Abstract
Theoretical models of mortality selection have great utility in explaining otherwise puzzling phenomena. The most famous example may be the Black-White mortality crossover: at old ages, Blacks outlive Whites, presumably because few frail Blacks survive to old ages while some frail Whites do. Yet theoretical models of unidimensional heterogeneity, or frailty, do not speak to the most common empirical situation for mortality researchers: the case in which some important population heterogeneity is observed and some is not. I show that, when one dimension of heterogeneity is observed and another is unobserved, neither the observed nor the unobserved dimension need behave as classic frailty models predict. For example, in a multidimensional model, mortality selection can increase the proportion of survivors who are disadvantaged, or “frail,” and can lead Black survivors to be more frail than Whites, along some dimensions of disadvantage. Transferring theoretical results about unidimensional heterogeneity to settings with both observed and unobserved heterogeneity produces misleading inferences about mortality disparities. The unusually flexible behavior of individual dimensions of multidimensional heterogeneity creates previously unrecognized challenges for empirically testing selection models of disparities, such as models of mortality crossovers.
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Notes
The multidimensional mortality selection considered here differs from the multivariate mortality selection analyzed elsewhere, as in “shared frailty models” (e.g., Guo and Rodriguez 1992; Henderson and Oman 1999; Vaupel 1988; Wienke 2010:131–160). The former deals with multiple independent variables; the latter, multiple (correlated) survival-time outcomes.
In the classical mortality selection literature about crossovers, disadvantage has been operationalized in different ways, including greater mortality for Blacks than Whites at all levels of frailty, with Black and White frailty equal at baseline (Vaupel et al. 1979); greater mortality for Blacks than Whites specifically among the frail (Vaupel and Yashin 1985); and greater mortality for Blacks than Whites among the frail and a larger initial proportion of frailty among Blacks at baseline (Lynch et al. 2003). The model I employ is consistent with the approach of Vaupel et al. (1979), which offers the neatest fit with the general preference for proportional hazard models (by assuming Black disadvantage for all cohort members, not just the frail) and allows me to highlight how multidimensional selection produces flexible crossover results even without differences in Black and White initial frailty distributions. An alternative model specification is considered in section 4 of the online appendix.
I assume the same baseline distributions of frailty in both populations in order to focus analysis cleanly on the dynamics of mortality selection, rather than other potential sources of racial difference in mortality. The main substantive points do not depend on this assumption. Nonetheless, if Black and White frailty composition differed at birth, some aspects of the presentation of results would differ, as I remark later in footnote 7.
These two derivations from unidimensional crossover models (e.g., Vaupel et al. 1979) follow from the widely known fact that in a proportional hazards context with one unmeasured (e.g., frailty) and one measured (e.g., race) covariate, the unmeasured covariate leads to an underestimation of the effect of the measured covariate (see, e.g., Aalen 1988; Henderson and Oman 1999; Hougaard et al. 1994). The second assumption is the defining assumption of fixed-frailty models (Finkelstein 2012), which have wide application beyond the crossover. The first assumption is particular to crossover models. The great achievement of selection models of the crossover is to make this first assumption compatible with the existence of a crossover (Vaupel et al. 1979; Vaupel and Yashin 1985).
The arrows in Fig. 1 represent multiplicative effects; thus, the overall sign of a path is the product of the signs on each arrow.
Increasing the frailty multiplier can increase as well as decrease mortality in each racial population. The total effect of f on aggregate mortality in a population depends on which path dominates. Thus, there can be spans of ages at which population-level mortality would be lower with a larger frailty multiplier because fewer frail survivors would remain.
A Black-White crossover can occur regardless of the signs of the total effect of f on aggregate mortality in the Black and White populations. When the total effect of frailty on mortality is less positive, or more negative, for the White population than for the Black population at a given age, a crossover can occur. (Whether a crossover occurs additionally depends on whether the effect of frailty outweighs the Black mortality disadvantage at the subpopulation level, as suggested by panel a of Fig. 1.)
The former fact depends on the assumption that Blacks and Whites are equally likely to be frail at birth. If Blacks were more likely than Whites to be frail at birth, then this term would become negative only if the greater selection against frailty among Blacks eventually outweighed their initial excess frailty.
The formula for πk, i(a) in the multidimensional model is analogous to the formula for πk(a) in the unidimensional model given in Eq. (4), replacing the subpopulation-level survivorships Sk, j(a) in Eq. (4) with the corresponding group-level survivorships Sk, i, j(a) for the ith (exposed or non-exposed) subpopulation.
In the multidimensional model, I use uppercase Greek letters for composition defined at the population level (the (observed) proportion of each racial population that is exposed, aggregated over residual frailty, Tk(a), and the (unobserved) proportion of each racial population that is residually frail, ∏k(a), aggregated over tobacco exposure) and lowercase Greek letters for composition defined at the subpopulation level (the (unobserved) proportion of each exposure subpopulation that is residually frail, πk, i(a), and the (unobserved) proportion of each residual frailty subpopulation that is tobacco-exposed, τk, j(a)).
One could instead decompose population-level mortality into the aggregate proportion of each racial population that is residually frail (∏k(a), unobserved) and the proportion of each of those subpopulations that is exposed (τk, j(a), unobserved). Regardless of how population-level mortality is decomposed, it reflects the distribution of each race along both dimensions of heterogeneity simultaneously.
This formulation is not necessarily a socially coherent counterfactual: in reality, if the social treatment of people designated as Black were substantially altered, presumably so would the social treatment of people designated as White. (For example, White mortality might rise if Whites were no longer protected by racism from meritocratic competition from Blacks; or White mortality might fall if social welfare programs were not highly racialized and denigrated.) In causal terms, the assumption beneath this simplification is the stable unit treatment value assumption (SUTVA) (Morgan and Winship 2015:48–52).
I have argued elsewhere (Wrigley-Field and Elwert 2017) that the crossover literature should more seriously attend to how selection dynamics interact with racial disadvantages that may shrink or grow over the life course. Incorporating dynamic frailty alongside multiple dimensions of frailty is mathematically complex.
The alternative model in which Black-White inequality is implicit in the heterogeneity distribution, rather than explicit in a mortality multiplier, also allows for a conceptually neat distinction between individual inequality and population disparity. In that alternative model, though, this distinction can only be defined with a more complex counterfactual (Wrigley-Field and Elwert 2017) rather than with a single, simple parameter.
The dimensions of heterogeneity explored in the empirical literature are traits that—unlike “frailty”—are acquired and lost by individuals over time. This extension of the classic mortality selection models to time-varying dimensions of heterogeneity can introduce significant complications (see Manton et al. 1994, 1995; Rogers 1992; Vaupel et al. 1988; Woodbury and Manton 1983; Wrigley-Field 2013) that are not considered in those studies or in this one. Here, I focus solely on how fixed dimensions of heterogeneity interact in the selection process.
This criterion is explicit in Dupre et al. (2006:146):
To investigate whether religious involvement operates as a source of heterogeneity, two conditions must be satisfied and are hypothesized separately. First, in accordance with prior research that shows that religious involvement is more protective for blacks, the following hypothesis must be true: the effect of religious involvement will have a greater impact among blacks on the risk of dying […]. Thus, blacks who attend services weekly or more will have a larger reduction in mortality than whites. Second, to support the claim that religion contributes to why hazard rates invert, the effect of religion must vary with age.
In Sautter et al. (2012), this criterion is implicit but undergirds the empirical analysis.
It is well known that frailty can increase in populations in which individuals can newly acquire frailty during their lives; for one systematic exploration of population dynamics that can result from such dynamic frailty, see Vaupel et al. (1988). It is specifically in the context of frailty fixed in individuals that the frailty increases and frailty reversals illustrated here are deeply surprising.
In the language of causal inference, the association occurs because mortality is a collider for its risk factors. Conditional on survival, those risk factors become associated (see Elwert and Winship 2015). In the classical mortality selection model of the crossover, mortality is a collider for race and frailty. In the multidimensional model, mortality is a collider for race, observed exposure, and residual frailty, producing three-way associations between them over age.
Whether the disadvantage associated with residual frailty, f∗, has a larger effect on the mortality of the non-exposed or the exposed (and whether these effects have the same sign) depends on whether the increased mortality of the residually frail groups outweighs the increased selection of the residually frail groups in each exposure subpopulation. Both effects are greater among the exposed, making the total effects of their competing signs unpredictable a priori.
Whether the effects of covariates have a priori predictable or unpredictable signs is determined by the level of aggregation, not the dimension of heterogeneity. The mortality penalty associated with residual frailty, f∗, always has a negative effect on the proportion of survivors who are residually frail at the subpopulation level, \( {f}^{\ast}\overline{\to}{\uppi}_{k,i}(a) \). But because the dimensions of heterogeneity interact at the population level, as illustrated in Fig. 5, f∗ can have either a negative or a positive effect on the proportion of survivors who are residually frail at the population level, \( {f}^{\ast}\overset{+/-}{\to }{\prod}_k(a) \). Hence, the effect of f* on population mortality via its effect on the population-level proportion residually frail could be positive or negative, \( {f}^{\ast}\overset{+/-}{\to }{\prod}_k(a)\overset{+/-}{\to }{\overline{\upmu}}_k(a) \).
Simulations show that the aggregate can cross simultaneously with either the exposed or the non-exposed subpopulation. Simultaneous crossovers are defined as crossovers occurring at the same survivorship of the robust non-exposed Whites (thus, the same age), to three decimal places. The simulation procedure is described in section 3 of the online appendix.
One might suspect that the aggregate crossover is nearly simultaneous with the non-exposed crossover because by the time the aggregate crossover occurs, virtually all survivors are non-exposed. But this is not the case. At age 82, when the aggregate and non-exposed crossovers occur, 27% of Black survivors and 24% of White survivors are exposed. (Thus, there has been a “frailty” reversal in observed exposure.)
Any such covariates would need to be studied in a setting in which, or operationalized such that, they are fixed in individuals. For example, chronic illnesses acquired by middle adult or early elderly ages might be used as a strong predictor of mortality at older ages.
Frailty reversals and increases might still occur along the residual frailty dimension, complicating the interpretation of the observed associations between tobacco exposure and mortality. See also the discussion in section 4 of the online appendix, which qualifies this criterion for alternative models in which one dimension of heterogeneity is more consequential for Whites and the other is more consequential for Blacks.
Because frailty reversals and frailty crossovers will not always occur along the dimension of heterogeneity that less strongly increases mortality, only the presence (not the absence) of these phenomena constitute tests of this model.
Because subpopulation crossovers will not always occur even in subpopulations whose frailest Whites have higher mortality than the most robust Blacks, only the presence (not the absence) of subpopulation crossovers constitute a test of this model.
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Acknowledgments
This article benefitted from extensive discussion with Felix Elwert, along with helpful comments from James Montgomery, Alberto Palloni, Jenna Nobles, James Walker, Erik Olin Wright, Kirkwood Adams, Jenny Conrad, Michal Engelman, Josh Goldstein, Kathryn Grace, Sarah Grey, Jeffrey Grigg, Paul Hanselman, Anna Haskins, Vida Maralani, Jude Mikal, Phyllis Moen, Michelle Niemann, Sarah Thomas, and Rob Warren; valuable feedback and generous assistance from Andreas Wienke; and valuable feedback from five anonymous reviewers, two editorial teams, and the copyeditor. Funding for this research was provided by the Robert Wood Johnson Foundation Health and Society Scholars; an NICHD training grant (T32 HD07014); graduate fellowships from the National Science Foundation, the Ford Foundation, and the Institute for Research on Poverty at the University of Wisconsin–Madison; and core grants to the Minnesota Population Center (P2C HD041023) at the University of Minnesota, to the Center for Demography and Ecology at the University of Wisconsin–Madison (P2C HD047873), and to the Center for Demography of Health and Aging (P30 AG017266) at the University of Wisconsin–Madison.
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Wrigley-Field, E. Multidimensional Mortality Selection: Why Individual Dimensions of Frailty Don’t Act Like Frailty. Demography 57, 747–777 (2020). https://doi.org/10.1007/s13524-020-00858-8
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DOI: https://doi.org/10.1007/s13524-020-00858-8