Mortality Crossovers from Dynamic Subpopulation Reordering

Abstract

Mortality crossovers are often understood to be the result of differential mortality selection. Models of mortality selection commonly assume a single dimension of heterogeneity, which stratifies populations into homogenous frail and robust subpopulations with proportional hazards. We propose a more realistic mortality selection model in which black and white populations are stratified by multiple crosscutting dimensions of heterogeneity, resulting in heterogeneous subpopulations. In the multidimensional model, in contrast to the conventional unidimensional model, the rank order of subpopulation mortalities is dynamic over age. As a result, a crossover can arise in either of two ways: from a change in the share of subpopulations in the black and white populations (analogous to the crossover in the standard, unidimensional mortality selection model), or alternatively, from a change in the rank order of subpopulation mortalities, regardless of subpopulation shares. The latter possibility has no analogue in the standard, unidimensional model. Our results therefore identify a new mechanism by which mortality selection can create mortality crossovers.

The erratum to this chapter is available at: http://dx.doi.org/10.1007/978-3-319-26603-9_18

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-26603-9_18

Appendix: Notes on the Simulation Parameters

This chapter identified a new mechanism by which mortality selection can generate aggregate mortality crossovers. In models with multidimensional heterogeneity, mortality selection can generate a dynamic subpopulation mortality ordering, and changes in the subpopulation mortality ordering can, in turn, generate one or more aggregate crossovers. Our numerical examples in Figs. 9.3, 9.4, and 9.5 focused on demonstrating these theoretical possibilities.

Calibrating these possibilities against data is challenging because the parameters in mortality selection models, whether unidimensional or multidimensional, refer to latent constructs (e.g., frailty). Nevertheless, it may be informative to consider how common the patterns shown in Figs. 9.3, 9.4, and 9.5 are within a larger universe of simulated cohorts. Table 9.A1 presents the parameter values for the four simulated cohorts used as illustrations in the chapter.

Table 9.A1 Parameter values for simulated cohorts

We simulated cohorts with mortality multipliers on being black at values of $b=1.2,\;1.5,2$, and mortality multipliers on being frail at values of $f=2,4,6,\;8$,⁷ with values of the initial proportion frail in each race, ${\pi}_k(0),$ ranging from .55 to .95 in units of .05. The mortality multiplier on being chronically ill, $c$, and the initial proportion chronically ill within each subpopulation defined by race and frailty, ${\tau}_{k,i}(0)$, were simulated over the same ranges as those for frailty. (The shared mortality intercept $\alpha$ and slope $\beta$ are irrelevant to whether a crossover occurs, although they influence the age at which it occurs in cohorts that have a crossover.)

In the unidimensional model, crossovers occur in 39 of the 108 simulated cohorts (36 %). Each of these cohorts has a frailty multiplier of at least 4. In the multidimensional model, a crossover occurs in 18,889 out of 34,992 simulated cohorts (54 %). In these cohorts, the mortality multipliers on being black, frail, or chronically ill can take any of the simulated values, but a crossover never occurs unless either the multiplier on being frail or the multiplier on being chronically ill exceeds 2. In other words, it is easier to produce an aggregate mortality crossover in the multidimensional model than in the unidimensional model.

In the simulation universe we explored, racial segregation of the subpopulations is quite rare. An interval in which subpopulation mortality (defined along the frail-robust dimension) is racially segregated in the direction of higher white mortality occurs in 90 of the simulated multidimensional cohorts (0.48 % of those with a crossover). These cohorts uniformly have a small mortality multiplier on being frail (f = 2) and a large mortality multiplier on being chronically ill (c = 6 or c = 8). They also have high initial values of the proportion chronically ill (.8–.9 for the robust and .85–.95 for the frail).

Cohorts experiencing two distinct crossover intervals are more common; this occurs in 3,374 multidimensional simulated cohorts (18 % of those with at least one crossover). These cohorts each have mortality multipliers on being frail and being chronically ill of at least 4, and initial proportion chronically ill among the frail of at least .65, but otherwise occur across the range of parameter values examined in these simulations.

The relative rarity of these crossover outcomes in the simulation universe we explored partly reflects that generating aggregate crossovers from a small number of subpopulations requires fairly extreme parameter values. For example, the unidimensional heterogeneity model in “Heterogeneity’s Ruses” (Vaupel and Yashin 1985) worked with an extreme black-frailty interaction: whereas frail whites had only 1.25 times the mortality of robust whites, frail blacks had mortality 5 times larger than the mortality of robust blacks. This black-frailty interaction helps to produce a crossover in two ways: it exaggerates the extent to which mortality selection against frailty occurs more extremely for blacks than whites, and it also means that the same percentage decline in frailty share reduces white mortality by only a small amount, but black mortality by a large amount. Our model eschews a black-frailty interaction: the frailty multiplier is the same for blacks and for whites. Consequently, the mortality multipliers required to generate a crossover are larger.