State Space Models for Survival Analysis

doi:10.1016/S0169-7161(03)23029-7

Handbook of Statistics

Volume 23, 2003, Pages 519-535

https://doi.org/10.1016/S0169-7161(03)23029-7 Get rights and content

Publisher Summary

To ease the estimation problem and to estimate the survival probabilities, this chapter proposes a state-space modeling approach by combining stochastic models with statistical models. The Gibbs sampling method and the Markov Chain and Monte Carlo approach (MCMC) can be readily applied to estimate the unknown parameters and the state variables. By using these estimates, the model can be validated and the survival probabilities can be estimated. The chapter illustrates the model and the method by using a birth–death–immigration–illness–cure process that involves stochastic birth–death processes with immigration and the illness and cure processes for a disease such as tuberculosis. It extends this modeling approach to other human diseases such as tuberculosis. This type of modeling approach is definitely useful for other diseases, such as heart disease, and the risk assessment of environmental agents.

Introduction

Many diseases such as AIDS, cancer and infectious diseases are often very complicated biologically. Most of these diseases are complex stochastic processes where it is often very difficult to estimate the unknown parameters, especially in cases where not many data are available. In these cases, it would be very difficult to estimate the survival probabilities. To ease the estimation problem and to estimate the survival probabilities, in this article we propose a state space modeling approach by combining stochastic models with statistical models. Then one can readily apply the Gibbs sampling method and the Markov Chain and Monte Carlo approach (MCMC) to estimate the unknown parameters and the state variables. By using these estimates, one can validate the model and estimate the survival probabilities. We will illustrate the model and the method by using a birth–death–immigration–illness–cure process which involves stochastic birth–death processes with immigration and the illness and cure processes for a disease such as tuberculosis.

Section snippets

The state space models and the generalized Bayesian approach

To illustrate, consider a disease such as tuberculosis which is curable by drugs. Let $X ∼ (t)$ be the vector of stochastic processes for key responses of the disease. Then, $X ∼ (t)$ is the stochastic model for this disease and in many cases, one can derive stochastic equations for the state variables of the system by using basic biological mechanism of the disease; for some illustrations in cancer and AIDS, see Tan, 2000, Tan, 2002, Tan and Chen, 1998, Tan et al., 2001. If some observed data are

Stochastic modeling of the birth–death–immigration–illness–cure processes

Consider a population of individuals who are at risk for a disease and suppose that some drugs are available to treat the disease. One example is the tuberculosis. In this population, then there are two types of people: Normal healthy people (denote by N₁) who do not have the disease and sick people (denote by N₂) who have contracted the disease. When the population is at risk for the disease, normal people may contract the disease to become sick via contacts with sick people or disease agents.

A state space model for the birth–death–immigration–illness–cure processes

The state space model of a system is a stochastic model consisting of two sub-models: The stochastic system model which is the stochastic model of the system and the observation model which is the statistical model relating available data from the system to the model. For the birth–death–immigration–illness–cure process, the state variables are ${N_{i} (t), i=1,2}$ and the stochastic system model is represented by the stochastic differential equations given by (5), (6). Assuming that the number of

The multi-level Gibbs sampling procedures for the birth–death–immigration–illness–cure processes

To implement the multi-level Gibbs sampling method, denote by $U ∼ (t)={F_{i} (t),R_{i} (t),B_{i} (t), i=1,2}$ and put $U ={U ∼ (t), t=0,1,…,t_{M} −1}$ . Then, given $X ∼ (0)$ and Θ, $P X, U ∣ X ∼ (0) = ∏ i=1 t_{M} P X ∼ (i)∣ X ∼ (i−1), U ∼ (i−1) P U ∼ (i−1)∣ X ∼ (i−1) .$

By the above distribution results with $i≠j; i,j=1,2$ : $P U ∼ (t)∣ X ∼ (t) =C_{1} (t) ∏ i=1 2 g_{i} R_{i} (t);t α_{i} (t)^{F_{i}(t)} b_{i} (t)^{B_{i}(t)} 1−α_{i} (t)−b_{i} (t)^{N_{i}(t)−F_{i}(t)−B_{i}(t)},$ where $C_{1} (t)=∏_{i=1}^{2} N_{i} (t) F_{i} (t) N_{i} (t)−F_{i} (t) B_{i} (t),$ $P X ∼ (t+1)∣ X ∼ (t), U ∼ (t) =∏_{i=1}^{2} N_{i} (t)−F_{i} (t)−B_{i} (t) η_{i} (t) d_{i} (t) 1−α_{i} (t)−b_{i} (t)^{η_{i}(t)} 1− d_{i} (t) 1−α_{i} (t)−b_{i} (t)^{ζ_{i}(t)},$ where $η_{i} (t)=N_{i}$

The survival probabilities of normal and sick people

Let S_i(t) denote the survival probability that an N_i person at time 0 will survive at least t months when the population is at risk for the disease. Then, under the conditions given in Section 3, one has, to order of o(Δt), $S_{i} (t+ Δ t)=α_{i} (t) Δ tS_{j} (t)+ 1−d_{i} (t) Δ t−α_{i} (t) Δ t S_{i} (t), for all i≠j, i,j=1,2.$

It follows that the S_i(t)'s satisfy the following system of equations: $d d t S_{1} (t)=α_{1} (t)S_{2} (t)− α_{1} (t)+d_{1} (t) S_{1} (t),$ $d d t S_{2} (t)=α_{2} (t)S_{1} (t)− α_{2} (t)+d_{2} (t) S_{2} (t).$

The initial condition is ${S_{i} (0)=1, i=1,2}$ .

Let $S ∼ (t)={S_{1} (t),S_{2} (t)}′$ .

Some illustrative examples

To illustrate the above methods, consider the disease – tuberculosis (TB) which is curable by drugs. Given in Table 1 are the numbers of TB cases in US from 1980 to 1992 reported by CDC together with the total US population sizes over these years (CDC Report, 1993). In this data set, it is clear that the curve of TB cases in US is declining to the lowest level in 1985 and then increases due presumably to the effects of HIV (CDC Report, 1993). To fit this data, we thus assume α₁(t)=α₁(1) before

Conclusions

In this article, we have developed a state space model for the birth–death–immigration–illness–cure process. We have developed a generalized Bayesian method to estimate the unknown parameters and the state variables, and hence the survival probabilities. The numerical examples indicate that the methods are useful and promising. Of course, more studies are needed to further confirm the usefulness of the method and to check the efficiency of the method.

In the past 5 years, we have developed some

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State Space Models for Survival Analysis

Publisher Summary

Introduction

Section snippets

The state space models and the generalized Bayesian approach

Stochastic modeling of the birth–death–immigration–illness–cure processes

A state space model for the birth–death–immigration–illness–cure processes

The multi-level Gibbs sampling procedures for the birth–death–immigration–illness–cure processes

The survival probabilities of normal and sick people

Some illustrative examples

Conclusions

Math. Comput. Modeling

Math. Comput. Modeling

Math. Biosci.

J. Statist. Inference Planning

Math. Biosci.

Math. Biosci.

State Space Modeling of Time Series

AIDS Epidemiology: A Quantitative Approach

Estimation, Control and Discrete Kalman Filter

The Jackknife, the Bootstrap and Other Resampling Plans

Applied Optimal Estimation

Forecasting, Structural Time Series Models and the Kalman Filter

A new approach to linear filter and prediction problems

J. Basic Engrg.