may also be expressed in terms of the ordering types of quintuples
of data points.
Its efficient O(n log n) computation seems to be new in R.
1 | hoeffding.D.test(xs, ys, na.rm = TRUE, collisions = TRUE, precision = 1e-05)
|
xs, ys |
Same-length numeric vectors, containing paired samples. |
na.rm |
Logical: Should missing values, |
collisions |
Logical: Warn of repeating values in |
precision |
of p-value, between 0 and 1. Otherwise p-value= |
A list, of class "indtest"
:
method | the test's name |
n | number of data points used |
Tn /Dn /Rn /... | the test statistic, measure of dependence |
scaled | the test statistic rescaled for a standard null distribution |
p.value |
the asymptotic p-value, by TauStar::pHoeffInd |
The null distribution of the test statistic was described by Hoeffding.
The p-value is approximated by calling the function
pHoeffInd
from the package TauStar
by
Luca Weihs.
By default, the p-value's precision
parameter is set to 1e-5
.
It seems that better precision would cost a considerable amount of time,
especially for large values of the test statistic.
It is therefore recommended to modify this parameter only upon need.
In case that TauStar
is unavailable, or to save time in repeated use,
set precision = 1
to avoid computing p-values altogether.
The scaled
test statistic may be used instead.
Its asymptotic distribution does not depend on any parameter.
Also the raw test statistic may be used, descriptively,
as a measure of dependence.
Only its accuracy depends on the sample size.
This package currently assumes that the variables under consideration are non-atomic, so that ties are not expected, other than by occasional effects of numerical precision. Addressing ties rigorously is left for future versions.
The flag collisions = TRUE
invokes checking for ties in xs
and in ys
, and produces an appropriate warning if they exist.
The current implementation breaks such ties arbitrarily, not randomly.
By the averaging nature of the test statistic, it seems that a handful of ties should not be of much concern. In case of more than a handful of ties, our current advice to the user is to break them uniformly at random beforehand.
The test statistic is computed in almost linear time, O(n log n), given a sample of size n. Its computation involves integer arithmetics of order n^4 or n^5, which should fit into an integer data type supported by the compiler.
Most 64-bit compilers emulate 128-bit arithmetics. Otherwise we use the standard 64-bit arithmetics. Find the upper limits of your environment using
max_taustar()
max_hoeffding()
Another limitation is 2^31-1, the maximum size and value of
an integer vector in a 32-bit build of R.
This is only relevant for the tau star statistic in 128-bit mode,
which could otherwise afford about three times that size.
If your sample size falls in this range, try recompiling the
function .calc.taustar
according to the instructions in the cpp source file.
Hoeffding, Wassily. "A non-parametric test of independence."
The annals of mathematical statistics (1948): 546-557.
Luca Weihs (2019). TauStar: Efficient Computation and Testing of the
Bergsma-Dassios Sign Covariance. R package version 1.1.4.
https://CRAN.R-project.org/package=TauStar
Frank E Harrell Jr, with contributions from Charles Dupont and many
others. (2020). Hmisc: Harrell Miscellaneous. R package version
4.4-0. https://CRAN.R-project.org/package=Hmisc
Even-Zohar, Chaim. "independence: Fast Rank Tests."
arXiv preprint arXiv:2010.09712 (2020).
independence
,
tau.star.test
,
hoeffding.refined.test
1 2 3 4 5 6 7 | ## independent, $p.value is 0.2582363
set.seed(123)
hoeffding.D.test(rnorm(10000),rnorm(10000))
## dependent, even though uncorrelated, $p.value is 0.0002891223
set.seed(123)
hoeffding.D.test(rnorm(10000,0,3001:13000), rnorm(10000,0,3001:13000))
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.