A Family of Survival Models for Longevity and Concentration
As noted in the previous section, and in particular in Figure 6.1, in human populations an increase in life expectancy at birth (i.e., longevity) is often associated with a decrease in the concentration of survival times. Gigliarano et al. (2017) propose to model the joint behavior of expected length of life and lifespan concentration through the parametric family of log-scale-location survival models.
Following the authors' notation, consider a nonnegative random variable T indicating the survival time with survival function S(f). The log-scale-location model is defined by:
where So(f) is an arbitrary baseline survival function, u(x) e R is the location parameter, b(x) > 0 is the scale parameter and x is a continuous covariate (Lawless, 2003). Common distributions that belong to the family of log-scale- location models are the Weibull, the log-normal and log-logistic distributions.
Gigliarano et al. (2017) study the behavior of the Gini index of the survival time T as function of the covariate *
where x represents the cohort year.
One of the most common models used in survival analysis for estimating covariate effects is the well-known Cox proportional hazards model, which assumes the following structure for the conditional (on x) survival function:
where So(f) is an arbitrary baseline survival function and r(x) = eax (see Cox, 1972 and Lawless, 2003). However, Gigliarano et al. (2017) show that such model is not suitable for capturing changes in the Gini index as a function of the covariate x. The authors show moreover that, in contrast to the proportional hazards model, within the log-scale-location family the Gini index of concentration is an increasing (decreasing) function of x, if the shape parameter b(x) is increasing (decreasing) function. This means that the family of models in (6.5) allows for a complete ordering of survival distributions in terms of concentration, as the cohort year * changes.
Moreover, the authors show that under the log-scale-location model life expectancy has the following expression as function of the covariate x (for example, the year of birth):
where R is a random variable following the baseline distribution So(r) in (6.5). They also obtain the expression of both Gini index and life expectancy for most parametric distributions belonging to the class (6.5)(6.5), revealing that the log-scale-location family allows for modeling different kinds of relationships between expected value and concentration as a function of the year of birth. This class of regression models enlarges the toolbox of statisticians, demographers and actuaries interested in modeling the evolution of the shape and pace of mortality.
In particular, when considering the Weibull baseline distribution for survival time T, the Gini index has the expression G(x) = 1 - (0.5),,(л), while life expectancy is given by:
where T( ) is the gamma function. Hence, under this parametric distribution the Gini index of lifetimes is always decreasing in x for decreasing b(x), while life expectancy increases as the cohort year x increases, assuming the location parameter u(x) to be constant and the shape parameter b(x) to be decreasing and ranging between 0 and 0.4616. In contrast, in the case of log-normal or log-logistic baseline distributions, as cohort year x increases, both lifetimes concentration and life expectancy increase.
Lastly, the family of log-scale-location models has been extended in that same article in order to take into account situations where a fraction of the population may experience short-term survival, e.g., infant mortality. In such case the authors propose to consider a mixture model that captures the shortterm survival as well as the mortality conditionally on not being subject to early death. Also, in this scenario conditions have been studied under which the mixture model can detect the joint phenomenon of increasing life expectancy and decreasing Gini index.
Final Comment
As we have seen, the Gini index allows one to study inequality in mortality. It is not surprising that the concepts of concentration and inequality are of particular interest when one studies how long we live.
However, one may wonder about the meaning of redistribution of a total amount of lifetime among different subjects in a population. The concept of redistribution is readily accepted in the study of income and wealth inequality, from which it is inherited. We feel that this concept is not too strange also in the survival setting. One should think, for example, of the effect of funding research to cure some diseases and not others, which may produce a prolonging of life for only some subpopulations (or countries). More generally, the effect of decision making in public health interventions and delivery, and in general the form of the economic and social development of our societies, all should be expected to have redistribution effects also on how long different groups of people live.
Therefore, from this point of view Corrado Gini's contribution has clearly been seminal and, over 100 years later, it helps us focus on the features of our lives that are most relevant.
