Cramer: Cramér Two-Sample Test

In DataSimilarity: Quantifying Similarity of Datasets and Multivariate Two- And k-Sample Testing

Cramér Two-Sample Test

Description

Performs Two-Sample Cramér Test (Baringhaus and Franz, 2004). The implementation here uses the cramer.test implementation from the cramer package.

Usage

Cramer(X1, X2, n.perm = 0, just.statistic = (n.perm <= 0), sim = "ordinary", 
        maxM = 2^14, K = 160, seed = 42)

Arguments

X1	First dataset as matrix or data.frame
X2	Second dataset as matrix or data.frame
n.perm	Number of permutations for permutation or Bootstrap test, respectively (default: 0, no permutation test performed)
just.statistic	Should only the test statistic be calculated without performing any test (default: TRUE if number of permutations is set to 0 and FALSE if number of permutations is set to any positive number)
sim	Type of Bootstrap or eigenvalue method for testing. Possible options are "ordinary" (default) for ordinary Boostrap, "permutation" for permutation testing, or "eigenvalue" for bootstrapping the limit distribution (especially good for datasets too large for performing Bootstrapping). For more details see cramer.test
maxM	Maximum number of points used for fast Fourier transform involved in eigenvalue method for approximating the null distribution (default: 2^14). Ignored if sim is either "ordinary" or "permutation". For more details see cramer::cramer.test.
K	Upper value up to which the integral for calculating the distribution function from the characteristic function is evaluated (default: 160). Note: when K is increased, it is necessary to also increase maxM. Ignored if sim is either "ordinary" or "permutation". For more details see cramer.test.
seed	Random seed (default: 42)

Details

The Cramér test (Baringhaus and Franz, 2004) is a specialcase of the test of Bahrinhaus and Franz (2010) where the kernel function \phi is set to

\phi_{\text{Cramer}}(x) = \sqrt{x} / 2

and can be recommended for location alternatives. The test statistic simplifies to

T_{n_1, n_2} = \frac{n_1 n_2}{n_1+n_2}\left(\frac{1}{n_1 n_2}\sum_{i=1}^{n_1}\sum_{j=1}^{n_2} ||X_{1i} - X_{2j}|| - \frac{1}{2n_1^2}\sum_{i,j=1}^{n_1} ||X_{1i} - X_{1j}|| - \frac{1}{2n_2^2}\sum_{i,j=1}^{n_2} ||X_{2i} - X_{2j}||\right).

This is equal to the Energy statistic (Székely and Rizzo, 2004).

The theoretical statistic underlying this test statistic is zero if and only if the distributions coincide. Therefore, low values of the test statistic incidate similarity of the datasets while high values indicate differences between the datasets.

This implementation is a wrapper function around the function cramer.test that modifies the in- and output of that function to match the other functions provided in this package. For more details see the cramer.test.

Value

An object of class htest with the following components:

method	Description of the test
d	Number of variables in each dataset
m	Sample size of first dataset
n	Sample size of second dataset
statistic	Observed value of the test statistic
p.value	Boostrap/ permutation p value (only if n.perm > 0)
sim	Type of Boostrap or eigenvalue method (only if n.perm > 0)
n.perm	Number of permutations for permutation or Boostrap test
hypdist	Distribution function under the null hypothesis reconstructed via fast Fourier transform. $x contains the x-values, $Fx contains the corresponding distribution function values. (only if n.perm > 0)
ev	Eigenvalues and eigenfunctions when using the eigenvalue method (only if n.perm > 0)
data.name	The dataset names
alternative	The alternative hypothesis

Applicability

Target variable?	Numeric?	Categorical?	K-sample?
No	Yes	No	No

Note

The Cramér test (Baringhaus and Franz, 2004) is equivalent to the test based on the Energy statistic (Székely and Rizzo, 2004).

References

Baringhaus, L. and Franz, C. (2010). Rigid motion invariant two-sample tests, Statistica Sinica 20, 1333-1361

Bahr, R. (1996). Ein neuer Test fuer das mehrdimensionale Zwei-Stichproben-Problem bei allgemeiner Alternative, German, Ph.D. thesis, University of Hanover

Franz, C. (2024). cramer: Multivariate Nonparametric Cramer-Test for the Two-Sample-Problem. R package version 0.9-4, https://CRAN.R-project.org/package=cramer.

Stolte, M., Kappenberg, F., Rahnenführer, J., Bommert, A. (2024). Methods for quantifying dataset similarity: a review, taxonomy and comparison. Statist. Surv. 18, 163 - 298. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/24-SS149")}

See Also

Energy, Bahr, BF

Examples

# Draw some data
X1 <- matrix(rnorm(1000), ncol = 10)
X2 <- matrix(rnorm(1000, mean = 0.5), ncol = 10)
# Perform Cramer test 
if(requireNamespace("cramer", quietly = TRUE)) {
  Cramer(X1, X2, n.perm = 100)
}

DataSimilarity documentation built on April 3, 2025, 9:39 p.m.