help for conindex Erasmus University Rotterdam & NUI Galway

Calculation of rank-dependent inequality indices (Concentration Indices) for bounded and unbounded variables.

conindex [varlist] [if] [in] [fweight aweight pweight] ,
RANKvar(varname)
[CLUSter(varname)]
[robust]
[truezero]
[generalized]
[v(#1)]
[beta(#1)]
[bounded]
[LIMits(#1 #2)]
[WAGstaff]
[ERReygers]
[graph]
[loud]
[COMPare(varname)]
[KEEPrank(string)]
[svy]

Description

conindex computes a range of rank-dependent inequality indices, including the Gini coefficient, the concentration index, the generalized (Gini) concentration index, the modified concentration index, the Wagstaff and Erreygers normalised concentration indices for bounded variables, and the distributionally sensitive extended and symmetric concentration indices (and their generalized versions). O'Donnell et al. (2016) offer an accessible introduction to the various concentration indices that have been proposed to suit different measurement scales and normative positions. conindex is not an official Stata command. It is a free contribution to the research community. Please include a citation to O'Donnell et al. (2016) in your work.

There is no default index. Options define the index to be computed. (Generalized) Lorenz and (generalized) concentration curves can be obtained using the graph option. The default axis labels can be replaced by using the xtitle(string) and ytitle(string) options.

For unbounded variables (i.e. those with at least one infinite bound),the option truezero should be specified if the variable of interest is ratio-scale (or fixed) and has a zero lower limit, in which case the standard concentration index is calculated. If instead the variable of interest is cardinal (with the zero point fixed arbitrarily), then the theoretical lower limit must be specified using the limits(#) option where # is the minimum value. Note that one should not use the lowest value observed in the sample if this does not correspond to the theoretical lower bound. Specification of this option results in calculation of the modified concentration index.

The generalized concentration index derives from specifying the option generalized in conjunction with the option truezero.

For bounded variables (i.e. those with both a finite lower and upper bound) the option bounded can be specified in conjunction with limits(#1 #2) where #1 and #2 denote theoretical minimum and maximum values of the variable of interest. The inequality indices are then calculated based on the standardised version of the variable of interest, h* (h*=(h-#1)/(#2-#1), and hence will be scale invariant.

The normalised concentration indices proposed by Wagstaff (2005) and Erreygers (2009a) may be obtained by specifying the wagstaff and erreygers option respectively in conjunction with the bounded and limits(#1 #2) options.

When a ranking variable is not provided using the rankvar option, conindex defaults to use varname to rank observations, leading to the calculation of uni-dimensional inequality indices (e.g. the Gini coefficient).

The extended concentration index, which allows for alternative attitudes to inequality (Periera, 1998; Wagstaff, 2002), is computed with the addition of the options truezero and v(#), where # is the distributional sensitivity parameter. With v(2) the extended concentration index is equivalent to the standard concentration index.

The symmetric concentration index, an alternative distributionally sensitive concentration index proposed by Erreygers et al. (2012), is obtained with the options truezero and beta(#). With beta(2), the symmetric concentration index is equivalent to the standard concentration index.

The generalized version of the extended and symmetric concentration indices are obtained by combining the options v(#) and beta(#) with the options truezero and generalized.

All indices are calculated using the so-called 'convenient covariance' approach (Kakwani, 1980; Jenkins, 1988; Kakwani et al, 1997). Robust and clustered corrected standard errors can be obtained with the usual options. Standard errors for the extended and symmetric indices are not calculated by the current version of conindex.

The value of an index can be compared across groups, and the null of homogeneity tested, using the compare option. This option cannot be combined with the prefix by varlist:.

The fractional rank may be preserved using the option keeprank(string) where string is the name given to the rank variable created.

conindex can be used on data from complex survey designs by including the option svy if svyset has been used to identify the survey design characteristics prior to running conindex.

Options

rankvar variable by which individuals are ranked. Must be at least an ordinal variable.

cluster(varname) requests clustered standard errors that allow for intragroup correlation.

robust requests Huber/White/sandwich standard errors.

truezero declares that the variable of interest is ratio scaled (or fixed), leading to computation of the standard concentration index.

generalized requests the generalized concentration (Gini) index measuring absolute inequality. This option can only be used in
conjunction with truezero.

v(#) requests the extended concentration index be computed. This option can only be used in conjunction with truezero. When v(2)
the standard concentration index is computed. If the options v(#), truezero and generalized are specified, one obtains the
generalized extended concentration index. In the later case, v(2) leads to the Erreygers index.

beta(#) requests the symmetric concentration index be computed. This option can only be used in conjunction with truezero. When beta(2)
the standard concentration index is computed. If the options beta(#), truezero and generalized are specified, one obtains the
generalized symmetric concentration index. In the later case, beta(2) leads to the Erreygers index.

bounded specifies that the dependent variable is bounded. This option must be used in conjunction with the limits option.

limits(#1 #2) must be used to specify the theoretical minimum (#1) and maximum (#2) for bounded variables. If the options bounded and truezero
are not specified then limits(#1) should be used to specify the minumum value to obtain the modified concentration index.

wagstaff in conjunction with bounded and limits(#1 #2) requests the Wagstaff Index.

erreygers in conjunction with bounded and limits(#1 #2) requests the Erreygers Index.

graph graph requests that a concentration curve be displayed. If no ranking variable is specified, a Lorenz curve is produced. In
conjunction with generalized, one obtains the generalized Lorenz or concentration curve. conindex draws the Lorenz and concentration curves
using the user-written lorenz command by Ben Jann. This must be installed prior to using the graph option. Note that aweights cannot be used with this command.

loud shows the output from the regression used to generate the inequality indices.

keeprank(string) creates a new variable which contains the fractional ranks, where string is the name of the variable to be created. When used
in conjunction with compare, the variable string will contain the fractional rank for the full sample and the suffix k is added to
string to indicate the fractional rank for group k.

compare(varname) computes indices specific to groups specified by varname. Two tests of the null of equality of the index values across groups are
produced: an F-test that is valid in small samples but requires an assumption of equal variances across groups and a z-test that
relaxes the assumption of equal variances but is valid only in large samples. If varname is binary, then only the F-test is given.

svy can be specified when using complex survey designs if svyset has been used to identify the survey design characteristics prior to running conindex.

Measurement scales:

Fixed requiring that the measurement scale is unique (or fixed) with the zero point corresponding to a situation of complete absence.
(Example: Visits to the hospital).

Ratio requiring that ratios between individuals have meaning with the zero point corresponding to a situation of complete absence, such that the
measurement scale is unique up to a proportional scaling factor.
(Example: Life Expectancy).

Cardinal requiring that differences between individuals make sense, but ratios do not, such that the zero point is fixed arbitrarily.
(Example: Health Utility Index).

Ordinal requiring that it be possible to order individuals, but with the differences between individuals being meaningless.
(Example: Self Assessed Health).

Nominal implying that one can classify individuals without being able to order them.
(Example: Type of Illness).

Saved Results

r(N) Number of observations
r(Nunique) Number of unique observations for rankvar
r(CI) Concentration index
r(CIse) Standard error of concentration index
r(SSE_unrestricted) Unrestricted sum of squared errors (with compare option)
r(SSE_restricted) Restricted sum of squared errors (with compare option)
r(F) F- statistic for joint hypothesis that concentration index is same for all groups (with compare option)
r(CI0) Concentration index for group 0 (with compare option if only two groups)
r(CI1) Concentration index for group 1 (with compare option if only two groups)
r(CIse0) Standard error of concentration index for group 0 (with compare option if only two groups)
r(CIse1) Standard error of concentration index for group 1 (with compare option if only two groups)
r(Diff) Difference in concentration index between groups (with compare option if only two groups)
r(Diffse) Standard error of difference in concentration index between groups (with compare option if only two groups)
r(z) z- statistic for hypothesis that concentration index is same for both groups (with compare option if only two groups)

Formulae

where N is the number of observations, Y represents income, Y_i is an individual's income, R_i is the deviation of the rank from the mean
(or median) rank, h_i is another variable of interest, mu_Y is the mean of Y, mu_h is the mean of h and the minimum and maximum of the variable
of interest are the scalars lowerlimit and upperlimit.

Gini Coefficient
= 2/N^2*mu_Y * Sum[Y_i*R_i]
conindex Y , truezero

Generalized Gini
= 2/N^2 * Sum[Y_i*R_i]
conindex Y , truezero generalized

Concentration index
= 2/N^2*mu_h * Sum[h_i*R_i]
conindex h , rankvar(Y) truezero

Generalized concentration index
= 2/N^2 * Sum[h_i*R_i]
conindex h , rankvar(Y) truezero generalized

Modified concentration index
= 2/(N^2(mu_h - min(h_i)) * Sum(h_i*R_i)
conindex h , rankvar(Y) limits(10)

Extended concentration index
= 1/mu_h * Sum[(1-v(1-R_i)^(v-1))/N] * h_i v>1
conindex h, rankvar(Y) v(3) truezero

Generalized extended concentration index
= [v^(v/(v-1))]/(v-1) * Sum[(1-v(1-R_i)^(v-1))/N] * h_i v>1
conindex h, rankvar(Y) v(3) truezero generalized bounded limits(lowerlimit upperlimit)

Symmetric concentration index
= 1/mu_h * Sum{[beta2^(beta-2)[(R_i-1/2)^2]^((beta-2)/2) * (R_i-1/2)]/N}*h_i
conindex h, rankvar(Y) beta(3) truezero

Generalized symmetric concentration index
= 4 * Sum{[beta*2^(beta-2)[(R_i-1/2)^2]^((beta-2)/2) * (R_i-1/2)]/N}*h_i
conindex h, rankvar(Y) beta(3) truezero generalized bounded limits(lowerlimit upperlimit)

Wagstaff index
= [2/N^2*mu_h * Sum[h_i*R_i]]/(1-mu_h)
conindex h , rankvar(Y) limits(lowerlimit upperlimit) bounded wagstaff

Erreygers index
= 4*mu_h*[2/N^2*mu_h * Sum[h_i*R_i]]
conindex h , rankvar(Y) limits(lowerlimit upperlimit) bounded erreygers

Comparing standard concentration indices across groups defined by the variable "group":
Note: Other indices may be compared by specifying options as above.
conindex h, rankvar(Y) truezero compare(group)

Concentration index for complex survey designs:
svyset [pw=weight], psu(psu_id) strata(strata_id)
conindex h, rankvar(Y) truezero svy

Author

Owen O’Donnell, Erasmus School of Economics, Erasmus University Rotterdam, the Netherlands; Tinbergen Institute, the Netherlands; and University of Macedonia, Greece.

Stephen O’Neill (corresponding author), Department of Health Services Research and Policy, London School of Hygiene and Tropical Medicine, UK. stephen.oneill@lshtm.ac.uk

Tom Van Ourti, Erasmus School of Economics, Erasmus University Rotterdam, the Netherlands; Tinbergen Institute, the Netherlands.

Brendan Walsh, HSR and Management, School of Health Sciences, and the City Health Economics Centre, City University London, UK.

Also see

Online: help for concindc, glcurve, concindexi, concindexg, ineqdeco, inequal, povdeco, ineqerr, lorenz if installed.

References

Erreygers, G., Clarke, P. & Van Ourti, T. 2012. "Mirror, mirror on the wall, who in this land is fairest of all?" Distributional sensitivity in the measurement of socioeconomic inequality in health. Journal of Health Economics 31, 257-270.

Jenkins SP. 1988. Calculating income distribution indices from micro-data. National Tax Journal, 139-142.

Kakwani NC. 1980. Income inequality and poverty: methods of estimation and policy applications. New York. Oxford University Press.

Kakwani, N., Wagstaff A., & van Doorslaer E. 1997. Socioeconomic inequalities in health: Measurement, computation, and statistical inference. Journal of Econometrics 77, 87-103.

O'Donnell, O., O'Neill, S., Van Ourti, T. & Walsh, B. 2016. conindex: Estimation of concentration indices. The Stata Journal (Forthcoming).

O'Donnell, O., van Doorslaer, E., Wagstaff, A. & Lindelow, M. 2008. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation. Washington, D.C: World Bank Institute.