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\begin{document}
\bibliographystyle{apalike}
{\Large Exponential  and Weibull Models}\\
\vspace{0.5cm}\\
We consider the  data of times to infection of kidney dialysis patients.
In a study \cite{art504}, (given in the book by \cite{book501})  designed to assess 
the time to first exitsite
infection (in months) in patients with renal insufficiency, 43 patients
utilized a surgically placed catheter (Group 1), and 76 patients utilized
a percutaneous placement of their catheter (Group 2), a total of 119 patients.\\
The variables represented in the data set are $\tt{time}$ to infection in months/$10$
denoted by $t$, infection indicator or $\tt{event}$ ($0$=no, $1$=yes) denoted by  $\delta$ and
 catheter $\tt{placement}$ (1=surgically, 2=percutaneously) denoted by  $trt$. We analyse the data set using exponential model and Weibull model.\\
The \textbf{exponential model} for this example can be specified as:
\begin{displaymath}
t_{i}\sim \mathit{E}(\lambda_{i})
\end{displaymath}
Where each survival time follows an exponential distribution with parameter $\lambda_{i}$  and  $i$ is from $1$ to $119$. 
For this example we have only one covariate, catheter placement ($trt$) 
and therefore $ \vect{\beta} = (\beta_{0},\beta_{1})^{'} $, where $\beta_{0}$ denotes the
intercept term and $\beta_{1}$ denotes the coefficient for the placement covariate ($trt$). Here, the latent field is
\begin{displaymath}
\lambda_{i}= \exp(\eta_{i}) 
\end{displaymath}
with
\begin{displaymath} 
\eta_{i}= \beta_{0}+ trt_{i}\beta_{1} 
\end{displaymath}
where both $\beta_{0}$ and $\beta_{1}$ are assign the following priors distributions
\begin{displaymath}
\beta_{0}\sim N(0,0.001)
\end{displaymath}
\begin{displaymath}
\beta_{1}\sim N(0,0.001)
\end{displaymath} 
There is no hyperparameter used in this model.\\
\\
The \textbf{Weibull model} for this example can be specified as:
\begin{displaymath}
t_{i}\sim \text{Weibull}(\alpha,\lambda_{i})
\end{displaymath}
Here also , the latent field is
\begin{displaymath}
\lambda_{i}= \exp(\eta_{i}) 
\end{displaymath}
with
\begin{displaymath} 
\eta_{i}= \beta_{0}+ trt_{i}\beta_{1} 
\end{displaymath}
where $\beta_{0}$ and $\beta_{1}$ are assign the following priors distributions
\begin{displaymath}
\beta_{0}\sim N(0,0.001)
\end{displaymath}
\begin{displaymath}
\beta_{1}\sim N(0,0.001)
\end{displaymath} 
The model has one hyperparameter, $\alpha$ , we assign the following  prior distribution
\begin{displaymath}
\alpha \sim \text{Gamma}(1, 0.001)
\end{displaymath} 
 
 
 
\bibliography{rupalibib} 
\end{document}
