MEASURING GENDER GAP FROM A POSET PERSPECTIVEcomplexity.stat.unipd.it/system/files/Di_Brisco_Farina.pdf · Interpretation of gap measure: Measure the severity of deprivation; Is basically

IntroductionBasic elements of partial order theory

Data analysisConclusions

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MEASURING GENDER GAPFROM A POSET PERSPECTIVE

Agnese M. Di Brisco1∗, Patrizia Farina2

∗[email protected]

1University of Milano Bicocca - Department of Economics, Management andStatistics,

2University of Milano Bicocca - Department of Sociology and Social Research

Dealing with Complexity in Society: from Plurality of Data to Synthetic Indicators

17th-18th September 2015, Padua (Italy)

Agnese M. Di Brisco 17th-18th September 2015 Measuring gender gap from a POSET perspective 1 / 26



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Outline

Introduction

Basic elements of partial order theory

Data analysis

Conclusions




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ORDER, ORDER, ORDER!

Order is not a property of a single object. It concerns comparisonbetween pairs of objects.




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Partial order

The use of partial ordering has two different types of justificationin interpersonal comparison or in inequality evaluation. First, ashas been just discussed, the ideas of well-being and inequality mayhave enough ambiguity and fuzziness to make it a mistake to lookfor a complete ordering of either. This may be called thefundamental reason for incompleteness. Second, even if it is not amistake to look for one complete ordering, we may not be able inpractice to identify it.The pragmatic reason for incompleteness is to sort outunambiguously, rather than maintaining complete silence untileverything has been sorted out and the world shines in dazzlingclarity

( A. Sen, Inequality reexamined, 1998)




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Partial order

• Dealing with ordinal variables: a problematic issue in syntheticindexes computation;

• The mathematical theory of partial order allows to respect theordinal nature of the data, avoiding any aggregation or scalingprocedures.

Aims of the presentation:

1 Introduce some basic concepts of partial order theory;

2 Apply a poset approach to gender sensitive data




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Definition of POSET

A finite partially ordered set P = (X ,≤) (POSET) is a finite set Xwith a partial order relation ≤ that is a binary relation satisfyingthe following properties:

1 Reflexivity: x ≤ x ∀x ∈ X ;

2 Antisymmetry: if x ≤ y and y ≤ x then x = y for x , y ∈ X ;

3 Transitivity: if x ≤ y and y ≤ z then x ≤ z for x , y , z ∈ X .

Two elements of set X are comparable if x ≤ y or y ≤ x .If any two elements of X are comparable then the poset P is said achain or a linear order.If any two elements of X are not comparable then the poset P issaid an antichain.




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Upset and Downset

An upset U of a poset is a subset of P such that if x ∈ U andx ≤ z then z ∈ U.A downset D of a poset is a subset of P such that if x ∈ D andy ≤ x then y ∈ D.

Proposition

Given a finite poset P and an upset U then ∃u antichain such thatu ⊆ P.Then z ∈ U if and only if ∃u ∈ u such that u ≤ z .The upset U, then, is generated by an antichain u:

U = u ↑




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Linear extentions

An extention of a poset P is a poset defined on the same set Xwhose set of comparabilities comprises that of P

Definition

A linear extension is an extension of P that is a linear order or achain.

Theorem, Neggers and Kim, 1988

The set of linear extensions of a finite poset P uniquely identifies P




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Ordinal variables

Let us consider k ordinal variables each with jk levels.Then we can compute all the possible profiles and provide a partialorder with the following :

Rule

Let s and t two profiles over v1, . . . , vk ordinal variables. Then tdominates s if and only if

vi (s) ≤ vi (t) ∀i = 1, . . . , k

How many profiles:

#p =k∏

i=1

jk




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Hasse Diagram

An Hasse diagram is a graph in which:

• if s ≤ t the node t is placed above the node s

• if s ≤ t and 6 ∃w : s ≤ w ≤ t then an edge is inserted

Example

Let consider three binary variables on a 0-1 scale. There are 8possible profiles represented in the following graph:




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Threshold

Objective: classification of the profiles in disadvantaged withrespect to the other gender.

• Define a threshold as a profile or a list of profiles that is ananti chain

• The threshold generates a down set D

• All profiles belonging to D are certainly under the threshold

• Some profiles can not be ordered with respect to the chosenthreshold




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Evaluation function

η : T → [0, 1]

where T is the finite collection of all possible profiles.

η(p) =

0 if p ∈ D{l∈E(P):∃d∈d :d≤p∈l}

|E(P)| otherwise

1 if p ∈ U

where E (P) is the set of all possible linear extensions of the posetd is the anti chain selected as threshold.

=⇒ complete order




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Evaluation Function

Probability that a profile is disadvantaged, with respect to thechosen threshold.

P(p ∈ disadvantage set) =∑p

η(p)wp

where wp is the weight assigned to profile p.




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Gap measure

A synthetic measure of gap is computed as follows:

G =

#p∑p=1

wpdist(p)

dist(p) =|p − p|M

where

• wp is a weight assigned to each profile, equal to the relativefrequency of subjects sharing the same profile

• p is the first profile greater than the threshold

• M is the absolute distance of the first profile greater than thethreshold from the minimal element.




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Gap measure

Note: The absolute distance is set equal to 0 when a profile isgreater than the threshold.Interpretation of gap measure:

• Measure the severity of deprivation;

• Is basically a measure of the fraction of people a woman mustovertake to exit deprivation;

• It depends upon the distribution on the graph of profiles.




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Case Study

Data Source: DHS dataset concerning women’s status andgender disparitiesCountries: 16 African states (surveys carried out between 2007and 2012)




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Responses Structure

Who has the final say on ... ?

1. Partner alone

2. Woman and Partner

3. Woman alone




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Selected Variables

Own health care

Who has the final say on woman’s health care?

Social Isolation

Who has the final say on visits to family or relatives?

Money administration

Who has the final say on making large household purchases?




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Hasse Diagram

Hasse Diagram of the 27 possible profiles




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Partial Order

Figure: Certainly disadvantaged




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Ranking

Country Rank Score1 Zimbabwe 0.14852 Egypt 0.28633 Kenya 0.30804 Mozambique 0.30935 Ghana 0.31206 Ethiopia 0.31487 Zambia 0.34078 Uganda 0.3716

Country Rank Score9 Benin 0.412310 Malawi 0.503911 Tanzania 0.546912 Guinea 0.601913 Nigeria 0.604914 Cote d’Ivoire 0.624515 Burkina Faso 0.660716 Mali 0.8047




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Comparisons




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Comparisons

EducationalAttainment

Health andSurvival

EconomicParticipation

PoliticalEmpowerment




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Comparisons




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Truly multidimensional

Figure: Multiple index VS poset




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Conclusions

Advantages

1 Respect of the ordinalnature of the data

2 Truly multidimensional

Disadvantages

1 Computationally intensive

2 Individual data are needed


Documents

MEASURING GENDER GAP FROM A POSET PERSPECTIVEcomplexity.stat.unipd.it/system/files/Di_Brisco_Farina.pdf · Interpretation of gap measure: Measure the severity of deprivation; Is basically