### Excerpt

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

JOURNAL/epide/04.02/00001648-201705000-00026/math_26MM1/v/2017-07-26T080259Z/r/image-tiff

and

JOURNAL/epide/04.02/00001648-201705000-00026/math_26MM2/v/2017-07-26T080259Z/r/image-tiff

, 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.