Applied Statistics Using SPSS, STATISTICA, MATLAB and R

10.6 Non-Parametric Tests 399
Example 10.18
Q: Consider the measurements R4, R5 and R6 of the negative gradient of the Soil
Pollution dataset, performed in similar conditions. Assess whether the mean
gradients above and below 20 m are significantly different at 5% level.
A: We establish two groups of measurements according to the value of variable z
(depth) being above or below 20 m. The mean directions of these two groups are:
Group 1: (156.17º, 117.40º);
Group 2: (316.99º, 116.25º).
Assuming that the groups are rotationally symmetric and since the sizes are
n1 = 45 and n2 = 30, we apply the Watson test at a significance level of 5%,
2
obtaining an observed test statistic of 44.9. Since χ 0.
95,
2 =5.99, we reject the null
hypothesis of equality of means.
10.6.3 Testing Two Paired Samples
The previous two-sample tests assumed that the samples were independent. The
two-paired-sample test can be reduced to a one-sample test using the same
technique as in Chapter 4 (see section 4.4.3.1), i.e., employing the differences
between pair members. If the distributions of the two samples are similar, we
expect that the difference sample will be uniformly distributed. The function
dirdif implemented in MATLAB (see Commands 10.3) computes the
directional data of the difference set in standard format.
Example 10.19
Q: Consider the measurements M2 and M3 of the Soil Pollution dataset.
Assess, at the 5% significance level, if one can accept that the two measurement
methods yield similar distributions.
A: Let soil denote the data matrix containing all measurements of the Soil
Pollution dataset. Measurements M2 and M3 correspond to the column pairs
3-4 and 5-6 of soil, respectively. We use the sequence of R commands shown
below and do not reject the hypothesis of similar distributions at the 5% level of
significance.
> m2 m3 d p p
[1] 0.1772144