Up to this point, the analysis of survival data has been considered in a frequentist framework with the development of likelihood formulations under differing scenarios. In this section, some discussion is provided regarding the fitting of models in a Bayesian framework. Initially, the special case of the parametric exponential model with no covariates and a gamma prior distribution is considered. Though somewhat limited, this special case is used to re-enforce the basic concepts of Bayesian methodology. Following this, an illustration is given regarding the difficulties encountered when the model is extended to include covariates. Finally, an exploration of sampling methods is given, in particular the Gibbs sampler, for providing solutions to more complex models.
2.4.1 Exponential model with gamma priors - no covariates
Considering the parametric Exponential model with no covariate and dataDi= (ti,‹i), the likelihood is defined as
L(⁄|D) = n Ÿ i=1 ⁄‹iexp{≠⁄t i}.
In this simple case it is shown that the total observed survival time, ’ = qni ti and the total number of events ÷ = qni ‹i are sufficient for estimating ⁄ with a solution provided by ˆ⁄=÷/’.
Consider the special case of a prior distribution for the hazard parameter⁄,P r(⁄),
which follows a gamma distribution with scale and shape parameters (Ê,›). The density
function is given by
Gamma(Ê,›)≥ ›
Ê⁄Ê≠1exp{≠›⁄}
(Ê) .
Here (.) is the gamma function. This prior distribution is shown to be conjugate
for the exponential distribution as the posterior distribution itself also has a gamma distribution. To see this, evaluate the posterior distribution as
P r(⁄|D)ÃP r(D|⁄)P r(⁄)
=!⁄÷exp{≠⁄’}"⁄Ê≠1exp{≠›⁄} =⁄÷+Ê≠1exp)≠⁄(’+›)*,
and thus P r(⁄|D) follows a gamma distribution given by Gamma(÷+Ê,’ +›). The posterior distribution can be directly summarised using
E(⁄|D) = ÷+Ê
’+› and
V ar(⁄|D) = ÷+Ê
It is shown therefore that the posterior summaries are a direct combination of the observed data and prior beliefs. As an example, an exponential model is applied to the ductal patients in the ESPAC-3 data and the hypothetical situation where prior information is available which takes the formGamma(300,15000) is considered.
Prior Likelihood Posterior
Figure 2.4: Prior, likelihood and posterior densities for an Exponential model fitted to the ESPAC-3 dataset
The fit of the exponential model to the ESPAC-3 data is shown in Figure 2.2. In Figure 2.4 the densities of the hazard parameter ⁄from the prior, likelihood and the posterior are shown. This shows how the inclusion of an informative prior compliments the information from the likelihood. In this situation, there is a large amount of infor- mation in the data and a highly informative prior is required to have any noticeable effect on the posterior distribution. What is notable here is not only the shift in the point estimation but also the increase in the precision when comparing the posterior distribution to the estimated obtained from the likelihood alone.
2.4.2 Exponential model with gamma priors - with covariates
Here it is demonstrated that difficulties ensue when the exponential model is extended to include a single covariate,z.
Formally define the likelihood for parameters ◊= (⁄,—) and dataD= (ti,‹i, zi) as
L(◊|D) = n
Ÿ
i=1
!
⁄exp{—Tzi}"‹iexp)≠⁄exp{—Tzi}ti*.
Following the laws of conditional probability define
P r(◊|D)ÃP r(D|◊)P r(◊)
=P r(D|⁄,—)P r(⁄|—)P r(—)
=P r(D|⁄,—)P r(⁄)P r(—),
if it is assumed a-priori that the parameters that describe the baseline hazard and the parameters that describe effects of covariates are independent.
Following the example of Section 2.4.1, set a prior for ⁄ based on a gamma dis- tribution and further set the prior for — to be a normal distribution. Note from (2.3) that to evaluate posterior distributions, only the terms directly dependent upon ◊are required and define
P r(⁄)Ã⁄Ê≠1exp{≠›⁄} P r(—)Ãexp ; ≠(—≠µ) 2 2‡2 < .
A full form for the posterior distribution is written as
P(◊|D)à n Ÿ i=1 ⁄‹i+Ê≠1!exp{—Tz i}"‹iexp ; ≠ 3 ⁄!exp{—Tzi}ti+›"+(—≠ µ)2 2‡2 4< .
Of primary interest are the marginal densities for ⁄ and — from which posterior inferences can be made, denote as
P r⁄(◊|D) = ⁄ P r(◊|D)d— P r—(◊|D) = ⁄ P r(◊|D)d⁄.
Given the form of the likelihood this is not straightforward even in this relatively simple situation. Estimation can be achieved by approximating the posterior distribution for instance via a Laplace transformation or via simulation techniques which shall be explained in the following Section.
2.4.3 Monte Carlo Markov Chain simulation
Markov Chain Monte Carlo (MCMC) simulation is a class of stochastic algorithms for sampling from probability densities. They follow the Monte Carlo property that states that given some probability distributionfi(◊), a sequence of random observations ◊1,◊2... can be drawn where at any point m, the distribution of ◊m depends only upon ◊m≠1. Using techniques such as this in a Bayesian framework facilitated the construction of marginal posterior distributions using a large number of samples taken directly from the joint posterior distribution. These samples can then be directly used to obtain inferences upon key parameters of interest.
Whilst there are a number of differing MCMC routines that can be applied, con- sideration is given here to the Gibbs sampler first proposed by Gemen and Gemen [67] and applied in a Bayesian setting by Gelfand and Smith [41] as well as the Metropolis Hastings rejection sampling routine [68, 69]. Greater details of both these methods are described by Gelmen et al. [70]. A direct application of the Gibbs sampler for proportional hazards models is given by Dellaportas and Smith [71].
With respect to the Gibbs sampler, suppose there exists a posterior distribution
P r( |D) where the parameter vector can be divided into P components = ( 1, ..., P). The Gibbs sampler is a routine which, at each iteration, draws a sample from each com- ponent of conditional on the values of all other components. Allow the sequence of iterations m to be denoted as superscripts and the parameter components P to be
denoted as subscripts. At each iteration a sample is randomly generated from the probability distribution given by
P r(◊p| m≠p≠1, D). where m≠1
≠p gives the most recent state of each component other thanpat iteration m such that
m≠1
≠p = ( m1 , ..., mp≠1, pm+1≠1, ..., mP≠1). Under this routine each draw of component m
p is updated dependent upon both the previous value m≠1 and the most recent states of all other components of m≠1
≠p . It is sometimes the case that for a given posterior density, marginal distributions for each p will be known analytically in which case the sampling procedure can be used directly. When it is not, rejection algorithms are required, such as the Metropolis Hastings algorithm of which the Gibbs sampler is a special case as shown by [70].
The Metropolis Hasting algorithm, [68, 69] works on the basis that an evaluation of the state of m
p with respect to m≠p≠1 can be made by drawing a proposed future state
ú
p from a ‘jumping’ distributionJm( ú| m≠1). Note that the jumping distribution is conditional on m≠1
p only and not on the previous state for the other parameters being evaluated m≠1
≠p .
Taking a draw from the jumping distribution, the posterior distribution is evaluated under the proposal state and the previous state for p. Formally evaluate
r= P r( ú|D)/Jm( ú| m≠1)
P r( m≠1|D)/Jm( m≠1| ú).
The inclusion of the jumping distribution in the numerator and denominator is required only if they are asymmetric. In the case of a symmetric distribution, that is
r= P r(◊ú|D) P r(◊m≠1|D),
can be used. The estimate r is used to evaluate the proposal state. Under the Metropolis Hastings routine, if ú is a state which provides a more likely solution (i.e. is closer to
some local maximum within the neighbourhood of m≠1) a value of r Ø1 is obtained
and it is accepted with probability 1, otherwise it is accepted with probabilityr. It is
defined as
◊mp =
I
◊pú with probability min(r,1)
◊m≠1
p otherwise.
Computationally, for each draw of each parameter in the sampling procedure, the following routine is defined:
1. Draw a single value from the jumping distribution ú
p ≥Jm( ú| m≠1) 2. Calculate r based on ú
p and mp ≠1
3. Draw a single value ‘ruÕ from a uniform distribution with limits (0,1) • ifr >‘ruÕ then set mp = úp
• otherwise set m
p = mp ≠1
An illustration of the use of the Gibbs sampler is given for the exponential model with a single covariate as described in Section 2.4.2. For notation purposes, set = ◊= (“,—). To start the procedure, starting values are required and set to ◊0 = (“0 =
≠3.4,—0 = 0.1). Figure 2.5 gives a graphical representation of the Gibbs sampler
illustrating how the estimates from each parameter converge upon some solution. Also illustrated are the marginal distributions for “ and —.
Once the process has converged, samples are continued to be drawn in order to construct the joint and marginal posterior densities from which inferences can be made. Results in Table 2.4 give the posterior summaries that are obtained.
Parameter Mean Median Std. Dev 95% Cred. Int
“ -3.52 -3.52 0.03 (-3.57, -3.46)
— -0.09 -0.09 0.06 (-0.20, 0.02)
Table 2.4: Summaries of posterior distribution obtained via Gibbs sampling Algorithms for the Metropolis Hasting algorithm can be easily coded in many sta- tistical packages and examples for the piecewise exponential model are included in the Appendix. In many situations however, the BUGS (Bayesian inference Under Gibbs Sampling) suite of packages such as WinBUGS and OpenBUGS are utilised. Issues on convergence and model fit are covered by Gelman et al. [70] and are discussed as necessary within the thesis.
−3.60 −3.55 −3.50 −3.45 −3.40 − 0.25 − 0.20 − 0.15 − 0.10 − 0.05 0.00 0.05 0.10 γ β
Joint Density of γ and β
Marginal Density for γ
Marginal Density f
or
β
Figure 2.5: An illustration of the Gibbs sampler for the exponential model with a single covariate