State Space Models for Survival Analysis

1
State Space Models for Survival Analysis

• Weiming Ke
• The University of Memphis
• (Joint work with Dr. Wai-yuan Tan)
• June 29, 2004

2
Outline

• Introduction
• State Space Model
• Estimation of Parameters
• ? Multi-level Gibbs Sampling Procedure
• ? Weighted Bootstrap Procedure
• Estimation of the Survival Probabilities
• An Illustrative Example
• Computer Simulation
• Summary

3
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.

4
Introduction (cont.)

• In this article we propose a state space modeling
  approach by combining stochastic models with
  statistical models to describe the process of a
  disease.
• Then we will apply the Gibbs sampling method and
  the weighted bootstrap method to estimate the
  unknown parameters and the state variables of the
  model.
• By using these estimates, we can validate the
  model and then estimate the survival
  probabilities.

5
State Space Model

• the state space model of a system consists of
  two sub-models
• 1. The stochastic system model which is the
• stochastic model of the system.
• 2. The observation model which is a
  statistical
• model relating some available data to the
  system.
• It extracts biological information from the
  system via its stochastic system model and
  integrates this information with those from the
  data through its observation model.

6
State Space Model (cont.)

• The state space model was originally proposed by
  Kalman in the early 60s for engineering control
  and communication (Kalman, 1960). Since then it
  has been successfully used as a powerful tool in
  aerospace research.
• It was first proposed by Tan and his associates
  for AIDS and cancer research (Tan et al., 1998,
  1999, 2000, 2001, 2002).
• Apparently state space models can be extended to
  other diseases as well, such as infectious
  diseases.

7
Advantages of the State Space Models

• The state space model of a system is advantageous
  over the stochastic model of the system alone or
  the statistical model of the system alone since
  it combines information and advantages from both
  of these models.
• The followings are some specific advantages of
  the state space models

8
Advantages of the State Space Models (cont.)

• State space model provides an optimal procedure
  to update the model by new data which may become
  available in the future. This is the smoothing
  step of the state space models.
• The state space model provides an optimal
  procedure via the Gibbs sampling method and the
  weighted bootstrap method to estimate
  simultaneously the unknown parameters and the
  state variables of interest.

9
The stochastic system model

• In many cases, we can derive stochastic equations
  for the state variables of a system by using
  basic biological mechanism of the disease.
• We will illustrate the state space modeling
  approach by using a birthdeathillnesscure
  process for a disease such as tuberculosis.

10
The stochastic system model (cont.)

• Consider a population of individuals who are at
  risk for a disease, such as tuberculosis.
• In this population, there are two types of
  people the normal healthy people who do not have
  the disease and the sick people who have
  contracted the disease.
• Let N1(t) and N2(t) denote the numbers of the
  normal people and the sick people respectively in
  the population at time t.

11
The stochastic system model (cont.)

• To derive stochastic differential equations for
  the state variables N1(t) and N2(t) , the
  following transition variables are used
• F1(t) Number of normal healthy people who
  become sick
• during t, t ?t),
• F2(t) Number of sick people who are cured by
  the drug
• during t, t ?t),
• B1(t),B2 (t) Numbers of births of N1, N2
  people during t, t ?t),
• D1(t),D2(t) Numbers of deaths of N1, N2 people
  during t, t ?t)
• R1(t),R2 (t) Numbers of immigrants of N1 and
  N2 people during
• t, t ?t),

12
The stochastic system model (cont.)

• Let a1 denote the disease rate, and a2 the cure
  rate. It means that a1 and a2 are the transition
  rates of N1?N2 and N2?N1 respectively.
• Let b1, d1, ?1 denote the birth rate, death
  rate and immigration rate of the N1 people.
• Let b2, d2, ?2 denote the birth rate, death
  rate and immigration rate of the N2 people.

13
The stochastic system model (cont.)

• Assume that during the time interval t, t ?t),
  the birthdeathillnesscure processes follow the
  multinomial distributions with parameters b1?t,
  d1?t, a1?t, and b2?t, d2?t, a2?t.
• Assume that the immigration follow the Poisson
  distributions with means ?1(t)?t and ?2 (t)?t .

14
The stochastic system model (cont.)

• Then, given N1(t) and N2(t) , the conditional
  probability distributions of
• B1(t), D1(t), F1(t) and B2(t), D2(t),
  F2(t) are given by
• B1(t), D1(t), F1(t) N1(t)
• Multinomial N1(t) b1?t, d1?t, a1?t,
• B2(t), D2(t), F2(t) N2(t)
• Multinomial N2(t) b2?t, d2?t, a2?t.

15
The stochastic system model (cont.)

• The conditional probability distributions of
• R1(t) and R2(t) are given respectively by
• R1(t) N1(t) Poisson with mean N1(t)?1(t)?t
  ,
• R2(t) N2(t) Poisson with mean N2(t)?2(t)?t
  .

16
The stochastic system model (cont.)

• Then, we have the following stochastic
  differential equations for N1(t) and N2(t)
• N1(t ?t) N1(t)R1(t)F2(t)B1(t)-F1(t)-D1(t)
  
• N2(t ?t) N2(t)R2(t)F1(t)B2(t)-F2(t)-D2(t)

17
The observation model

• If some observed data are available from this
  system, then we can derive some statistical
  models to relate the data to the system.
• For the observation model, assume that observed
  numbers of N1 and N2 people at times tk , k 1,
  . . . , n are available.
• Let Y1(k) and Y2(k) be the observed numbers of N1
  and N2 people at time tk .

18
The observation model (cont.)

• Assume that and
  are normally
  distributed with mean 0 and variances s12 and
  s22, independently for k 1, . . . , n. The
  observation models are represented by the
  statistical models that are given by
• Y1(k) N1(tk) N1(tk)1/2e1 ,
  
• Y2(k) N2(tk) N2(tk)1/2e2 ,
• for k 1, . . . , n
• where e1 and e2 are independently
  distributed as normal with mean 0 and variances
  s12 and s22 .

19
Estimation of Parameters (cont.)

• From the state space models, we can estimate
  simultaneously the unknown parameters and the
  state variables through the multi-level Gibbs
  sampling method and the weighted bootstrap
  method.
• For the above example, birth rates b1, b2,
  death rates d1, d2, and immigration rates
  ?1, ?2 of the normal and sick people, and the
  transition rates a1, a2 of N1?N2 and N2?N1 are
  the parameters to be estimated.

20
Estimation of Parameters (cont.)

• To illustrate, let X be the collection of all the
  state variables N1(t), N2(t), T the collection
  of all unknown parameters b1, b2, d1, d2, ?1,
  ?2, a1, a2, and Y the collection of vectors of
  observed data sets Y1(k), Y2(k), k 1, . . . ,
  n.
• Let P(T) be the prior distribution of the
  parameters T, P(X T) the conditional
  probability density of X given the parameters T,
  and P(Y X, T) the conditional probability
  density of Y given X and T.

21
Estimation of Parameters (cont.)

• The joint probability density function of (X, Y,
  T) is
• P(X, Y, T) P(T) P(X T) P(Y X, T)
  
• From above, we can derive the conditional
  probability density function P(X T, Y) of X
  given (T, Y) and the conditional probability
  density function P(T X, Y) of T given (X,Y),
  respectively, as
• P(X T, Y) ? P(X T) P(Y X, T),
• and P(T X, Y) ? P(T) P(X T) P(Y X, T).

22
Gibbs sampling procedure

• The multi-level Gibbs sampling method is an
  extension of the Gibbs sampling method to the
  multivariate cases.
• The method was first proposed by Sheppard
  (1994).
• This method was a useful tool for estimating the
  unknown parameters and state variables through a
  sequential procedure.

23
Gibbs sampling procedure (cont.)

• The multi-level Gibbs sampling procedures for
  estimating the unknown parameters T and the state
  variables X are given by the following loop
• (1) Given initial values of T and observed data
  Y, generate X from P(X Y, T) through the
  weighted Bootstrap method due to Smith and
  Gelfand (1992) .

24
Gibbs sampling procedure (cont.)

• (2) Generate T from the conditional distribution
• PT X, Y where X is the value obtained
• in step (1).
• (3) Using T obtained from (2) as initial values,
• go back to (1) to generate X, and repeat
  the
• (1), (2) loop until convergence.

25
Gibbs sampling procedure (cont.)

• At convergence, we then generate a random sample
  of X from the conditional distribution PX Y,
  and a random sample of T from the posterior
  distribution PT Y.
• Repeating these procedures we then generate a
  random sample of X and a random sample of T. we
  may then use the sample means as the estimates of
  X and T, and use the sample variances as the
  variances of these estimates.

26
Weighted bootstrap procedure

• Because in practice it is often very difficult to
  derive P(X Y, T), whereas it is easy to
  generate X from P(X T).
• We developed an indirect method by using the
  weighted bootstrap method due to Smith and
  Gelfand (1992) to generate X from P(X Y, T)
  through the generation of X from P(X T).
• The proof of this algorithm has been given in Tan
  (2002).

27
Weighted bootstrap procedure (cont.)

• The algorithm of the weighted bootstrap method is
  given by the following steps
• (a) Given T and X(i), generate a large
  random sample of size m on X(i1) by using
  PX(i1)X(i) from the stochastic system model
  denoted it by
• X(1)(i1), , X(m)(i1)
• (b) Computing ?k PY(i1) X(k)(i1), T by
  using the statistical model.
• Computing qk ?k ? (?1 ?2 ?m )
• for k 1, , m.

28
Weighted bootstrap procedure (cont.)

• (c) Construct a population ? with element E1,
  , Em and with P(Ek) qk .
• Draw an element randomly from ?. If the
  outcome is Ek, then X(k)(i1) is the element of
  X(i1) generated from the conditional
  distribution of X given the observed data Y and
  the parameter T.
• (d) Start with i 1 and repeat (a) ? (c) until
  i tM to generate a random sample of X from P(X
  Y, T).

29
Estimating Parameters

• Now we explain how to generate parameters
• T T1, T2, , Tk from P(T X, Y) by using
  the multi-level Gibbs sampling method.
• With no loss of generality, we illustrate the
  method with k 3 and write T T1, T2, T3.
• The procedure for estimating the parameters goes
  through the following loop

30
Estimating Parameters (cont.)

• Given initial values of T and X, generate
• T1 from P(T1 Y, X, T2, T3), and
• denote it by T1(1).
• (2) Generate T2 from P(T2 Y, X, T1(1), T3) and
  denote it by T2(1), where T1(1) is the value
  obtained in step (1).

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Estimating Parameters (cont.)

• Generate T3 from PT3Y, X, T1(1), T2(1) and
  denote it by T3(1), where T1(1) and T2(1) are the
  values obtained in (1) and (2).
• Using T1(1), T2(1), T3(1) obtained from
• (1)?(3) as initial values, generate X from
• P(X Y, T) and denote it by X(1).

32
Estimating Parameters (cont.)

• (5) Using T T1(1), T2(1), T3(1) and X X(1)
• obtained from (1)?(4) as initial values, go
  
• back to repeat (1)?(4) to generate T1(2),
• T2(2), T3(2) and X(2), and repeat the loop
• until convergence.

33
Estimating Parameters (cont.)

• At convergence, we can generate a random sample
  of X from the conditional distribution P(X Y)
  of X given Y, and a random sample of T from the
  posterior distribution P(T Y) of T given Y.
• Repeating these procedures we then generate a
  random sample of X and a random sample of T.

34
Estimating Parameters (cont.)

• We may then use the sample means to derive the
  estimates of the parameters T and the state
  variables X, and use the sample variances as the
  variances of these estimates.
• The convergence of these procedures is proved by
  using the basic theory of homogeneous Markov
  chains see (Tan, 2002, Chapter 3).

35
Estimation of the survival probabilities

• By using the estimates of the parameters, we can
  estimate the survival probabilities.
• We will illustrate the procedure by using the
  same example.
• Let d1 and d2 denote the death rates of the
  normal and sick people, respectively a1 and a2
  are the transition rates of N1?N2 and N2?N1,
  respectively.

36
Estimation of the survival probabilities (cont.)

• Let S1(t) and S2(t) denote the survival
  probabilities that normal and sick people will
  survive at time t when the population is at risk
  for the disease.
• Then we have the following system of equations

37
Estimation of the survival probabilities (cont.)

• The solution of the above equations are given by
• where w a1 a2 d1 d22 4a2 d2
  d1 1/2.
• d1 a1 a2 d1 d2 w.
  
• d2 a1 a2 d1 d2 ? w.
  

38
An illustrative example

• 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
  USA from 1980 to 1992 reported by CDC together
  with the total USA population sizes over these
  years (CDC Report, 1993).
• Given in Figure 1 are the numbers of TB people.
• In this data set, it is clear that the number of
  TB cases in USA is declining to the lowest level
  in 1985 and then increases due presumably to the
  effects of HIV (CDC Report, 1993).

39
Table 1. Observed numbers of total people, TB
people and normal people
40
Figure 1. Observed numbers of TB people
41
An illustrative example (cont.)

• To fit this data, we thus assume that the
  infection rate a1 a1(1) before January 1985 and
  assume that a1 a1(2) after January 1985.
  
• Assume that other parameters are not affected by
  HIV and other factors. Because the TBs are rare
  in children, we may ignore birth so that the
  unknown parameters are T ?1, ?2 d1, d2 a1(1),
  a1(2), a2.
• Let t0 0 denote January 1980 so that
• N1(0) 226517805 and N2(0) 28000.

42
An illustrative example (cont.)

• Using the data given in Table 1, we apply the
  Gibbs sampling method and the weighted bootstrap
  method to estimate the unknown parameters and
  the state variables.
• Because we do not have previous knowledge about
  the parameters, we assumed a non-informative
  uniform prior for the parameters.
• The estimates of the parameters are given in
  Table 2.
• The estimates of the numbers of TB people are
  plotted in Figure 2, together with the observed
  numbers.
• The estimates of the survival probabilities
  of the sick people are plotted in Figure 3.

43
Table 2 Estimates of parameters and standard
errors
44
Figure 2. Estimated and Observed Numbers of TB
people(--- Estimated --- Observed)
45
Figure 3. Estimates of the survival probabilities
of TB people
46
An illustrative example (cont.)

• From Figure 2, apparently the estimated numbers
  of the sick people are close to its observed
  numbers.
• From results in Table 2, it turned out that the
  estimate of a1(2) is slightly greater than a1(1),
  indicating that since 1985 HIV and/or other
  causes have increased the infection rate of TB
  slightly.

47
Computer Simulation

• To further examine this approach, we have assumed
  some parameter values and generated some computer
  Monte Carlo data.
• The generated numbers are given in Table 3
• The values of the parameters for generating these
  data are ?1 0.05, ?2 0.03,
• d1 0.04, d2 0.1, a1 0.2, a2 0.4,
  
• N1(0) 1000, N2(0) 10.

48
Table 3. Generated Numbers of Normal People and
Sick People
49
Computer Simulation (cont.)

• Using the data in Table 3 and assuming a
  non-informative uniform prior for the parameters,
  we have applied the Gibbs sampling procedure and
  the weighted bootstrap procedure to estimate the
  parameters and the state variables.
• Given in Table 4 are the estimates of the
  parameters and their true values.
• Plotted in Figure 4 are the estimates of the
  numbers of the sick people together with the
  generated numbers.
• Plotted in Figure 5 are the estimates of the
  survival probabilities.

50
Table 4. Estimates of Parameters and Standard
Errors
51
Figure 4. Estimated and Observed Numbers of Sick
people (--- Estimated --- Observed)
52
Figure 5. Estimates of the Survival Probabilities
of Sick People
53
Computer Simulation (cont.)

• From the results in Table 4, apparently, the
  estimates are very close to its true values.
• From Figure 4, it is also apparent that the
  estimates of the state variables are very close
  to the generated numbers.
• These results indicate that the methods proposed
  in this article are very effective.

54
Summary

• In this article, we have developed a state space
  model for a birthdeathillnesscure process.
• We have developed a procedure (the Gibbs
  sampling method together with the weighted
  bootstrap method) to estimate the unknown
  parameters and the state variables, and hence the
  survival probabilities.
• The numerical example and the computer simulation
  indicate that the methods are quite useful and
  promising.

55
Discussion

• In recent years, Tan and his associates have
  developed some state space models for cancers and
  AIDS (Tan and Chen, 1998 Tan and Xiang, 1998
  Tan and Xiang, 1999 Tan and Ye, 2000 Wu and
  Tan, 2000 Tan et al., 2001 Tan et al., 2002).
• The present article extends this modeling
  approach to other human diseases such as
  tuberculosis. This type of modeling approach is
  definitely useful for other diseases such as the
  heart disease and to risk assessment of
  environmental agents as well. In this respect,
  more research works are needed.

56

• Thank You !