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Limitations of using the Lorenz curve framework to understand the distribution of population viral load

Tsai, Alexander C.

doi: 10.1097/QAD.0000000000001388
Correspondence
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SDC

Chester M. Pierce, MD, Division of Global Psychiatry, Massachusetts General Hospital, Boston, Massachusetts, USA.

Correspondence to Alexander C. Tsai, MGH Global Health, 125 Nashua Street, 7th floor, Boston, MA 02114, USA. E-mail: actsai@partners.org

Received 12 November, 2016

Accepted 20 December, 2016

I read with interest the research letter by Christopoulos et al.[1] on applying the Lorenz curve to understand the distribution of population viral load. The authors suggest that this framework could be used to inform targeting a small group of patients with the aim of reducing population viral load. Presumably, the same reasoning might be applied to inform targeting across populations or subpopulations (e.g. targeting clinics or counties most responsible for concentrated ‘viral load wealth’ with the aim of reducing population viral load). To do so, one might use the population Lorenz curves to calculate the corresponding Gini coefficients (i.e. the area between the Lorenz curve and the 45° line of perfect equality, divided by the total area under the 45° line) and then direct resources across populations based on their coefficient values. The authors may or may not have had such an idea in mind when writing the article, but I thought it prudent to point out some potential limitations of the framework should such an idea someday take hold.

First, in some circumstances, Lorenz curves can cross, yielding identical or similar Gini coefficient values. These scenarios would complicate the comparison of Gini coefficients across populations.

Second, computation of the Gini coefficient is most sensitive to inequalities in the middle portion of the population viral load distribution [2,3].

Third, there are a number of other measures that could be used to measure the concentration of viral load wealth [4]; namely, the decile ratio: viremia copy-months among the ‘wealthiest’ 10% of patients divided by viremia copy-months among the ‘poorest’ 10% of patients; the Robin Hood index: the maximum vertical distance measured from the Lorenz curve to the 45° line of perfect equality; and the Atkinson index: the Gini coefficient modified so that different portions of the population viral load distribution are given different weights. There are, of course, a number of other measures that could be used, all with different advantages and disadvantages.

Fourth, the Gini coefficient is downward biased in small samples [5]. In such situations, the asymptotic distribution of the Gini would be of little use.

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Acknowledgements

Conflicts of interest

There are no conflicts of interest.

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References

1. Christopoulos K, Hartogensis W, Glidden DV, Pilcher CD, Gandhi M, Geng EH. The Lorenz curve: a novel method for understanding viral load distribution at the population level. AIDS 2017; 31:309–310.
2. Atkinson AB. On the measurement of inequality. J Econ Theory 1970; 2:244–263.
3. Hey JD, Lambert PJ. Relative deprivation and the Gini coefficient: comment. Q J Econ 1980; 95:567–573.
4. Kawachi I, Kennedy BP. The relationship of income inequality to mortality: does the choice of indicator matter?. Soc Sci Med 1997; 45:1121–1127.
5. Deltas G. The small-sample bias of the Gini coefficient: results and implications for empirical research. Rev Econ Stat 2003; 85:226–234.
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