Skip to contents

Equal \(\kappa\) Test

Source: R/stat_tests.R
equal_kappa_test.Rd

This function tests for the homogeneity of concentration parameters for multiple samples of directional data.

Usage

equal_kappa_test(samples, method = NULL, alpha = 0.05, axial = TRUE)

Arguments

samples

list of samples of orientations

method

character. One of

"f"

(default for two samples): \(F = ((n_1 - R_1) / \nu_1) / ((n_2 - R_2) / \nu_2)\), two-sided, \(F(\nu_1, \nu_2)\) (Mardia and Jupp, 2000 eq. 10.5.3; Watson, 1956).

"bartlett"

(default for more samples): Bartlett's test of the dispersions \((n_i - R_i) / \nu_i\), chi-square with \(q - 1\) degrees of freedom (Mardia and Jupp, 2000 eq. 10.6.21; McFadden & Lowes, 1981, eq. 64-66).

alpha

Significance level of the test ranging between 0 and 1. For dist = "vonmises" it must be one of 0.1, 0.05, 0.01 or 0 (only the tabulated p-value range is returned).

axial

logical. Whether the data are axial, i.e. \(\pi\)-periodical (TRUE, the default) or directional, i.e. \(2 \pi\)-periodical (FALSE). In case of axial data, the angles will be doubled for the test.

Value

A list with class "htest"

Details

The samples are assumed to have been drawn from von Mises populations. The null hypothesis tested is that all populations sampled have the same concentration parameter, kappa \(\kappa\).

For concentrated data \(2 \kappa (n_i - R_i)\) is approximately chi-square with \(\nu_i = n_i - 1\) degrees of freedom on the circle, where \(R_i\) is the resultant length of sample \(i\).

Note

Both need concentrated data (\(\bar{R} \ge 0.7\)); MJ eq. 10.6.24-10.6.27 and 7.3.24-7.3.25 give variance-stabilised tests for dispersed data.

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 5.3, World Scientific Press, Singapore.

Mardia, K. and Jupp, P. (1999). Directional Statistics, Section 7.4, John Wiley and Sons, England.

See also

Other Two-Sample-Tests: ar_test(), uniform_scores_test(), walraff_test(), watson_two_sample, watson_wheeler_test_perm(), watson_williams_test()

Examples

set.seed(20250411)
x1 <- c(35, 45, 50, 55, 60, 70, 85, 95, 105, 120)
x2 <- c(75, 80, 90, 100, 110, 130, 135, 140, 150, 160, 165)
equal_kappa_test(list(x1, x2))
#> 
#> 	F test of equal concentrations
#> 
#> data:  1 and 2
#> F = 0.78175, num df = 9, denom df = 10, p-value = 0.7216
#> alternative hypothesis: true ratio of concentrations is not equal to 1
#> sample estimates:
#> ratio of concentrations 
#>                0.781748 
#> 

x1 <- rvm(100, mean = 90, kappa = 20)
x2 <- rvm(100, mean = 90, kappa = 10)
x3 <- rvm(100, mean = 90, kappa = 1)
equal_kappa_test(list(x1, x2, x3))
#> 
#> 	Bartlett test of equal concentrations
#> 
#> data:  1 and 2 and 3
#> Bartlett's K-squared = 133.57, df = 2, p-value < 2.2e-16
#> sample estimates:
#>   kappa_1   kappa_2   kappa_3 
#> 5.6313939 2.5644762 0.5629104 
#> 

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
equal_kappa_test(list(sa.por$azi.PoR, rvm(100, 135, 10)))
#> 
#> 	F test of equal concentrations
#> 
#> data:  1 and 2
#> F = 1.3227, num df = 1125, denom df = 99, p-value = 0.0764
#> alternative hypothesis: true ratio of concentrations is not equal to 1
#> sample estimates:
#> ratio of concentrations 
#>                 1.32274 
#>