Chapter 3 Frailty Models Based on the Gompertz Baseline ...shodhganga.inflibnet.ac.in/bitstream/10603/99055/11/11_chapter3.pdf · Chapter 3 Frailty Models Based on the Gompertz Baseline

Chapter 3

Frailty Models Based on the

Gompertz Baseline Distribution

3.1 Introduction

Mostly Weibull distribution is considered for baseline hazard. It is well known that the Weibull

model is not suitable for modeling survival data in all situations. Gompertz distribution is one of

the most important growth models. It has many applications in, for example, medical, biological

and actuarial studies. The Gompertz distribution plays an important role in modeling human

mortality and fitting actuarial tables. It has been used as a growth model and also used to fit

the tumor growth.

This Chapter consider shared frailty models with Gompertz baseline distribution. The mate-

rial of this Chapter i.e. Shared gamma frailty model with Gompertz baseline has been accepted

by the journal Statistics and Probability Letters, 82, 1310-17, (2012) and the remaining content

of this Chapter i.e. Shared inverse Gaussian frailty model with Gompertz baseline has been

submitted for publication in the well-known international journal.

57

Gompertz Baseline Distribution 58

3.2 Gompertz Baseline Distribution

Gompertz model, used most frequently by medical researchers and biologists in modeling the

mortality ratio data, was formulated by Gompertz (1825). Gompertz distribution is a growth

model and has been used in relation with tumor development. Ahuja and Nash (1979) showed

that Gompertz distribution, with a simple conversion, related to some distributions in the Pear-

son distributions family. According to Jaheen (2003), Garg et al. (1970) obtained the maximum

likelihood estimations of the Gompertz distribution parameters. Osman (1987), used a Gompertz

distribution with two parameters, worked on the features of the distribution and offered that it

should be used in modeling the lifespan data analyzing the survival ratio in heterogenic masses.

It has been widely used, especially in actuarial and biological applications and in demography.

A continuous random variable T is said to follow a Gompertz distribution with parameters

λ > 0 and γ > 0 (T ∼ Gompertz(λ, γ)), if the baseline hazard function corresponds to

h0(t) =

λ exp(γ t) ; t > 0, λ > 0, γ ∈ R

0 ; otherwise.(3.1)

and the cumulative hazard function is

H0(t) =

λγ−1(exp(γ t)− 1) ; t > 0, λ > 0, γ ∈ R

0 ; otherwise.(3.2)

where λ and γ are scale and shape parameters, respectively. For γ = 0 the baseline hazard (3.1)

reduces to the exponential hazard.

The corresponding survival function and probability density function of Gompertz distribu-

tion are

S0(t) =

exp[−λγ−1(exp(γt)− 1)] ; t > 0, λ > 0, γ ∈ R

0 ; otherwise.(3.3)

and

f0(t) =

λ exp(γ t) exp[−λγ−1(exp(γt)− 1)] ; t > 0, λ > 0, γ ∈ R

0 ; otherwise.(3.4)

We note that for γ > 0, S0(t) goes to zero for t→∞. With γ < 0, S0(t) goes to 0 < exp(λγ−1) <

1 for t → ∞. Therefore the event never occurs for a proportion exp(λγ−1) of the population.

We therefore consider the case γ > 0.


In this Chapter, the two-parameter Gompertz distribution is considered. Let us assume that

the independent random variables T1 and T2 have Gompertz distribution with parameters λ1, γ1

and λ2, γ2, respectively. In short we say Gomp(λj , γj), (j = 1, 2).

Here, T1 and T2 are independent, thus, H(ti1, ti2) = H1(ti1) + H2(ti2), where Hj(tij) is the

integrated hazard of Tij , (i = 1, 2, ..., n; j = 1, 2).

Under this independency assumption, using equation (1.1), the resulting regression model

H(tij |Ui, Xi) = uiλjγ−1

j (exp(γj tij)− 1) exp(x′iβ), (i = 1, .....n; j = 1, 2) (3.5)

is indeed an extended cumulative proportional hazards model conditional on frailty and fixed

factors. The conditional survival function of the jth individual in the ith cluster is given by

S(tij |Ui, Xi) = exp[−uiλjγ−1

j exp(x′iβ)(exp(γj tij)− 1)], tij ≥ 0, λj > 0, γj > 0 (3.6)

Here value of Ui is common to two components in a group. When there is no variability in the

distribution of Ui, that is, when Ui has a degenerate distribution then there is no dependency.

When the distribution is not degenerate, the dependence is positive. The value of Ui can be

considered as generated from unknown values of some explanatory variables. Conditional on

Ui = ui, the bivariate survival function is

S(ti1, ti2|Ui, Xi) = S(ti1|Ui, Xi)S(ti2|Ui, Xi)

= exp[−ui exp(x′iβ){λ1γ−11 (exp(γ1ti1)− 1) + λ2γ

−12 (exp(γ2ti2)− 1)}] (3.7)

where Ui (i = 1, 2, · · · , n) follows gamma distribution and inverse Gaussian distribution given

in (1.7) and (1.18), respectively.

3.2.1 Proposed Models

Substituting cumulative hazard function for Gompertz distribution in equations (1.9) and (1.20),

we get the unconditional bivariate survival functions based on gamma frailty and inverse Gaus-

sian frailty as,

Sθ(ti1, ti2|Xi) = [1 + θ exp(x′iβ){λ1γ−11 (exp(γ1ti1)− 1) + λ2γ

−12 (exp(γ2ti2)− 1)}]−1/θ (3.8)


and

Sθ(ti1, ti2|Xi) = exp[

1− (1 + 2θ exp(x′iβ){λ1γ−11 (exp(γ1ti1)− 1) + λ2γ

−12 (exp(γ2ti2)− 1)})1/2

θ

](3.9)

Here onwards we call equations (3.8) and (3.9) as Model-I and Model-II, respectively. Thus,

Model-I is the shared gamma frailty model under Gompertz baseline hazard and Model-II is the

shared inverse Gaussian frailty model under the same Gompertz baseline hazard.

Once we have unconditional survival function of bivariate random variable (Ti1, Ti2) we can

obtain likelihood function and estimate the parameters of the model. Here onwards in this

Chapter we represent Sθ(ti1, ti2|Xi) as Sθ(ti1, ti2).

3.3 Fitting of Models

Inferential procedures for frailty models have almost exclusively focused on likelihood based

approaches (Liang et al., 1995). The likelihood function in (2.7) for the random censorship

model can be used for the estimation procedure. To estimate the parameters of the model firstly

we must have to obtain likelihood function given by (2.7) for which we need to obtain the

functions, fi1, fi2, fi3, and Fi. Differentiating survival function for each of the model given in

equations (3.8) and (3.9) for obtaining these functions.

For Model-I, we have

fi1 = (1 + θ)λ1λ2 exp(2x′iβ) exp(γ1ti1 + γ2ti2)S(1+2θ)θ (ti1, ti2)

fi2 = λ1 exp(x′iβ) exp(γ1ti1)S(1+θ)θ (ti1, wi)

fi3 = λ2 exp(x′iβ) exp(γ2ti2)S(1+θ)θ (wi, ti2)

Fi = Sθ(wi, wi)

where Sθ(., .) is given by equation (3.8).


For Model-II, we have

fi1 =λ1λ2 exp(γ1ti1 + γ2ti2)Sθ(ti1, ti2)φ2(ti1, ti2)

[φ1(ti1, ti2)]32

exp(2x′iβ)

fi2 =λ1 exp(γ1ti1)Sθ(ti1, wi)

[φ1(ti1, wi)]12

exp(x′iβ)

fi3 =λ2 exp(γ2ti2)Sθ(wi, ti2)

[φ1(wi, ti2)]12

exp(x′iβ)

Fi = Sθ(wi, wi)

where φ1(ai, bi) = 1 + 2θ{λ1γ−11 (exp(γ1ai − 1)) + λ2γ

−12 (exp(γ2bi − 1))} exp(x′iβ); φ2(ai, bi) =

θ + [φ1(ai, bi)]12 and Sθ(., .) is given by equation (3.9).

Now we apply MCMC procedure discussed in Section (2.5) to estimate the parameters of

the models. Firstly, we consider a simulation study to evaluate the performance of the Bayesian

estimation procedure and then we apply the procedure to litters of rat data.

3.4 Simulation Study

To evaluate the performance of the Bayesian estimation procedure we carried out a simulation

study. For the simulation purpose we have considered only one covariate X = X1 which we

assume to follow binomial distribution for Model-I and normal distribution for Model-II. With

one covariate, the shared gamma frailty model and the shared inverse Gaussian frailty model

given in equation (3.8) and (3.9), respectively, has six parameters. The frailty variable U is

assumed to have gamma distribution and inverse Gaussian distribution with variance θ. Lifetimes

(Ti1, Ti2) for ith pair are conditionally independent for given frailty Ui = ui. We assume that

Tij (i = 1, . . . , n; j = 1, 2) follows the Gompertz baseline distribution. As the Bayesian methods

are time consuming, we generate only fifty, seventy-five and one hundred pairs of lifetimes using

inverse transform technique.

According to the assumption, for given frailty U , lifetimes of individuals are independent.

So, the conditional survival function for an individual for given frailty U = u and a covariate X

at time t > 0 is,

S(t | U,X) = exp[−uH0(t) exp(x′iβ)] (3.10)


Equating S(t | U,X) to a random number say r (0 < r < 1) over t > 0, using (3.6), we get,

t = γ−1ln

[1 +

γA

λ

](3.11)

where A = − ln(r)u exp(xβ) . Equations (3.11) is a generator to generate lifetimes for model (3.8) and

(3.9). We have generated different random samples of size n = 50, 75 and 100 for lifetimes Ti1

and Ti2 using (3.11) and for frailty from gamma and inverse Gaussian density functions f(u)

given in (1.7) and (1.18). To generate frailty values from gamma distribution we have used direct

commands in R package but for generating frailty values from inverse Gaussian distribution we

have used different packages like ‘SuppDists’, ‘statmod’ etc. in R.

Here we have generated different random samples of size n = 50, 75 and 100 for lifetimes Ti1

and Ti2 using (3.11). But we are giving procedure for sample generation of only one sample size,

say, n = 50. Samples are generated using following procedure for Model-I and Model-II:

1. Generate a random sample of size n = 50 from gamma and inverse Gaussian distributions

having density as given in equation (1.7) and (1.18), with θ = 3.6 as shared frailties (ui)

for ith (i = 1, 2, . . . , 50) cluster.

2. Generate 50 covariate values for X from binomial distribution for Model-I and from normal

distribution for Model-II.

3. Compute exp(Xiβ) with regression coefficient β = 2 and β = 0.7 for Model-I and Model-II,

respectively.

4. Generate 50 pairs of lifetimes (ti1, ti2) for given covariate (xi) using following generators,

ti1 = γ−11 ln

[1 +

γ1Ai1λ1

](3.12)

ti2 = γ−12 ln

[1 +

γ2Ai2λ2

](3.13)

for shared frailty models Model-I and Model-II, where Ai1 = − ln(ri1)ui exp(xiβ) and Ai2 =

− ln(ri2)ui exp(xiβ) ; r1 and r2 are random variables having U(0, 1) distribution and γ1, λ1 are

respectively shape and scale parameters of baseline distribution of first survival time and

γ2, λ2 are that of second survival time.


5. Generate censoring time wi from exponential distribution with failure rate 0.9 for Model-I

and 1.5 for Model-II.

6. Observe jth individual survival time t∗ij = min(tij , wi) and censoring indicator δij for ith

cluster (i = 1, 2, . . . , 50 and j = 1, 2), where

δij =

1, ; tij ≤ wi

0, ; tij > wi

Thus we have data consists of 50 pairs of survival times (t∗i1, t∗i2) and censoring indicators

δij .

Prior distributions that we have assumed for the parameters of Model-I are respec-

tively, G(0.0001, 0.0001) for baseline parameters λ1, γ1 and γ2 and G(0.01, 0.01) for λ2;

G(0.0001, 0.0001) for frailty parameter θ and N(0, 1000) for regression parameter β. Also,

prior distributions that we have assumed for the parameters of Model-II are respectively,

G(0.0001, 0.0001) for baseline parameters λ1, λ2, γ1 and γ2; G(0.0001, 0.0001) for frailty pa-

rameter θ and N(0, 1000) for regression parameter β. Here G(a, b) is gamma distribution with

shape parameter a and scale parameter b and N(µ, σ2) represents normal distribution with mean

µ and variance σ2.

We run two parallel chains for the proposed model with the different starting points using

Metropolis-Hastings algorithm within Gibbs sampler based on normal transition kernels. We

have iterated both the chains for 95,000 times using gamma prior. For both the chains the

results were somewhat similar so we present here the analysis for only one chain (i.e. chain I)

and for only one sample size for the resulting shared frailty models.

Trace plots and coupling from past plots are presented in Figures 3.1 and 3.3 for Model-I and

Figures 3.2 and 3.4 for Model-II with gamma prior assumption. These plots for other sample sizes

also have same pattern. So, we have not shown any graphs for other sample sizes, only for n = 75

with gamma prior has been shown to get the idea about these graphs. Trace plots are having

zigzag pattern which says that parameters are moving freely and appropriate for both models.

Sample autocorrelation plots are presented in Figures 3.5 and 3.6 for both models. Generated

chains have slightly higher autocorrelation lag which is decided by using sample autocorrelation

function plot.


Autocorrelation lag for all the parameters have different values so for uniqueness we have taken

Figure 3.1: Trace plots of the posterior samples for the simulation study of

Model-I

lag as maximum of lag of all parameters of a model. To have a single value of lag we have

taken lag value k (given in Table 3.5 for Model-I and Table 3.6 for Model-II) as maximum of

autocorrelation lag of all the parameters of a model.

Gelman-Rubin scale reduction factor and Geweke test with corresponding p-values for both


Figure 3.2: Trace plots of the posterior samples for the simulation study of

Model-II

the models are provided in Tables 3.1 and 3.3 for Model-I and Tables 3.2 and 3.4 for Model-II,

respectively. From these Tables, we can observe that Gelman-Rubin scale reduction factor values

are quite near to one and also p-values for Geweke test values are large enough to say the chains

attain stationary distribution.

The posterior summaries including posterior mean, standard error, and 95% credible intervals

along with burn in period values, sample autocorrelation lag values and sample sizes are provided


in Table 3.5 and Table 3.6 for both shared frailty models, respectively. From these Tables, it

Figure 3.3: Coupling from past plots for the simulation study of Model-I

can be observed that estimated values of parameters reach quite close to true values of the

parameters with decreasing standard errors as the sample size goes on increasing. We have used

R statistical software to perform this simulation study.


Figure 3.4: Coupling from past plots for the simulation study of Model-II


Figure 3.5: Sample autocorrelation function plots for the simulation study of

Model-I


Figure 3.6: Sample autocorrelation function plots for the simulation study of

Model-II


Table 3.1: Gelman-Rubin convergence statistic values for simulation for Model-I

Parameters λ1 λ2 γ1 γ2 θ β

True values 3.0 3.0 1.0 2.0 3.6 2.0

n = 50

stat. value 1.010111 1.003496 1.002179 1.000078 1.001579 1.000022

n = 75

stat. value 1.000276 1.000163 1.000015 0.9999592 1.000463 1.006026

n = 100

stat. value 1.020548 1.007764 1.000666 1.000023 1.001366 1.000979

Table 3.2: Gelman-Rubin convergence statistic values for simulation for Model-

II


True values 3.0 3.0 1.0 2.0 3.6 0.7

n = 50

stat. value 1.000161 0.9999772 1.000717 1.003983 1.006078 1.000492

n = 75

stat. value 1.001183 1.000145 0.9999587 1.001571 1.000021 1.002332

n = 100

stat. value 1.010941 1.004914 1.000031 1.003994 1.001281 1.009831


Table 3.3: Geweke test values and corresponding p-values for simulation for

Model-I

Sample Size→ n = 50 n = 75 n = 100

True test test test

Parameters values value p− value value p− value value p− value

λ1 3.0 -0.00129 0.49948 -0.00153 0.49938 -0.00056 0.49977

λ2 3.0 -0.00702 0.49719 0.01471 0.50586 0.01807 0.50721

γ1 1.0 -0.00737 0.49705 0.00462 0.50184 0.01586 0.50633

γ2 2.0 -0.00127 0.49949 0.00209 0.50083 0.000401 0.50016

θ 3.6 0.01427 0.50569 0.01133 0.50452 -0.00404 0.49838

β 2.0 -0.00439 0.49824 0.01168 0.50466 -0.00316 0.49873

Table 3.4: Geweke test values and corresponding p-values for simulation for

Model-II

Sample Size→ n = 50 n = 75 n = 100

True test test test

Parameters values value p− value value p− value value p− value

λ1 3.0 0.01211 0.50483 -0.00019 0.49992 -0.00857 0.49658

λ2 3.0 -0.00649 0.49740 0.00726 0.50289 -0.00525 0.49790

γ1 1.0 0.00969 0.50386 0.00742 0.50296 0.01601 0.50639

γ2 2.0 -0.00701 0.49720 0.00629 0.50251 -0.00275 0.49890

θ 3.6 0.00287 0.50114 -0.00808 0.49677 0.01192 0.50475

β 0.7 -0.00152 0.49939 -0.00951 0.49620 0.00921 0.50367


Table 3.5: Parameters estimates for simulation of shared gamma frailty model

(Model-I) with Gompertz baseline hazard


True values 3.0 3.0 1.0 2.0 3.6 2.0

n = 50

sample size = 870 burn in period = 8000 autocorrelation lag = 100

Estimates 3.339832 3.58423 0.8029933 1.898675 3.728842 1.784984

S.E. 0.5166535 0.5047603 0.2513277 0.2350126 0.3168721 0.2564108

LowerLt. 2.17248 2.030812 0.51023 1.603307 3.039895 1.519136

UpperLt. 3.935928 3.862703 1.464501 2.375263 4.159701 2.412618

n = 75


Estimates 3.275938 3.352028 0.9781125 1.990647 3.670445 1.807583

S.E. 0.4715845 0.4150472 0.2474298 0.1986308 0.2753138 0.2123236

LowerLt. 2.235856 2.462795 0.550843 1.609894 3.114257 1.521822

UpperLt. 3.927783 3.967307 1.498642 2.360427 4.147788 2.286009

n = 100


Estimates 2.818765 2.805313 1.019907 1.997276 3.601107 1.829282

S.E. 0.4173838 0.3207179 0.2032705 0.1980163 0.2688608 0.2091362

LowerLt. 2.283129 2.876453 0.5884597 1.621737 3.180849 1.516968

UpperLt. 3.783039 3.992017 1.409367 2.37215 4.156142 2.259177


Table 3.6: Parameters estimates for simulation of shared inverse Gaussian frailty

model (Model-II) with Gompertz baseline hazard


True values 3.0 3.0 1.0 2.0 3.6 0.7

n = 50


Estimates 2.8591 2.601739 0.9236211 1.910175 3.508175 0.6428016

S.E. 0.5856733 0.4414507 0.2899155 0.2864638 0.3244146 0.04853413

LowerLt. 1.849004 2.015302 0.5109697 1.413235 3.042288 0.3450716

UpperLt. 3.923291 3.887658 1.459524 2.592098 4.192036 0.8330701

n = 75


Estimates 2.955741 2.846793 0.974943 1.95329 3.624358 0.7514799

S.E. 0.3480802 0.2487879 0.277412 0.2842537 0.3186504 0.03493911

LowerLt. 2.236078 2.111656 0.5273721 1.520869 3.051731 0.3770092

UpperLt. 3.479806 3.380265 1.456028 2.527003 4.17338 0.8168667

n = 100


Estimates 2.959122 3.089771 0.9826119 1.993055 3.600546 0.6761037

S.E. 0.2529426 0.2006675 0.265877 0.2802394 0.3161122 0.03123944

LowerLt. 2.400494 2.161371 0.5331393 1.533585 3.097311 0.4168464

UpperLt. 3.330827 3.35982 1.451178 2.519502 4.131662 0.7792813


3.5 Analysis of Tumorigenesis Experiment Data

In this Section the method is applied to the animal tumorigenesis data described by Mantel

et al. (1977) (given in Appendix A). The experiment involved 50 male and 50 female litters,

consist of three rats within each litter. A subset of the data, for the female rats litter, was

analyzed by Hougaard (1986) who used a Cox type model and a parametric model with Weibull

margins. The Bayesian approach using Gibbs sampling was given by Clayton (1991). We also

use this subset of data but with two controlled rats to illustrate the application of our proposed

models because there should be homogeneity within cluster. One of the main research aims is to

investigate whether some unobserved factors like genetic or environmental factors significantly

increases the failure rate of rats while allowing for common covariates for both rats within the

same litter. Here, we fit the proposed shared frailty models (defined in 3.8 and 3.9) to the female

rats’ tumor time data, without any covariate as we consider only controlled female rats. First,

we check goodness-of-fit of the data for Gompertz baseline distribution and then apply the

Bayesian estimation procedure. In the analysis of this study, we have used the R package. To

(a) C1 (b) C2

Figure 3.7: Graphs for goodness-of-fit for litter matched tumorigenesis data for

Gompertz distribution

check goodness-of-fit of data set, firstly we consider graphical procedure to check appropriateness

of the model. To assess model graphically, we plot the graph of ln[1−{γln(S(t))/λ}] versus t for


proposed model. If the resulted plot is roughly a straight line then we can say that the underline

baseline model is appropriate. The graph of ln[1−{γln(S(t))/λ}] versus t for Gompertz baseline

distribution for both controlled rats C1 and C2, using R-program, are shown in the Figures 3.7(a)

and 3.7(b), respectively. From the Figures, we can observe that many of the points are nearly

on the straight line.

(a) C1 (b) C2

Figure 3.8: Comparison of non parametric survival function with Gompertz

survival function for litter matched tumorigenesis data

Figures 3.8(a) and 3.8(b) which plot the Kaplan-Meier estimates and hypothesized theo-

retical distribution S0(t) for the model demonstrate a close agreement between the two i.e.

non-parametric and parametric survival curves.

Table 3.7: p-values for Kolmogorov-Smirnov test statistic for litter matched

tumorigenesis data set

Test C1 C2

K-S 0.9804 0.9753


We have considered a statistical test also viz. Kolmogorov-Smirnov (K-S) test for testing

composite goodness-of-fit hypotheses to the Gompertz baseline distribution for models (3.8)

and (3.9). Test procedure is developed for testing goodness-of-fit with data subjected to random

right censoring. Here, we assume that survival times of rats are conditionally independent so

we apply K-S test to the survival times of both rats separately. The p-values of K-S test for

baseline distribution are presented in Table 3.7 which are very large approximately close to 1.

Thus, from different goodness-of-fit graphs (see Figures 3.7(a) and 3.7(b)) and p-values of K-S

test (see Table 3.7) we can say that there is no statistical evidence to reject the hypothesis that

data are from Gompertz distribution.

In the analysis, we have used the R program. Given the model assumptions, this program

performs the Gibbs sampler by simulating from the full conditional distributions. We run two

parallel chains for both shared frailty models using two sets of prior distributions with the

different starting points using Metropolis-Hastings algorithm within Gibbs sampler based on

normal transition kernels. In our study of this data, the prior distribution we use for frailty

parameter θ is gamma distribution with mean one and large variance, say Γ(φ, φ), with a small

choice of φ. Since, we do not have any prior information about baseline parameters, λ1, γ1, λ2,

and γ2, therefore, prior distributions of these parameters are assumed to be flat. We consider

two different sets of prior distributions for baseline parameters, one is Γ(a, b) and another is

U(c, d). All the hyper-parameters φ, a, b, c, and d are known. Here Γ(a, b) is gamma distribution

with shape parameter a and scale parameter b and U(c, d) represents uniform distribution over

the interval c to d. We set hyper-parameters as

• for Model-I

φ = 0.00001, a = 1; b = 0.0001, c = 0 and d = 100.

• for Model-II

φ = 0.00001; a = 1, b = 0.0001 for λ1, λ2, γ1, and γ2; and c = 0, d = 100 for λ1, λ2, γ2,

and c = 0, d = 60 for γ1.

We implemented 95,000 iterations of the algorithm. To diminish the effect of the starting

distribution, we generally discard the early iterations of each sequence and focus attention on

the remaining.


Figure 3.9: Posterior for Model-I: time series plots of the posterior samples for

litter matched tumorigenesis data

We described the first 2,700 and 4,000 iterations as a burn-in for Model-I and Model-II,

respectively. There is no effect of prior distributions on posterior summaries because estimates

of parameters are nearly same for both sets of priors. Here, results for chain I and chain II are

similar so we present results for only one chain (i.e. chain I) with Γ(a, b) as prior for baseline


parameters.

Figure 3.10: Posterior for Model-II: time series plots of the posterior samples

for litter matched tumorigenesis data

Graphs for time series (or trace) plot, coupling from past plots and sample autocorrelation

plot with thinning (i.e. after considering lag shown in Figures) for the parameters are shown in

Figures 3.9, 3.11, and 3.13 and Figures 3.10, 3.12, and 3.14 for Model-I and Model-II, respectively.


Figure 3.11: Posterior for Model-I: coupling from past plots of the posterior

samples for litter matched tumorigenesis data

Trace plots for all the parameters show zigzag pattern which indicates that parameters

move and mix more freely. Thus, it seems that the Markov chain has reached the stationary

state. However, a sequence of draws after burn-in period may have autocorrelation. Because

of autocorrelation consecutive draws may not be random, but values at widely separated time

points are approximately independent.


So, a pseudo random sample from the posterior distribution can be found by taking values from

a single run of the Markov chain at widely spaced time points (autocorrelation lag) after burn-in

period.

Figure 3.12: Posterior for Model-II: coupling from past plots of the posterior

samples for litter matched tumorigenesis data

Simulated values of parameters have autocorrelation of lag k (value given below each plot

in Figure 3.13 and Figure 3.14), so every kth iteration is selected as sample to thin the chain

and discarding the rest. The autocorrelation of parameters become almost negligible after the


defined lag, given in Figures.

Figure 3.13: Plots of autocorrelations with thinning number lag (given below

the Figure) of the posterior samples for litter matched tumorigenesis data for

Model-I

We can also use running mean plots to check how well our chains are mixing. A running

mean plot is a plot of the iterations against the mean of the draws up to each iteration. In fact

running mean plots display a time series of the running mean for each parameter in each chain,


as shown in Figure 3.15 and Figure 3.16 for Model-I and Model-II, respectively.

These plots should be converging to a value. Running mean plot for each parameter is converging

Figure 3.14: Plots of autocorrelations with thinning number lag (given below

the Figure) of the posterior samples for litter matched tumorigenesis data for

Model-II

to the posterior mean of the parameter, thus, represents a good mixing of chain. Thus, our

diagnostic plots suggest that the MCMC chains are mixing very well.


(a) (b)

(c) (d)

(e)

Figure 3.15: Graphs of running mean plots for litter matched tumorigenesis

data for Model-I


(a) (b)

(c) (d)

(e)

Figure 3.16: Graphs of running mean plots for litter matched tumorigenesis

data for Model-II


Tables 3.8 and 3.9 present the posterior mean, standard deviation, and 95% credible inter-

vals for all baseline parameters, frailty variance along with Gelman-Rubin convergence statistic

values and Geweke test statistic values with corresponding p-values for Model-I and Model-II,

respectively. Monitoring convergence of the chains has been done via the Brooks and Gelman

(1998) convergence-diagnostic.

Hence, once convergence has been achieved, 92,300 and 91,000 observations, for Model-I and

Table 3.8: Posterior summary for litter matched tumorigenesis data set for

Model-I

Credible Interval

Parameters BGR Geweke test Estimate S.E. Lower Upper

stat. value test value p-value limit limit

λ1 1.0016 0.0141 0.5056 0.0006 0.0003 0.0002 0.0014

λ2 1.0007 -0.0137 0.4945 0.0009 0.0004 0.0003 0.0020

γ1 1.0005 -0.0107 0.4956 0.0244 0.0046 0.0156 0.0316

γ2 1.0012 0.0128 0.5051 0.0242 0.0043 0.0160 0.0315

θ 1.0028 0.0041 0.5016 1.9875 0.3127 1.5209 2.6236

Model-II, respectively, are taken from each chain after the burn-in period. On inspection of the

Brooks and Gelman’s diagnostic, we find the BGR (Brooks and Gelman Ratio) convergent to

one, this shows that the convergence for variance of frailty θ and other parameters has been

obtained. Also, the Geweke test statistic values are quite small and corresponding p-values are

large enough to say the chains attain stationary distribution.

There is little difference between the posterior estimates of baseline parameters γ1, γ2 and

frailty parameter θ for Model-I and Model-II, presented in Tables 3.8 and 3.9. The posterior

estimate of dispersion of random effects i.e. θ = 1.9875 for Model-I and θ = 1.9958 for Model-II,

shows that there exists highly significant heterogeneity in population of rats. Thus, there is a

dependence within the litter.


Table 3.9: Posterior summary for litter matched tumorigenesis data set for

Model-II

Credible Interval

Parameters BGR Geweke test Estimate S.E. Lower Upper

stat. value test value p-value limit limit

λ1 1.1623 0.0059 0.5023 0.0005 0.0002 0.0001 0.0011

λ2 1.0895 0.0079 0.5031 0.0007 0.0003 0.0003 0.0016

γ1 1.0168 -0.0060 0.4975 0.0250 0.0042 0.0159 0.0308

γ2 1.0123 -0.0105 0.4957 0.0248 0.0039 0.0165 0.0306

θ 1.0377 -0.0041 0.4983 1.9958 0.3182 1.5183 2.6637

3.5.1 Testing of Significance of Frailty

Therneau and Grambsch (2000) analyzed this data using gamma and Gaussian frailty models.

They observed the estimate of the variance of the random effect is highest when Gaussian frailty

with AIC method is used. Hanagal (2011) also considered the same data for Weibull distribution

as the baseline model to analyze and test for no frailty in frailty models. From the Gaussian and

gamma frailty models, he observed from the R output that frailty variable (litter) is very highly

significantly related to the recurrence time. Now we test for no-frailty in shared gamma frailty

model and shared inverse Gaussian model. To test significance of frailty we compared two models

M0 and M1. One with shared gamma frailty as model M1,G, shared inverse Gaussian as model

M1,IG and without frailty as model M0. As it is observed from Table 3.10 that AIC, BIC and

DIC are not able to discriminate between the models with frailty and without frailty for Model-I

and Model-II because the difference in information criterion between M0 and M1,G for Model-I

and M0 and M1,IG for Model-II are very small i.e. approximately less than 3. So, we use Bayes

factor for this comparison. Bayes factor value which is 6.363487 and lies in (6, 10) for Model-I

and 2.443901 and lies in (2, 6) for Model-II (see Table 3.11) gives strong positive and positive

evidence against the model M0 i.e. without frailty for both shared frailty models. Therefore, M0


is worse than M1 based on Bayes factor and all other criterion. Thus, it is observed from the

results that frailty variable i.e litter effect is very highly significantly related to the recurrence

time of rats.

Table 3.10: AIC, BIC and DIC values for test of significance for frailty under

Model-I & Model-II fitted to litter matched tumorigenesis data set

Models Frailty AIC AICc BIC DIC log-likelihood

I with (M1,G) 267.649 269.014 279.114 261.457 -128.825

II with (M1,IG) 266.3979 267.7615 278.2191 260.9202 -128.1989

I&II without (M0) 265.2342 266.123 275.2299 261.9294 -128.6171

Table 3.11: Bayes factor values and decision for test of significance for frailty

under Model-I & Model-II fitted to litter matched tumorigenesis data set

numerator model

Models against 2loge(B10) range Evidence against

denominator model model in denominator

I M1,G against M0 6.363487 ≥ 6 and ≤ 10 Strong Positive

II M1,IG against M0 2.443901 ≥ 2and ≤ 6 Positive

3.6 Discussion

In this Chapter, shared gamma frailty model and shared inverse Gaussian frailty model with

Gompertz baseline distribution has been introduced. We have developed the Bayesian estimation

procedure to estimate the parameters of the models. Firstly we have considered simulation study

to see the performance of the models and then applied to the litter matched tumorigenesis data.

In case of simulation, we have chosen the gamma as prior distribution to check the effect of sample


size on the posterior estimates. It was observed from all results of Tables and Figures that the

posterior summary estimates with gamma prior assumptions were close to true parameter values.

In this Chapter, we have considered the litter matched tumorigenesis experiment study. Here,

we have discussed the Bayesian estimation procedure including Gibbs sampling for computing

the estimation of the unknown parameters for litter matched tumorigenesis data set. For this

data set we have no covariate as we have considered only controlled rats and also, we were

mainly interested to see the effect of frailty. We have discussed Kolmogorov-Smirnov statistical

tests for testing goodness-of-fit to Gompertz distribution. K-S test gives the statistical evidence

of not rejecting the hypotheses that data follows Gompertz baseline distribution.

For estimation purpose, we have run two parallel chains with two sets of priors from different

starting points and considered the “burn-in” interval for each chain. We have provided 95,000

iterations to perform the analysis. We have clearly written the steps involved in the iteration

procedure, using R-program, given in Appendix B. The quality of convergence was checked

graphically by trace plots and running mean plots and statistically by Geweke test statistics

and Gelman-Rubin statistics (see Brooks and Gelman, 1998). The values of the Gelman-Rubin

statistics in this case are quite close to one. Thus, the sample can be considered to have arisen

from stationary distribution. Thus, all convergence diagnostics suggest that the MCMC chains

are mixing very well, converging to a value, and attain stationary distribution.

We have given the complete posterior summary of the data set including posterior mean,

standard error, and 95% credible interval for both shared frailty models. We have compared

the model with frailty and model without frailty using some information criteria such as, log-

likelihood, AIC, AICc, BIC, DIC, and Bayes factor for both shared frailty models. From the

value of all these criteria we conclude that the shared frailty model provide a suitable choice

for modeling the life time model of litter matched tumorigenesis data as compared to without

frailty model. From the tests for heterogeneity and posterior estimates of frailty parameter set

we can conclude that there is highly significant heterogeneity among the rats. Data analysis of

tumorigenesis experiment indicates that the performance of Bayesian estimation method is quite

satisfactory.

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