Assessing Subjective Well-being in Wide Populations. A Posetic Approach. . .
253
those respondents who have experienced negative emotions for most of the time
during the reporting period and positive emotions sometimes or never. We consider
them as people in a low emotional state. After setting the threshold, an identification
function 16 is defined, to assign deprivation membership scores in [0,1] to the poset
elements (Fattore 2016) 17 . In particular, it expresses the quote of events in which the
profile falls into the area of discomfort, considering the different linear extensions.
The severity is the average of the graph distance of the profile from the first profile
above all threshold elements, and it is equal to 0 for profiles above the threshold.
Both the identification function and the severity function are meaningful to describe
the deprivation in the poset: the first one describes the deprivation ambiguity,
the second one its intensity. We cannot use only the identification function as an
absolute value expressing the ES level, because not all profiles with the same level
of it are equally deprived. Indeed, their deprivation can show very different levels
of severity. Taking into account the two information, we can assess the (probable)
deprivation condition of a profile and consequently of the subject expressing it.
In order to construct a synthetic indicator of SWB, we need to know the specific
value of the ES of each respondent. In this way, we can treat it as an input variable
for the second stage of synthesis. To do this, we decided to summarize the three
functions explained above in a single index, so as to have a measure that takes into
account the position of profiles in the general order and the intensity and ambiguity
of their deprivation with respect to the chosen threshold. We could have calculated
the different functions upon the theoretical set of profiles (i.e. 243 profiles) or upon
the actual dataset (in which we observe 230 profiles of the 243 possible ones).
We decided to consider the theoretical set, so as to obtain values of the functions
not influenced by the respondents’ population. To obtain a synthesis, we decided
to use the Mazziotta-Pareto Index 18 (hereinafter: MPI), a composite index for
summarizing a set of indicators that are assumed not fully substitutable (Mazziotta
and Pareto 2016). Our composite (constructed by synthesizing the average rank, the
identification function and the severity function 19 ) is positive, i.e., increasing values
of the index correspond to positive variations of the phenomenon. The composite
assumes values between 87 (profile [1,1,1,1,1]) and 121 (profile [3,3,3,3,3]). Table
16 For a complete definition of the identification function and its computation, please see: Fattore
et al. 2012.
17 All the operations were carried out using the R package PARSEC (Arcagni and Fattore 2018).
18 MPI is a partially non-compensatory composite indicator based on a non-linear function which,
starting from the arithmetic mean of the normalized indicators (indicators are standardized by
means of a variant of z-scores, which transforms the indicators into distributions with mean 100 and
standard deviation 10) introduces a penalty for the units with unbalanced values of the indicators.
We chose this method because various analyses have shown that it is more robust than others are
(for instance, see: Mazziotta and Pareto 2015; Alaimo 2020). For more information on the MPI,
please see: Mazziotta and Pareto 2016 and 2017.
19 In the normalization, it is necessary to define the polarity of the basic indicators, i.e. the sign of
the relation between the indicator itself and the phenomenon to be measured. Therefore, the type of
composite we want to construct defines polarity. In our case, the average rank has positive polarity,
while the identification and the severity functions negative.
253
those respondents who have experienced negative emotions for most of the time
during the reporting period and positive emotions sometimes or never. We consider
them as people in a low emotional state. After setting the threshold, an identification
function 16 is defined, to assign deprivation membership scores in [0,1] to the poset
elements (Fattore 2016) 17 . In particular, it expresses the quote of events in which the
profile falls into the area of discomfort, considering the different linear extensions.
The severity is the average of the graph distance of the profile from the first profile
above all threshold elements, and it is equal to 0 for profiles above the threshold.
Both the identification function and the severity function are meaningful to describe
the deprivation in the poset: the first one describes the deprivation ambiguity,
the second one its intensity. We cannot use only the identification function as an
absolute value expressing the ES level, because not all profiles with the same level
of it are equally deprived. Indeed, their deprivation can show very different levels
of severity. Taking into account the two information, we can assess the (probable)
deprivation condition of a profile and consequently of the subject expressing it.
In order to construct a synthetic indicator of SWB, we need to know the specific
value of the ES of each respondent. In this way, we can treat it as an input variable
for the second stage of synthesis. To do this, we decided to summarize the three
functions explained above in a single index, so as to have a measure that takes into
account the position of profiles in the general order and the intensity and ambiguity
of their deprivation with respect to the chosen threshold. We could have calculated
the different functions upon the theoretical set of profiles (i.e. 243 profiles) or upon
the actual dataset (in which we observe 230 profiles of the 243 possible ones).
We decided to consider the theoretical set, so as to obtain values of the functions
not influenced by the respondents’ population. To obtain a synthesis, we decided
to use the Mazziotta-Pareto Index 18 (hereinafter: MPI), a composite index for
summarizing a set of indicators that are assumed not fully substitutable (Mazziotta
and Pareto 2016). Our composite (constructed by synthesizing the average rank, the
identification function and the severity function 19 ) is positive, i.e., increasing values
of the index correspond to positive variations of the phenomenon. The composite
assumes values between 87 (profile [1,1,1,1,1]) and 121 (profile [3,3,3,3,3]). Table
16 For a complete definition of the identification function and its computation, please see: Fattore
et al. 2012.
17 All the operations were carried out using the R package PARSEC (Arcagni and Fattore 2018).
18 MPI is a partially non-compensatory composite indicator based on a non-linear function which,
starting from the arithmetic mean of the normalized indicators (indicators are standardized by
means of a variant of z-scores, which transforms the indicators into distributions with mean 100 and
standard deviation 10) introduces a penalty for the units with unbalanced values of the indicators.
We chose this method because various analyses have shown that it is more robust than others are
(for instance, see: Mazziotta and Pareto 2015; Alaimo 2020). For more information on the MPI,
please see: Mazziotta and Pareto 2016 and 2017.
19 In the normalization, it is necessary to define the polarity of the basic indicators, i.e. the sign of
the relation between the indicator itself and the phenomenon to be measured. Therefore, the type of
composite we want to construct defines polarity. In our case, the average rank has positive polarity,
while the identification and the severity functions negative.