A Bayesian via Laplace Approximation on Log-gamma Model with Censored Data

Log-gamma distribution is the extension of gamma distribution which is more flexible, versatile and provides a great fit to some skewed and censored data. Problem/Objective: In this paper we introduce a solution to closed forms of its survival function of the model which shows the suitability and flexibility towards modelling real life data. Methods/Analysis: Alternatively, Bayesian estimation by MCMC simulation using the Random-walk Metropolis algorithm was applied, using AIC and BIC comparison makes it the smallest and great choice for fitting the survival models and simulations by Markov Chain Monte Carlo Methods. Findings/Conclusion: It shows that this procedure and methods are better option in modelling Bayesian regression and survival/reliability analysis integrations in applied statistics, which based on the comparison criterion log-gamma model have the least values. However, the results of the censored data have been clarified with the simulation results.


Introduction
Bayesian method approach is applied to model censored Survival data analysis its increasingly active research in the last few decades in response to a more refined statistical tools to analysed complex data structures and parameters (Lindley & Smith, 1994).This method is applied to the log-gamma model analytically simulates the model parameters which approximates generally by obtaining the posterior summaries of the density parameters using "LaplacesDemon" package in R software.
The shorthand X~log-gamma (a,b) is used to indicate that the random variable.The Log-gamma distribution (Consul & Jain, 1971) is defined in the following way having a probability density function (PDF) given as: (1) Survival function is given as: (2) Corresponding the reliability function: he obtaining the aplace's meth Approximation ha is given by: Otherwise, "a" tion, as a weak ended ution on a standard deviation parameter.Value, The half-Cauchy distribution does not has mean and variance, but its mode is equal to 0 having the" a=25"as a default.(Akhtar, 2014;Bernardo, 1980;Bilal, Khan, Hasan & Khan, 2003) Suggested, the uniform prior distribution, where its compulsory in estimation but half-Cauchy is a better option used as a non-informative prior (Polson & Scott) showing its graph below as follows.

Bayesian Analysis: Simulation with Laplace's Demon
Based on some reviews in the area of approximating a Laplace distribution in the literature which has a very effective response for decades and also, in recent years based on Log-gamma estimation of parameters using different approach like Bayes estimate, MLE, Lindley, Newton Raphson's method of optimization etc. (Akhtar, 2014;Bilal, Khan, Hasan & Khan, 2003;Khan & Bhat, 2002;Khan & Khan, 2013).Actually to find the posterior results summaries of such functions with their mean and variances, it is a very intricate case to handle, more especially when more covariate were involved as incorporate variables.In such cases, we use the Bayesian frame-work approach using the Metropolis-Hastings sampling algorithm in MCMC methods to solve and find the posterior result.
As an alternative method to solve intricate integrals using simulation technique by direct method of simulation suggested by (Buckley & James, 1979, Kimber, 1990), in intricate purposes where by MCMC methods is used.

Application of Censored Data
The Log-gamma distribution as a parametric family is however used in censored survival modelling, with two parameters, shape and scale parameter.We analyze a data from (R Development Core Team, 2012), known as the leukaemia data having 23 observations with three (3) variables of observation namely: time, status and group (R Development Core Team, 2012).

Model Specification
The Log-gamma model with two parameters alpha and beta is also has almost same properties with the original gamma model stated as its suit the continuous and skewed data having a Weibull model property which is one of its sub-model and also fits a wide range failure-time data quite well.On the other side, it has a very good relationship with its sub-models and also enhances the use of its advantages in-terms of the identically independently distributed (iid) for some exponential variables in inferential statistics (Collet, 1997;Koul, Susarla & Van, 1981).Akaike (1974) suggested and introduced a suitable vast criterion (AIC) with some assumptions attached: (a) A parametric distribution encompasses a true model.
(b) Its estimate using MLE and other methods, where the least value becomes the best model for selection (Akaike, 1974), which is given given by: Schwarz ( 1978) also proposed the BIC criterion following some assumptions that render great impact to statistical methodology as:  The above table 4 shows the result of comparison between the sub models which indicates the Log-gamma model is having the smallest value among them clearly not prove to be the best model but based on the survival data used it makes it superior and better fit.

Conclusion
In this research we proposed an Rcode base on simulating and estimating censored survival data and initiate the use of R package Laplace's Demon (Khan & Bhat, 2002) that makes a great impact in Bayesian statistical inference.The log-gamma distribution was used as a Bayesian model to fit the censored data and simulation, where by important techniques were used like: Asymptotic approximation and direct simulation were implemented using the R package LaplacesDemon (Khan & Bhat, 2002).Also, the simulation results shows that the Mean square error of log-gamma model is least compare to other sub-models like (Weibull and gamma models) as well as the AIC and BIC with (42.140 and 41.134) making it the smallest and great choice for fitting the survival models and simulations by Markov Chain Monte Carlo Methods.It shows that this procedure and methods are better option in modelling Bayesian regression and survival/reliability analysis integrations in applied statistics (Lindley & Smith,
(a) It has a constant independent prior vague.(b) It checks the efficiency and complexity of the parameterized model in terms of intricacy.(c) BIC [21], has a very close relation to AIC [2], in terms of model selection.The Bayesian Information Criterion is formally defined as = −2ln + ln (13) Where, L= the likelihood function of the estimated model.x= the observe dataset.n= the number of samples.k= the number of free parameters to be estimated.

Table 1 .
A t.org

Table 3 .
Posterior Mean Summaries for the Parameter Estimated By Simulation Using the Sampling Technique and Stationary Samples

Table 4 .
Comparison of Parametric sub-models with Log-gamma based on(AIC and BIC)