Dissertation - HQ

The statistical analysis of vertical distributions 99
where n ′ is the number of non-zero values of a i (i.e. the number of
nets with catches; nets with no catches have a weight of zero). It is easy
to check that if all a i are equal and non-null, equation (5.4) becomes
equation (5.3). All other descriptors, such as standard deviation or
quantiles, can be computed from weighted variance.
Brodeur & Rugen 210 used the z cm to describe the vertical descrip- Definitions of weighted
tion of ichthyoplankton in Alaska, but their formula for the standard variance differ
deviation erroneously added a square factor to the weights compared
to equation (5.4). While their formula also reduces to (5.3) in the case
of equal weights, it emphasises nets in which abundances are high.
As those are generally closer to the z cm, it diminishes variance. An
alternative equation for the weighted variance, often used in software
packages, is
s 2 w =
P ai(z i − ¯z w) 2
P
ai − 1
(5.5)
which conceptually corresponds to considering weights as “repeats”,
hence the number of observations is the sum of weights (n ′ ↔ P i ai).
When the weights are normalised (i.e. their mean is made equal to one)
equations (5.4) and (5.5) are equivalent.
Confidence of the mean and hypothesis testing
While the z cm is an efficient way to summarise vertical distributions
for further analyses, it can also be used by itself, and two z cms may
be compared. Testing hypothesis on means involves a measure of the
confidence in the value of those means (i.e. the standard deviation of
the mean, also called standard error). However, there is no analytical
equivalent to the standard error for weighted data. Gatz & Smith 211
discuss the validity of several estimates of the weighted standard error
by comparing them to a bootstrap method. The best suited is an
approximate ratio variance by Cochran 212
se 2 w =
n ′
(n ′ − 1)( P a i) 2 hX
(aiz i − ā ¯z w) 2
− 2¯z w
X
(ai − ā)(a iz i − ā¯z w)
(5.6)
No analytical definition
of the weighted
standard error
+¯z 2 w
X
(ai − ā) 2i .
The weighted standard error allows to compute weighted confidence
intervals, perform weighted t-tests, and everything that would normally
be available for non weighted data.
The problem of unequal variances
Dealing with z cms means dealing with non-normal data, hence using
non-parametric tests. It is common belief that the popular Wilcoxon-
Mann-Whitney test, and its multi-sample extension, the Kruskal-Wallis
test, do not require the variances to be equal, because they are nonparametric.
This is wrong 213,214 . As, under the null hypothesis, the two
Non-parametric tests
require homogeneity
of variances