Analyzing Health Equity Using
Household Survey Data
Lecture 8
Concentration Index
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Can you compare the degree of inequality
in child mortality across these countries?
100%
Equality
Brazil
Cote d'Ivoire
Ghana
Nepal
Nicaragua
Pakistan
Cebu
S Africa
Vietnam
cumul %
under-5 deaths
80%
60%
40%
Brazil is most unequal,
but how do the rest compare?
20%
0%
0%
20%
40%
60%
80%
100%
cumul % live births,
ranked by equiv consumption
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Concentration index (CI)
cum. % of under 5 deaths
100%
75%
CI = 2 x area
between 450 line and
concentration curve
50%
CI < 0 when variable
is higher amongst
poor
25%
0%
0%
25%
50%
75%
100%
cum. % live births ranked by equivalent consumption
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Concentration indices for U5MR
C and 95% conf interval
0.1
0.0
-0.1
-0.2
-0.3
-0.4
Brazil (NE &
SE) 1987-92
Nicaragua
1983-88
Phillipines
(Cebu) 198191
South Africa
1985-89
Nepal 198596
Cote d’Ivoire
1978-89
Ghana 197889
Pakistan
1981-90
Vietnam
1982-93
-0.5
“Analyzing
Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
.
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Concentration index defined
C = 2 x area
between 450 line and concentration
curve
= A/(A+B)
cum. % of health variable
100%
75%
C>0 (<0) if health variable is
disproportionately concentrated on
rich (poor)
50%
Lh  p 
A
C=0 if distribution in proportionate
B
25%
C lies in range (-1,1)
0%
0%
25%
50%
75%
100%
cum. % population ranked by income
C=1 if richest person has all of the
health variable
C=-1 of poorest person has all of
the health variable
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Some formulae for the concentration
index
1
C 1  2 Lh  p  dp
0
If the living standards variable is discrete:
2 n
1 where n is sample size, h the
C
hi ri  1 

n i 1
n health variable, μ its mean and
r the fractional rank by income
For computation, this is more convenient:
C
2

cov  h, r 
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Properties of the concentration index
• depend on the measurement characteristics of the health
variable of interest.
• Strictly, requires ratio scaled, non-negative variable
• Invariant to multiplication by scalar
• But not to any linear transformation
• So, not appropriate for interval scaled variable with
arbitrary mean
• This can be problematic for measures of health that are
often ordinal
• If variable is dichotomous, C lies in the interval (μ-1, 1-μ)
(Wagstaff, 2005):
– So interval shrinks as mean rises.
– Normalise by dividing C by 1-μ
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Erreygers (2006) modified
concentration index
E h  4

bh  ah
C h
Where bh and ah are the max and min
of the health variable (h)
• This satisfies the following axioms:
– Level independence: E(h*)=E(h), h*=k+h
– Cardinal consistency: E(h*)=E(h), h*=k+gH,
k>0, g>0
– Mirror: E(h)=-E(s), s=bh-h
– Monotonicity
– Transfer
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Interpreting the concentration index
• How “bad” is a C of 0.10?
• Does a doubling of C imply a doubling of
inequality?
• Koolman & van Doorslaer (2004) –
– 75C = % of health variable that must be
(linearly) transferred from richer to poorer half
of pop. to arrive at distribution with a C of zero
– But this ensures equality of health predicted by
income rank and not equality per se
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Inequality is not simply correlation
• Milanovic (1997) decomposition for Gini
can be adapted for concentration index:
2
12 r  h
C 
  h, r 
3 
• C is (scaled) product of coefficient of
variation     and correlation   h, r 
h
– C captures both association and variability
– C is a covariance scaled in interval [-1,1]
– same association can imply different inequality
depending on variability
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Total inequality in health and
socioeconomic-related health inequality
100%
By definition, the
health Lorenz curve
must lie below the
concentration curve.
cum % of health
80%
60%
That is, total health
inequality is greater
than income-related
health inequality.
40%
20%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
cum % of pop, ranked by health or income
diagonal
Lorenz curve
Conc curve
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Total inequality in health is larger than
socioeconomic-related health inequality
Gini index of total health inequality
2
G  cov(h , rh )
rh is rank in health distribution

Then G  cov(h , rh )  1
C
cov(h , r )
Thus, G = C + R, where R>=0 and measures the outward move from
the health concentration curve to the health Lorenz curve, or the
re-ranking in moving from the SES to the health distribution
“even if the social class gradient was magically eliminated,
dispersion in health outcomes in the population would remain
very much the same”
Smith J, 1999, Healthy bodies and thick wallets”, J Econ Perspectives
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Computing concentration index with
grouped data
C  ( p1L2  p2 L1 )  ( p2 L3  p3 L2 )  ...  ( pT 1LT  pT LT 1 )
Under-5 deaths in India
pt
Wealth
group
Poorest
2nd
Middle
4th
Richest
Total/average
No. of
births
29939
28776
26528
24689
19739
129671
rel %
births
23%
22%
20%
19%
15%
cumul % U5MR
No. of
births per 1000 deaths
0%
23%
45%
66%
85%
100%
154.7
152.9
119.5
86.9
54.3
118.8
4632
4400
3170
2145
1072
15419
Lt (pt-1Lt-ptLt-1)
rel % cumul %
deaths deaths
30%
29%
21%
14%
7%
0%
30%
59%
79%
93%
100%
Conc.
Index
-0.0008
-0.0267
-0.0592
-0.0827
0.0000
-0.1694
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Estimating the concentration index
from micro data
• Use “convenient covariance” formula C=2cov(h,r)/μ
– Weights applied in computation of mean, covar and rank
• Equivalently, use “convenient regression”
2  hi 
2 r       ri   i

– Where the fractional rank (r) is calculated as follows if there are weights (w)
i 1
wi
ri   w j  ,
2
j 0
w0  0
– OLS estimate of β is the estimate of the concentration index
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Standard error of the estimate of the
concentration index
• Kakwani et al (1997) provide a formula for deltamethod SE
– But formula does not take account of weights or sample
design
• Could use the SE from the convenient regression
– Allows adjustment for weights, clustering, serial
correlation, etc
– But that does not take account of the sampling variability
of the estimate of the mean
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Delta method standard error from
convenient regression
To take account of the sampling variability of the
estimate of the mean, run this regression hi 1  1ri  ui
2

 ˆ
2

r
Estimate the concentration index from ˆ 
 ˆ  1
  
Or using the properties of OLS

 2 2
r
ˆ  
 ˆ  ˆ1
 1
2



 ˆ1



This estimate is a non-linear
function of the regression
coeffs and so its standard error can be obtained by the
delta method.
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Demographic standardization of the
concentration index
• Can use either method of standardization
presented in lecture 5 & compute the C index
for the standardized distribution
• If want to standardized for the total correlation
with demographic confounding variables (x),
then can do in one-step
• OLS estimate of β2 is indirectly standardized
concentration index 2 2  hi     r   x  
r

 
2
2 i

j
ji
j
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
i
Sensitivity of the concentration
index to the living standards measure
• C reflects covariance between health and rank in
the living standards distribution
• C will differ across living standards measures if
re-ranking of individuals is correlated with health
(Wagstaff & Watanabe, 2003)
 hi 
From OLS estimate of 2       ri   i
 
where ri  r1i  r2i is the re-ranking and  2r its variance,
2
r
the difference in concentration indices is
C1  C2  ˆ
“Analyzing Health Equity Using Household Survey Data” Owen O’Donnell, Eddy van Doorslaer, Adam Wagstaff and
Magnus Lindelow, The World Bank, Washington DC, 2008, www.worldbank.org/analyzinghealthequity
Evidence on sensitivity of
concentration index
Wagstaff & Watanabe (2003) – signif. difference b/w C estimated
from consumption and assets index in only 6/19 cases for
underweight and stunting
But Lindelow (2006) find greater sensitivity in concentration
indices for health service utilization in Mozambique
Consumption
Asset index
t-value for
difference
CI
t-value
CI
t-value
Difference
CIC – CIAI
Hospital visits
0.166
8.72
0.231
12.94
-0.065
-3.35
Health center visits
0.066
3.85
-0.136
-8.49
0.202
9.99
Complete immunizations
0.059
8.35
0.194
34.69
-0.135
-19.1
Delivery control
0.063
11.86
0.154
35.01
-0.091
-15.27
Institutional delivery
0.089
11.31
0.266
43.26
-0.176
-20.06