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 = NULL,
  dist = c("uniform", "vonmises"),
  axial = TRUE,
  quiet = FALSE
)

Arguments

x: numeric vector. Values in degrees
alpha: Significance level of the test. Valid levels are 0.01, 0.05, and 0.1. This argument may be omitted (NULL, the default), in which case, a range for the p-value will be 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

list containing the test statistic statistic, the significance level p.value, the critical value critical.value, whether to reject the null hypothesis, the significance level alpha, the tested distribution dist, and the number of data n

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. non-parametric, rank-based alternative to one-sample

References

Mardia and Jupp (1999). 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
#> $statistic
#> [1] 0.1153633
#> 
#> $p.value
#> [1] NA
#> 
#> $critical.value
#> [1] 0.187
#> 
#> $reject
#> [1] FALSE
#> 
#> $alpha
#> [1] 0.05
#> 
#> $dist
#> [1] "uniform"
#> 
#> $n
#> [1] 10
#> 

# 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
#> $statistic
#> [1] 31.68687
#> 
#> $p.value
#> [1] NA
#> 
#> $critical.value
#> [1] 0.187
#> 
#> $reject
#> [1] TRUE
#> 
#> $alpha
#> [1] 0.05
#> 
#> $dist
#> [1] "uniform"
#> 
#> $n
#> [1] 1126
#> 
watson_test(sa.por$azi.PoR, alpha = .05, dist = "vonmises")
#> Reject Null Hypothesis
#> $statistic
#> [1] 0.5084434
#> 
#> $p.value
#> [1] NA
#> 
#> $critical.value
#> [1] 0.101
#> 
#> $reject
#> [1] TRUE
#> 
#> $alpha
#> [1] 0.05
#> 
#> $dist
#> [1] "vonmises"
#> 
#> $n
#> [1] 1126
#> 
watson_test(sa.por$azi.PoR, alpha = .05, dist = "vonmises")
#> Reject Null Hypothesis
#> $statistic
#> [1] 0.5084434
#> 
#> $p.value
#> [1] NA
#> 
#> $critical.value
#> [1] 0.101
#> 
#> $reject
#> [1] TRUE
#> 
#> $alpha
#> [1] 0.05
#> 
#> $dist
#> [1] "vonmises"
#> 
#> $n
#> [1] 1126
#>