Health decline, aging and mortality: how are they related?

Abstract

The deterioration of human health with age is manifested in changes of thousands of physiological and biological variables. The contribution of some of such changes to the mortality risk may be small and cannot be reliably detected by existing statistical methods. A cumulative index of health/well-being disorders, which counts changes in observed variables on the way of losing health, may be an appropriate way to take the effects of such variables into account. In this paper we investigate regularities of the aging-related changes in human health/well-being/survival status described by such an index using the new version of the quadratic hazard model of human aging and mortality. We found that the shape and the location of the mortality risk, considered as a function of the introduced health-related index, changes with age reflecting the decline in stress resistance and the age-dependence of the “optimal” health/well-being status. Comparison of these results with findings from early studies using the Cox’s-like model of risk function indicates that the results are likely to describe regularities of deterioration in human health during the aging process.

Appendix

General model

Let X _t , Y _t be two stochastic processes describing the life history of an individual. The process ${X_{t_k}}$ is equal to zero if an individual died in the age interval [ t _k ,t _k+1), and it is equal to one if he/she survived until age t _k+1. The process Y _t is a discrete time stochastic process describing observations of a health-related index (covariate). Assume that this process satisfies the following equation:

$$ Y_{t_{k+1}} =Y_{t_k}+a_{t_k}\left({f_{t_k}^1 -Y_{t_k}} \right)\left({t_{k+1}-t_k}\right)+\sigma_1 \sqrt {t_{k+1}-t_k} \varepsilon _{t_k},k > 1,Y_{t_1},$$

(A1)

where ${\varepsilon _{{t}_{k}}}$ ∼ N(0,1), ${Y_{{t}_{1}}}$ ∼ N(${\tilde{f}_{t_1}^{1}}$,σ ²₀ ). Let ${\tilde{Y}_{t_1}^{t_k}=Y_{t_1},\ldots,Y_{t_k}}$ , k = 1... n be a random vector of observations of the process Y _t at ages t ₁ ,..., t _k . Denote by ${Q\left({t_k,\tilde{Y}_{t_1}^{t_k}}\right)}$ the conditional probability of death at the interval [t _k ,t _k+1) of an individual given an observed trajectory ${\tilde{Y}_{t_1}^{t_k}}$ , i.e., ${Q\left({t_k,\tilde{Y}_{t_1}^{t_k}} \right)=P\left({X_{t_k} =0\left|{\,\tilde{Y}_{t_1}^{t_k},X_{t_{k-1}} =1} \right.} \right)}.$ Assume that this probability depends only on values of ${Y_{{t}_{k}}}$as follows:

$$ Q\left({t_k,\tilde {Y}_{t_1}^{t_k}} \right)=1-e^{-\mu \left( {t_k,Y_{t_k}} \right)\left({t_{k+1}-t_k}\right)}, $$

(A2)

with μ (t _k ,${Y_{{t}_{k}}}$) given by (A1). For the likelihood function we need conditional distributions of ${Y_{{t}_{k}}}$ given the observations ${\tilde{Y}_{t_1}^{t_{k-1}}}.$ From (A1),

$$p\left({Y_{t_k} \left|{\tilde {Y}_{t_1}^{t_{k-1}}}\right.}\right)=\frac{1}{\sqrt{2\pi \left({t_k -t_{k-1}}\right)}\sigma_1}e^{-\frac{\left({Y_{t_k}-\bar {Y}_{t_{k-1}}}\right)^2}{2\left({t_k -t_{k-1}}\right)\sigma _1^2}},$$

(A3)

where

$$\bar{Y}_{t_{k-1}}=Y_{t_{k-1}}+a_{t_{k-1}}\left({f_{t_{k-1}}^1 -Y_{t_{k-1}}}\right)\left({t_k-t_{k-1}}\right),$$

(A4)

for k > 2, and

$$p\left({Y_{t_1}} \right)=\frac{1}{\sqrt {2\pi} \sigma _0}e^{−\frac{\left({Y_{t_1}-f_{t_1}^1}\right)^2}{2\sigma _0^2}}.$$

(A5)

Consider N independent observations of individuals in the above described scheme. Denote by ${\tilde {Y}_{t_1^i}^{t_{n_i}^i}}$ the observed trajectories of the process Y _t for i ^th individual, where n _i is the number of observations of the process Y _t for i ^th individual. Let δ _i = 1 if i ^th individual died in the interval (${t}_{{n_i}^i},$ ${t_{n_{i}+1}^i}$), δ _i  = 0 if he/she survived until age ${t_{n_i+1}^i}$ and δ _i  = 2 if an individual is lost to follow up at the last observation (censored at age ${t}_{{n_i}^i}$). The contribution of i ^th individual into the likelihood function is

$$ \begin{aligned} L_i \left({\tilde{Y}_{t_1^i}^{t_{n_i}^i},\tilde{X}_{t_1^i}^{t_{n_i }^i},\delta _i} \right)=P\left({\tilde{Y}_{t_1^i}^{t_{n_i}^i} ,\tilde{X}_{t_1^i}^{t_{n_i}^i},\delta _i} \right)=p\left({\tilde {Y}_{t_1^i}^{t_{n_i}^i}} \right)P\left({\tilde{X}_{t_1^i}^{t_{n_i}^i } \left|{\tilde{Y}_{t_1^i}^{t_{n_i}^i},\delta _i} \right.} \right)=\\ p\left({Y_{t_1^i}} \right)\prod\limits_{k=2}^{n_i}{p\left({Y_{t_k^i} \left|{\tilde{Y}_{t_1^i}^{t_{k-1}^i}} \right.} \right)} \prod\limits_{k=1}^{n_i -1}{\left({1-Q\left({t_k^i,\tilde{Y}_{t_1^i }^{t_k^i}} \right)} \right)\left({1-Q\left({t_{n_i}^i,\tilde {Y}_{t_1^i}^{t_{n_i}^i}} \right)} \right)} ^{I\left({\delta _i =0} \right)}Q\left({t_{n_i}^i,\tilde{Y}_{t_1^i}^{t_{n_i}^i}} \right)^{I\left({\delta _i =1} \right)}, \end{aligned} $$

(A6)

where the respective probabilities are given by (A2)–(A5). The likelihood function is a product of ${L_i \left({\tilde {Y}_{t_1^i }^{t_{n_i}^i},\tilde {X}_{t_1^i}^{t_{n_i}^i},\delta _i} \right)},$ i = 1... N.

Application to the NLTCS data on the cumulative indices of deficits (DIs)

We applied different variants of the general model to the DIs calculated from the NLTCS data for males and females. In all models, we assumed that ${a_{t_k}}$ = a _Y and ${\mu_t^{1j} =a_{\mu ^{1j}} +b_{\mu ^{1j}}\left({t-t_{\rm min}}\right)},$ j = 1, 2, t _min = 65. We calculated the models for one- and 2-year follow-up (the results are shown for the 1-year follow-up). That is, the observed value of the DI is assumed to be constant during the respective interval after the observation. Note also that this model assumes that we consider the fact of death only during the respective (1- or 2-year) time interval after the observation (i.e., if an individual dies within the specified time interval then he/she is considered to be dead, otherwise the individual is considered to be censored). The following models denoted as QH0-QH10, QH100-QH110 use different specifications of functions f ¹ _t and f _t .

Model QH0

This is the model with the Gompertz mortality ${\mu _t^0=a_{\mu ^0} e^{b_{\mu ^0}\left({t-t_{\rm min}}\right)}}$ without the quadratic hazard term and observations of the DI. Parameters to be estimated are: ${a_{\mu ^0}}$ and ${b_{\mu ^0}}.$

Model QH1

${f_t^1}$ and f _t are linear functions of age, ${f_t^1 =a_{f^1}+b_{f^1}\left({t-t_{\rm min}}\right)}$, ${f_t =a_f +b_f\left({t-t_{\rm min}}\right)}$. In all models QH1-10, we use the Gompertz mortality ${\mu _t^0 =a_{\mu ^0} e^{b_{\mu^0} \left({t-t_{\rm min}}\right)}}$. In models QH1-10 and QH101-110, we estimated parameters ${a_{\mu ^0}}$, ${b_{\mu^0}}$, ${a_{\mu^{11}}}$, ${b_{\mu ^{11}}}$, ${a_{\mu ^{12}}}$, ${b_{\mu^{12}}}$, a _Y , σ₀, and σ ₁. The QH1 model-specific parameters are: ${a_{f^1}}$, ${b_{f^1}}$, a _f , and b _f .

Model QH2

The same as QH1, but with equal ${f_t^1}$ and f _t : ${f_t^1 =f_t =a_f +b_f \left({t-t_{\rm min}}\right)}.$ The QH2 model-specific parameters are: a _f and b _f .

Model QH3

${f_t^1}$ is a quadratic and f _t is a linear function of age, ${f_t^1 =a_{f^1} +b_{f^1} \left({t-t_{\rm min}} \right)+c_{f^1} \left({t-t_{\rm min}} \right)^2},$ ${f_t =a_f +b_f \left({t-t_{\rm min}}\right)}.$ The QH3 model-specific parameters are: ${a_{f^1}},$ ${b_{f^1}},$ ${c_{f^1}},$ a _f , and b _f .

Model QH4

The same as QH1, but with fixed f _t : ${f_t =a_f^\ast +b_f^\ast \left({t-t_{\rm min}} \right)},$ where the parameters a ^* _f and b ^* _f were empirically estimated from the data on mortality. The QH4 model-specific parameters are: ${a_{f^1}}$ and ${b_{f^1}}.$

Model QH5

f ¹ _t is a linear function of age,${f_t^1 =a_{f^1} +b_{f^1}\left({t-t_{\rm min}}\right)},$ and ${f_t =\hat{Y}_t},$ where ${\hat {Y}_t}$ is the trajectory of mean values of Y at ages t estimated from the data and smoothed using the moving average method with window 7. The QH5 model-specific parameters are: ${a_{f^1}}$ and ${b_{f^1}}.$

Model QH6

The same as QH3, but with fixed zero f _t . The QH6 model-specific parameters are: ${a_{f^1}},$ ${b_{f^1}}$ and ${c_{f^1}}.$

Model QH7

f _t is a linear function of age,${f_t =a_f +b_f \left({t-t_{\rm min}}\right)},$ and ${f_t^1 =\hat{Y}_t},$ where ${\hat {Y}_t}$ is the trajectory of mean values of Y at ages t estimated from the data and smoothed using the moving average method with window 7. The QH7 model-specific parameters are: a _f and b _f .

Model QH8

${f_t^1}$ and f _t are fixed:${f_t^1 =f_t =a_f^\ast +b_f^\ast \left({t-t_{\rm min}} \right)},$ where the parameters a ^* _f and b ^* _f were estimated from the data on mortality.

Model QH9

f ¹ _t and f _t are fixed:${f_t =a_f^\ast +b_f^\ast \left({t-t_{\rm min}} \right)},$ where the parameters a ^* _f and b ^* _f were estimated from the data on mortality, and ${f_t^1 =\hat {Y}_t},$ where ${\hat {Y}_t}$ is the trajectory of mean values of Y at ages t estimated from the data and smoothed using the moving average method with window 7.

Model QH10

f ¹ _t and f _t are fixed: ${f_t^1 =f_t =\hat {Y}_t},$ where ${\hat {Y}_t}$ is the trajectory of mean values of Y at ages t estimated from the data and smoothed using the moving average method with window 7.

Model QH100–110

Similar to QH0-QH10 but use the logistic baseline mortality rate

$$ \mu _t^0 ={a_{\mu ^0}e^{b_{\mu ^0}\left({t-t_{\rm min}}\right)}} /{\left({1+\sigma _2^2 \frac{a_{\mu ^0}}{b_{\mu ^0}}\left( {e^{b_{\mu^0}\left({t-t_{\rm min}}\right)}-1}\right)} \right)} $$

(A7)

in place of the Gompertz mortality rate ${\mu _t^0 = a_{\mu ^0} e^{b_{\mu^0}\left({t-t_{\rm min}}\right)}}.$ Here σ ₂ is an additional parameter to be estimated.

Extended Cox’s model

Here we briefly describe the extended Cox’s model analyzed in Yashin et al. (2006) and cited in this paper. Generally, this is the model given by (A1)–(A6) with the Cox-like proportional hazards instead of (1):

$$ \mu\left({t,Y_t}\right) = \mu_0 (t)e^{\left({\beta_2+\beta _4 t}\right)\left({Y_t -f_t}\right)I\left({Y_t \ge f_t} \right)+\left({\beta _3+\beta _5 t} \right)\left({f_t -Y_t} \right)I\left({Y_t < f_t}\right)}. $$

(A8)

Here the notations for Y _t and f _t are the same as above, ${I\left( \cdot \right)}$ is an indicator function, which equals 1, if the inequality in the parentheses is true, and 0 otherwise, β₂, β₃, β₄, and β₅ are regression coefficients. Possible asymmetry of the risk function is captured by the different regression coefficients β₂ andβ₃, which measure the contribution of the covariates’ deviations to the one or the other side from the optimal trajectory f _t . The linear dependence of regression coefficients on age allows for capturing age-related changes in the shape of the risk function.

In Yashin et al. (2006), we analyzed models Cox0–Cox10 and Cox100–Cox110, which are the respective analogues of QH0–QH10 and QH100–QH110, i.e., the models with similar specifications of ${f_t^1},$ f _t , a _t , and mortality rates (A8) with the Gompertz baseline hazards ${\mu_0 \left(t \right)=\mu_0 e^{\beta _1 t}}$ for Cox0–Cox10 and the logistic baseline hazards ${\mu_0(t)={\mu_0 e^{\beta_1 t}}\mathord{\left/{\vphantom {{\mu_0 e^{\beta _1 t}} {\left({1+\sigma _2^2 \mu_0 \left({e^{\beta _1 t}-e^{\beta_1 t_{\rm min}}}\right)/\beta _1} \right)}}}\right. \kern-\nulldelimiterspace} {\left({1+\sigma _2^2 \mu_0 \left( {e^{\beta _1 t}-e^{\beta_1 t_{\rm min}}}\right)/\beta_1} \right)}}$ for Cox100–Cox110.