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

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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 t0 and X + δX and Y + δY be the life expectancies at time t1 for populations group 1 and group 2, respectively (t1 > t0). 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 t0 to t1 are given by (δY − δX). The trends in attainment-absolute and shortfall-absolute inequalities move in the same direction from t0 to t1. The attainment- and shortfall-relative inequalities at time t0 are given by (Y − X)/X and [(100 − X) − (100 − Y)]/(100 − Y), respectively. The attainment- and shortfall-relative inequalities at time t1 are given by
, respectively. It can be shown that the attainment-relative inequality increases or declines from t0 to t1 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 t0 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 t1 are given by (83 − 74) = 9 and (26 − 17) = 9. Both attainment-absolute and shortfall-absolute inequalities decline by 1 from time t0 to t1. On the other hand, the attainment- and shortfall-relative inequalities at time t0 are given by (80 − 70)/70 = 0.14 and (30 − 20)/20 = 0.5 respectively, and at time t1 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.
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