The Wasserstein Distance for Mortality Comparisons: Absolute versus Net Differences in Survival

Abstract

It is well known that the gap in life expectancy at birth can be seen as the net difference between two survivorship functions. When calculating the absolute difference between the two survivorship functions instead, we derive a distributional inequality measure which is called the Wasserstein distance. The measure quantifies how far apart two probability distributions are by the minimal cost of transporting one into the other. This paper relates the Wasserstein distance to the difference in life expectancy at birth. Both measures correspond to each other, whenever the survivorship functions do not cross, i.e., the net difference equals the absolute difference in the no crossing case. Whether the curves cross is driven by the accumulated difference in age-specific death rates. Crossing occurs in situations where an accumulated survival advantage for given populations is reversed because elevated death rates at later ages outweigh the earlier survival advantage. These cases are particularly interesting because the net difference in survival (or the life expectancy gap) may be small even though the two populations show very different mortality schedules. In our empirical analysis, we search for these cases using life table data from Human Mortality Database life tables for the period 1990 to 2020. Across 69 188 population pairs, survivorship functions cross in 59.8 % of comparisons. Yet, the magnitude of the reversal in survivorship is usually very small. We therefore conclude that in most cases, the gap in life expectancy at birth is not only a comparison of means but quantifies the overall difference between the two mortality regimes