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The Demography and Epidemiology of Human Health and Aging pp 731–773Cite as

Models of Aging, Health, and Mortality, and Mortality/Health Projections

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Abstract

As we saw, the life span of individuals is determined by a complex interplay of evolutionary, genetic, epigenetic, social/individual/ecological, and stochastic factors operating roughly within the limits of the genetic predispositions of these individuals. The stochastic factor represents both a residual category that jointly encompasses those influences that cannot be assigned to the other categories and those influences that occur by chance. Chance events are a major influence in the determination of the life span and, like evolutionary and genetic influences, may be considered at least in part beyond society’s or the individual’s control. Pending the personalization in health planning of the genomic characteristics of each individual and developments in gene therapy, therefore, the focus should be on social/personal/ecological factors as areas where major changes in health status can now be largely effected.

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Notes

  1. 1.

    It is not practical to describe the methods of preparing mortality and population projections in detail here. The reader is referred to other publications in which these methods are discussed more fully. See, for example, Siegel and Swanson (2004); Ahlburg and Land (1992); Smith et al. (2001); and Siegel (2002).

  2. 2.

    The implementation of this matrix equation and the Leslie matrix requires an abridged life table, a set of age-specific maternal birth rates, and a population age-sex distribution. Five-year survival rates are shown in the subdiagonal and the birth rates are shown in row one; the age distribution for one of the sexes is placed on the right as a column vector. The birth rates in row one are modified so as to allow for the sex of the births, the shift in the age distribution of the female population over a 5-year period, and the survival of the births to ages under 5 at the end of the 5-year period. For further explanation, see ?  (? :31).

  3. 3.

    ARIMA is the acronym for autoregressive integrated moving average. The terms refer to the various processes by which the time series is transformed for the purpose of preparing it for projection. In autoregression, the data points are “regressed” against a previous data point or points in the same time series; “integration” refers to the order of differencing of the data points (e.g., first or second differences); and “moving average” refers to the substitution of moving-average values for the original values (defined by the number of years included in the moving average).

  4. 4.

    The original series may be transformed by taking first or second differences, calculating natural logarithms or square roots, or fitting a straight line. Moving averages may be calculated from the transformed series or the series may be “regressed on itself” with a specified time lag (1 or 2 years).

  5. 5.

    A simulation is a form of model testing, i.e., a calculation, normally carried out by computer, in which a set of assumptions or conditions, either theoretical or realistic, are applied to the model in order to generate one or more representative scenarios where it is not possible or desirable to test all conditions pertaining to the model.

  6. 6.

    The equation for a random walk with drift is,

    $${\mathrm{k}}_{\mathrm{t}} = \mathrm{c} +{ \mathrm{k}}_{\mathrm{t}-1} +{ \mathrm{E}}_{\mathrm{t}}$$

    where c is the drift term and E t is a normally distributed random variable with mean zero and variance σ2. A random walk with drift refers to the way the ARIMA model is applied. Initially, the first differences of the series are examined to see if a predictable pattern can be discerned. The transformed series tends to appear stationary and quite random. The random-walk model assumes that the projected figures take a random step from their last recorded level equal to the average of a random walk-with-drift-difference in the past. If this average difference is zero, it is a random walk without drift. In a random walk with drift, the projected series includes a non-zero constant term, assuming an upward or downward trend. In ARIMA terms, this is a “(0,1,0) model with constant,” where the 1 refers to first differences.

  7. 7.

    In exponential smoothing the members of the series are averaged with shifting weights, the greater weights being given to the more recent ratios. Specifically, the weights for the observations decrease exponentially as the observations recede farther into the past:

    $$\mathrm{P} ={ \mathrm{aX}}_{\mathrm{t}} + \mathrm{a}(1 -\mathrm{a}){\mathrm{X}}_{\mathrm{t}-1} + \mathrm{a}{(1 -\mathrm{a})}^{2}{\mathrm{X}}_{\mathrm{ t}-2} + \mathrm{a}{(1 -\mathrm{a})}^{3}{\mathrm{X}}_{\mathrm{ t}-3} + \ldots \ldots $$

    where P is the projected ratio, a is the smoothing constant, and X t is the past observation at time t. Determining the smoothing constant requires some experimentation but a value of 0.4 may serve the purpose. It must be between zero and 1. Once the smoothing ratios are determined by the formula, they are held constant for all future years.

  8. 8.

    The series had been examined for stationarity and, though found to be nonstationary, was modeled without being corrected. The reason for this choice was that it was assumed that the historical series fluctuated on the basis of changes in the law rather than on the basis of demographic or socioeconomic factors.

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Siegel, J.S. (2012). Models of Aging, Health, and Mortality, and Mortality/Health Projections. In: The Demography and Epidemiology of Human Health and Aging. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1315-4_14

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