Rayleigh Test of Circular Uniformity

A test to determine whether a sample of circular or directional data is evenly spread out or clustered around a single specific direction. The test assesses the significance of the mean resultant length. rayleigh_test_perm() estimates p-values by Monte Carlo simulation of uniform samples.

Usage

rayleigh_test(x, mu = NULL, axial = TRUE, alpha = 0.05, quiet = FALSE)

rayleigh_test_perm(x, mu = NULL, axial = TRUE, n_perm = 1000L)

Arguments

x: numeric vector. Values in degrees
mu: (optional) The specified or known mean direction (in degrees) in alternative hypothesis.
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.
alpha: Significance level of the test ranging between 0 and 1.
quiet: logical. Prints the test's decision.
n_perm: integer. Number of simulated uniform samples.

Value

a list with the components:

R or C: mean resultant length or, if mu is specified, the mean cosine of the deviations from mu. Large values of R (large positive values of C) reject uniformity. Negative values of C indicate that the vectors point away from mu; the one-sided V-test does not reject them.
statistic: test statistic
p.value: significance level of the test statistic

Details

Hypotheses

Null Hypothesis \(H_0\): The population is distributed uniformly (randomly) around the circle with no preferred direction.

Alternative Hypothesis \(H_1\): The population is not uniform and has a unimodal (single-peaked) concentration in a preferred direction. When mu is specified (V-test), the alternative is a unimodal distribution centred on the specified direction; the test is one-sided.

Mean Resultant Length (\(\bar{R}\) or R): A value between 0 and 1 that measures how concentrated the data points are.

\(\bar{R}\) = 0: The data is completely spread out around the circle
\(\bar{R}\) = 1: All data points point in the exact same direction.

p-value The probability of seeing data this clustered purely by chance under the assumption of uniformity.

Interpretation

Small p-value (p < 0.05): Reject the null hypothesis. The length of the mean resultant differs significantly from zero, and the angles are not randomly distributed. You have strong evidence that the data points share a significant preferred or mean direction (unimodal clustering).
Large p-value (p \(\ge\) 0.05): Fail to reject the null hypothesis. There is not enough evidence to claim a preferred direction, meaning the data looks random or uniform around the circle.

Note

Although the Rayleigh test is consistent against (non-uniform) von Mises alternatives, it is not consistent against alternatives with a mean resultant length \(\rho = 0\) (in particular, distributions with antipodal symmetry, i.e. undoubled axial data). Tests of non-uniformity which are consistent against all alternatives include Kuiper's test (kuiper_test()) and Watson's \(U^2\) test (watson_test()).

Limitations

The test assumes a unimodal alternative (one main peak).
If your data has two opposite clusters (bimodal or axial data, like a bi-directional line trend), the Rayleigh test can yield a high/non-significant p-value because the opposing vectors cancel each other out.

References

Fisher, N. I. (1993) Statistical Analysis of Circular Data, Cambridge University Press.

Examples

# Example data from Mardia and Jupp (1999), pp. 93
rayleigh_test(homing, axial = FALSE) # Do not reject null hypothesis.
#> Do Not Reject Null Hypothesis
#> 
#> 	Rayleigh test of uniformity
#> 
#> data:  homing
#> S = 0.49672, n = 10, p-value = 0.6201
#> alternative hypothesis: unimodal distribution with an unknown mean direction
#> sample estimates:
#>         R      Rbar 
#> 2.2287169 0.2228717 
#> 
# R = 0.22; stat = 0.497, p = 0.62
rayleigh_test_perm(homing, axial = FALSE)
#> 
#> 	Rayleigh test of uniformity (Monte Carlo p-value based on 1000 uniform
#> 	samples)
#> 
#> data:  homing
#> Rbar = 0.22287, n = 10, n_perm = 1000, p-value = 0.6653
#> alternative hypothesis: unimodal distribution
#> sample estimates:
#>         R      Rbar 
#> 2.2287169 0.2228717 
#> 

# Example data from Davis (1986), pp. 316
rayleigh_test(striae, axial = FALSE) # reject null hypothesis
#> Reject Null Hypothesis
#> 
#> 	Rayleigh test of uniformity
#> 
#> data:  striae
#> S = 32.67, n = 51, p-value = 6.479e-15
#> alternative hypothesis: unimodal distribution with an unknown mean direction
#> sample estimates:
#>          R       Rbar 
#> 40.8188388  0.8003694 
#> 
rayleigh_test_perm(striae, axial = FALSE) # reject null hypothesis
#> 
#> 	Rayleigh test of uniformity (Monte Carlo p-value based on 1000 uniform
#> 	samples)
#> 
#> data:  striae
#> Rbar = 0.80037, n = 51, n_perm = 1000, p-value = 0.000999
#> alternative hypothesis: unimodal distribution
#> sample estimates:
#>          R       Rbar 
#> 40.8188388  0.8003694 
#> 

rayleigh_test(striae, mu = 105, axial = FALSE) # reject null hypothesis
#> Reject Null Hypothesis
#> 
#> 	Rayleigh V-test of uniformity
#> 
#> data:  striae
#> S = 7.3735, n = 51, p-value = 2.131e-13
#> alternative hypothesis: unimodal distribution with the specified mean direction 105
#> sample estimates:
#>         C 
#> 0.7300887 
#> 
rayleigh_test_perm(striae, mu = 105, axial = FALSE) # reject null hypothesis
#> 
#> 	Rayleigh V-test of uniformity (Monte Carlo p-value based on 1000
#> 	uniform samples)
#> 
#> data:  striae
#> C = 0.73009, n = 51, n_perm = 1000, p-value = 0.000999
#> alternative hypothesis: unimodal distribution with mean direction 105
#> 

# Example data from Mardia and Jupp (1999), pp. 99
atomic_weight <- c(
  rep(0, 12), rep(3.6, 1), rep(36, 6), rep(72, 1),
  rep(108, 2), rep(169.2, 1), rep(324, 1)
)
rayleigh_test(atomic_weight, 0, axial = FALSE) # reject null hypothesis
#> Reject Null Hypothesis
#> 
#> 	Rayleigh V-test of uniformity
#> 
#> data:  atomic_weight
#> S = 5.0142, n = 24, p-value = 5.348e-08
#> alternative hypothesis: unimodal distribution with the specified mean direction 0
#> sample estimates:
#>         C 
#> 0.7237434 
#> 

# San Andreas Fault Data:
data(san_andreas)
rayleigh_test(san_andreas$azi) # reject null hypothesis
#> Reject Null Hypothesis
#> 
#> 	Rayleigh test of uniformity
#> 
#> data:  san_andreas$azi
#> S = 524.18, n = 1126, p-value < 2.2e-16
#> alternative hypothesis: unimodal distribution with an unknown mean direction
#> sample estimates:
#>           R        Rbar 
#> 768.2592226   0.6822906 
#> 
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
rayleigh_test(sa.por$azi.PoR, mu = 135) # reject null hypothesis
#> Reject Null Hypothesis
#> 
#> 	Rayleigh V-test of uniformity
#> 
#> data:  sa.por$azi.PoR
#> S = 33.264, n = 1126, p-value < 2.2e-16
#> alternative hypothesis: unimodal distribution with the specified mean direction 135
#> sample estimates:
#>         C 
#> 0.7009544 
#> 
rayleigh_test_perm(sa.por$azi.PoR, mu = 135, n_perm = 1e3) # reject null hypothesis
#> 
#> 	Rayleigh V-test of uniformity (Monte Carlo p-value based on 1000
#> 	uniform samples)
#> 
#> data:  sa.por$azi.PoR
#> C = 0.70095, n = 1126, n_perm = 1000, p-value = 0.000999
#> alternative hypothesis: unimodal distribution with mean direction 135
#>