Evaluating the Effects of Observed and Unobserved Diffusion Processes in Survival Analysis of Longitudinal Data

In analyses of human survival often explicit consideration of the dynamics of physiological processes governing survival is not made. Failure to consider the influence of such processes can lead to incorrect inferences about the operation of such processes and the inability to forecast future changes in survival. An explicit model of such processes has been presented by Woodbury and Manton (1977). Myers (1981) developed another approach based on the appropriate extension of the Cameron-Martin method. We show that estimation can be conducted using a conditional Gaussian strategy and that the conditional Gaussian approach offers several substantive and computational advantages

Mortality and Aging in a Heterogeneous Population: A Stochastic Process Model with Observed and Unobserved Variables

A number of multivariate stochastic process models have been developed to represent human physiological aging and mortality. In this paper, we extend those efforts by considering the effects of unobserved state variables on the age trajectory of physiological parameters. This is accomplished by deriving the Kolmogorov-Fokker-Planck equations for the distribution of the state variables conditionally on the process of the observed state variables. Proofs are given that this form of the process will preserve the Gaussian properties of the distribution. Strategies for estimating the parameters of the distribution of the unobserved variable are suggested based on an extension of the theory of Kalman filters to include systematic mortality selection. Implications of individual differences on the trajectories of the unobserved process for observed aging changes are discussed as well as the consequences of such modeling for dealing with other types of processes in heterogeneous populations

Debilitation's Aftermath: Stochastic Process Models of Mortality

The paper is devoted to the analysis of stochastic process models of mortality which can explain both selection and debilitation processes in the evolution of cohort mortality. The relative importance of each process is analyzed. The examples of various regimes of mortality evolution are demonstrated

The Propogation of Uncertainty in Human Mortality Processes

Human mortality and aging have frequently been modeled as stochastic diffusion processes. Estimates of the parameters of these processes have been made from various longitudinal studies. This paper shows how the stochasticity intrinsic to those processes will propagate through time and generate uncertainty about the future physiological state of the population. Variance expressions are derived for the future values of the physiological variables; and for the conditional survival functions and conditional life expectancies which reflect the uncertainty in the future values of the physiological variables. The results show that a major component of uncertainty is due to mortality. This suggests that the limits to forecasting may be different in physiological systems subject to systematic mortality than in physical systems such as weather

Mortality and aging in a heterogeneous population: A stochastic process model with observed and unobserved variables

Various multivariate stochastic process models have been developed to represent human physiological aging and mortality. These efforts are extended by considering the effects of observed and unobserved state variables on the age trajectory of physiological parameters. This is done by deriving the Kolmogorov-Fokker-Planck equations describing the distribution of the unobserved state variables conditional on the history of the observed state variables. Given some assumptions, it is proved that the distribution is Gaussian. Strategies for estimating the parameters of the distribution are suggested based on an extension of the theory of Kalman filters to include systematic mortality selection. Various empirical applications of the model to studies of human aging and mortality as well as to other types of “failure” processes in heterogeneous populations are discussed

The propagation of uncertainty in human mortality processes operating in stochastic environments

This paper presents a model describing how the uncertainty due to influential exogenous processes combines with stochasticity intrinsic to physiological aging processes and propagates through time to generate uncertainty about the future physiological state of the population. Variance expressions are derived for (a) the future values of the physiological variables under the assumption that external factors evolve under a linear stochastic diffusion process, and (b) the cohort survival functions and cohort life expectancies which reflect the uncertainty in the future values of the physiological variables. The model implies that a major component of uncertainty in forecasts of the physiological characteristics of a closed cohort is due to differential rates of survival associated with different realizations of the external process. This suggests that the limits to forecasting may be different in physiological systems subject to systematic mortality than in physical systems such as weather where the concepts of closed cohorts and of mortality selection have no simple analog

Dependent competing risks: a stochastic process model

Analyses of human mortality data classified according to cause of death frequently are based on competing risk theory. In particular, the times to death for different causes often are assumed to be independent. In this paper, a competing risk model with a weaker assumption of conditional independence of the times to death, given an assumed stochastic covariate process, is developed and applied to cause specific mortality data from the Framingham Heart Study. The results generated under this conditional independence model are compared with analogous results under the standard marginal independence model. Under the assumption that this conditional independence model is valid, the comparison suggests that the standard model overestimates by 4% the effect on life expectancy at age 30 due to the hypothetical elimination of cancer and by 7% the effect for cardiovascular/cerebrovascular disease. By age 80 the overestimates were 11% for cancer and 16% for heart disease. These results suggest the importance of avoiding the marginal independence assumption when appropriate data are available - especially when focusing on mortality at advanced ages

Debilitation's aftermath: Stochastic process models of mortality

A stochastic differential equation model is developed to clarify the interaction of debilitation, recuperation, selection and aging. The model yields various insights about lingering mortality consequences of disasters such as wars, famines and epidemics that may weaken the survivors. A key result is that debilitation and selection are interdependent: debilitation that increases population heterogeneity will result in subsequent selection; selection, by altering the distribution of population heterogeneity, will influence the impact of debilitating events

Evaluating the effects of observed and unobserved diffusion processes in survival analysis of longitudinal data

In biostatistical, epidemiological and demographic studies of human survival it is often necessary to consider the dynamics of physiological processes and their influences on observed mortality rates. The parameters of a stochastic covariate process can be estimated using a conditional Gaussian strategy based on the mortality model presented in M.A. Woodbury and K.G. Manton, A random walk model of human mortality and aging. Theor. Popul. Biol. 11, 37–48 (1977) and A.I. Yashin, K.G. Manton, and J.W. Vaupel, Mortality and aging in a heterogeneous population: A stochastic process model with observed and unobserved variables. Theor. Popul. Biol., in press. (1985). The utility of this approach for modeling survival in a longitudinally followed population is discussed—especially in the context of conducing coordinated analyses of multiple similarly constituted databases. Furthermore, the conditional Gaussian approach offers several substantive and computational advantages over the Cameron- Martin approach R.H. Cameron and W.T. Martin, The Wiener measure of Hilbert neighborhoods in the space of real continuous functions