Watson's \(U^2\) Test for Goodness-of-Fit against known Distribution

A non-parametric statistical test used for circular data to determine whether a sample fits a specified theoretical distribution

Usage

watson_test(
  x,
  alpha = 0.05,
  dist = c("uniform", "vonmises"),
  axial = TRUE,
  quiet = FALSE
)

Arguments

x: numeric vector. Values in degrees
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).
dist: Distribution to test for. The default, "uniform", is the circular uniform distribution. "vonmises" tests the von Mises distribution.
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.

Value

A list with class "htest" containing the following components:

statistic: the value of the U2 statistic (modified for dist = "uniform")
parameter: the number of data
p.value: the p-value for the test (dist = "uniform" only)
p.range: the range of the p-value from the tabulated critical values (dist = "vonmises" only)
estimate: the MLE of the kappa value for the von Mises distribution
alternative: a character string describing the alternative hypothesis.
method: a character string indicating what type of Watson test was performed.
data.name: a character string giving the name(s) of the data.
critical: the critical value at the significance level
alpha: the significance level
reject: logical. whether to reject the null-hypothesis based on the significance level.

Details

Hypotheses

Null Hypothesis (\(H_0\)): The circular sample comes from a specified theoretical distribution (such as a uniform distribution or a specific von Mises distribution).

Alternative Hypothesis (\(H_1\)): The circular sample does not follow the specified theoretical distribution.

Interpretation

To interpret the output of Watson's \(U^2\) test, compare your calculated \(U^2\) test statistic to the critical value from Watson's goodness-of-fit/homogeneity tables at your chosen significance level (\(\alpha\), commonly set to 0.05), or check the resulting p-value:

If \(U^2_{\text{calculated}} > U^2_{\text{critical}}\) (or p < \(\alpha\)): Reject the null hypothesis (\(H_0\)). Conclude that the data significantly deviates from the theoretical distribution.
If \(U^2_{\text{calculated}} \le U^2_{\text{critical}}\) (or \(p \ge \alpha\)): Fail to reject the null hypothesis (\(H_0\)). There is not enough evidence to claim the data deviates from the expected model.

Note

Watson's test statistic is a rotation-invariant Cramer - von Mises test.

References

Mardia and Jupp (2000). Directional Statistics. John Wiley and Sons.

Examples

# Example data from Mardia and Jupp (1999), pp. 93
watson_test(homing, axial = FALSE, alpha = .05)
#> Do Not Reject Null Hypothesis
#> 
#> 	Watson's U2 test against uniformity
#> 
#> data:  homing
#> U2 = 0.11536, n = 10, p-value = 0.2049
#> alternative hypothesis: non-uniform distribution
#> 

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
watson_test(sa.por$azi.PoR, alpha = .05)
#> Reject Null Hypothesis
#> 
#> 	Watson's U2 test against uniformity
#> 
#> data:  sa.por$azi.PoR
#> U2 = 31.687, n = 1126, p-value < 2.2e-16
#> alternative hypothesis: non-uniform distribution
#> 
watson_test(sa.por$azi.PoR, alpha = .05, dist = "vonmises")
#> Reject Null Hypothesis
#> 
#> 	Watson's U2 test against von Mises distribution (P-value < 0.01)
#> 
#> data:  sa.por$azi.PoR
#> U2 = 0.50844, n = 1126
#> alternative hypothesis: Sample does not follow a von Mises distribution
#> sample estimates:
#> Kappa MLE 
#>  2.184977 
#>