Rayleigh Test of Circular Uniformity

Performs a Rayleigh test for uniformity of circular/directional data by assessing the significance of the mean resultant length.

Usage

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

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).
quiet: logical. Prints the test's decision.

Value

a list with the components:

R or C: mean resultant length or the dispersion (if mu is specified). Small values of R (large values of C) will reject uniformity. Negative values of C indicate that vectors point in opposite directions (also lead to rejection).
statistic: test statistic
p.value: significance level of the test statistic

Details

\(H_0\):: angles are randomly distributed around the circle.
\(H_1\):: angles are from non-uniformly distribution with unknown mean direction and mean resultant length (when mu is NULL. Alternatively (when mu is specified), angles are non-uniformly distributed around a specified direction.

If statistic > p.value, the null hypothesis is rejected, i.e. the length of the mean resultant differs significantly from zero, and the angles are not randomly distributed.

Note

Although the Rayleigh test is consistent against (non-uniform) von Mises alternatives, it is not consistent against alternatives with p = 0 (in particular, distributions with antipodal symmetry, i.e. 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()).

References

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

Examples

# Example data from Mardia and Jupp (1999), pp. 93
pidgeon_homing <- c(55, 60, 65, 95, 100, 110, 260, 275, 285, 295)
rayleigh_test(pidgeon_homing, axial = FALSE) # Do not reject null hypothesis.
#> Do Not Reject Null Hypothesis
#> $R
#> [1] 0.2228717
#> 
#> $statistic
#> [1] 0.4967179
#> 
#> $p.value
#> [1] 0.6201354
#> 
# R = 0.22; stat = 0.497, p = 0.62

# Example data from Davis (1986), pp. 316
finland_striae <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
rayleigh_test(finland_striae, axial = FALSE) # reject null hypothesis
#> Reject Null Hypothesis
#> $R
#> [1] 0.8003694
#> 
#> $statistic
#> [1] 32.67015
#> 
#> $p.value
#> [1] 6.479397e-15
#> 
rayleigh_test(finland_striae, mu = 105, axial = FALSE) # reject null hypothesis
#> Reject Null Hypothesis
#> $C
#> [1] 0.7300887
#> 
#> $statistic
#> [1] 7.373534
#> 
#> $p.value
#> [1] 2.130845e-13
#> 

# 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
#> $C
#> [1] 0.7237434
#> 
#> $statistic
#> [1] 5.014241
#> 
#> $p.value
#> [1] 5.348331e-08
#> 

# San Andreas Fault Data:
data(san_andreas)
rayleigh_test(san_andreas$azi) # reject null hypothesis
#> Reject Null Hypothesis
#> $R
#> [1] 0.6822906
#> 
#> $statistic
#> [1] 524.1761
#> 
#> $p.value
#> [1] 2.255452e-228
#> 
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$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
#> $C
#> [1] 0.7009544
#> 
#> $statistic
#> [1] 33.26396
#> 
#> $p.value
#> [1] 1.420092e-239
#>