`dcov.test(x, y, index = 1.0, R = 199)`

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]

- method
- description of test
- statistic
- observed value of the test statistic
- estimate
- dCov(x,y)
- estimates
- a vector: [dCov(x,y), dCor(x,y), dVar(x), dVar(y)]
- replicates
- replicates of the test statistic
- p.value
- approximate p-value of the test
- data.name
- description of data

`dcov.test`

returns a list with class `htest`

containing
`dcov.test`

performs a nonparametric
test 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. Arguments
`x`

, `y`

can optionally be `dist`

objects;
otherwise these arguments are treated as data.
The statistic is
$nV_n^2$ where
$V_n(x,y)$ = dcov(x,y),
which is based on interpoint Euclidean distances
$||x_{i}-x_{j}||$. The `index`

is an optional exponent on Euclidean distance. 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 $R$ generalizes the idea of correlation in two fundamental ways:

(1) $R(X,Y)$ is defined for $X$ and $Y$ in arbitrary dimension. (2) $R(X,Y)=0$ characterizes independence of $X$ and $Y$.

Characterization (2) also holds for powers of Euclidean distance $|x_i-x_j|^s$, where $0

Distance correlation satisfies $0 \le R \le 1$, and
$R = 0$ only if $X$ and $Y$ are independent. Distance
covariance $V$ provides a new approach to the problem of
testing the joint independence of random vectors. The formal
definitions of the population coefficients $V$ and
$R$ 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 $nV_n^2$ 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 $n V_n^2$ can be implemented as a permutation test. See (SRB 2007) for theoretical properties of the test, including statistical consistency.

Szekely, G.J. and Rizzo, M.L. (2009),
Brownian Distance Covariance,
*Annals of Applied Statistics*,
Vol. 3, No. 4, 1236-1265.
http://dx.doi.org/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 `

`DCOR`

`dcor.ttest`

```
x <- iris[1:50, 1:4]
y <- iris[51:100, 1:4]
set.seed(1)
dcov.test(x, y)
set.seed(1)
dcov.test(dist(x), dist(y)) #same thing
set.seed(1)
dcov.test(x, y, index=.5)
set.seed(1)
dcov.test(dist(x), dist(y), index=.5) #same thing
## Example with dvar=0 (so dcov=0 and pval=1)
x <- rep.int(1, 10)
y <- 1:10
dcov.test(x, y, R=199)
```

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