# PermutationTest: Permutation test of dependence between X and Y using (Gini)... In GiniDistance: A New Gini Correlation Between Quantitative and Qualitative Variables

## Description

Perform permutation test using various dependence measures, in which Xs are quantitative, Y are categorical, alpha is an exponent on Euclidean distance, sigma is kernel parameter in kernel methods and return the test statistic, critical value, p-value and decision of the test.

## Usage

 1 PermutationTest(x, y, method, sigma, alpha, M = 200, level = 0.05)

## Arguments

 x data y label of data or univariate response variable method name of permutation test method and is chosen from one of the method list: dCov, dCor, KdCov, KdCor, gCov, gCor, KgCov, Kgcor sigma kernel parameter for kenerl methods alpha exponent on Euclidean distance, in (0,2), the default value = 1 M number of permutations level significance level of the test, the default value = 0.05

## Details

H_0: X and Y are independent \Longleftrightarrow H_0: F(x|y=1)=F(x|Y=2)=...=F(x|Y=K)

PermutationTest compute the p-value value of a permutation test of a (Gini) distance covariance or correlation statistics. It is a self-contained R function the measure of dependence statistics.

The p-value is obtained by a permutation procedure. Let \hat{ρ}(ν) be the sample dependnce measure based on the orginal sample indexed by ν=\{1,2,...,n\}. Let π(ν) denote a permutation of the elements of ν and the corresponding \hat{ρ}(π) is computed for the permutated data on y labels. Under the {\cal H}_0, \hat{ρ}(ν) and \hat{ρ}(π) are identically distributed for every permutation π of ν. Hence, based on M permutations, the critical value q_{γ} is estimated by the (1-γ)100\% sample quantile of \hat{ρ}(π_m), m=1,...,M and the p-value is estimated by the proportion of \hat{ρ}(π_m) greater than \hat{ρ}(ν). Usually 100≤q M≤q 1000 is sufficient for a good estimation on the critical value or p-value. The default value is M=200.

## Value

PermutationTest returns the p-value, critical value and decision of the permutation test of a specified method.