State Space Models for Survival Analysis
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 be the vector of stochastic processes for key responses of the disease. Then, 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 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 and put . Then, given and Θ,
By the above distribution results with : where where
The survival probabilities of normal and sick people
Let Si(t) denote the survival probability that an Ni 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),
It follows that the Si(t)'s satisfy the following system of equations:
The initial condition is .
Let .
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
References (28)
- et al.
Stochastic modeling of carcinogenesis: Some new insight
Math. Comput. Modeling
(1998) - et al.
Stochastic modeling of carcinogenesis by state space models: A new approach
Math. Comput. Modeling
(2001) - et al.
State space models of the HIV epidemic in homosexual populations and some applications
Math. Biosci.
(1998) - et al.
Modeling the HIV epidemic with variable infection in homosexual populations by state space models
J. Statist. Inference Planning
(1999) - et al.
A state space model of HIV pathogenesis under treatment by anti-viral drugs in HIV-infected individuals
Math. Biosci.
(1999) - et al.
Estimation of HIV infection and HIV incubation via state space models
Math. Biosci.
(2000) State Space Modeling of Time Series
(1990)- et al.
AIDS Epidemiology: A Quantitative Approach
(1994) Estimation, Control and Discrete Kalman Filter
(1989)- CDC Report (1993). Tuberculosis morbidity – United States, 1992. MMWR Morb. Mortal. WKLY Rep. 1993; 42, pp. 696–697,...
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.
Cited by (1)
A stochastic and state space model for tumour growth and applications
2009, Computational and Mathematical Methods in Medicine