where the multiplier already depends on the stress at time t and on the age x.
Different environments can be defined not necessarily by the switching point or by considering changing in time stresses. Let the stress zo(to) be a baseline stress at a baseline (fixed) time instant to. Denote the corresponding mortality rate, as previously, by ¡io(x,to). Then the stress z(t) and the mortality rate ¡(x,t) characterize the current instant of time t. Note that in this approach populations can be different and t — to can be reasonably large (e.g., 10 or 20 years). The PH model for this case is naturally defined as (compare with (27) )
¡(x,t) = wp(z(t),x)^o(x,to), x > 0, t > to, (34)
where ¡o(x, to) plays the role of a baseline mortality rate. The analogue of the ALM, however, is not straightforward, as there should be a pair wise comparison between the corresponding cohorts of the same age x, using expression (32) for both instants of time. This topic needs further study.
Example 3. Gompertz shift model. As stated in Bongaarts and Feeney [BF02], the mortality rate in contemporary populations with high level of life expectancy tends to improve over time by a similar factor at all adult ages which results in our notation in the following Gompertz shift model (similar to equations (21) and (22)):
This model was verified using contemporary data for different developed countries. Equations (35) and (36) define formally the age independent PH model. We do not have a switching in stress here which could help in verifying plasticity.
Most researchers agree that the process of human aging is the process of accumulation of damage of some kind (e.g., accumulation of deleterious mutations). Given the reasoning of the previous sections it means that the PH model (35), (36) is not suitable for this case unless it describes the lifesaving model. On the other hand, as it was stated in Example 2, it is really unnatural trying to explain (35)-(36) via some degradation model. Therefore, if the linear trend takes place (or, equivalently, the logarithms of mortality rates at different time instants are practically parallel), this can be explained by lifesaving in a general sense and not by slowing down the degradation processes, for instance. In other words: lifesaving is likely to be the main source of lifespan extension at present time. In fact, it is not a strict statement, but just a reasoning that seems to be true.
The most popular models that account for an impact of environment on a lifetime are the PH model and the ALM. The first one is the simplest way to describe the memory-less property, whereas the second describes the simplest dependence on a history in a form of accumulated damage. Various generalizations of these models are considered in Bagdonavicius and Nikulin [BN02]. In survival analysis these models were traditionally defined for the cohort setting.
The conventional demographic definition of the observed in a period (from t to t + At) age-specific and time-dependent mortality rate is given by equation(26). The generalization of the cohort PH model to this case is given by equations (27) and (28). The corresponding generalization of the ALM is explicitly performed for a specific case of the step stress zs. Therefore, the cohort ALM is applied to each cohort with varying age x (0 < x < x) at time t, which results in equations (29)-(31) defining the age specific mortality rate.
Although human aging is definitely a process of damage accumulation, the contemporary demographic data supports the Gompertz shift model (33)-(34), which is, at least formally, the PH model. In line with our reasoning of the previous section this means that lifesaving (versus the decrease in the rate of degradation) can explain the decrease in mortality rates with time.
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