m. fattore, f. maggino - qualità della vita in italia: vent'anni di studi attraverso...

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

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13 22 31

12 21

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Hasse diagram

3. POSet and inequality evaluation procedure

Downsetof 22

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Downset

3. POSet and inequality evaluation procedure

Upset of 22

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Upset

3. POSet and inequality evaluation procedure

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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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1. Setting the threshold

3. POSet and inequality evaluation procedure

Example

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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)

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33 42

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13 22 31

12 21

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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

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?

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

filomena.maggino@unifi.it

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