Appraising inequality in situation where one group is advantaged as to some outcomes and another group is advantaged as to other outcomes will always be problematic even when one can effectively quantify the inequality between the advantaged and disadvantaged group as to each outcome.

The additional problem in almost all studies of inequality is they rely on a measure of inequality without consideration of the way the measure tends to be affected by the prevalence of an outcome.[1.2] This study measures inequality in terms of relative differences between favorable outcome rates (as to its first indicator) without consideration of the pattern by which the rarer an outcome, the greater tends to be the relative difference in experiencing it and the smaller tends to be the relative difference in avoiding it. For example, as reflected in Table 1 of references 1 and 2, a situation where favorable outcome rates are 80% for an advantaged group (AG) and 63% for a disadvantaged group (DG) is essentially the same as a situation where the rates are 95% for AG and 13% for DG. Yet the authors’ approach would find greater inequality in the former case based on the AG/DG ratio of 1.27 compared with the AG/AG ratio of 1.09 in the latter case. Thus, the approach will tend to show less inequality in countries where favorable outcome rates are comparatively high (mainly advantaged countries) for reasons unrelated to any actual differences in inequality from country to country. Also, determination of which of the three educational outcomes shows the greatest inequality in a particular country will commonly turn on which of the outcomes is least common.

On the other hand, if the authors examined the corresponding adverse outcomes, they would find a larger inequality in the second situation based on the DG/AG ratio of 2.60 (13%/5%) compared with a DG/AG ratio of 1.85 (37%/20%) in the first situation. And they would tend to find larger inequalities in countries where adverse outcomes are less common and largest educational inequalities for the educational outcomes with highest favorable (lowest adverse) outcome rates. See references 3-4 regarding the tendency to ponder the comparatively large inequalities in adverse outcomes in advantaged countries (or subpopulations) without recognizing the reason to expect such pattern (and a corresponding opposite pattern as the corresponding favorable outcomes) where adverse outcomes are comparatively uncommon.

With regard to drawing of inferences about underlying processes, see references 2 at 339-341 regarding the way one often draws opposite inferences about processes depending on whether one examines relative differences in favorable outcome or relative differences in the corresponding opposite outcome. Probably some part, and perhaps all, of the reason for the comparatively large relative difference in education favoring girls in Lesotho (discussed at page 13 with regard to boys’ involvement in herding) has to do with generally low levels of favorable educational outcomes in the country.

These points go only to quantifying an inequality, not determinations as which group is disadvantaged. Appraisals of which gender is disadvantaged as to obesity will be the same regardless of whether one examines the favorable of the adverse outcome. But one cannot effectively quantify the gender differences in obesity (or the corresponding favorable outcome of avoiding obesity) without consideration of the patterns by which the two relative differences tend to be affected by the prevalence of the outcome. The same holds for other measures that tend to be affected by the prevalence of an outcome.

I don’t know enough about the satisfaction index to know whether the above points at all pertain to the index (which may involve an average unaffected by any dichotomy). But life expectancy and some other seemingly continuous variables do tend to be affected by the overall level of an outcome or component outcomes, though in a very complicated way. See references 5 (at 6-7) and 6.

It is also important to keep in mind that a ratio of rates (proportions) is something very different from a ratio of average values for a continuous variable. In my view, both differences in rates and differences in average values should be quantified in terms of percentage of a standard deviation (a) between the hypothesized underlying means in the case of rates (as discussed in reference 1) and (b) between the actual means in the case of truly continuous variables.

I do not know whether those differences can then be usefully averaged to derive an overall figure but suspect that there can be some serious problems in doing so.

1. Scanlan JP. Race and mortality revisited. Society 2014;51:327-346
http://link.springer.com/...
2. Scanlan JP. The mismeasure of health disparities. J Public Health Manag Pract 2016;22(4):416-19.
https://journals.lww.com/...
3. Scanlan JP. “It’s easy to misunderstand gaps and mistake good fortune for a crisis,” Minneapolis Star Tribune 2014 (Feb. 8)
http://www.startribune.co...
4. Scanlan JP. The mismeasure of health disparities in massachusetts and less affluent places. Quantitative Methods Seminar, Department of Quantitative Health Sciences, University of Massachusetts Medical School (Nov. 18, 2015)
http://jpscanlan.com/imag...
5. The misinterpretation of health inequalities in the United Kingdom. British Society for Populations Studies Conference 2006, Southampton, England (Sept. 18-20, 2006)
http://www.jpscanlan.com/...
6. Recognizing why dichotomous and continuous measures may yield contrary results. BMJ June 11, 2007 (responding to (Chandola T, Ferrie J, Sacker A, Marmot M. Social inequalities in self reported health in early old age: follow-up of prospective cohort study. BMJ 2007:334:990-996) http://www.BMJ.com/cgi/el...