m. fattore, f. maggino - qualità della vita in italia: vent'anni di studi attraverso...
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Qualità della vita in Italia: vent'anni di studi attraverso l'indagine Multiscopo
Roma, ISTAT – Aula Magna
27-28 gennaio 2015
Intensità e struttura della
disuguaglianza nel benessere
individuale in Italia.
Intensità e struttura della
disuguaglianza nel benessere
individuale in Italia.
Marco FattoreUniversità degli Studi di Milano-Bicocca (Italy)
Filomena MagginoUniversità degli Studi di Firenze (Italy)
Two-fold goal
a.outlining the features and the recent dynamics of inequality in Italy;
b.introducing and applying an innovative data analysis methodology, drawing on the concepts of partial order and partially ordered sets.
The methodology may be applied to inequality evaluation and, more generally, to well-being evaluation, when available data are of a multidimensional ordinal kind.
Introduction
The study
We apply the methodology to data about inequality in Italy, for years 2007 and 2010.
Introduction
Introduction
Data source
Multipurpose survey on Families, held by the Italian National Institute of Statistics (ISTAT).
Time span
year 2007 (before the crisis) and year 2010 (in the middle of the crisis)
Index
1. Defining subjective well-being
2. Data and sample
3. POSet and inequality evaluation procedure
4. Subjective inequality in Italy, before and within the economic crisis
Index
1. Defining subjective well-being
2. Data and sample
3. POSet and inequality evaluation procedure
4. Subjective inequality in Italy, before and within the economic crisis
1. Defining subjective well-being
One of the most adopted definitions
1. Defining subjective well-being
Subjective well-being
Cognitive componentssatisfaction
• “global level” overall satisfaction (satisfaction with life)• “specific levels” satisfaction in different domains
1. Households and families2. Housing3. Transport4. Leisure and culture5. Participation6. Education7. Labour market and working condition
8. Income, standard of living and consumption patterns9. Health 10. Environment11. Social security12. Crime and safety13. Total life situation
1. Defining subjective well-being
Subjective well-being
Affective components • Pleasant affects
- happiness- feelings of self-determination
• Unpleasant affects- worries- losing self-confidence- insurmountable difficulties- constantly under strain
Index
1. Defining subjective well-being
2. Data and sample
3. POSet and inequality evaluation procedure
4. Subjective inequality in Italy, before and within the economic crisis
“Multipurpose survey about families: aspects of daily life”
held by the Italian National Institute of Statistics
The survey investigates a number of different aspects of daily life at individual and familiar level.
2. Data and sample
Four subjective indicators:
Are you satisfied about your economic situation?Are you satisfied about your health?Are you satisfied about with familiar relations?Are you satisfied about relations with friends?
Scored on a four-degree scale: 1 – very 2 – enough3 – little4 – not at all
2. Data and sample
2. Data and sample
Considered at national and macro-regional level
Year Available records
After removing missing
2007 48253 40665
2010 48336 40859
Missing data non systematic
Index
1. Defining subjective well-being
2. Data and sample
3. POSet and inequality evaluation procedure
4. Subjective inequality in Italy, before and within the economic crisis
3. POSet and inequality evaluation procedure
1. Data involved in the study are ordinal
Cannot be aggregated
2. In general terms, well-being dimensions are not highly interrelated
Applying dimension reduction approaches looses too much information
Some methodological issues
3. POSet and inequality evaluation procedure
A new mathematical language is needed
Partial order theory
3. POSet and inequality evaluation procedure
Consider the four well-being indicators. To each statistical unit we can associate the set of degrees on the variables:
Individual → (d1, d2, d3, d4)
e.g., [3,4,4,2] means:•Satisfaction about economic situation little•Satisfaction about health not at all•Satisfaction with familiar relations not at all•Satisfaction with relations with friends enough
«profile»
3. POSet and inequality evaluation procedure
At first sight, we can say very little:
•some profiles are better than others, for example (2,3,2,3) is better than (4,3,3,1)
•some profiles are incomparable, for example (3,3,4,2) and (4,3,3,2). In fact the first profile is better on the first component, but it is worse on the third. So, which is the best of the two?
3. POSet and inequality evaluation procedure
So, some profiles can be ordered, others cannot. What we get is a
PARTIAL ORDER
(«partial» since not any pair of profiles can be ordered)
We can represent the partial order of the well-being profiles very effectively, by means of a graph, called «Hasse diagram» (from the
name of the German mathematician Helmut Hasse)
ordinal variable v wscale four-degree three-degreecode [1, 2, 3, 4] [1, 2, 3]
A simple example: two ordinal variables
3. POSet and inequality evaluation procedure
ordinal variable v wscale four-degree three-degreecode [1, 2, 3, 4] [1, 2, 3]
each unit (x, y) profile12 possible combinations/profiles
A simple example: two ordinal variables
3. POSet and inequality evaluation procedure
12 possible combinations/profiles
43
33 42
3223 41
13 22 31
12 21
11
Hasse diagram
3. POSet and inequality evaluation procedure
Downsetof 22
43
33 42
3223 41
13 22 31
12 21
11
Downset
3. POSet and inequality evaluation procedure
Upset of 22
43
33 42
3223 41
13 22 31
12 21
11
Upset
3. POSet and inequality evaluation procedure
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33 42
3223 41
13 22 31
12 21
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Incomparable profiles
Incomparabilities
3. POSet and inequality evaluation procedure
3. POSet and inequality evaluation procedure
Why posets are useful for inequality evaluation?
3. POSet and inequality evaluation procedure
Why posets are useful for inequality evaluation? Since they provide a natural description of the data and allow multidimensional comparisons to be able to answer the following question:
Given a set of profiles considered as a “inequality threshold”, which is the degree of inequality of any other profile in the poset?
And, more important, this is achieved without aggregating profile scores on single inequality variables.
3. POSet and inequality evaluation procedure
THE LOGIC:
1.Identify a inequality threshold
2.Compare profiles to the threshold and assess a inequality degree/score for each profile
3.Assign each statistical unit the inequality score of the corresponding profile
4.Compute synthetic indicators
3. POSet and inequality evaluation procedure
THE LOGIC:
1.Identify a inequality threshold
2.Compare profiles to the threshold and assess a inequality degree/score for each profile
3.Assign each statistical unit the inequality score of the corresponding profile
4.Compute synthetic indicators
In classical studies (e. g. pertaining to poverty) a numerical threshold is identified.
How is the threshold defined in the poset approach?
1. Setting the threshold
3. POSet and inequality evaluation procedure
In classical studies (e. g. pertaining to poverty) a numerical threshold is identified.
How is the threshold defined in the poset approach?
A threshold is a set of profiles that can be considered as representing situations «on the edge» of inequality.
N.B. in a multidimensional setting, the threshold is usually composed of more than one profile: this is sensible, since inequality may show different patterns.
1. Setting the threshold
3. POSet and inequality evaluation procedure
Threshold
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33 42
3223 41
13 22 31
12 21
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1. Setting the threshold
3. POSet and inequality evaluation procedure
Example
43
33 42
3223 41
13 22 31
12 21
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1. Setting the threshold
3. POSet and inequality evaluation procedure
ExampleSuffering profiles(they are equal or aboveat least one elementof the threshold)
43
33 42
3223 41
13 22 31
12 21
11
Suffering profiles(they are equal or aboveat least one elementof the threshold)
Non – inequality profiles (they are below all
profiles of the threshold)
1. Setting the threshold
3. POSet and inequality evaluation procedure
Example
43
33 42
3223 41
13 22 31
12 21
11
?
1. Setting the threshold
3. POSet and inequality evaluation procedure
Suffering profiles(they are equal or aboveat least one elementof the threshold)
Non – inequality profiles (they are below all
profiles of the threshold)
Example
We then assign:
1.Suffering score equal to 1 to profiles of the threshold or above
2.Suffering score equal to 0 to profiles below all elements of the threshold
3.Suffering scores in (0,1) to other profiles, since they are «ambiguously inequality». Which scores?
The computation of inequality scores for ambiguously inequality profiles is based on a combinatory approach, which avoids any aggregation of ordinal degrees.
The procedure quantifies the degree of inequality, based on the «relational position» of profiles with respect to the threshold, in the Hasse diagram.
Details can be found in the references.
2. Suffering scores
3. POSet and inequality evaluation procedure
Index
1. Defining subjective well-being
2. Data and sample
3. POSet and inequality evaluation procedure
4. Subjective inequality in Italy, before and within the economic crisis
4. Subjective inequality in Italybefore and within the economic crisis
We now apply this procedure to the analysis of inequality in Italy.
1.Construction of the satisfaction poset
2.Selection of the threshold
3.Evaluation of the inequality degree of each profile (and thus of each statistical unit sharing it)
4.Computation of synthetic indicators
4. Subjective inequality in Italybefore and within the economic crisis
1. Construction of the satisfaction poset
Four indicators on four-degree scales
256 partially ordered profiles
We do not represent graphically the poset, since the Hasse diagram is too cumbersome.
4. Subjective inequality in Italybefore and within the economic crisis
2. Selection of the threshold
We consider the economic and health attributes as the most relevant
The threshold is composed by two profiles:
3323 and 3332 (the digits refer respectively to economic situation, health, family and
friendship)
to be considered as “unambiguously inequality”, a profile must
comprise at least three attributes scored “little”, two of which must pertain to economy and health, and the fourth attribute cannot be
scored higher than “enough”
4. Subjective inequality in Italybefore and within the economic crisis
3. Evaluation of the inequality degree
Applying the evaluation algorithm (R-package PARSEC), each profile is assigned the corresponding inequality degree ( score).
Some profiles get scores between 0 and 1, as expected.
4. Subjective inequality in Italybefore and within the economic crisis
Remark. Profiles has been ordered according to their inequality degree
Notice jumps and non-linearities
4. Subjective inequality in Italybefore and within the economic crisis
4. Computing synthetic indicators
a.Each statistical unit is assigned the inequality score of the corresponding profile.
b.The average score over the population ( overall inequality level) gives a measure of the level of inequality in the country (a fuzzy generaliztion of the Head Count Ratio of classical poverty studies).
c.The average score excluding non-inequality individuals (i.e., with score 0) is called «specific inequality level» and provides information on the severity of inequality.
4. Subjective inequality in Italybefore and within the economic crisis
Data analysis and interpretation
SUFFERING LEVEL
2007 2010 REGION OVERALL SPECIFIC OVERALL SPECIFIC
Italy 0.102 0.410 0.101 0.405 North-West 0.080 0.364 0.083 0.377 North-East 0.077 0.359 0.079 0.366
Centre 0.099 0.427 0.100 0.399 South 0.132 0.445 0.132 0.440
Islands 0.144 0.450 0.128 0.439
4. Subjective inequality in Italybefore and within the economic crisis
Data analysis and interpretation
OVERALL SUFFERING LEVEL
2007 2010 REGION MALES FEMALES MALES FEMALES
Italy 0.086 0.117 0.088 0.115 North-West 0.065 0.094 0.075 0.091 North-East 0.065 0.089 0.068 0.088
Centre 0.087 0.110 0.081 0.117 South 0.107 0.154 0.112 0.150
Islands 0.130 0.156 0.113 0.142
Summary
The analysis reveals the existence of different inequality levels across the country and between males and females.
Interestingly, the temporal dynamics of subjective inequality suggests that people living in economically more developed regions worsen their self-perception, across the
beginning of the economic crisis.
Final remarks
Davey, B. A., & Priestley, B. H. (2002) Introduction to lattices and order, Cambridge: Cambridge University Press.
Fattore M. & F. Maggino (forthcoming 2015) “A New Method For Measuring and Analyzing Suffering–Comparing Suffering in Italian Society”, in Anderson R. (ed.) World Suffering and Quality of Life, Springer.
Fattore M., Maggino F., Colombo E. (2012) “From composite indicators to partial orders: evaluating socio-economic phenomena through ordinal data”, in Maggino F., Nuvolati G. (eds.) Quality of life in Italy: researches and reflections, Social Indicators Research Series, Springer.
Fattore M., Maggino F., Greselin F. (2011) “Socio-economic evaluation with ordinal variables: integrating counting and poset approaches”, in Statistica & Applicazioni, Partial Orders in Applied Sciences, Special Issue 2011.
Fattore M., Brueggemann R., Owsiński J. (2011) “Using poset theory to compare fuzzy multidimensional material deprivation across regions”, in Ingrassia S., Rocci R., Vichi M. (eds.) New Perspectives in Statistical Modeling and Data Analysis, Springer-Verlag.
References
Computation procedure
Extending a poset turning some incomparabilities into comparabilities (i.e. enlarging the
subset of elements that can be compared).
If all the incomparabilities of a poset are turned into comparabilities, one gets a so-called linear extension, that is an extension that is also a complete order.
Computation procedure
A simple but fundamental result of partial order theory states that the set of all possible linear extensions of a poset characterizes the poset itself, i.e. different posets have different sets of linear extensions and given the set of linear extensions of a poset, one can reconstruct it.
1. If the poset would be a linear order, chosen a threshold (which would comprise just one element) we could identify inequality and non-inequality profiles with no ambiguity: either a profile would be scored 1 or it would be scored 0.
2. As seen in the previous slide, any finite poset can be described in terms of linear extensions
3. For any of these linear orders we can unambiguously assess whether a profile is scored to 1 or to 0.
4. The fraction of linear extensions where a profile is scored 1 is taken as the degree of inequality of that profile.
The mathematical details can be found in the papers cited in the References.
Computation procedure
POSET ITS LINEAR EXTENSIONS
A small example
POSET ITS LINEAR EXTENSIONS
Threshold
A small example
POSET ITS LINEAR EXTENSIONS
Threshold
Non-inequality profile
Suffering profile
A small example
A small example
Profile Suffering degree
a 1
b 1
c 2/5
d 0
e 1
A comment
1. We do not aggregate variables! Yet we get a synthetic indicator
2. This can be achieved since we have set the evaluation problem as a «multidimensional comparison problem», not as a problem of measurement against a natural scale
3. Since the number of linear extensions of the inequality poset is huge, ona cannot count them all, on the contrary, one has to sample linear extensions, thorugh the Bubley-Dyer algorithm