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Electronic copy available at: http://ssrn.com/abstract=1920336

Guido Erreygers, Philip Clarke and Tom van Ourti

“Mirror, Mirror, on the Wall, Who in this

Land is Fairest of All”

Distributional Sensitivity in the Measurement of

Socioeconomic Inequality of Health

DP 08/2011-074

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Electronic copy available at: http://ssrn.com/abstract=1920336

1

“Mirror, mirror, on the wall, who in this land is fairest of all?” –

Distributional sensitivity in the measurement of socioeconomic

inequality of health

Guido ERREYGERS, Philip CLARKE, and Tom VAN OURTI

Addresses: Guido Erreygers, Department of Economics, University of Antwerp, City

Campus, Prinsstraat 13, 2000 Antwerpen, Belgium; guido.erreygers@ua.ac.be

Philip Clarke, School of Public Health, University of Sydney, Edward Ford Building,

NSW 2006 Australia; philipc@health.usyd.edu.au

Tom Van Ourti, Erasmus School of Economics, Erasmus University Rotterdam, PO

Box 1738, 3000 DR Rotterdam, The Netherlands; Tinbergen Institute and NETSPAR;

vanourti@ese.eur.nl

Abstract:

This paper explores four alternative indices for measuring health inequalities in a way

that takes into account attitudes towards inequality. First, we revisit the extended

concentration index which has been proposed to make it possible to introduce changes

into the distributional value judgements implicit in the standard concentration index.

Next, we suggest an alternative index based on a different weighting scheme. In contrast

to the extended concentration index, this new index has the ‘symmetry’ property. We

also show how these indices can be generalized so that they satisfy the ‘mirror’

property, which may be seen as a desirable property when dealing with bounded

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Electronic copy available at: http://ssrn.com/abstract=1920336

2

variables. We empirically compare the different indices for under-five mortality rates

and the number of antenatal visits in developing countries.

JEL Classification Codes: I19, I32, D63

Keywords: health inequality, socioeconomic inequality, extended concentration index,

distributional sensitivity, small-sample bias

Acknowledgments: Tom Van Ourti is supported by the NETSPAR project ‘Income,

health and work across the life cycle II’, and acknowledges support by the National

Institute on Ageing, under grant R01AG037398-01. We have benefited from the

comments and suggestions of two anonymous referees, Ramses Abul Naga, Paul

Allanson, Clément de Chaisemartin, Gustav Kjellsson, Andreas Knabe, Ann Lecluyse,

Dennis Petrie, and participants of seminars at Erasmus University Rotterdam,

University of Antwerp, Lund University, the Health, happiness and inequality

conference at the Technische Universität Darmstadt, the 2010 Irdes workshop on

applied health economics and policy evaluation in Paris, the LowLands Health

Economics Study Group in Egmond aan Zee, and the 2011 iHEA conference in

Toronto. We also thank Ellen Van de Poel for assistance with the DHS data. The usual

caveats apply and all remaining errors are our responsibility.

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1. Introduction

Pereira (1998) and more recently Wagstaff (2002) have proposed to extend the

concentration index by including a distributional judgement parameter. The extension is

seen as a device which makes it possible to formally incorporate attitudes towards

inequality into the calculation of the index of socioeconomic inequality of health. It

builds on suggestions of Kakwani (1980) developed by Yitzhaki (1983), who shows

how a similar extension of the Gini coefficient allows the expression of distributional

judgements in the context of income inequality measurement.

The extended concentration index can be applied to a broad range of health and

health care variables. Following Pereira (1998) and Wagstaff (2002), who have used the

index to calculate the degree of socioeconomic inequality in child mortality in

developed as well as developing countries, there are now a growing number of

empirical studies which have applied the index to various health variables. Examples

include health limitations within eight European countries across time (Hernández-

Quevedo et al. 2006), child malnutrition in Nigeria (Uthman, 2009), immunization

ratios in developing countries (Gaudin and Yazbeck 2006; Meheus and Van Doorslaer

2008), and child mortality and child malnutrition in India (Arokiasamy and Pradhan

2010).

In line with recent research on health inequality measurement, we make a clear

distinction between bounded and unbounded variables, and hence treat them separately.

The main reason for this different treatment is that bounded variables, in contrast to

unbounded variables, can be looked at from two points of view: the positive side, where

the focus is on ‘good health’ (e.g. the proportion of children without malnutrition), and

the negative side, where the emphasis is on ‘ill health’ (e.g. the proportion of children

with malnutrition).

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In this paper we first of all explore whether the extended concentration index is

an appropriate tool to take into account attitudes to inequality when measuring the

socioeconomic inequality of health. Our initial focus is on understanding the precise

way the extended concentration index incorporates distributional sensitivity when it is

applied to unbounded health variables (section 3). Next, we identify a property which

the index does not have, and suggest an alternative index – the symmetric index – based

upon a different distributional weighting scheme (section 4). We then move to bounded

variables, and generalize both the extended concentration index and the symmetric

index (section 5). An empirical study serves to illustrate the differences between the

indices (section 6). We also enclose an appendix specifying how we deal with small-

sample bias, ties in the ranking variable, and differences in (ex-post) sampling

probabilities when doing empirical work using finite samples.

2. Preliminaries

In the first part of this paper we consider unbounded ratio-scale health variables. These

are variables which have no natural upper bound and vary between 0 and +∞; health

expenditure is an example. In section 5 we will turn our attention to bounded variables,

which occur very frequently in the domain of health.

Suppose the population N consists of n individuals, where n is a finite,

positive natural number. Let

= {1,2,..., }

Nn , and assume that individuals are ranked

according to their socioeconomic position, in ascending order (i.e. individual 1 is the

poorest, and individual n the richest person). If individual i is not tied to any other, his

rank

iρ coincides with his number i (

=

i

i

ρ

); if he is tied to other persons, all persons of

the tied group have a rank equal to the average number of the members of this group.

The fractional rank

iR of individual i is equal to (21)/(2 )

i

n

ρ −

, and varies between

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5

1/(2 ) n and 1 1/(2 ) n

−

(if there are no ties). The average rank

ρ

µ is equal to (1)/ 2

n +

,

and the average fractional rank

R

µ equal to 1/2 .

When n becomes very large, the fractional rank can be approximated by a

continuous variable p defined over the interval [0,1]. The interval [0, ] p then

represents the 100 %

p

poorest individuals of the population, just as those with

fractional ranks 1/(2 ),3/(2 ),...,(2 1)/(2 )

nnin

−

represent the 100( / )%

i n

poorest

individuals. The function ( )

h p expresses the health status of an individual as a function

of where this individual is located in the interval [0,1]. Clearly, (0)

h

is the health status

of the poorest individual and (1)

h

that of the richest.

In case of a finite number of individuals, the average health status is defined as:

=1

1

=

µ

∑

i

n

hi

h

n

(1)

For the continuous case we have:

1

0

= ( )

h p dp

µ

∫

h

(2)

3. The Extended Concentration Index

Following suggestions by Kakwani (1980) and Yitzhaki (1983), both Pereira (1998) and

Wagstaff (2002) have introduced the following extended concentration index:

1

=1

( , ) =1

C h

ν

(1)

n

ii

i

h

Rh

n

ν−

ν

µ∑

−−

(3)

where

1

ν ≥ is a distributional sensitivity parameter. Expression (3) can be formulated

in many equivalent ways. Using the definitions of the previous section, (3) can be

transformed into a weighted sum of health shares:

()

1

=1

11

1

( , ) =

C h

n

i

i

i

h

R

h

n

ν−

−ν−

ν

µ

∑

(4)

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Since p corresponds to

iR, the continuous counterpart of (4) is:

1

1

0

1

( , ) =

C h

[1(1)] ( )

h p dp

h

p

ν−

ν−ν −

µ∫

(5)

The extended concentration index (5), in a similar fashion to income inequality

measures such as the extended Gini index (Yitzhaki 1983), assigns weights to

individuals based upon their fractional rank p modified by the distributional sensitivity

parameter ν . A good way to understand how the extended concentration index is

influenced by ν is to focus on the weighting function, which expresses how the weight

of a person depends on her fractional rank p and the distributional sensitivity

parameter ν :

1

( , ) 1

w p

ν = −ν −

(1)

p

ν−

(6)

Figure 1 plots the weighting function for a range of values of ν .

[Insert Figure 1 somewhere here]

What we will term the standard concentration index is simply a special case of

(5) with ν being set equal to 2. In this case the weighting function is ( ,2)21

w pp

=− , a

linear function of p which goes from 1

− to 1

+ as the individual’s position increases

from the lowest to the highest in the population. Those above the median have positive

weights, and those below it negative ones. In case all have the same health level, the

standard concentration index is zero; a negative

( ,2)

C h

indicates that health is

concentrated more among the poor than the rich, and a positive one the reverse.

With regard to other values of the distributional sensitivity parameter, if we take

1

ν = the weighting function is constant and equal to 0. Therefore the index will always

have the value of 0 and so inequalities are not taken into account in the extended index.

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From now on we assume that

1

ν > ; the weighting function is then a strictly increasing

function of the fractional rank, with some individuals having a negative weight and

some a positive. (The cut-off point between positive and negative values can be

determined by searching for the individual whose fractional rank p is such that

1/(1)

1 (1/ )

= −

p

ν−

ν

.) For values 12

< ν < only the individuals at the higher end of the

income distribution have positive weights. For values

2

ν > also individuals below the

median receive positive weights, but those at the bottom of the income distribution have

quite large negative weights. As the value of ν increases, gradually more and more

individuals have positive weights (which will all tend to 1) and in the end only the

poorest individual has a negative (and very large) weight.1 In the most extreme case

when ν → +∞, the extended concentration index in equation (5) tends to

(0)

h

h

h

µ −

µ

. So

unless we take

2

ν = the weighting scheme is asymmetric, and the more so the higher

the value of ν .

The bounds of the extended concentration index can be derived by assuming that

either the poorest or the richest individual is the only one with a positive level of health.

Using the continuous version of the index we obtain:

1( , )

C h

1

−ν ≤ν ≤ (7)

Except for the case when

2

ν = , these bounds are not symmetric. An intuitive

interpretation of these bounds is that they provide the weights given to the poorest and

the richest person when calculating the extended index. In the standard case these

weights are

1

− and

1

+ , which means that the absolute distance between the two is

1 For ν → +∞, the value of the index based on a finite number of individuals becomes distorted and

needs to be adjusted. The appendix provides more details.

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equal to 2. The choice of a particular value of ν can therefore be made dependent on

the desired distance between the two.

4. Symmetry and Distributional Sensitivity

4.1. Univariate vs. Bivariate Inequality

The extended concentration index has been obtained by applying a concept used for the

measurement of the inequality of income – the extended Gini coefficient – to the

measurement of the socioeconomic inequality of health. Basically, a one-dimensional

construction is transplanted into a two-dimensional context. It cannot be taken for

granted, however, that anything which works well in a univariate environment is

automatically suited for a bivariate world.

Here we propose a simple test to check whether an indicator is a good measure

of the degree of association between socioeconomic status and health. Imagine that we

turn the world upside down: for a brief moment of time the poor and the rich switch

roles (one may think of Carnival). More specifically, let us assume that the poorest

person and the richest person switch their health levels, that the second poorest and the

second richest person switch their health levels, etc. In formal terms, this leads to a new

health function ( )(1)

≡−

g php , which is the health function ( )

h p turned upside down.

Our test consists of looking at the reaction of the indicator when the health function

( )

h p is replaced by ( )

g p . We say that an indicator I passes the ‘upside down’ test if

( )

I h and ( )

I g are always of the opposite sign (or both equal to zero). In other terms, if

the indicator states that distribution ( )

h p is pro-poor (c.q. pro-rich), then it must always

state that distribution ( )

g p is pro-rich (c.q. pro-poor).

It is not difficult to verify that the extended concentration index does not pass

this test, except when

1

ν = , a case we excluded, or when

2

ν = , the standard

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concentration index. The reason for this lies in the asymmetric nature of the weighting

function. This can be illustrated by looking at the case where the chances of having high

or low health levels are symmetrically distributed over the rich and the poor. An

extreme example of such a symmetric distribution is the one in which only the richest

and the poorest individuals have a very high health level, and all others the minimum

level. This is of course a very unequal distribution, but it may be argued that since there

is no systematic bias in favour of either the rich or the poor, the index of socioeconomic

inequality should therefore be equal to zero. This is exactly what we find if we use the

standard concentration index, but not if we use the extended concentration index with ν

different from 1 or 2.

4.2. A General Formulation

When looking for an alternative, we will try to remain as close as possible to the

extended concentration index. Let us consider indices of the following type:2

1

0

( , )

I h

( , )( , ) ( )

w p

ε

h

f h p dp

ε = µ ε

∫

(8)

where

( , )

h

f µ ε is a normalization function,

( , )

w p ε a weighting function, and ε an

distributional sensitivity parameter. These indices belong to the class of Mehran (1976)

measures, applied to socioeconomic inequality. It is customary to restrict the attention

to indices for which the weighting function is an increasing function of p and for

which the weights sum up to zero, i.e.

1

0

( , )0

ε =

∫w p

.3 In addition, we assume that the

2 This is the continuous version; we introduce the version for a finite number of individuals in the

appendix.

3 These conditions ensure that the property of income-related health transfers holds (Bleichrodt and van

Doorslaer 2006), and that the value of the index is equal to zero when health is distributed equally.

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weighting function is continuous and not identically equal to zero (because then the

index would always be equal to zero, as in the case

1

ν = above). With regard to the

normalization function we assume that it is positive-valued, which implies that there is

no level of average health or of the distributional sensitivity parameter such that

( , )0

h

f µ ε = . We call

( , ) ( , )

w p

µ ε

h

f

ε the normalized weighting function. It is

immediately clear that (5) is a special case of (8), with ε =ν,

( , ) 1/

hh

f µ ν =µ and

1

( , ) 1

w p

ν = −ν −

(1)

p

ν−

.

4.3. The Symmetry Property

The following result specifies for which type of weighting function an indicator passes

the ‘upside down’ test, i.e. is such that ( , )

I h ε and ( , )

I g ε are always of the opposite

sign (or both equal to zero).

Theorem

The index ( , )

I h ε passes the ‘upside down’ test if and only if the weighting function is

inversely symmetric around 1

2, i.e. if and only if we have

( , )

w p

(1, )

ε for any

wp

ε = −−

01

p

≤≤ .

Proof. (i) Let us rewrite (8) as

1/21

01/2

∫

( , )

I h

(, )( , ) ( )

w p

ε

( , ) ( )

w p

ε

h

fh p dp h p dp

ε =µ ε+

∫

. If

( , )

w p

(1 , )

ε for

wp

ε = −−

any 01

p

≤≤ , then obviously we have

11/2

1/2

∫

0

( , ) ( )

w p

ε

( , ) (1

w p

ε

)

h p dphp dp

= −−

∫

.

Hence we obtain

1/2

0

( , )

I h

(, )( , )[ ( )

w p

ε

(1)]

h

fh php dp

ε =µ ε−−

∫

. Likewise we derive

that

1/2

0

( , )

I g

(, )

ε

( , )[ ( )

w p

ε

(1)]

g

fg pgp dp

ε =µ−−

∫

. Since

( )

g p

(1)

hp

=−

and

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11

gh

µ = µ , it follows that ( , )( , )

I g I h

ε = −ε . This proves sufficiency. (ii) Suppose that the

weighting function is not inversely symmetrical around

1

2

p = . Then we can always find

an interval [ , ]

a b , where

1

2

0

ab

≤< ≤ , such that

1

1

( , )

w p

( , )

w p

ba

ab

dp dp

−

−

ε ≠ −ε

∫∫

. Since at

least one of these integrals is different from zero, we can assume without loss of

generality that

( , )0

b

aw p

∫

dp

ε≠

, so that we can write

1

1

( , )

w p

( , )

w p

ab

ba

dp dp

−

−

−ε = θε

∫∫

,

where

1

θ ≠ . Consider a health distribution characterized by a function ( )

h p with the

following properties:

( )

h p

0

c

=> for apb

≤≤

and 11

bpa

−≤≤ − , and

( )

h p =

0

elsewhere. For this distribution we have

1

1

( , )

I h

(, )( , )

w p

( , )

w p

(, ) (1

µ ε

)( , )

w p

bab

hh

aba

fc dpc dpfcdp

−

−

ε =µ εε+ε= −θε

∫∫∫

and

also

1

1

( , )

I g

(, ) ( , )

w p

( , )

w p

(, ) (1

µ ε

)( , )

w p

bab

gg

aba

fcdp c dpfcdp

−

−

ε = µ εε+ε= −θε

∫∫∫

.

Since

gh

µ = µ , we obtain ( , )( , )

I g

0

I h

ε =ε ≠

, which means that the indicator does not

pass the upside down test. This proves necessity.■

The sufficiency part of the proof shows that indicators which pass the upside down test

are always such that ( , )( , )

I gI h

ε = −ε . This is what we call the symmetry property.

Although the symmetry property is at first sight a stronger requirement than passing the

upside down test, Theorem 1 reveals that they are equivalent. The theorem also shows

that if we want the symmetry property to hold, then we are obliged to abandon the

weighting function of the extended concentration index.

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4.4. The Symmetric Index

In order to construct an index with the symmetry property, we have to replace the

asymmetric weighting scheme of the extended concentration index by an inversely

symmetric weighting scheme. This implies that if we want to maintain relatively high

negative weights for the poorest individuals, we need to give relatively high positive

weights to the richest individuals. The symmetric index we propose here is defined as

follows:

2

1

2

22

1

2

1

2

0

1

( , )

S h

2()() ( )

h p dp

h

pp

β−

β−

β =β−−

µ∫

(9)

with

1

β > .4 In terms of expression (8), we have ε = β and:

2

22

1

2

1

2

2

1

(, ),( , )

w p

2()()

h

h

fpp

β−

β−

µ β = β =β−−

µ

(10)

One can check that for

2

β = we have

( ,2)

w p

21

p

=− , which means that for this value

the symmetric index coincides with the extended concentration index with

2

ν = .

The weighting scheme has been devised in such a way that those with fractional

ranks above the median always have positive weights, and those below the median

always negative weights. As can be seen from Figure 2, by taking 12

< β <

we give

relatively higher weights to those with a fractional rank close to the median, while by

taking

2

β > we give relatively higher weights to those at the upper and lower end of

income distribution. In the most extreme case (β → +∞ ) the symmetric index tends to

4 For

1

β< the weighting function is decreasing in p , and for

1

β= it has a discontinuity at

1/2

p =

(it

jumps from 1/2

−

to 1/2

+

), so we exclude these values

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(1) (0)

4

h

hh

−

µ

. The value of the index varies between

/ 2

−β

and

/ 2

+β

.5 Just as for the

extended concentration index, the distance between the bounds is equal to the value of

the distributional sensitivity parameter, β , and coincides with the distance between the

weights of the richest and the poorest individual.

[Insert Figure 2 somewhere here]

4.5. Comparing the Extended Concentration Index and the Symmetric Index

Because of the symmetry property, the individual who occupies the median position in

the income distribution plays a pivotal role in the calculation of the symmetric index.

Suppose there is a ceteris paribus increase of the health level of one person located at

position p in the socioeconomic distribution. What would be the effect of such a

change upon the value of the index measuring socioeconomic health inequalities? Let us

start at

0

p = (the poorest individual). Obviously this is a pro-poor change, and we

expect the index to become more pro-poor, i.e. to decrease in value. This implies that

we always have

(0, )0

w

ε < . Next, let us increase p and wonder from what value of p

the change becomes pro-rich, i.e. at which point

( , )

w p ε turns positive. If we think that

this threshold value

*

p should be lower than the median, we could opt for the extended

concentration index: given

*

1/2

p <

, if we choose the value of the distributional

sensitivity parameter

*

ν , where

*

ν is such that

*

** 1/(

)

1)

1 (1/

= −

p

ν −

ν

, we obtain the

5 By changing the normalization function into

2

(, )

h

h

f µ β =βµ

, we would obtain an index which always

varies between –1 and +1.

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14

desired result. If, however, we decide that the threshold value should always be equal to

the median, the symmetric index seems a more appropriate choice.

The threshold value

*

p demarcates the group of the poor from the group of the

non-poor. We believe that the choice of

*

0.5

p =

is a reasonable point of departure as

0.5 is the expected location of a person. In other words, the lower half of the population

is considered as poor, and the upper half as rich. We do not exclude that another value,

say

*

0.25

p =

, might be more appropriate than our a priori choice, but without

additional information (e.g. on income levels) we think it is very hard to make a case for

such an alternative boundary. By construction, rank-dependent inequality measures

leave that kind of information out of consideration, and therefore naturally lead us to

take

*

0.5

p =

, at least as a starting point.

Another issue concerns the reaction of the index of socioeconomic health

inequality to health transfers at different locations in the distribution. Suppose there is a

transfer of health ∆ from a person located at position

jp to a person located at position

ip , with

ji

ppd

=+

and

0

d > (i.e. the first person is richer than the second). We can

compare the effect of such a transfer for different equidistant individuals. A good

measure of where the transfer takes place is given by the number

/2 /2

ji

zpdpd

=−=+

, i.e. the location halfway between

ip and

jp . If we believe

that the effect of such a transfer should become smaller and smaller as z increases, we

have to opt for the extended concentration index with

2

v > . If, however, we think that

the effect should be smaller the closer z lies to the centre (i.e.

*

0.5

p =

), then the

symmetric index with

2

β > seems more appropriate. The first property is that of

sensitivity strictly increasing with poverty, in short ‘sensitivity to poverty’; the second