14 
A. Kolmogoroff-Smirnoffs test on the 
goodness of fit of population distributions. 
###It is a powerful test and its use is 
recommendable instead of classical chi- 
square goodness of fit of population 
distribution test. 
###The test criterion compares the largest 
absolute difference between cumulative 
sample frequencies 7/ and population 
distributions P[ with Kolmogoroff- 
Smirnoffs extremal distribution critical 
values, where the degrees of freedom are 
the size of the samples: 
H 0 : 
P(KS v=n {a/ 2) < max17) - P\ < KS v=n ( 1 - a/2)) = 1-a 
B-l ,###Kolmogoroff-Smirnoffs test on the 
independence against interdependence. 
###The same test performed for the goodness 
of fit of population distributions can be 
used to the test the independence against 
interdependence. Indeed because the 
goodness of fit hasn't limitations 
concerning the dimension of domain of 
population distributions, these could be 
derived by multiplying marginal 
frequencies. 
###Thus comparing each multidimensional 
frequency with the product of the 
corresponding marginal frequencies, the 
test on the independence against 
interdependence is performed. The test 
criterion is, obviously, the same explained 
for the test on the goodness of fit of 
population distributions, moreover because 
the uniqueness of the test, its power is 
conserved. 
B-2. Modified Wilcoxon-Wilcox's test on 
the independence against dependence (e.g., 
collinearity). 
This test must be used instead of classical 
Hotelling's test, when population 
distributions aren't normal. It is a 
generalisation of Spearman's rank test for 
one Spearman's rank correlation 
coefficient under the same hypotheses. 
Let remember that the Spearman's rank 
correlation coefficient is a function of the 
sum of rank difference squares Df : 
r s = 1—- 
-1) /=1 
I Df 
where the ranks are assigned separately to 
each component from the smallest 
argument to the largest one and then 
compared, element by element, in terms of 
their difference. Furthermore when the null 
hypothesis is the independence, the test 
criterion compares the standardised value 
of the Spearman's correlation coefficient 
with the Student's t distribution critical 
values, where the degrees of freedom are 
the size of the sample minus two: 
Therefore the modified Wilcoxon-Wilcox's 
test on independence against dependence 
(e.g., collinearity) requires for two separate 
applications of a multiple test on a 
quadratic combination of independent 
Spearman's correlation coefficients. If the 
answer is two times positive the null 
hypothesis is accepted; on the contrary if 
the answer is two times negative the null 
hypothesis is rejected. Note that the third 
case, where the null hypothesis is one time 
accepted and another time rejected, or vice 
versa, should be considered rare or, at 
least, a priori known (e.g., the variables are 
by physical or geometrical reasons, two by 
two, correlated). 
The test criterion compares the sums H 
and K of standardised Spearman's rank 
correlation coefficient squares with C-\ 
Fischer's F distribution critical values, 
where the degrees of freedom are the 
number of terms of the quadratic 
combinations and the size of the samples 
minus two: 
H 0 : 
. : (a/2)< H < F, 
.A> 
-«/2))= 
Hq- 
. : {al2)< K < F„ 
..»O- 
-a ll))= 
where: