Performs Watson's test for homogeneity on two samples of circular data.
watson_two_test_perm() uses permutation to estimate p-values.
Usage
watson_two_test(x, y, alpha = 0.05, axial = TRUE, quiet = FALSE)
watson_two_test_perm(x, y, axial = TRUE, n_perm = 1000L, alpha = NULL)Arguments
- x, y
numeric vectors. Angles in degrees
- alpha
Significance level of the test ranging between 0 and 1. For
dist = "vonmises"it must be one of0.1,0.05,0.01or0(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.- quiet
logical. Prints the test's decision.
- n_perm
integer. Number of permutations
Details
A two-sample Watson's \(U^2\) test determines whether two independent groups of circular data (angles or directions) come from the same underlying distribution. Ties (e.g. azimuths rounded to whole degrees) are handled by evaluating the difference of the empirical distribution functions only at distinct values, weighted by the number of tied observations.
Hypotheses
Null Hypothesis (\(H_0\)): The two samples come from the same circular distribution (the two groups of angles share a common distribution around the circle).
Alternative Hypothesis (\(H_1\)): The two samples come from different circular distributions (the groups are oriented or dispersed differently around the circle).
Interpretation
The Test Statistic (\(U^2\)) measures the distance between the cumulative distribution functions of the two circular samples. A larger \(U^2\) value means the two sets of angles look more different from each other.
The p-value of watson_two_test_perm() represents the probability of getting
a U² value as large as (or larger than) your observed value purely by chance,
assuming the null hypothesis is true. It is calculated by shuffling the group
labels across your data many times to build a reference permutation distribution.
Making a Decision
Low p-value (\(p \le \alpha\), usually 0.05): Reject the null hypothesis. Conclude that the two groups have significantly different circular distributions.
High p-value (\(p > \alpha\)): Fail to reject the null hypothesis. There is not enough evidence to say the two groups are distributed differently around the circle.
Note
Critical values and p-values of watson_two_test() are obtained using
the asymptotic distribution of the test statistic. They should only be used
for combined sample sizes in excess of 17. Tables are available for smaller
sample sizes and can be found in Mardia (1972) for instance; alternatively
use watson_two_test_perm().
See also
Other Two-Sample-Tests:
ar_test(),
equal_kappa_test(),
uniform_scores_test(),
walraff_test(),
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)
watson_two_test(x1, x2, axial = FALSE)
#>
#> Watson two-sample U2 test
#>
#> data: x1 and x2
#> U2 = 0.14574, n = 21, p-value = 0.1126
#> alternative hypothesis: Samples come from different distributions
#>
watson_two_test_perm(x1, x2, axial = FALSE)
#>
#> Watson two-sample U2 test based on 1000 permutations
#>
#> data: x1 and x2
#> U2 = 0.14574, n = 21, n_perm = 1000, p-value = 0.1199
#> alternative hypothesis: Samples come from different distributions
#>
data1 <- rvm(n=20, mean = 0, kappa=3)
data2 <- rvm(n=20, mean = 90, kappa=2)
watson_two_test(data1, data2, axial = FALSE)
#>
#> Watson two-sample U2 test
#>
#> data: data1 and data2
#> U2 = 0.61775, n = 40, p-value = 1.012e-05
#> alternative hypothesis: Samples come from different distributions
#>
watson_two_test_perm(data1, data2, axial = FALSE)
#>
#> Watson two-sample U2 test based on 1000 permutations
#>
#> data: data1 and data2
#> U2 = 0.61775, n = 40, n_perm = 1000, p-value = 0.000999
#> alternative hypothesis: Samples come from different distributions
#>
# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
watson_two_test(sa.por$azi, rvm(100, 135, 10), alpha = 0.05)
#>
#> Watson two-sample U2 test
#>
#> data: sa.por$azi and rvm(100, 135, 10)
#> U2 = 0.20134, n = 1226, p-value = 0.03759
#> alternative hypothesis: Samples come from different distributions
#>
watson_two_test_perm(sa.por$azi, rvm(100, 135, 10), alpha = 0.05)
#>
#> Watson two-sample U2 test based on 1000 permutations
#>
#> data: sa.por$azi and rvm(100, 135, 10)
#> U2 = 0.21981, n = 1226, n_perm = 1000, p-value = 0.02498
#> alternative hypothesis: Samples come from different distributions
#>