tau.star.test: The Bergsma-Dassios tau* independence test
In independence: Fast Rank-Based Independence Testing
Description
Usage
Arguments
Value
P-value
Ties
Big Data
References
See Also
Examples
Description
The function tau.star.test provides an
independence test for two continuous numeric variables,
that is consistent for all alternatives.
It is based on the Yanagimoto-Bergsma-Dassios coefficient,
which compares the frequencies of concordant and discordant
quadruples of data points.
The test statistic is efficiently computed in O(n log n) time,
following work of Even-Zohar and Leng.
Usage
1
tau.star.test(xs, ys, na.rm = TRUE, collisions = TRUE, precision = 1e-05)
Arguments
xs, ys
Same-length numeric vectors, containing paired samples.
na.rm
Logical: Should missing values, NaN, and Inf be removed?
collisions
Logical: Warn of repeating values in xs or ys.
precision
of p-value, between 0 and 1. Otherwise p-value=NA.
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
P-value
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.
Ties
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.
Big Data
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.
References
Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based
on a sign covariance related to Kendall's tau. Bernoulli 20 (2014), no.
2, 1006–1028.
Yanagimoto, Takemi. "On measures of association and a related problem."
Annals of the Institute of Statistical Mathematics 22.1 (1970): 57-63.
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
Even-Zohar, Chaim, and Calvin Leng. "Counting Small Permutation Patterns."
arXiv preprint arXiv:1911.01414 (2019).
Even-Zohar, Chaim. "independence: Fast Rank Tests."
arXiv preprint arXiv:2010.09712 (2020).
See Also
independence,
hoeffding.D.test,
hoeffding.refined.test
Examples
1
2
3
4
5
6
7
## independent, $p.value is 0.2585027
set.seed(123)
tau.star.test(rnorm(10000),rnorm(10000))
## dependent, even though uncorrelated, $p.value is 0.000297492
set.seed(123)
tau.star.test(rnorm(10000,0,3001:13000), rnorm(10000,0,3001:13000))
independence documentation built on Jan. 14, 2021, 5:20 a.m.
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Description Usage Arguments Value P-value Ties Big Data References See Also Examples
Description
The function tau.star.test provides an
independence test for two continuous numeric variables,
that is consistent for all alternatives.
It is based on the Yanagimoto-Bergsma-Dassios coefficient,
which compares the frequencies of concordant and discordant
quadruples of data points.
The test statistic is efficiently computed in O(n log n) time,
following work of Even-Zohar and Leng.
Usage
1 | tau.star.test(xs, ys, na.rm = TRUE, collisions = TRUE, precision = 1e-05)
|
Arguments
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= |
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 |
P-value
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.
Ties
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.
Big Data
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.
References
Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based
on a sign covariance related to Kendall's tau. Bernoulli 20 (2014), no.
2, 1006–1028.
Yanagimoto, Takemi. "On measures of association and a related problem."
Annals of the Institute of Statistical Mathematics 22.1 (1970): 57-63.
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
Even-Zohar, Chaim, and Calvin Leng. "Counting Small Permutation Patterns."
arXiv preprint arXiv:1911.01414 (2019).
Even-Zohar, Chaim. "independence: Fast Rank Tests."
arXiv preprint arXiv:2010.09712 (2020).
See Also
independence,
hoeffding.D.test,
hoeffding.refined.test
Examples
1 2 3 4 5 6 7 | ## independent, $p.value is 0.2585027
set.seed(123)
tau.star.test(rnorm(10000),rnorm(10000))
## dependent, even though uncorrelated, $p.value is 0.000297492
set.seed(123)
tau.star.test(rnorm(10000,0,3001:13000), rnorm(10000,0,3001:13000))
|
R Package Documentation
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