Page 384 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

RESULTS—EXPERIMENT ONE 359

To test the null hypothesis H 0 against the alternative hypothesis H 1 ,weuse
the rank sum statistics by Wilcoxon and by Mann and Whitney [125]. The test
statistics by Wilcoxon is
n

W = R i (7.4)
i=1
That is, W is the sum of ranks for the sample observations Y when ranked among
all (m + n) observations. The test statistics by Mann and Whitney is

m n

U = (Y j − X i ) (7.5)
i=1 j=1
where (t) = 1for t> 0, otherwise (t) = 0for t ≤ 0. It represents the total
number of times a Y observation is larger than an X observation. W and U are
linearly related,
n(n + 1)
W = U +
2
Therefore, the discrete distribution of W or U under the null hypothesis H 0 is
something that we might know or can tabulate from permutations of the sequence
(1,..., N). The test will be of the form

reject H 0 : = 0 in favor of H 1 : ≤ 0 if and only if W ≥ w(α, m, n)

where w(α, m, n) is some accepted critical value that is dependent on a desired
significance level α and sample sizes m and n. In other words, the Mann–Whitney
U-test is based on rank sums rather than sample means.

Implementation. As mentioned in Section 7.3.1, Experiment One includes a
total of six group data sets, with 94 to 96 sample size each, related to three
independent variables: direction of motion, visibility, and interface. Each variable
has two levels. The data satisfy the statistical requirement that the observations
appear from their populations randomly and independently. The objectives of the
Mann–Whitney U-test here are as follows:

• Compare the left-to-right group data with the right-to-left group data, thereby
testing the effect of the direction variable.
• Compare the visible group data with the invisible group data, testing the
effect of the visibility variable.
• Compare the virtual group data with the physical group data, testing the
effect of the interface variable.
The null hypothesis H 0 here is that each of the two group data were drawn
from the same population distribution, for each variable test, respectively. The
alternative hypothesis H 1 for the corresponding test is that the two group data
were drawn from different population distributions.