hoeffding.refined.test: The refined Hoeffding independence test

Description Usage Arguments Value P-value Ties Big Data References See Also Examples

View source: R/independence.R


The function hoeffding.refined.test provides independence testing for two continuous numeric variables, that is consistent for all alternatives. The test statistic is a variant of the classical Hoeffding's D statistic. In terms of CDFs, it estimates the integral of (Fxy-Fx*Fy)^2 dFx dFy, based on the ordering types of quintuples of data points. This test statistic is efficiently computed via a new O(n log n)-time algorithm, following work of Even-Zohar and Leng.


  na.rm = TRUE,
  collisions = TRUE,
  precision = 1e-05


xs, ys

Same-length numeric vectors, containing paired samples.


Logical: Should missing values, NaN, and Inf be removed?


Logical: Warn of repeating values in xs or ys.


of p-value, between 0 and 1. Otherwise p-value=NA.


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.

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

  1. max_taustar()

  2. 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.

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. "Patterns in Random Permutations." arXiv preprint arXiv:1811.07883 (2018).

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, tau.star.test, hoeffding.D.test,


## independent, $p.value is 0.258636

## dependent, even though uncorrelated, $p.value is 0.0003017679
hoeffding.refined.test(rnorm(10000,0,3001:13000), rnorm(10000,0,3001:13000))

independence documentation built on Jan. 14, 2021, 5:20 a.m.