energy (version 1.7-11)

# dcov.test: Distance Covariance Test and Distance Correlation test

## Description

Distance covariance test and distance correlation test of multivariate independence. Distance covariance and distance correlation are multivariate measures of dependence.

## Usage

dcov.test(x, y, index = 1.0, R = NULL)
dcor.test(x, y, index = 1.0, R)

## Value

dcov.test or dcor.test returns a list with class htest containing

method

description of test

statistic

observed value of the test statistic

estimate

dCov(x,y) or dCor(x,y)

estimates

a vector: [dCov(x,y), dCor(x,y), dVar(x), dVar(y)]

condition

logical, permutation test applied

replicates

replicates of the test statistic

p.value

approximate p-value of the test

n

sample size

data.name

description of data

## Arguments

x

data or distances of first sample

y

data or distances of second sample

R

number of replicates

index

exponent on Euclidean distance, in (0,2]

## Author

Maria L. Rizzo mrizzo@bgsu.edu and Gabor J. Szekely

## Details

dcov.test and dcor.test are nonparametric tests of multivariate independence. The test decision is obtained via permutation bootstrap, with R replicates.

The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

The index is an optional exponent on Euclidean distance. Valid exponents for energy are in (0, 2) excluding 2.

Argument types supported are numeric data matrix, data.frame, or tibble, with observations in rows; numeric vector; ordered or unordered factors. In case of unordered factors a 0-1 distance matrix is computed.

Optionally pre-computed distances can be input as class "dist" objects or as distance matrices. For data types of arguments, distance matrices are computed internally.

The dcov test statistic is $$n \mathcal V_n^2$$ where $$\mathcal V_n(x,y)$$ = dcov(x,y), which is based on interpoint Euclidean distances $$\|x_{i}-x_{j}\|$$. The index is an optional exponent on Euclidean distance.

Similarly, the dcor test statistic is based on the normalized coefficient, the distance correlation. (See the manual page for dcor.)

Distance correlation is a new measure of dependence between random vectors introduced by Szekely, Rizzo, and Bakirov (2007). For all distributions with finite first moments, distance correlation $$\mathcal R$$ generalizes the idea of correlation in two fundamental ways:

(1) $$\mathcal R(X,Y)$$ is defined for $$X$$ and $$Y$$ in arbitrary dimension.

(2) $$\mathcal R(X,Y)=0$$ characterizes independence of $$X$$ and $$Y$$.

Characterization (2) also holds for powers of Euclidean distance $$\|x_i-x_j\|^s$$, where $$0<s<2$$, but (2) does not hold when $$s=2$$.

Distance correlation satisfies $$0 \le \mathcal R \le 1$$, and $$\mathcal R = 0$$ only if $$X$$ and $$Y$$ are independent. Distance covariance $$\mathcal V$$ provides a new approach to the problem of testing the joint independence of random vectors. The formal definitions of the population coefficients $$\mathcal V$$ and $$\mathcal R$$ are given in (SRB 2007). The definitions of the empirical coefficients are given in the energy dcov topic.

For all values of the index in (0,2), under independence the asymptotic distribution of $$n\mathcal V_n^2$$ is a quadratic form of centered Gaussian random variables, with coefficients that depend on the distributions of $$X$$ and $$Y$$. For the general problem of testing independence when the distributions of $$X$$ and $$Y$$ are unknown, the test based on $$n\mathcal V^2_n$$ can be implemented as a permutation test. See (SRB 2007) for theoretical properties of the test, including statistical consistency.

## References

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007), Measuring and Testing Dependence by Correlation of Distances, Annals of Statistics, Vol. 35 No. 6, pp. 2769-2794.
tools:::Rd_expr_doi("10.1214/009053607000000505")

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1236-1265.
tools:::Rd_expr_doi("10.1214/09-AOAS312")

Szekely, G.J. and Rizzo, M.L. (2009), Rejoinder: Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3, No. 4, 1303-1308.

dcov  dcor  pdcov.test pdcor.test dcor.ttest

## Examples

Run this code
 x <- iris[1:50, 1:4]
y <- iris[51:100, 1:4]
set.seed(1)
dcor.test(dist(x), dist(y), R=199)
set.seed(1)
dcov.test(x, y, R=199)


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