\hfill \thepage} %} \input{tcilatex} \begin{document} \title{Weibull regression in censored survival analysis; The kidney data} \author{} \maketitle A typical setting in survival analysis is that we observe the time point $t$ at which the death of a patient occurs. Patients may leave the study (for some reason) before they die. In this case the survival time is said to be censored, and $t$ refers to the time point when the patient left the study. The indicator variable $\delta $ is used to indicate whether $t$ refers to the death of the patient ($\delta =1$) or a censoring event ($\delta =0$). The key quantity in modelling the probability distribution of $t$ is the hazard function $h(t)$, which measures the instantaneous death rate at time $% t$. We also define the cumulative hazard function $\Lambda (t)=\int_{0}^{t}h(s)ds$, implicitly assuming that the study started at time $% t=0$. The loglikelihood contribution from our patient is $\delta \log (h(t))-H(t)$. A commonly used model for $h(t)$ is Cox's proportional hazard model, in which the hazard rate for the $i$th patient is assumed to be on the form% \[ h_{i}(t)=h_{0}(t)\exp (\eta _{i}\mathbf{),\qquad }i=1,\ldots n. \]% Here, $h_{0}(t)$ is the ``baseline'' hazard function (common to all patients) and $\eta _{i}=\mathbf{X}_{i}\mathbf{\beta }$, where $\mathbf{X}% _{i}$ is a covariate vector specific to the $i$th patient and $\mathbf{\beta }$ is a vector of regression parameters. In this example we shall assume that the baseline hazard belongs to the Weibull family: $h_{0}(t)=rt^{r-1}$ for $r>0$. In the collection of examples following the distribution of WinBUGS this model is used to analyse a dataset on times to kidney infection for a set of $n=38$ patients (`Kidney:~Weibull~regression~with~random~effects', Examples Volume 1, WinBUGS 1.4). The dataset contains two observations per patient (the time to first and second recurrence of infection). In addition there are three covariates:\ `age' (continuous), `sex' (dichotomous) and `type of disease' (categorical, four levels), and an individual spesific random effect $u_{i}\sim N(0,\sigma ^{2})$. Thus, the linear predictor becomes% \begin{equation} \eta _{i}=\beta _{0}+\beta _{\text{sex}}\,\cdot \text{sex}_{i}+\beta _{\text{% age}}\,\cdot \text{age}_{i}+\mathbf{\beta }_{\text{D}}\,\mathbf{x}_{i}+u_{i}, \label{eta_survival} \end{equation}% where $\mathbf{\beta }_{\text{D}}=(\beta _{1},\beta _{2},\beta _{3})$ and $% \mathbf{x}_{i}$ is a dummy vector coding for the disease type. Parameter estimates are shown in Table \ref{tab:survival}. Posterior means as calculated by BUGS are also shown in the table, and are similar to the maximum likelihood estimates. \begin{center} $% \begin{tabular}{lllllllll} & $\beta _{0}$ & $\beta _{\text{age}}$ & $\beta _{1}$ & $\beta _{2}$ & $% \beta _{3}$ & $\beta _{\text{sex}}$ & $r$ & $\sigma $ \\ \hline Laplace appr. & -4.344 & 0.003 & 0.1208 & 0.6058 & -1.1423 & -1.8767 & 1.1624 & 0.5617 \\ Std. dev. & 0.872 & 0.0137 & 0.5008 & 0.5011 & 0.7729 & 0.4754 & 0.1626 & 0.297 \\ BUGS & -4.6 & 0.003 & 0.1329 & 0.6444 & -1.168 & -1.938 & 1.215 & 0.6374 \\ Std. dev. & 0.8962 & 0.0148 & 0.5393 & 0.5301 & 0.8335 & 0.4854 & 0.1623 & 0.357% \end{tabular}% $ \end{center} \end{document}