Distance Covariance Test

Description

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

Usage

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

Arguments

Details

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<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 returns a list with class htest containing

Note

For the 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)
 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)
 

Results