R: Energy-tests of Independence

indep.test {energy} | R Documentation

## Energy-tests of Independence

### Description

Computes a multivariate nonparametric test of independence. The default method implements the distance covariance test `dcov.test`.

### Usage

    indep.test(x, y, method = c("dcov","mvI"), index = 1, R)

### Arguments

`x` | matrix: first sample, observations in rows
`y` | matrix: second sample, observations in rows
`method` | a character string giving the name of the test
`index` | exponent on Euclidean distances
`R` | number of replicates

### Details

`indep.test` with the default `method = "dcov"` computes the distance covariance test of independence. `index` is an exponent on the Euclidean distances. Valid choices for `index` are in (0,2], with default value 1 (Euclidean distance). The arguments are passed to the `dcov.test` function. See the help topic `dcov.test` for the description and documentation and also see the references below.

`indep.test` with `method = "mvI"` computes the coefficient I_n and performs a nonparametric E-test of independence. The arguments are passed to `mvI.test`. The `index` argument is ignored (`index = 1` is applied). See the help topic `mvI.test` and also see the reference (2006) below for details.

The test decision is obtained via bootstrap, with `R` replicates. The sample sizes (number of rows) of the two samples must agree, and samples must not contain missing values.

These energy tests of independence are based on related theoretical results, but different test statistics. The `dcov` method is faster than `mvI` method by approximately a factor of O(n).

### Value

`indep.test` returns a list with class `htest` containing

`method` | description of test
`statistic` | observed value of the test statistic n V_n^2 or n I_n^2
`estimate` | V_n or I_n
`estimates` | a vector [dCov(x,y), dCor(x,y), dVar(x), dVar(y)] (method dcov)
`replicates` | replicates of the test statistic
`p.value` | approximate p-value of the test
`data.name` | description of data

### Note

As of energy-1.1-0, `indep.etest` is deprecated and replaced by `indep.test`, which has methods for two different energy tests of independence. `indep.test` applies the distance covariance test (see `dcov.test`) by default (`method = "dcov"`). The original `indep.etest` applied the independence coefficient I_n, which is now obtained by `method = "mvI"`.

### Author(s)

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

### References

Szekely, G.J. and Rizzo, M.L. (2009), Brownian Distance Covariance, Annals of Applied Statistics, Vol. 3 No. 4, pp. 1236-1265. (Also see discussion and rejoinder.)
http://dx.doi.org/10.1214/09-AOAS312

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

Bakirov, N.K., Rizzo, M.L., and Szekely, G.J. (2006), A Multivariate Nonparametric Test of Independence, Journal of Multivariate Analysis 93/1, 58-80,
http://dx.doi.org/10.1016/j.jmva.2005.10.005

### See Also

`dcov.test` `mvI.test` `dcov` `mvI`

### Examples

     ## independent multivariate data
     x <- matrix(rnorm(60), nrow=20, ncol=3)
     y <- matrix(rnorm(40), nrow=20, ncol=2)
     indep.test(x, y, method = "dcov", R = 99)
     indep.test(x, y, method = "mvI", R = 99)

     ## Not run:
     ## dependent multivariate data
     if (require(MASS)) {
       Sigma <- matrix(c(1, .1, 0, 0 , 1, 0, 0 ,.1, 1), 3, 3)
       x <- mvrnorm(30, c(0, 0, 0), diag(3))
       y <- mvrnorm(30, c(0, 0, 0), Sigma) * x
       indep.test(x, y, R = 99)    #dcov method
       indep.test(x, y, method = "mvI", R = 99)
      }

    ## End(Not run)

     ## Not run:
     ## compare the computing time
     x <- mvrnorm(50, c(0, 0, 0), diag(3))
     y <- mvrnorm(50, c(0, 0, 0), Sigma) * x
     set.seed(123)
     system.time(indep.test(x, y, method = "dcov", R = 1000))
     set.seed(123)
     system.time(indep.test(x, y, method = "mvI", R = 1000))

    ## End(Not run)

[Package energy version 1.7-6 Index]