% \documentclass[serif]{beamer} % Serif for Computer Modern math font. \documentclass[serif, handout]{beamer} % Handout mode to ignore pause statements \hypersetup{colorlinks,linkcolor=,urlcolor=red} \usefonttheme{serif} % Looks like Computer Modern for non-math text -- nice! \setbeamertemplate{navigation symbols}{} % Supress navigation symbols \usetheme{AnnArbor} % CambridgeUS Blue and yellow, Shows current section title % \usetheme{Berlin} % Blue: Displays section titles on top % \usetheme{Frankfurt} % Displays section titles on top: Fairly thin but still swallows some material at bottom of crowded slides \usepackage[english]{babel} % \definecolor{links}{HTML}{2A1B81} % \definecolor{links}{red} \setbeamertemplate{footline}[frame number] \mode % \mode{\setbeamercolor{background canvas}{bg=black!5}} \title{Censoring and Likelihood\footnote{See last slide for copyright information.}} \subtitle{STA312 Fall 2023} \date{} % To suppress date \begin{document} \begin{frame} \titlepage \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[fragile] \frametitle{Censored Data} %\framesubtitle{} \begin{itemize} \item Let $T^*$ represent time to failure. \item Let $U$ represent censoring time. \item We observe $T=\min(T^*,U)$. \item And an indicator for whether failure occurred: $\delta = 1$ if uncensored, and zero if censored. \item All this is $i=1, \ldots,n$, so we observe $n$ pairs $(T_1,\delta_1), (T_2,\delta_2), \ldots, (T_n,\delta_n)$. \pause \item A simple data file might look like this: \end{itemize} \begin{verbatim} Patient Time Uncensored (delta) 1 5 1 2 6 0 3 8 1 4 3 1 5 22 1 \end{verbatim} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[fragile] \frametitle{Simulation code} \framesubtitle{To illustrate the process} {\footnotesize % or scriptsize \begin{verbatim} # Exponential distribution, true parameter lambda, right censoring # Censoring times will be uniform(0,top) # Observation will be censored if censoring time is less than lifetime. rm(list=ls()); options(scipen=999) lambda = 1/5; top=20 # True parameters # Simulate set.seed(9999); n = 200 delta = numeric(n) # Indicator for uncensored, initially zero lifetime = rexp(n,rate=lambda) censortime = runif(n,0,top) # If censoring time is greater than lifetime, then it's NOT censored. delta[censortime>lifetime] = 1 # Minumum of censortime and lifetime is what we can observe. T = pmin(censortime,lifetime) # pmin is parallel minimum. round(cbind(lifetime,censortime,T,delta)[1:10,],2) # Take a look expodata = cbind(T,delta) # This is all you can see in practice. \end{verbatim} } % End size Note that censoring time is independent of lifetime, and they share no parameters. \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[fragile] \frametitle{Output} %\framesubtitle{} {\footnotesize % or scriptsize {\color{blue} \begin{verbatim} round(cbind(lifetime,censortime,T,delta)[1:10,],2) # Take a look \end{verbatim} } % End color \begin{verbatim} lifetime censortime T delta [1,] 2.29 1.38 1.38 0 [2,] 10.24 4.31 4.31 0 [3,] 1.89 13.23 1.89 1 [4,] 3.45 14.61 3.45 1 [5,] 9.77 17.82 9.77 1 [6,] 2.41 16.46 2.41 1 [7,] 3.18 13.83 3.18 1 [8,] 20.50 7.41 7.41 0 [9,] 8.64 8.53 8.53 0 [10,] 2.65 19.61 2.65 1 \end{verbatim} } % End size \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Likelihood function for censored data} \framesubtitle{Possibly right censored, with random censoring} {\LARGE \begin{eqnarray*} L(\theta) & = & \prod_{i=1}^n f(t_i|\theta)^{\delta_i} \, S(t_i|\theta)^{1-\delta_i} \\ \pause && ~\strut \\ \ell(\theta) & = & \sum_{i=1}^n \left(\strut \delta_i \log f(t_i|\theta) + (1-\delta_i) \log S(t_i|\theta) \right) \\ \pause &=& \sum_{i=1}^n \delta_i \log f(t_i|\theta) + \sum_{i=1}^n (1-\delta_i) \log S(t_i|\theta) \end{eqnarray*} } % End size \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Example: $T_1, \ldots, T_n \stackrel{i.i.d}{\sim} \exp(\lambda)$} \framesubtitle{$F(t|\lambda) = 1-e^{-\lambda t}$, Ordinary $\widehat{\lambda}_n = 1/\overline{T}_n$} \pause {\LARGE \begin{displaymath} L(\theta) = \prod_{i=1}^n \left(\lambda e^{-\lambda t_i}\right)^{\delta_i} \, \left(e^{-\lambda t_i}\right)^{1-\delta_i} \end{displaymath} \pause } % End size \vspace{3mm} As an exercise, you will show that {\Large $\displaystyle \widehat{\lambda}_n = \frac{\sum_{i=1}^n \delta_i}{\sum_{i=1}^n t_i}$. } \pause \vspace{2mm} \begin{itemize} \item So the estimated expected value is $\frac{\sum_{i=1}^n t_i}{\sum_{i=1}^n \delta_i} \pause > \frac{\sum_{i=1}^n t_i}{n} = \overline{T}_n$. \pause \item Ignoring the censoring would cause you to under-estimate average survival time. \end{itemize} \end{frame} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Copyright Information} This slide show was prepared by \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner}, Department of Statistics, University of Toronto. It is licensed under a \href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US} {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: \href{http://www.utstat.toronto.edu/brunner/oldclass/312f23} {\footnotesize \texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f23}} \end{frame} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{}\pause %\framesubtitle{} \begin{itemize} \item \pause \item \pause \item \end{itemize} \end{frame} %%%%%%%%%% I dropped this because it was not clear enough. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{What happened to the distribution of censoring time?} \pause %\framesubtitle{} {\LARGE \begin{displaymath} L(\theta) = \prod_{i=1}^n f(t_i|\theta)^{\delta_i} \, S(t_i|\theta)^{1-\delta_i} \end{displaymath} \pause } % End size \begin{itemize} \item $\delta_i$ depends on survival time and censoring time, which are both random. \pause \item We saw this in the simulation code. \pause \item So $\delta_i$ is a random variable. \pause \item Its distribution must depend on $\theta$ \pause and the parameters of censoring time. \pause \item But there is nothing like this in the likelihood function. \pause \item What is going on? \end{itemize} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Hidden assumptions} \framesubtitle{That make everything okay} \pause \begin{itemize} \item Let $\theta_1$ be the parameters of survival time \pause and $\theta_2$ be the parameters of censoring time. \pause \item $\theta = (\theta_1,\theta_2)$ \pause \item For an ordinary likelihood without censoring, $L(\theta) = \prod_{i=1}^n p(t_i|\theta) \pause = p_\theta(\mathbf{x})$. \pause \item With censoring, $L(\theta) = p_{(\theta_1,\theta_2)}(\mathbf{x},\boldsymbol{\delta})$ \pause \end{itemize} \begin{eqnarray*} L(\theta) & = & p_{(\theta_1,\theta_2)}(\mathbf{x},\boldsymbol{\delta}) \\ & = & p_{(\theta_1,\theta_2)}(\mathbf{x}|\boldsymbol{\delta}) \, p_{(\theta_1,\theta_2)}(\boldsymbol{\delta}) \\ & = & p_{\theta_1}(\mathbf{x}|\boldsymbol{\delta}) \, p_{\theta_2}(\boldsymbol{\delta}) \\ \end{eqnarray*} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%