There is no consensus regarding conditions and circumstances where each individual rank-dependant indicator of socio-economic inequality is to be used. What emerges from this paper is that the concentration index approach needs to be confined to situations where the health variable is of ratio-scale type.

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DEPARTMENT OF SOCIAL POLICY
Topic
Do Concentration Indice(s) “CI”
measure Horizontal (IN)equity “HI” in
health care?
Author
Kaouthar Lbiati (MD, MSc)
Year
2016
Objectives
The purpose of this paper is threefold. First, to describe the relationship
between the distribution of health need and health care use and how the
imbalance between these two leads to inequity. Second, to compare the
properties of the most commonly used rank-dependent inequality indices
and discuss their respective suitability, according to their intrinsic
properties, under a variety of socioeconomic inequalities’ measurements
circumstances; including when health variables are qualitative and in
presence of different measurement scales. Third, to provide “guidelines” on
the most relevant indicator’s choice against these same circumstances.

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I/ Relation between Health and Equity
In most of OECD countries, the main concept of the health care systems is
egalitarian. This implies that health care is allocated according to individual
need and not willingness to pay. Need for care is defined as a person’s
expected use of medical care on the basis of actual need characteristics
(age, gender), with the effects of all other variables, such as income,
“neutralized” by their being set at their sample means in the prediction
stage of the performed linear regression. Need then indicates the amount of
medical care a person would have received had that person been treated the
same as others with the same need characteristics, on average.
Research on income inequality has inspired much of the research on the
horizontal equity. According to Wagstaff et al (1) “individuals with the
same levels of need consume the same amount of resources”. The
assumption here is that the average treatment differences between people in
unequal need, reflect the accepted overall “norm”. To measure the degree
of horizontal inequity in health care use by income, we compare the actual
observed distribution of medical care use by income with the need-
standardized expected distribution of use. This method looks only at
relative inequalities in mean use levels by income after any need
differences have been standardized for. In our paper, inequities in use of
care denotes of the number of visits to GPs and/or specialists. When the
horizontal inequity index equals zero, it indicates horizontal equity: people
in equal need (but at different incomes) are treated equally. When the index

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is positive, it indicates pro-rich inequity, and when it is negative, it
indicates pro-poor inequity.
In the next section we will describe the most commonly used tool for
measuring rank-dependent inequalities, discuss its limitations and explain
how it has evolved, under a variety of socioeconomic circumstances, into
alternative rank-dependent inequalities indices which satisfy better a full set
of requirements which we will detail.
II/ Measurement of health equity/INequity
The degree of inequality in health care use can be measured using the
concept of a concentration curve. This plots the cumulative distribution of
care use as a function of the cumulative distribution of the population
ranked by its income. A distribution is equal if its concentration curve
coincides with the diagonal. A curve that lies above the diagonal indicates
that use is more concentrated among the poor.
Wagstaff et al (2) have reviewed and compared the properties of the
concentration curves and indices with alternative measures of health
inequality. They argue that the main advantages of the concentration curves
and indices are the following:
(i) they measure the socioeconomic inequality of health by taking into
account every individual’s level of health and every individual’s rank in the
socioeconomic domain; (ii) they use information from the whole income
distribution rather than just the extremes; (iii) they give the possibility of

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visual representation through the concentration curve, (iv) they allow
checks of dominance relationships.
1- Concentration Curve - Concentration Index
A Concentration Index measures the degree of inequality in actual use
as the area between the concentration curve and the diagonal.
Concentration Index is equal to the covariance between individual health
(hi) and the individual’s relative rank (Ri) according to socioeconomic
status, scaled by the mean of health in the population (µ). Then the whole
expression is multiplied by 2, to ensure the Concentration Index ranges
between −1 and +1. CI is defined as follows:
The concentration index is usually standardised by age and gender to
account for the role that demographic factors play in generating such
inequality (3)
Of note, a value of zero for the concentration index does not mean absence
of inequality, but an absence of the socioeconomic gradient in the
distribution, that is, an absence of inequality associated with the
socioeconomic characteristics.

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2- Properties of an Ideal Health Index
As a matter of fact, an ideal health index should satisfy four key
requirements – transfer, level independence (which is closely related to
monotonicity), cardinal invariance and mirror.
The transfer property states that any mean-preserving change of the
distribution which favours the better-off is translated into a pro-rich
change of the index, and one in favour of the worse-off into a pro-poor
change.
Level independence: For any given health distribution “h”, if the health
levels of all persons change by the same absolute amount and if both the
lower and the upper limit of the health variable shift by the same amount
the value of health index remains constant.
Monotonicity for any given distribution h, if the health of a “rich” person
increases, the value of the index increases; if the health of a “poor” person
increases, the value of index decreases. For rank-dependant socioeconomic
indicators of health, level independence implies monotonicity.
Cardinal invariance: For any given health distribution h, a positive linear
transformation of the health variable does not affect the value of the health
index

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Mirror: For any given health distribution “h”, the health index and the
associated ill health index are equal in absolute value, but have opposite
signs.
As an ad-hoc, it is useful to mention Linearity or linear trajectory present
in the event of an equi-proportional reduction of all individual levels; it
implies that when all individual levels are halved, the measured degree of
inequality is also cut in half.
3- Limitations of Concentration Index
The health Concentration Index has got some shortcomings. The first issue
is related to the bounds of the Concentration Index which may depend
upon the mean of the health variable and hence make a comparison of
populations with different mean health levels problematic. The bounds’
character is a phenomenon which has been described by Wagstaff back in
2005 (4), in the case of a binary variable. This holds more generally for any
bounded variables (5).
Moreover, the bounded character of the health variable raises the question
of the relation between health and ill health inequality. Since health and ill
health are mirrors of one another, it seems reasonable to expect that the two
Concentration Indices C(h) and −C(s) give mirror images of the degree of
inequality unless the corresponding respective distributions have different
means. Clarke et al. (6), have shown that, when comparing inequality
between countries, using CI can lead to different rankings according to
whether we look at health or ill health.

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Second, the value of the index is to a large extent arbitrary if the health
variable is of a qualitative nature (7, 8). Indeed, Rank-dependent
inequality indices cannot be applied to nominal and ordinal health
indicators since nominal and ordinal measurement scales do not allow
differences between individuals to be compared. Researchers have
projected cardinal scales upon these ordinal health indicators (9) for
example. In principle, these can also lead to ratio scaled variables.
Third, the property of invariance to measurement scale (or the absolute /
relative criterion) is indispensable as it makes the value of an index
independent of the specific unit in which a variable is expressed.
The value of the health Concentration Index remains unchanged only
under a positive proportional transformation (relative inequality) (7) but not
under a positive linear transformation (absolute inequality) that would be
needed for cardinal health variables. Implicitly, CI requires for health to be
measured on a ratio-scale.
The fourth issue is when trying to apply the notions of the absolute /
relative criterion to bounded variables. This concept is critical to ensure the
robustness of the index when changes to health happen. The difficulty is
that some of the changes are infeasible because the bounds of the variables
act as constraints. Hence, the change in the definitions. When referring to
bounded variables, we refer to ‘quasi-relative’ and ‘quasi-absolute’ indices.
As opposed to quasi absolute index, a quasi-relative scale-invariant rank-

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dependent health inequality index remains insensitive to any feasible
proportional change of health and takes into account only the relative
positions of individuals, not the absolute differences among persons.
4- Alternative Health Indices
4-1 The Modified Concentration Index
In order to satisfy simultaneously the Mirror condition and the Scale
Invariance, the standard Concentration Index needs to be modified as
proposed by Erreygers (10) into a corrected version of the standard CI
denoted as follows:
ax - lower bound; µ- the mean of health in the population n, λ rank
Also, one should apply the modified CI when the projection of an
ordinal variable gives rise to an unbounded cardinal variable.
4-2 The Wagstaff normalization

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Wagstaff et al (4) has suggested a normalization formula which addresses
the bounds issue, ill health and cardinality issues.
ax - lower bound; bh - upper bound
However, the Wagstaff procedure apparently exaggerates the correction by
forcing the bounds always to the same values -1/+1. Clearly,there might be
situations of changes in the distribution of health to which the Wagstaff
index remains insensitive, or reacts in a counter-intuitive way. For instance,
when it fails totranslate a pro-rich change of the health distribution into an
increase, and a pro-poor change into a decrease. Hence the introduction of
The Generalized Concentration Index V(h)
4-3 The Generalized Concentration Index V(h) is expressed as
It satisfies the mirror condition and also succeeds where W(h) fails in that it
translates a pro-rich change of the health distribution into an increase and a
pro-poor change into a decrease. It is sensitive to the scale or unit of the

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health variable.A positive linear transformation of the health variable can
change the value of the Generalized Concentration Index which makes it an
absolute measure of inequality, not a relative one. However, Erreygers (10)
argues that a simple correction, consisting of dividing index V(h) by the
range of the health variable bh – ah (upper bound-lower bound) and
multiplying the result by 4, suffices to turn this absolute indicator of
inequality into a relative one.
4-4 The Erreygers index E(h) or E(x) is expressed as follows:
The Erreygers index manages to remedy all shortcomings from which
suffer the previous ones (11). As such, it satisfies the Mirror condition and
Cardinal Invariance. E(h) has also the advantage to be both an absolute and
relative health inequality indicator (for bounded variables). Moreover, it is
the only index satisfying the property of Linearity which implies that when
all individual levels are halved, the measured degree of inequality is also
cut in half.
In summary, each index satisfies a set of properties. Thus, it can be adapted
to the inequalities measurement purpose of the analyst. In the absence of
consensus, Erreygers (10) has elaborated recommendations to serve as
guidance under specific socio-economic circumstances. We will detail
these guidelines in the next section.

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Finally, there seems to be correlations between indicators. A positive
correlation between CI(h) and E(h) has been reported by Van de Poel (12)
and a strong negative relationship between −CI(s) and W(h) by Wagstaff
and Watanabe (13)
III/ Guidelines on the choice between health indices
Different types of indicators among the rank-dependant family all with
different properties are available for use. Choosing the wrong indicator
might be misleading, in the sense that it may lead to a wrong assessment of
inequity. Therefore, the choice of the concentration Indices requires a more
scrupulous evaluation.
Although, there doesn’t seem to be a global agreement on the type of
indicator one should use to measure health inequalities. Yet, depending on
the type of health variable and dimension of inequality to be measured, one
may want to choose one index over another.
The purpose of this section is to link the indices properties to qualitative
characteristics and measurement scales of health variables, as previously
discussed, in order to provide guidelines on how to discriminate between
the rank-dependent inequality indices. This has been summarized by the
authors (10) in tables 2 & 3, annexe A and as follows:
The CI that measures relative inequalities – can be applied only to
unbounded variables with ratio- or fixed scale.

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The Generalized CI that measures absolute inequalities – can be applied
only to unbounded variables with fixed scale
In view of the findings on the Scale Invariance section, one should apply
the modified CI when the projection of an ordinal variable gives rise to an
unbounded cardinal variable, the CI when the projection leads to an
unbounded ratio-scale variable, and any member of the class of indices
defined above when the projection generates a bounded cardinal or ratio-
scale variable.
The modified CI emerges as the only index to satisfy the condition of
simultaneous Mirror and quasi-Relativity.
Erreygers Index E(h) is the only rank-dependent inequality measure that
has the properties of Mirror and quasi-Absoluteness.
For bounded variables, whether they be of the cardinal, ratio- or of the
fixed-scale type, the choice of indices depends on one’s judgements on how
the rank-dependent inequality index should react to changes in average
standardized health (or ill health).
The only case in which the CI and the Generalized CI should be combined
to measure both absolute and relative inequalities is when health is
unbounded and measured on a fixed scale (11).

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IV/ Conclusion
There is no consensus regarding conditions and circumstances where each
individual rank-dependant indicator of socio-economic inequality is to be
used.
What emerges from this paper is that the concentration index approach
needs to be confined to situations where the health variable is of ratio-scale
type. Alternatively, the set of desirable properties each rank-independent
indicator of socio-economic inequality offers (through the normalization
function), the properties of the health variable (whether it is bounded or
unbounded) and the scale of measurement (ratio-, fixed-, or cardinal-scale)
are ultimately what determines the choice of the indicator by the researcher
which in turn determines the perceived change of horizontal inequality.
Moreover, depending on the mono or bi-dimensionality (Relative vs
Absolute as these may rank health distribution differently) sought, one may
favour an index over another.
References
1. wagstaff and Van Doorslaer 2000
2. Wagstaff A, Paci P, van Doorslaer E. Social Science and Medicine
1991
3. Kakwani N, Wagstaff A, van Doorslaer E. Journal of Economy 1997
4. Wagstaff, 2005
5. Erreygers, 2009a
6. Clarke et al, 2002

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7. Erreygers, 2006
8. Zheng, 2006
9. van Doorslaer and Jones, 2003
10. G. Erreygers. Journal of Health Economics 2009
11. Erreygers and van ourti. J Health Econ 2011
12. Van de Poel et al. 2006
13. Wagstaff and Watanabe 2000, p. 33
Annexe A
Table 1: (10)

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Table 2 (10)