Heterogeneity's ruses: Some surprising effects of selection on population dynamics
The article begins with a brief description of hazard rates (i.e., a cohort's rate of death or exit) and the observation that only two homogeneous subpopulations are necessary to illustrate many of heterogeneity's ruses. For instance, given a situation in which the hazard rate for subpopulation 1 is constant (i1) and higher than subpopulation 2 (i2-also constant), the mean hazard rate for the entire population will decrease steadily over time and eventually converge with the lower hazard rate (i.e., i2). So, the observation that recidivism rates for released convicts decline over time may not in fact be due to a declining propensity to commit crime. Rather, if heterogeneity is present in the simple form of one group of "reformed" and another group of "incorrigibles", then declining recidivism rates might simply be a reflection of the higher "death rate" (i.e., imprisonment) of the incorrigibles. Another ruse of heterogeneity may occur when i(c)-the mean hazard for a cohort- increases steadily, drops suddenly for a brief period, and then resumes climbing, but at a slower rate. This trajectory is produced when two subcohorts have different but constantly increasing hazard rates. The cohort with the higher mortality rate is frailer, and hence becomes extinct more quickly. As the frail cohort dies out, the hazard rate drops toward the more robust cohort. When the cohort's hazard rate begins to approach the more robust cohort's hazard rate, it begins to rise again at a rate that is dominated by the robust cohort. Thus, the appearance of falling mortality within a cohort is in fact an illusion, a ruse.
The mover-stayer model presents another example of a ruse introduced by heterogeneity. Assume that one subpopulation is "immune" to divorce (i.e., stayers) while another has a constantly increasing hazard of divorce (i.e., movers). "If the hazard for the susceptible subpopulation is steadily increasing, then . . . the observed hazard for the entire population may rise and then fall. . . Does this imply that marriages are shakiest after a few years of marriage? Not necessarily . . . The same basic effect can be produced even if one group is not immune but simply at low risk. Indeed the rising-falling pattern can be produced if the hazard steadily increases for the high-risk group but steadily decreases for the low-risk group. For one group, marriages strengthen with duration, while for the other, marriages weaken-despite the appearance of the cohort curve, there is no 'seven-year itch'" (p. 178).
One final example: Mortality crossovers may occur when robust subcohorts from two populations (e.g., white and black robust) face equal mortality chances, but frail subcohorts face unequal chances (e.g., i-white frail < i-black frail). In such a scenario, the total black hazard is initially greater than the total white hazard due to the influence of frail blacks. But, since the frail black subcohort dies out relatively quickly, only the robust black subcohort remains. Since more of the frail white subcohort survives to later ages, it weights the total white hazard rate upward-resulting in a mortality crossover in which the total black hazard is lower than the total white hazard.
Conclusions: Heterogeneity may threaten research conclusions, even when care is taken to eliminate population differences (e.g., via randomization). "When should a researcher suspect substantial heterogeneity? A useful clue occurs when theory and evidence pertaining to individuals suggest a trajectory of mortality that diverges from the observed trajectory for the population. For instance, human mortality may increase exponentially, but observed mortality curves appear to level off at advanced ages: the discrepancy suggests heterogeneity" (p. 184).
Theoretical and practical relevance:
Be careful of heterogeneity!