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

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

`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

- 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]

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

`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.

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`

```
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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