R: Distance Covariance Test and Distance Correlation test

dcov.test {energy} | R Documentation

## 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)

### 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]

### 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. 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.

### 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)]
`replicates` | replicates of the test statistic
`p.value` | approximate p-value of the test
`n` | sample size
`data.name` | description of data

### Note

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

### Author(s)

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

### 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.
http://dx.doi.org/10.1214/009053607000000505

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.

### See Also

`dcov` `dcor` `DCOR` `dcor.ttest`

### Examples

     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)

[Package energy version 1.7-6 Index]