Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

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`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] |

`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. Arguments
`x`

, `y`

can optionally be `dist`

objects;
otherwise these arguments are treated as data.

The `dcov`

test 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.

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 *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<s<2*, but (2) does not hold when *s=2*.

Distance correlation satisfies *0 ≤ R ≤ 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.

`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)] |

` replicates` |
replicates of the test statistic |

` p.value` |
approximate p-value of the test |

` n` |
sample size |

` data.name` |
description of data |

For the dcov test of independence,
the distance covariance test statistic is the V-statistic
*n V_n^2* (not dCov).

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

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.

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.

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.

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