A non-parametric statistical test used to determine whether two independent samples of circular data (angles, directions, or periodic times) come from the same underlying population distribution. The difference between the samples can be in either the mean direction or the dispersion. P-values are estimated using permutation.
Arguments
- x, y
numeric vectors. Angles in degrees
- 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.- n_perm
integer. Number of permutations
- 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).
Details
The statistic is \(R^* = 2 (n - 1) (C_1^2 + S_1^2) / (n_1 n_2)\) on the uniform scores \(2 \pi r_i / n\) of the ranks \(r_i\) in the pooled sample (Mardia & Jupp, 2000, eq. 8.3.4). Tied values get their average rank.
Hypotheses
Null Hypothesis (\(H_0\)) The samples come from identical populations (meaning both the mean direction and the dispersion/variance are homogeneous across groups).
Alternative Hypothesis (\(H_1\)): The samples come from different population distributions, which can be due to a difference in the mean direction, a difference in variance/concentration, or both.
Interpretation
Test Statistic: Higher values indicate larger discrepancies between the angular distributions of your groups.
If the p-value is less than your significance level (commonly \(\alpha\) = 0.05), you reject the null hypothesis. This means you have strong evidence that the groups differ significantly in their central direction or spread around the circle.
If the p-value is greater than 0.05, you fail to reject the null hypothesis, meaning there is no statistically significant evidence of difference among the groups.
Note
Important Considerations & Limitations:
Sensitivity to both mean and variance: Because it detects differences in either mean or variance, a significant result does not automatically mean the mean angles are different; it could be driven entirely by differences in concentration (variance).
For more than two samples, or the asymptotic \(\chi^2\) version, see
uniform_scores_test().
See also
Other Two-Sample-Tests:
ar_test(),
equal_kappa_test(),
uniform_scores_test(),
walraff_test(),
watson_two_sample,
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_wheeler_test_perm(x1, x2, axial = FALSE)
#>
#> Watson-Wheeler test of homogeneity based on 1000 permutations
#>
#> data: x1 and x2
#> R* = 3.6783, n = 21, n_perm = 1000, p-value = 0.1489
#> alternative hypothesis: Samples come from different distributions
#>
data1 <- rvm(n=20, mean = 0, kappa=3)
data2 <- rvm(n=20, mean = 90, kappa=2)
watson_wheeler_test_perm(data1, data2, axial = FALSE)
#>
#> Watson-Wheeler test of homogeneity based on 1000 permutations
#>
#> data: data1 and data2
#> R* = 28.652, 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$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
watson_wheeler_test_perm(sa.por$azi.PoR, rvm(100, 135, 10))
#>
#> Watson-Wheeler test of homogeneity based on 1000 permutations
#>
#> data: sa.por$azi.PoR and rvm(100, 135, 10)
#> R* = 1.9205, n = 1226, n_perm = 1000, p-value = 0.3836
#> alternative hypothesis: Samples come from different distributions
#>