Applied Statistics Using SPSS, STATISTICA, MATLAB and R

10.6 Non-Parametric Tests
10.6 Non-Parametric Tests 397
The von Misessness of directional data distributions is difficult to guarantee in
many practical cases 3
. Therefore, non-parametric tests, namely those based on
ranking procedures similar to those described in Chapter 5, constitute an important
tool when comparing directional data samples.
10.6.1 The Uniform Scores Test for Circular Data
Let us consider q independent samples of circular data, each with nk cases. The
uniform scores test assesses the similarity of the q distributions based on scores of
the ordered combined data. For that purpose, let us consider the combined dataset
q
with n =∑ k = k n observations sorted by ascending order. Denoting the ith
1
observation in the kth group by θik, we now substitute it by the uniform score:
β
ik
2π
wik = , i = 1, …, nk, 10.23
n
where the wik are linear ranks in [1, n]. Thus, the observations are replaced by
equally spaced points in the unit circle, preserving their order.
Let rk represent the resultant length of the kth sample corresponding to the
uniform scores. Under the null hypothesis of equal distributions, we expect the βik
to be uniformly distributed in the circle. Using the test statistic:
q 2
rk
∑
k = 1 nk
W = 2 , 10.24
we then reject the null hypothesis for significantly large values of W.
2
The asymptotic distribution of W, adequate for n > 20, is χ 2( q−1)
. For further
details see (Mardia KV, Jupp PE, 2000). The uniform scores test is implemented
by function unifscores (see Commands 10.5).
Example 10.17
Q: Assess whether the distribution of the wind direction (WD) of the Weather
dataset (Data 2 datasheet) can be considered the same for all four seasons.
A: Denoting by wd the matrix whose first column is the wind direction data and
whose second column is the season code, we apply the MATLAB unifscores
function as shown below and conclude the rejection of equal distributions of the
wind direction in all four seasons at the 5% significance level (w > wc).
3
Unfortunately, there is no equivalent of the Central Limit Theorem for directional data.