# Trends in Relative Inequalities in Measures of Favorable and Adverse Population Health Outcomes

Office of Minority Health and Health Equity, Centers for Disease Control and Prevention, Atlanta, GA, rmoonesinghe@cdc.gov

National Center for Chronic Disease Prevention and Health Promotion, Centers for Disease Control and Prevention, Atlanta, GA

The authors report no conflicts of interest.

The findings and conclusions in this article are those of the authors and do not necessarily represent the official position of the Centers for Disease Control and Prevention.

## To the Editor:

The recent article by Kjellsson et al.^{1} discussed implications of the choice of attainment (“favorable”) and shortfall (“adverse”) relative inequalities using life expectancy and mortality as examples. They showed that these two relative inequalities could be different from each other and from absolute inequalities and concluded that choosing among the three measures requires a value judgment about their relative importance. However, it is important to illustrate the mathematical arguments that explain how and why attainment- and shortfall-relative measures give different answers to the question of interpreting whether the trends in these relative inequalities increased or decreased across a specific time period in some situations.

First, we use the example on life expectancy given by Kjellsson et al.^{1} Let *X* and *Y* be the life expectancies at time *t*_{0} and *X* + *δX* and *Y* + *δY* be the life expectancies at time *t*_{1} for populations group 1 and group 2, respectively (*t*_{1} > *t*_{0}). We assume *δX* > 0, *δY* > 0, *Y* > *X*, and the referent group for calculating inequalities as the group with the lower value of the outcome. The changes in both attainment-absolute inequality and shortfall-absolute inequality from *t*_{0} to *t*_{1} are given by (*δY* − *δX*). The trends in attainment-absolute and shortfall-absolute inequalities move in the same direction from *t*_{0} to *t*_{1}. The attainment- and shortfall-relative inequalities at time *t*_{0} are given by (*Y* − *X*)/*X* and [(100 − *X*) − (100 − *Y*)]/(100 − *Y*), respectively. The attainment- and shortfall-relative inequalities at time *t*_{1} are given by

and

, respectively. It can be shown that the attainment-relative inequality increases or declines from *t*_{0} to *t*_{1} depending on whether (*XδY* − *YδX*) > 0 or not (or whether *δY*/*δX* > *Y*/*X* or not). Similarly, it can be shown that the shortfall-relative inequality increases or decreases based on whether *δY*/*δX* > (100 − *Y*)/(100 − *X*) or not. Suppose the attainment-relative inequality increases. Since *Y/X* > 1, (100 − *Y*)/(100 − *X*) < 1 and *δY*/*δX* > (100 − *Y*)/(100 − *X*), the shortfall-relative inequality also increases. But if the attainment-relative inequality declines, it is possible for the shortfall-relative inequality to increase when (100 − *Y*)/(100 − *X*) < *δY*/*δX* < *Y*/*X*. In other words, when this condition is satisfied, the trends in the two relative measures move in opposite directions.

To illustrate, suppose that *X* = 70, *Y*= 80, *δX* = 4, and *δY* = 3. The attainment-absolute inequality at time *t*_{0} is (80 − 70) = 10, and the shortfall-absolute inequality is (30 − 20) =10 (note the change in referent group). The attainment-absolute and shortfall-absolute inequalities at time *t*_{1} are given by (83 − 74) = 9 and (26 − 17) = 9. Both attainment-absolute and shortfall-absolute inequalities decline by 1 from time *t*_{0} to *t*_{1}. On the other hand, the attainment- and shortfall-relative inequalities at time *t*_{0} are given by (80 − 70)/70 = 0.14 and (30 − 20)/20 = 0.5 respectively, and at time *t*_{1} are given by (83 − 74)/74 = 0.12 and (26 − 17)/17 = 0.52, respectively. The attainment-relative inequality declined, but the shortfall-relative inequality increased.

Not surprisingly, we can show similar results for other bounded health outcomes. For example, both prevalence and proportional mortality of a condition (such as a disease or cause of death) are bounded below by 0 and bounded above by 1. Let *p* and *q* be the prevalence of a condition for two population groups at time *t*_{0} and *p* + *δp* and *q* + *δq* be the prevalence of the condition for the two groups at time *t*_{1} with *δp* > 0, *δq* > 0 and *p* > *q*. The change in absolute inequality of prevalence of the condition and the prevalence of not having the condition from time *t*_{0} to *t*_{1} is given by *δp* − *δq*. Similar to the example on life expectancy, it can be shown that the trends in inequality in prevalence of the condition and inequality in prevalence of not having the condition could move in opposite directions when the condition, (1 − *p*)/(1 − *q*) < *δp*/*δq* < *p*/*q*, is satisfied. For example, suppose *p* = 0.20, *q* = 0.19, and *δp* = *δq* = 0.08. There is no increase or decline in absolute inequality of prevalence of the condition and prevalence of not having the condition (*δp* − *δq* = 0) from time *t*_{0} to *t*_{1}. The relative inequality in prevalence of the condition at time *t*_{0} and at time *t*_{1} are (0.2 − 0.19)/0.19 = 0.053 and (0.28 − 0.27)/0.27 = 0.037, respectively. The relative inequality in the prevalence of not having the condition at time *t*_{0} and at time *t*_{1} are (0.81 − 0.8)/0.8 = 0.012 and (0.73 − 0.72)/0.72 = 0.014, respectively. The relative inequality in the prevalence of the condition declined, but the relative inequality in the prevalence of not having the condition increased from *t*_{0} to *t*_{1}.

Situations in which trends in absolute and relative disparities in health outcomes between two population groups move in opposite directions are well known.^{2} This has led to discussion of the role of value judgments when choosing between absolute and relative measures. Some authors assert that the choice depends on which aspect of the inequality is more important.^{3} Other authors assert that the choice is more complex and still others avoid using relative measures because of this dilemma, as described by Kjellsson et al.^{1},^{4} Some authors have discussed the implications of relative inequality in adverse outcomes and relative inequality in favorable outcomes between two population groups moving in opposite direction.^{5} One way to avoid this situation when monitoring changes in disparities over time is to measure relative disparities of health only in terms of adverse events.^{6} Even though one could select a relative measure for trend analysis based on a value judgment, it is also informative to know under what conditions these situations could arise. Knowing the patterns of change under which these paradoxical results could occur may assist researchers to make implicit value judgments explicit.

Ramal Moonesinghe

Ana Penman-Aguilar

Office of Minority Health and Health Equity

Centers for Disease Control and Prevention

Atlanta, GA

rmoonesinghe@cdc.gov

Gloria L. A. Beckles

National Center for Chronic Disease Prevention and Health Promotion

Centers for Disease Control and Prevention

Atlanta, GA

## REFERENCES

*et al*. respond to “Measuring social disparities in health”. Am J Epidemiol. 2008;167:905–907.