Wellner's Two-Sample Test

Wellner's (1979, Example 1a) Rayleigh-style T-statistic, which quantifies the dissimilarity between two sets of vectors. The statistic increases with the degree of difference between the datasets.

Usage

wellner(x, y, n_perm, level)

# S3 method for class 'Ray'
wellner(x, y, n_perm = 1000L, level = 0.95)

# S3 method for class 'Line'
wellner(x, y, n_perm = 1000L, level = 0.95)

# S3 method for class 'Vec3'
wellner(x, y, n_perm = 1000L, level = 0.95)

# S3 method for class 'Plane'
wellner(x, y, n_perm = 1000L, level = 0.95)

Source

modified after geologyGeometry (J.R. Davis): http://www.joshuadavis.us/software/

Arguments

x, y

objects of class "Vec3", "Line", "Ray", or "Plane", where the rows are the observations and the columns are the coordinates.

n_perm

integer. Number of permutations.

level

numeric. Confidence level (\(1-\alpha\)). Default is 0.95

Value

object of class "Test"

Details

P-values are estimated using permutation-based inference to assess whether the two datasets are drawn from the same population.

#' wellner() computes Wellner's T-statistic, a non-negative measure of dissimilarity between two datasets. The value is zero when the datasets are identical.

Inference estimates the fraction of permutation tests in which the computed T-statistic exceeds the observed T for the original data (a number between 0 and 1). This value can be interpreted as a p-value for the null hypothesis that the two populations are identical (not merely that their means coincide). Thus, smaller p-values indicate stronger evidence that the two populations differ in a meaningful way.

References

Jon A. Wellner. "Permutation Tests for Directional Data." Ann. Statist. 7(5) 929-943, September, 1979. doi:10.1214/aos/1176344779

See also

Other tests: bingham_test(), fisher_ftest(), fisher_goodness_of_fit(), fisher_kent_test(), rayleigh_test3(), watson_symmetry_test()

Examples

set.seed(20250411)
test <- rvmf(100)
wellner(test, Line(120, 50))
#> 
#> 	Wellner permutation test
#> 
#> statistic = 8.2327, p-value = 0.03896
#> critical value (95% confidence level) = 8.2246
#> null hypothesis rejected at the 95% confidence level: yes
#> further statistics:
#>       n_perm   df_uniform    p_uniform 
#> 1000.0000000    5.0000000    0.1438671 
#> source: Wellner (1979), Ann. Statist. 7, 929-943 
#>