Page 215 - Contributed Paper Session (CPS) - Volume 4

CPS2192 Laurent D. et al.

2
̅
(10) ̃ = ∑ ∑ (( , )) .
̃
̃

•

=1 =1

As the expressions (8), (9) and (10) are now crisp, the well-known
decomposition of the sum of squares can be easily verified:
̃ = ̃ + ̃ .

and it follows that we can, analogously to the classical ANOVA case, derive a
͠ /(−1)
test statistic. Let be such a crisp test statistic, where = ͠ /(−) .

Under the classical assumption of normality, we have ~ −1,− .

4. Empirical analysis
The 2014 SILC data represents more or less 17,000 persons from a sample
of Swiss households. We take a subset of the database and keep only the
active population, i.e. employed persons which age greater or equal to 18. We
focus our analysis on the financial situation with a poverty attribute
consisting of the two sub-items "good deprivation" and "satisfaction". These
are of course only a part of the components of a multidimensional poverty
measure. The table 1 describe our variables.

4.1 Individual and global assessments
After modelling each of these sub-items by triangular fuzzy numbers, and
combining them by aggregation rules, we defuzzified the obtained
aggregated result by the signed distance measure, and proceeded to the
computation of the individual and global assessments. The distribution of the
(crisp) financial situation (finance) is sketched in figure 1. The global
assessment, in this case the mean, is equal to 8.587. It is a relatively high value,
which means a weak level of poverty according to finance. This measure is
8.668 for the Swiss citizens and 8.099 for the foreigners, i.e. a sightly lesser
(bad) value for the foreigners. As the distribution is crisp, we can without
problem perform traditional analysis with it. For instance, a T-test of the
difference between the latter two global assesments (Swiss vs. Foreigners) can
be done without other restrictions. In the particular case, we observe a
significant difference in global assessment of the financial situation between
the two sub-populations.

4.2 Fuzzy ANOVA
We propose to execute a fuzzy ANOVA in order to test in a fuzzy context
the factor foreign on the two sub-items deprivation and satisfaction.
Membership functions for these two latter variables have to be defined. We
opted for triangular isosceles fuzzy numbers (Table 2). The results are listed in
table 3 and show significant differences with respect to the factor foreign.

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