Inference on the Loglogistic Model with Right Censored Data

Abstract

Survival Analysis Methods are commonly used to analyze clinical trial data. In most clinical studies, the time until the occurrence of an event is the main outcome of significance. Clinical trials are conducted to assess the worth of new treatment regimens. The major events that the trial subjects seek to determine are either death, development of an undesirable reaction, relapse from remission, or the progress of a new disease entity. In order to model timeto- event data or clinical trials data, a parametric distribution can be assumed. We have in this study assumed that the data follow a log-logistic distribution. To estimate the parameters of this lifetime distribution, the Bayesian estimation approach is considered under the assumption of informative (gamma) priors as well as the frequentist estimation method. The Bayes estimators cannot be obtained in close forms; therefore, approximate Bayesian estimates are computed using the idea of Lindley. The clinical trial data considered in this study is either randomly or non-informatively censored. These types of data occur when each subject has a censoring time that is statistically independent of their failure times. A simulation study is carried out and also three different sets of real data have been analyzed in order to examine our methods. The Bayesian methods are considered under squared error and linear exponential loss functions.

Keywords: Bayesian Inference; Maximum likelihood; Squared Error and LINEX Loss Functions

Introduction

The log-logistic survival model is a lifetime distributional model which can be used as an alternative to the well-known and used Weibull distribution in lifetime or clinical trials data analysis. The shape parameter of the log-logistic distribution performs similar functions as that of the Weibull distribution. It is important that sometimes we model the survival or clinical trial data using a distribution that has a non-monotone hazard rate. According to [1], when the shape parameter is say p > 1, the hazard function becomes unimodal and when p = 1, the hazard decreases monotonically. The fact that the cumulative distribution function can be written in closed form unlike the lognormal distribution makes it useful for analyzing survival data. The loglogistic model has the distribution, density and survival functions respectively as

$F(t;\theta ,p)={\left[1+{\left(\frac{t}{\theta }\right)}^{-p}\right]}^{-1}$

$f(t;\theta p)=\frac{p}{\theta }{\left(\frac{t}{\theta }\right)}^{p-1}{\left[1+{\left(\frac{t}{\theta }\right)}^p\right]}^{-2}\forall t>0,\theta ,p>0$

$s(t;\theta ,p)={\left[1+{\left(\frac{t}{\theta }\right)}^p\right]}^{-1}$

where p is the shape parameter and θ the scale parameter.

The log logistic distribution is a continuous probability distribution which has non-negative random variables, hence, it can be used in survival analysis as a parametric model for events whose rate increases initially and decreases consequently, For instance, mortality of cancer patients following diagnoses or treatments. See for instance, [2-5].

According to [6], the log logistic distribution has been shown to be a suitable model in analyzing survival or clinical data was considered by Cox, Cox and Oakes, Bennet and others. [7], employed the log logistic distribution on lung cancer data and in their study, they estimated the mortality ratio at which it reached a maximum level. They determined the parameters of the log logistic model by making use of maximum likelihood estimate and bootstrap methods and observed the proximity of the results. A study conducted by [8], on the spread of HIV virus in San Francisco between 1978 and 1986 indicated that, the log logistic model was most suitable among other models to use with half censored data.

Under random or non-informative censoring, sample of say n elements are followed for a specified time say, T, the number of elements that is experiencing the event is considered to be random, but the entire length of study is fixed. Since the time is fixed, there are certain practical advantages with regards to designing a follow-up study. In a straightforward overview of this scheme, which is known as fixed time censoring, each element has a maximum inspection time say T_i, for i = 1 ,…, n, which may possibly vary from one situation to another. S(t)represents the probability that a unit i will be alive at the end of the inspection time. Consider an experiment where we start with an observation of 50 cancer patients that have died or survived at the specified time. The survival of the patients may be due to withdrawal, inadequate monitoring mechanism or deaths which are not related to the purpose of the study.

Maximum Likelihood Estimator (MLE) has been used frequently in determining the parameters of most of the lifetime distributions such as Weibull, lognormal, generalized exponential and others. Some of the works can be found in [11], they studied, generalized exponential distribution: Bayesian estimations. Other estimation procedures related to the above were considered by [12]. Determined the Bayes estimates of the reliability function and the hazard rate of the Weibull failure time distribution by employing squared error loss function [13]. Applied Bayesian to the parameter and reliability estimate of Weibull failure time distribution [14], studied the approximate Bayesian estimates for the Weibull reliability function and hazard rate from censored data by employing a new method that has the potential of reducing the number of terms in Lindley’s approximation procedure. Others include; [15-20].

The main objective of this study is to apply the Bayesian estimator’s procedure using Lindley’s approximation method with two loss functions for the unknown parameters of the log logistic distribution against the classical maximum likelihood estimator with different sample sizes and parameter values using simulation study. Since both parameters of the distribution are non-negative, we assume that both take on the gamma prior distributions which are not necessarily the conjugate priors for the parameters.

Maximum Likelihood Estimation

Consider a set of n independently and identically distributed random pairs of (t_i,δ_i), where t_i= min (X_i,T_i) and δ_i=I(X_i=T_i) indicating whether the observation is censored or not for I = 1, 2,…, n. in an independent random censored model, it is assumed that the survival time X_i and the censoring time T_i are independent and from the same distribution. The score vectors are

$h(\theta )=\frac{\partial \text{Log L(θ,p,t,δ)}}{\partial \theta }\text{ and }h(p)\text{=}\frac{\partial \text{Log L(θ,p,t,δ)}}{\partial p}$

where the score becomes a vector of the first partial derivatives of (θ, p). When using maximum likelihood to estimate unknown parameters that cannot be obtained in close form, one always requires that an iterative (eg, Newton-Raphson) procedure be implemented, such that, one can consider evaluating MLEs of $\widehat{\alpha },$ with a trial value say α₀ using a first order Taylor series as

$h(\widehat{\alpha })\approx h({\alpha }_0)+\frac{\partial h(\alpha )}{\partial \alpha }(\widehat{\alpha }-{\alpha }_0)$ (1)

Setting the left hand side of equation (1) to zero and solving for $\widehat{\alpha },$ we have

$\widehat{\alpha }={\alpha }_0-H^{-1}({\alpha }_0)h({\alpha }_0)$ (2)

where H(a₀) is the Hessian matrix and h(a₀) the score vector.

Considering the two parameters of the log logistic distribution, the Hessian matrix can be obtained as follows for the parameters estimates. The score vector of

$h(\theta )=-\frac{p{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{\theta }+\frac{p{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_i}{\theta }\right)}^p}}{\theta {\displaystyle {\sum }_{i=1}^n(1+{\left(\frac{t_i}{\theta }\right)}^p)}}$ (3)

$h(p)=\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{p}-In(\theta ){\displaystyle {\sum }_{i=1}^n{\delta }_i}+{\displaystyle {\sum }_{i=1}^n{\delta }_{i}In(t_i)-}$$\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_i}{\theta }\right)}^{p}In\left(\frac{t_i}{\theta }\right)}{{\displaystyle {\sum }_{i=1}^n(1+}{\left(\frac{t_i}{\theta }\right)}^p)}-\frac{{\displaystyle {\sum }_{i=1}^n\left(\frac{t_i}{\theta }\right)In\left(\frac{t_i}{\theta }\right)}}{{\displaystyle {\sum }_{i=1}^n(1+}{\left(\frac{t_i}{\theta }\right)}^p)}$ (4)

From above, the partial derivatives for both θ and p is

$\frac{{\partial }^2\log L(\theta ,p;t_i,{\delta }_i)}{\partial p\partial \theta }=\frac{{\partial }^2\log L(\theta ,p;t_i,{\delta }_i)}{\partial \theta \partial p}=$$\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{\theta }+\frac{p{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_i}{\theta }\right)}^p{\displaystyle {\sum }_{i=1}^{n}In{\left(\frac{t_i}{\theta }\right)}^p}}}{\theta {\displaystyle {\sum }_{i=1}^n(1+{\left(\frac{t_i}{\theta }\right)}^p)}}+$

$\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_i}{\theta }\right)}^p}}{\theta {\displaystyle {\sum }_{i=1}^n(1+{\left(\frac{t_i}{\theta }\right)}^p)}}-\frac{p{\displaystyle {\sum }_{i=1}^n{\left[{\delta }_i{\left(\frac{t_i}{\theta }\right)}^p\right]}^2{\displaystyle {\sum }_{i=1}^{n}In{\left(\frac{t_i}{\theta }\right)}^p}}}{\theta {\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}+$$\frac{p{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p{\displaystyle {\sum }_{i=1}^{n}In{\left(\frac{t_i}{\theta }\right)}^p}}}{\theta {\displaystyle {\sum }_{i=1}^n(1+{\left(\frac{t_i}{\theta }\right)}^p)}}+\frac{{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}}{\theta {\displaystyle {\sum }_{i=1}^n(1+{\left(\frac{t_i}{\theta }\right)}^p)}}$ (5)

Where$\frac{\partial h(p)}{\partial p}and\frac{\partial h(\theta )}{\partial \theta }$ are easy to obtain. Equations (3), (4) and (5) can be substituted into equation (2), from which an iterative procedure could be implemented to obtain the parameter estimates under maximum likelihood.

Bayesian Inference of the Unknown Parameters

In this section, we consider Bayesian inference of the unknown parameters of the log logistic distribution. In order to employ the Bayesian methods, a prior needs to be defined. A prior is simply one’s knowledge or an expert’s opinion on the parameters being estimated. We have little prior information for all the parameters being estimated, and so we want our data information to dominate the prior distribution by assuming reasonably non-informative priors for all the parameters in this model. It is assumed that the two parameters follow a vague Gamma (a, b) and Gamma (c, d) prior distributions. These prior models are chosen because both the scale and shape parameters of the log logistic distribution are non-negative.

π&sub>1(θ)aθ ^a-1 exp (-θb),θ > 0 (6)

π&sub>1(p)ap ^c-1 exp (-pd),θ > 0 (7)

The Bayesian posterior distribution based on which inferences are drawn is

${\pi }^{*}(\theta ,p|t)\alpha \frac{L(data|\theta ,p){\pi }_1(\theta ){\pi }_2(p)}{\int \int L(data|\theta ,p){\pi }_1(\theta ){\pi }_2(p)d\theta dp}$ (8)

Squared-error Loss

The squared error loss is the loss incurred by adapting action say, $\widehat{a}$ when the true value is say, a.

In other words, it implies the cost obtained by replacing the actual value of the parameter with the parameter estimate. Let the Bayesian estimator say, β_se be the posterior mean. If u (θ, p) is considered as the function of interest, then:

${\beta }_{se}=E\{u(\theta ,p)|t,\delta \}=\frac{\int \int u(\theta ,p)L(data|\theta ,p){\pi }_1(\theta ){\pi }_2(p)d\theta dp}{\int \int L(data|\theta ,p){\pi }_1(\theta ){\pi }_2(p)d\theta dp}$ (9)

Note; the function of interest in our study is the loss function which measures the distribution parameters of θ and p. It is observed that equation (9) cannot be computed explicitly even if we take some specific priors on the parameters, as a result [21] proposed an approximation procedure to compute the ratio of two integrals similar to equation (9). The approximation procedure is adopted in this paper.

Lindley Approximation

The posterior Bayes estimator of an arbitrary function u(a) given by [21] is

$E\{u(\alpha )|x\}=\frac{\int \omega (\alpha )\exp [l(\alpha )]d\alpha }{\int v(\alpha )\exp [l(\alpha )]d\alpha }$

Where l(a) is the log-likelihood and ω(a), v(a) are arbitrary functions of a. We assume that v(a) is the prior distribution for and ω(a)= u(a).v(a) with u(a) being some function of interest. The posterior expectation according to [12] is

$E\{u(a)|t\}=\frac{\int u(a)\exp \{l(a)+\rho (\alpha )\}d\alpha }{\int \exp \{l(a)+\rho (\alpha )\}d\alpha }$ (11)

Where ρ(a)=log{v(a)}.

An asymptotic expansion of Lindley’s approach of equation (11) according to [18] is

$\widehat{u}=u(\widehat{\theta ,}\widehat{p})+\frac{1}{2}[(u_{11}{\sigma }_{11})+(u_{11}{\sigma }_{11})]+u_1{\rho }_1{\sigma }_{11}+u_2{\rho }_2{\sigma }_{22}+\frac{1}{2}[(l_{30}{u}_1{\sigma }_{11}^2)+(l_{03}{u}_2{\sigma }_{22}^2)]$ (12)

where l stands for the log-likelihood function.

Considering the Bayesian estimator via Lindley, the following are obtained with u₁,u₁₁ and u₂,u₂₂ representing the first and second derivatives of θ and ρ respectively under the squared error loss which is referred to as the posterior mean.

$u(\theta )=\theta ,u_1=1,u_{11}=0,u(p)=p,u_2=1,u_{22}=0\rho =\text{In }{\pi }_1(\theta )+\text{In }{\pi }_2(p),$

${\rho }_1=\frac{a-1}{\theta }-b,{\rho }_2=\frac{c-1}{\theta }-d{\sigma }_{11}={(-l_{20})}^{-1},{\sigma }_{22}={(-l_{02})}^{-1}$

Let l₂₀ and l₃₀ represent the second and third derivatives of the loglikelihood function with respect to the scale parameter θ, then

$l_{20}=-\frac{p{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{{\theta }^2}-\frac{p^2{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_1}{\theta }\right)}^p}}{{\theta }^2{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_1}{\theta }\right)}^p\right]}}\frac{p{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_1}{\theta }\right)}^p}}{{\theta }^2{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_1}{\theta }\right)}^p\right]}}+$$\frac{p^2{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}{{\theta }^2{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}-\frac{p^2{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left(\frac{t_i}{\theta }\right)}^p}}{{\theta }^2{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}-$

$\frac{p{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}}{{\theta }^2{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+\frac{p{\left[{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}\right]}^2}{{\theta }^2{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}$

$l_{30}=-\frac{2p{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{{\theta }^3}+\frac{p^3{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_i}{\theta }\right)}^p}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+$$\frac{3p^2{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_1}{\theta }\right)}^p}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_1}{\theta }\right)}^p\right]}}-\frac{3p^3{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left[{\left(\frac{t_1}{\theta }\right)}^p\right]}^2}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_1}{\theta }\right)}^p\right]}^2}}+\frac{2p{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_1}{\theta }\right)}^p}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_1}{\theta }\right)}^p\right]}}-$

$\frac{3p^2{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}^2}}+\frac{2p^9{\displaystyle {\sum }_{i=1}^n{\delta }_i{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^3}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}^3}}+$

$\frac{p^3{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}}+\frac{3p^2{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}}-\frac{3p^3{\displaystyle {\sum }_{i=1}^n{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}^2}}+$

$\frac{2p{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}}{{\theta }^3{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}-\frac{3p^2{{\displaystyle {\sum }_{i=1}^n{\delta }_i\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}}^2}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}^2}}+\frac{2p^3{{\displaystyle {\sum }_{i=1}^n{\delta }_i\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}}^3}{{\theta }^3{\displaystyle {\sum }_{i=1}^n{\left[\left(1+{\left(\frac{t_i}{\theta }\right)}^p\right)\right]}^3}}$

If we let l₀₂ and l₀₃ represent the second and third derivatives of the log-likelihood function with respect to the shape parameter p, we will have

$l_{02}=-\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{p^2}-\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_i}{\theta }\right)}^p\text{In}{\left(\frac{t_i}{\theta }\right)}^2}{{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^2\text{In}{\left(\frac{t_i}{\theta }\right)}^2}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}-$$\frac{{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p\text{In}{\left(\frac{t_i}{\theta }\right)}^2}}{{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+\frac{{\left[{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}\right]}^2\text{In}{\left(\frac{t_i}{\theta }\right)}^2}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}$

$l_{03}=\frac{2{\displaystyle {\sum }_{i=1}^n{\delta }_i}}{p^3}-\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_i}{\theta }\right)}^p\text{In}{\left(\frac{t_i}{\theta }\right)}^3}{{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+\frac{3{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^2\text{In}{\left(\frac{t_i}{\theta }\right)}^2}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}-$

$\frac{2{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left[{\left(\frac{t_i}{\theta }\right)}^p\right]}^3\text{In}{\left(\frac{t_i}{\theta }\right)}^3}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^3}}-\frac{{\displaystyle {\sum }_{i=1}^n{\delta }_i}{\left(\frac{t_i}{\theta }\right)}^p\text{In}{\left(\frac{t_i}{\theta }\right)}^3}{{\displaystyle {\sum }_{i=1}^n\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}}+$$\frac{3{\left[{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}\right]}^2\text{In}{\left(\frac{t_i}{\theta }\right)}^3}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^2}}-\frac{2{\left[{\displaystyle {\sum }_{i=1}^n{\left(\frac{t_i}{\theta }\right)}^p}\right]}^3\text{In}{\left(\frac{t_i}{\theta }\right)}^3}{{\displaystyle {\sum }_{i=1}^n{\left[1+{\left(\frac{t_i}{\theta }\right)}^p\right]}^3}}$

Linear exponential loss function

This loss function measures the degree of overestimation and underestimation of the parameters being examined. Let k represent the shape parameter of the LINEX loss function. Refer to [13] for the posterior expectation of the LINEX loss function. The Bayes estimator $\widehat{u_{BL}}$ of a function u=u[exp(-kθ),exp(-kp)] under LINEX is given as

$\widehat{u_{BL}}=\frac{\int \int u{\pi }_1(\theta ){\pi }_2(p)L(\delta ,t_i;\theta ,p)d\theta dp}{\int \int {\pi }_1(\theta ){\pi }_2(p)L(\delta ,t_i;\theta ,p)d\theta dp}$

With Lindley’s approach, u₁,u₁₁ and u₂,u₂₂ are the first and second derivatives for θ and p respectively under the linear exponential loss function, hence

$u(\theta )=\exp (-k\theta ),u_1=\frac{\partial u}{\partial \theta }=-k\exp (-k\theta )$

$u_{11}=\frac{{\partial }^{2}u}{\partial {\theta }^2}=k^2\exp (-k\theta ),u_2=u_{22=0}$

$u(p)=\exp (-kp),u_2=\frac{\partial u}{\partial p}=-k\exp (-kp)$

$u_{22}=\frac{{\partial }^{2}u}{\partial p^2}=k^2\exp (-kp),u_1=u_{11}=0$

Real Data Analysis

Example 1

The data for this example are from survival of patients with cervical cancer recruited to a randomised clinical trial that was aimed at analysing the effect of an addition of a radio sensitizer to radiotherapy (New therapy- “treatment B”) compared to using radiotherapy alone (Control - “treatment A”). Treatment A and B were given to 16 and 14 patients respectively. The data are in days since the start of the study, the event of interest was death caused by this cancer. Our interest is on patients under treatment A to illustrate the proposed methods in this paper. The data is obtained from [22], and asterisked observations are censored.

Using the iterative procedure suggested in this paper and basing on comparison criterion on standard errors as well as their average confidence/credible lengths, we have for the MLEs of $\widehat{\theta }$ and $\widehat{p}$ to be 770.5429 and 1.90488 with their corresponding standard errors as 48.15893 and 0.11906 respectively. Since we do not have any prior information on the hyper-parameters, we assume a = b = c = d = 0.0001. The Bayes estimators under squared error loss for $\widehat{\theta }$ and $\widehat{p}$ have respectively the following parameters estimates and standard errors, 770.5429, 1.90206 and 48.15893, 0.11888.

Computing the Bayes estimates of $\widehat{\theta }$ and $\widehat{p}$ and that of their standard errors via the linear exponential loss function with a loss parameter of 0:7 we have, 859.7094, 1.78586 and 53.73182, 0.11162. With the loss parameter of 0:7, we have, 909.4092, 1.82677 and 56.83807, 0.11417 respectively.

What has been observed here is, both the maximum likelihood and Bayes under squared error loss function have the same scale parameter estimates and standard errors which are smaller than that of Bayes under the linear exponential loss function. For the shape parameter, Bayes under LINEX loss function with the loss parameter of 0.7 has the smallest standard error. This implies that overestimation is more serious than underestimation.

Considering a 95% confidence interval under MLE, we have $\widehat{\theta }$ = (679.1514,864.9344) and that of $\widehat{p}$ = (1.67153, 2.13823). The Bayesian credible intervals via the squared error loss function for $\widehat{\theta }$ and $\widehat{p}$ are (679.1514, 864.9344) and (1.66906, 2.13506) respectively. The Bayes credible intervals with respect to the linear exponential loss function with a loss parameter of 0.7 for $\widehat{\theta }$ and $\widehat{p}$ are (754.3950, 965.2380) and (1.56709, 2.00463) and that of the 0:7 are (798.0065, 1020.8120) and (1.60299, 2.05055) respectively.

Observing from above, LINEX loss function with a positive loss parameter had narrower credible intervals as compared to squared error loss function and maximum likelihood for the shape parameter. For the scale parameter, maximum likelihood’s confidence interval and Bayes credible interval with squared error loss were narrower than Bayes using LINEX.

Example 2

In this example, we analyse another data set which is considered moderate to obtain the parameter estimates and their standard errors in order to compare the methods employed in this paper. The data shown in Table 6, Example 2, are obtained from [22] and refer to remission times, in weeks, for a group of 30 patients with leukaemia who received similar treatment. Asterisks denote censoring times.