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Mortality Crossovers from Dynamic Subpopulation Reordering

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Part of the The Springer Series on Demographic Methods and Population Analysis book series (PSDE,volume 39)


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.


  • Mortality selection
  • Selective mortality
  • Heterogeneity
  • Frailty
  • Mortality crossover
  • Dynamic mortality model

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Fig. 9.1
Fig. 9.2
Fig. 9.3
Fig. 9.4
Fig. 9.5


  1. 1.

    For important prior work on mortality selection with multidimensional heterogeneity (albeit not analyzing mortality crossovers), see e.g., Manton et al. (1994, 1995) and Woodbury and Manton (1983).

  2. 2.

    Throughout, when we refer to the makeup of a larger aggregation in terms of its lower-level components, we use composition from the perspective of the aggregation and share from the perspective of the components. Thus, the population composition of subpopulations and the subpopulation shares of the population both refer to the proportions of the population that are in each subpopulation.

  3. 3.

    Gamma-Gompertz models replace the fixed binary frailty term of our model with a fixed gamma-distributed continuous frailty term. Vaupel and Yashin (1985) replace the frailty multiplier of Eq. 9.1 with an interacted multiplier: robust people have the same mortality regardless of whether they are in the advantaged (e.g., white) or disadvantaged (e.g., black) population, and the penalty for frailty is large among the disadvantaged and small among the advantaged.

  4. 4.

    The parameter values for all examples are presented and discussed in the Appendix.

  5. 5.

    It follows from Eq. 9.4b that racial segregation of the subpopulations along one dimension of heterogeneity determines the direction of aggregate mortality differentials: if black and white populations are exhaustively partitioned into non-overlapping subsets (e.g., subpopulations defined along one dimension of heterogeneity) such that all white subsets have higher mortality than all black subsets (or vice-versa), then aggregate white mortality must exceed aggregate black mortality (or vice-versa). (If subpopulations are racially segregated along one dimension of heterogeneity, they may or may not also be segregated along the other dimension of heterogeneity.)

  6. 6.

    Nothing new happens after age 100.

  7. 7.

    We simulated cohorts with small values of the black mortality multiplier and large values of the frailty mortality multiplier (and chronically ill mortality multiplier) in order to find cohorts that might have an aggregate black-white crossover.


  • Berkman, L., Singer, B., & Manton, K. (1989). Black-white differences in health status and mortality among the elderly. Demography, 26(4), 661–678. doi:10.2307/2061264.

    CrossRef  Google Scholar 

  • Dupre, M. E., Franzese, A. T., & Parrado, E. A. (2006). Religious attendance and mortality: Implications for the Black-White mortality crossover. Demography, 43(1), 141–164. doi:10.1353/dem.2006.0004.

    CrossRef  Google Scholar 

  • Fenelon, A. (2013). An examination of black/white differences in the rate of age-related mortality increase. Demographic Research, 29(17), 441–472. doi:10.4054/DemRes.2013.29.17.

    CrossRef  Google Scholar 

  • Gampe, J. (2010). Human mortality beyond age 110. In H. Maier, J. Gampe, B. Jeune, J.-M. Robine, & J. Vaupel (Eds.), Supercentenarians (Demographic Research Monographs 7, pp. 219–230). Berlin: Springer. doi:10.1007/978-3-642-11520-2.

    CrossRef  Google Scholar 

  • Horiuchi, S., & Wilmoth, J. R. (1998). Deceleration in the age pattern of mortality at older ages. Demography, 35, 391–412.

    CrossRef  Google Scholar 

  • Kestenbaum, B. (1992). A description of the extreme aged population based on improved medicare enrollment data. Demography, 29(4), 565–580. doi:10.2307/2061852.

    CrossRef  Google Scholar 

  • Lynch, S. M., Brown, J. S., & Harmsen, K. G. (2003). Black-white differences in mortality compression and deceleration and the mortality crossover reconsidered. Research on Aging, 25(5), 456–483. doi:10.1177/0164027503254675.

    CrossRef  Google Scholar 

  • Manton, K. G., Stallard, E., Woodbury, M. A., & Dowd, J. E. (1994). Time-varying covariates in models of human mortality and aging: Multidimensional generalizations of the Gompertz. Journals of Gerontology, 49(4), B169–B190. doi:10.1093/geronj/49.4.b169.

    CrossRef  Google Scholar 

  • Manton, K. G., Woodbury, M. A., & Stallard, E. (1995). Sex differences in human mortality and aging at late ages: The effect of mortality selection and state dynamics. Gerontologist, 35(5), 597–608. doi:10.1093/geront/35.5.597.

    CrossRef  Google Scholar 

  • Masters, R. K. (2012). Uncrossing the U.S. Black-white mortality crossover: The role of cohort forces in life course mortality risk. Demography, 49(3), 773–796. doi:10.1007/s13524-012-0107-y.

    CrossRef  Google Scholar 

  • Missov, T. I., & Finkelstein, M. S. (2011). Admissible mixing distributions for a general class of mixture survival models with known asymptotics. Theoretical Population Biology, 80, 64–70.

    CrossRef  Google Scholar 

  • Preston, S. H., Elo, I. T., Hill, M. E., & Rosenwaike, I. (2003). The demography of African Americans, 1930–1990. Norwell: Kluwer/Springer. doi:10.1007/978-94-017-0325-3.

    CrossRef  Google Scholar 

  • Sautter, J. M., Thomas, P. A., Dupre, M. E., & George, L. K. (2012). Socioeconomic status and the black-white mortality crossover. American Journal of Public Health, 102(8), 1566–1571. doi:10.2105/ajph.2011.300518.

    CrossRef  Google Scholar 

  • Steinsaltz, D. R., & Wachter, K. W. (2006). Understanding mortality rate deceleration and heterogeneity. Mathematical Population Studies, 13, 19–37.

    CrossRef  Google Scholar 

  • Vaupel, J. W., & Yashin, A. I. (1985). Heterogeneity’s ruses: Some surprising effects of selection on population dynamics. American Statistician, 39(3), 176–185. doi:10.1080/00031305.1985.10479424.

    Google Scholar 

  • Vaupel, J. W., Manton, K. G., & Stallard, E. (1979). Impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16(3), 439–454. doi:10.2307/2061224.

    CrossRef  Google Scholar 

  • Wilmoth, J. R., & Dennis, M. (2007). Social differences in older adult mortality in the United States: Questions, data, methods, and results. In J. M. Robine, E. I. Crimmins, S. Horiuchi, & Y. Zeng (Eds.), Human longevity, individual life duration, and the growth of the oldest-old population (pp. 297–332). Dordrecht: Springer. doi:10.1007/978-1-4020-4848-7_14.

    CrossRef  Google Scholar 

  • Woodbury, M. A., & Manton, K. G. (1983). A mathematical model of the physiological dynamics of aging and correlated mortality selection. 1: Theoretical development and critiques. Journals of Gerontology, 38, 398–405. doi:10.1093/geronj/38.4.398.

    CrossRef  Google Scholar 

  • Wrigley-Field, E. & Elwert, F. (2015). Multidimensional mortality selection and the black-white mortality crossover. Paper presented at the Population Association of America: April, San Diego.

    Google Scholar 

  • Yao, L., & Robert, S. A. (2011). Examining the racial crossover in mortality between African American and White older adults: A multilevel survival analysis of race, individual socioeconomic status, and neighborhood socioeconomic context. Journal of Aging Research, 1–8.

    Google Scholar 

  • Zeng, Y., & Vaupel, J. W. (2003). Oldest-old mortality in China. Demographic Research, 8(7), 215–244. doi:10.4054/demres.2003.8.7.

    CrossRef  Google Scholar 

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Appendix: Notes on the Simulation Parameters

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 \( f=2,4,6,\;8 \),Footnote 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.

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Wrigley-Field, E., Elwert, F. (2016). Mortality Crossovers from Dynamic Subpopulation Reordering. In: Schoen, R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham.

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