Letters

# The Authors Respond

Kjellsson, Gustav; Gerdtham, Ulf-G; Petrie, Dennis

Author Information
doi: 10.1097/EDE.0000000000000441
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We would like to thank Popham for highlighting an important issue. Our article1 claims that no “relative” measure can be symmetric, inequalities expressed as shortfalls are the same as inequalities expressed as attainments, but Popham shows that the odds ratio (OR) does have this property.2 The OR is, however, not a “pure” relative measure according to the terminology we use in our article, it is a relative measure in terms of the “odds” rather than the outcome itself. This discrepancy in terminology further stresses the importance of acknowledging the value judgments underlying any inequality index. Our article primarily discusses the inequality equivalence criterion of an index: what distribution of changes keeps inequalities constant. We define an absolute, attainment-relative, and shortfall-relative inequality equivalence criterion as being constant to uniform changes (red/yellow pill), proportional changes in attainment (blue pill), and proportional changes in shortfalls (green pill), respectively. It is clear from Popham’s table that the inequality expressed as an OR changes for all three pills, i.e., the inequality equivalence criterion corresponding to the OR is neither attainment-relative, absolute, nor shortfall relative.

To illustrate the distribution of changes needed in this particular example to keep inequalities constant according to the OR, we provide an additional “brown” pill. This brown pill increases life expectancy in total from 25 to 50 years (as the others), but distributes the additional years differently. To find the final life expectancies (a and b) that keep the OR constant but have a mean life expectancy of 50, we solve the following simultaneous equations: (a × (100 − b))/(b × (100 − a)) = 12/7 = initial OR and (a + b)/2 = 50 = final mean life expectancy. The life expectancies that solve these simultaneous equations are a = 56.7 years and b = 43.3 years which yield an absolute difference of 13.4 years. For each mean health value, we obtain the required absolute difference between the two groups that would keep the OR constant. We draw the line corresponding to these values in the Figure. To show how this OR inequality equivalence criterion relates to the shortfall-relative, absolute, and attainment-relative criteria, we also include similar lines corresponding to their criteria.

This graph clearly illustrates the inequality equivalence criterion of the OR—this is best described as a mixture of the attainment-relative and shortfall-relative inequality equivalence criteria. We provide a short discussion of such alternative inequality equivalence criteria in the supplementary online appendix of the original article.1,3,4 The graph further highlights that grouping a number of clearly “different” inequality measurements under a common title “relative inequality measures” is unhelpful for those trying to understand their implicit value judgments. The usefulness of the OR as an inequality measure depends on how easy it is for policy makers to understand its implicit value judgments and how closely its inequality equivalence criterion matches public perspectives of what changes in health would leave inequalities unchanged. Given that the inequality equivalence criterion of the OR is a mixture of the attainment-relative and shortfall-relative measures, it may not add much to the picture but instead overcomplicate things. This is an issue for further research.

Gustav Kjellsson

Department of Economics

University of Gothenburg

Centre for Health Economics at the University of Gothenburg (CHEGU)

Gothenburg, Sweden

gustav.kjellsson@economics.gu.se

Ulf-G Gerdtham

Department of Economics

Health Economics & Management

Institute of Economic Research Lund University

Lund, Sweden

Center for Primary Health Care Research

Lund University/Region Skåne

Lund, Sweden

Dennis Petrie

Centre for Health Policy

School of Population and Global Health

The University of Melbourne

Melbourne, VIC, Australia

## REFERENCES

1. Kjellsson G, Gerdtham UG, Petrie D.. Lies, damned lies, and health inequality measurements: understanding the value judgments. Epidemiology. 2015;26:673–680
2. Popham F.. Re: Lies, damned lies, and health inequality measurements: understanding the value judgments. Epidemiology. 2016;27:e14–e15
3. Wagstaff A.. The bounds of the concentration index when the variable of interest is binary, with an application to immunization inequality. Health Econ. 2005;14:429–432
4. Kjellsson G, Gerdtham U-G.. Lost in translation: rethinking the inequality equivalence criteria for bounded health variables. Research on Economic Inequality. 2013;Vol 21 Bingley, England Emerald Group Publishing:3–32 Available at: http://www.emeraldinsight.com/10.1108/S1049-2585(2013)0000021002. Accessed September 5, 2015