cramer.test {cramer}R Documentation

Cramer-Test for uni- and multivariate two-sample-problem

Description

Perform Cramer-test for two-sample-problem. Both univariate and multivariate data is possible. For calculation of the critical value bootstrap-methods are used. Ordinary and permutation methods can be chosen as well as the number of bootstrap-replicates taken.

Usage

cramer.test(x,y,conf.level=0.95,replicates=1000,
            sim="ordinary",just.statistic=FALSE) 

Arguments

x First set of observations. Either in vector form (univariate) or in a matrix with one observation per row (multivariate).
y Second set of observations. Same dimension as x.
conf.level Confidence level of test.
replicates Number of bootstrap-replicates taken to obtain critical value.
sim Type of bootstrap method. Possible values are "ordinary" for normal bootstrap and "permutation" for a permutation test.
just.statistic Boolean variable. If TRUE just the value of the Cramer-statistic is calculated and no bootstrap-replicates are produced.

Details

The Cramer-statistic is given by

T=mn/(m+n) ( 1/(mn) Sum[i=1..m,j=1..n] ||X_i-Y_j|| - 1/(2m^2) Sum[i=1..m,j=1..m] ||X_i-X_j|| - 1/(2n^2) Sum[i=1..n,j=1..n] ||Y_i-Y_j|| )

The critical value is obtained by using the bootstrap-method with this statistic. The proof that this procedure yields in a working test is given in the reference listed below.

The idea of using this statistic is due to L. Baringhaus, University of Hanover.

Value

The returned value is an object of class "cramertest", containing the following components:

method Describing the test in words.
d Dimension of the observations.
m Number of x observations.
n Number of y observations.
statistic Value of the Cramer-statistic for the given observations.
conf.level Confidence level for the test.
crit.value Critical value calculated by bootstrap method.
p.value Estimated p-value of the test.
result Contains 1 if the hypothesis of equal distributions should not be accepted and 0 otherwise.
sim Method used for obtaining the critical value.
replicates Number of bootstrap-replicates taken.

References

The test and its properties is described in :

Baringhaus, L. and Franz, C. (2001) On a new multivariate two-sample test, submitted

Franz, C. (2000) Ein statistischer Test fuer das mehrdimensionale Zweistichproben-Problem, German, Diploma thesis, University of Hanover.

See Also

print.cramertest

Examples

# comparison of two univariate normal distributions
x<-rnorm(20,mean=0,sd=1)
y<-rnorm(50,mean=0.5,sd=1)
cramer.test(x,y)

# comparison of two multivariate normal distributions with permutation test
library(MASS)
x<-mvrnorm(n=20,mu=c(0,0),Sigma=diag(c(1,1)))
y<-mvrnorm(n=50,mu=c(0.3,0),Sigma=diag(c(1,1)))
cramer.test(x,y,sim="permutation")

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