Drivers of Mortality Dynamics: Identifying Age/Period/Cohort Components of Historical U.S. Mortality Improvements

Abstract

The goal of this article is to obtain an age/period/cohort (A/P/C) decomposition of historical U.S. mortality improvement. Two different routes to achieving this goal are considered. In the first route, the desired components are obtained by fitting an A/P/C model directly to historical mortality improvement rates. In the second route, an A/P/C model is estimated to historical crude death rates and the desired components are then obtained by differencing the estimated model parameters. For each route, various possible A/P/C model structures are tested and evaluated on the basis of their robustness to several factors (e.g., changes in the calibration window) and their ability to explain historical changes in mortality improvement. Based on the evaluation results, an A/P/C decomposition for each gender is recommended. The decomposition will be examined in a follow-up project, in which the linkages between the A/P/C components and certain intrinsic factors will be identified.

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Notes

1 We base our conclusions on the SSA data instead of data from the Human Mortality Database because of the reasons discussed by Goss et al. (2015).

2 To facilitate identification of patterns, the mortality improvement rates are calculated from mortality rates that have been smoothed by a two-dimensional (age/cohort) P-spline (Currie, Durban, and Eilers 2004). The mortality improvement rate used here is defined in Equation (2)(2) $$Z_{x,t}=1-\frac{q_{x,t}}{q_{x,t-1}},$$(2) in Section 4.

3 We calculate the observed value of $Z_{x,t}$ using Equation (1)(1) $$q_{x,t}\approx 1-\hspace{0.17em}\text{exp}\hspace{0.17em}(-m_{x,t}),$$(1) with $m_{x,t}$ being set to the ratio of the death count to the exposure count at age x in year t.

4 If $\text{ln}\hspace{0.17em}{m}_{x,t}$ follows the Renshaw-Haberman structure exactly, then the Renshaw-Haberman structure for $Z_{x,t}$ would contain no stand-alone age-specific parameters. Still, when modeling $Z_{x,t},$ stand-alone age-specific parameters may be used to achieve a better goodness of fit.

5 Our data sample spans ages 20 to 95 and years 1968 to 2014. For the earliest cohort ($c=1968-95=1873$), we have $n_{1873}=1$ because there is only one data point (which corresponds to age 95 and year 1968) associated with this cohort. Likewise, for the second earliest cohort c = 1874, we have $n_{1874}=2$ because there are two data points (one of which corresponds to age 94 and year 1968 and the other of which corresponds to age 95 and year 1969) associated with this cohort. The values of n_c for other cohorts can be deduced in a similar manner.

6 We refer interested readers to Cairns et al. (2009) for an explanation as to why the identifiability constraints we use ensure that the resulting cohort component will fluctuate around zero and will exhibit no linear and/or quadratic trend.

7 We refer interested readers to section 8 of volume 1 of the full report of this study (Li, Zhou, and Liu 2018a) for a comparison between the two methods.

8 At the end of each iteration (when all of the parameters are updated), the parameters are rescaled so that the applicable identifiability constraints are satisfied.

9 In principle, a higher order polynomial can be used as the standard of smoothness instead. We have experimented with higher order polynomials, and found that the resulting A/P/C decompositions are similar.

10 The chosen values of ${\pi }_{\beta },{\pi }_{\kappa },$ and ${\pi }_{\gamma }$ are 10⁵, ${10}^{2.75},$ and ${10}^{2.75},$ respectively.

11 Typically, it takes less than 100 iterations to reach convergence.

12 At a 5% significance level, we reject the null hypothesis if the p value is smaller than .05; that is, we conclude that the standardized residuals do not follow the standard normal distribution if the p value is less than 0.05.

13 The logit transform of a quantity y is defined as $\text{ln}\hspace{0.17em}(y/(1-y\left)\right).$

14 For Model M2, Model M3, the full Plat model, the simplified Plat model, and the APCI model, we calculate the historical mortality improvement rates as $\text{ln}\hspace{0.17em}{m}_{x,t-1}-\hspace{0.17em}\text{ln}\hspace{0.17em}{m}_{x,t};$ for Models M6, M7, and M8, the historical mortality improvement rates are calculated as $\text{ln}\hspace{0.17em}\frac{q_{x,t-1}}{1-q_{x,t-1}}-\hspace{0.17em}\text{ln}\hspace{0.17em}\frac{q_{x,t}}{1-q_{x,t}}.$ This arrangement maximizes consistency with the way in which $M{I}_{x,t}$ Z$M{I}_{x,t}$ is defined for the models.

15 This definition of standardized residuals is different from that for Route A, which is based on the deviations between fitted and observed mortality improvement rates. Because the standardized residuals for Route A and Route B models are calculated in different ways, the standardized residuals presented in this section cannot be directly compared and contrasted with those shown in Section 5.4.

16 The additional analyses are documented in the full report of this study (section 7, Li, Zhou, and Liu 2018a; section 5, Li, Zhou, and Liu 2018b).