# The Gini Concentration Index for the Study of Survival in Demography

In this section we review the contributions of the Gini concentration index to the study of survival distributions in demography for human and nonhuman populations.

The unprecedented improvement in human longevity witnessed over the last two centuries is considered one of the most remarkable achievements of modern societies. Longevity is typically measured by life expectancy at birth, which is the mean age at death of the population. Life expectancy at birth has increased by more than 30 years from the 19th to the 21st century around the globe (Riley, 2001), with its record level rising at a steady pace of three months per year since 1840s (Oeppen and Vaupel, 2002).

Despite being the most meaningful and used indicator of population health, life expectancy at birth alone does not capture all the relevant features of the age-at-death distribution. Indeed, a specific mean value can originate from very different shapes of the underlying distribution. As such, scholars have recently started to go beyond the analysis of "central longevity indicators" (mean, median and modal age at death, Cheung et al., 2005) and to analyze other summary measures of the distribution.

In particular, the concentration of lifespan distributions in human populations, typically referred to as lifespan inequality, disparity or variability in the demographic literature, has received significant attention during recent decades. Inequality in length of life is indeed the most fundamental of all inequalities, as every other inequality is conditional upon being alive (van Raalte et al., 2018).

Several indices have been proposed to measure lifespan inequality, such as the standard deviation of the age-at-death distribution, computed above age 10 (Edwards and Tuljapurkar, 2005), above age 60 (Myers and Manton, 1984), above the modal age at death (Kannisto, 2000), or around the modal age at death (Canudas-Romo, 2008); the variance of the distribution above age 15 (Gillespie et al., 2014); the interquartile range of the distribution (Wilmoth and Horiuchi, 1999); the shortest age interval in which a given proportion of deaths take place (Kannisto, 2000); the average number of life years lost at birth due to death (Vaupel and Canudas-Romo, 2003); the entropy of the life table (Keyfitz, 1977; Demetrius, 1978); the Theil index (Smits and Monden, 2009; Permanyer et al., 2018); and the Gini concentration index.

These indices have been shown to be highly correlated across countries and times (see e.g., Wilmoth and Horiuchi, 1999; Vaupel et al., 2011; van Raalte and Caswell, 2013; Colchero et al., 2016). Nevertheless, several scholars have chosen the Gini index to analyze lifespan inequality, and here we provide a brief review of their work.

Smits and Monden (2009) compute the Gini index for the entire world in the year 2000s using a large set of life tables for 191 countries. Length of life was found to be more unequally distributed among men (Gini indices of 0.189 and 0.128 for the total and adult population, respectively) than women (Gini indices of 0.177 and 0.115 for the two groups).

One of the very first formulas to compute the Gini concentration index from life tables was proposed by Hanada (1983), who shows that both life expectancy at birth and the Drewnowski equality index, computed as D = 1 - G, had been increasing over time when analyzing Japanese life tables from 1955 to 1975. This increase occurred for both males and females. Furthermore, the equality was found to be higher in males for a given level of life expectancy at birth, although females were expected to become more egalitarian than males in the future due to a faster rate of increase of D.

A seminal contribution to the computation and analysis of the Gini index in survival times from life tables has also been made by Shkolnikov et al. (2003), who propose a different numerical approximation to estimate the integral *f _{n+} S^{2}(u)du* from discrete life-table data. Following Sen (1973) and Anand (1983), Shkolnikov et al. (2003) also summarize three desirable properties that an inequality index should satisfy:

a. *population-size independence:* the index should not change if the number of people at each income (or survival) level is changed by the same proportion;

b. *mean* (or *scale*) *independence:* the index should not change if everyone's income (or survival) is changed by the same proportion;

c. *Pigou-Dalton condition:* any transfer from a richer to a poorer person that does not reverse their relative ranks should reduce the value of the index.

Satisfying these three conditions guarantees that an inequality index will correctly reflect the Lorenz-dominance condition. Indeed, many authors favor the Gini index for the analysis of lifespan inequality over other indicators because: (/) it satisfies these three properties (Anand, 1983); *(it)* it is not too sensitive to redistributions at early ages of life; and *(iii)* it reflects changes at adult ages well. Conversely, measures such as the interquartile range, the variance and the VarLog (variance of the logarithm of length of life) do not satisfy all three properties (a)-(c).

The choice of the inequality measure can therefore influence the judgment on the direction of changes in the degree of inequality To illustrate this fact, Shkolnikov et al. (2003) investigate proportional changes in the Gini and in four other inequality measures in the United States over the period 1950-1995 and in Russia for the years 1959-2000, and they show that the measures respond differently to age-specific mortality changes. A more detailed illustration of this is provided by van Raalte and Caswell (2013), who investigate the sensitivity and elasticity of seven indices of lifespan inequality to changes in mortality at different ages. Although finding similar general age patterns among the indices, the authors report differences in their sensitivity to infant mortality, in the slope of decline from birth to late adulthood and in the "threshold age." The latter is the age separating positive from negative contributions to lifespan inequality. Averting deaths at any age indeed increases life expectancy; conversely, there exists a threshold age such that averting deaths before that age decreases inequality, while averting deaths after that age increases inequality (Zhang and Vaupel, 2009).