hpstat/docs/turnbull.tex

\documentclass[a4paper,12pt]{article}

\usepackage[math-style=ISO, bold-style=ISO]{unicode-math}
\setmainfont[RawFeature=-tlig]{TeX Gyre Termes}
\setmathfont{TeX Gyre Termes Math}

\usepackage{parskip}
\usepackage{microtype}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\frenchspacing
\setlength{\emergencystretch}{3em}

\usepackage[hidelinks]{hyperref}
\usepackage{mathtools}

\newcommand{\bbeta}{\kern -0.1em\symbf{β}}
\newcommand{\blambda}{\kern -0.1em\symbf{Λ}}
\newcommand{\nablasub}[1]{\nabla_{\kern -0.15em #1}}

\begin{document}
	{\centering\bfseries Supplemental documentation for hpstat \textit{turnbull} command\par}
	
	The hpstat \textit{turnbull} command implements Turnbull's nonparametric survival curve estimation for interval-censored observations [1]. This documentation discusses technical details of the implementation.
	
	Let $\hat{F}(t)$ be a maximum likelihood estimator for the cumulative distribution function for failure times. Turnbull [1] demonstrated that $\hat{F}(t)$ increases only on the set of what are now called ‘Turnbull intervals’, or ‘innermost intervals’, $T_j$ for $j = 1, 2, …, k$.
	
	Let $p_j$ be the probability of failure within the interval $T_j$. We seek a maximum likelihood estimator for the vector $\symbf{p} = (p_1, p_2, …, p_k)^\mathrm{T}$. We apply an efficient expectation maximisation–iterative convex minorant (EM-ICM) algorithm described by Anderson-Bergman [2] to find $\hat{\symbf{p}}$.
	
	Now take the $i$-th observation, $1 ≤ i ≤ n$, whose failure time falls in $O_i$, and let $α_{i,j} = \mathrm{I}\left(T_j \subseteq O_i\right)$. Let $\hat{F}_0 = 0 ≤ \hat{F}_1 ≤ \hat{F}_2 ≤ … ≤ \hat{F}_k = 1$ be the values of $\hat{F}(t)$ outside the Turnbull intervals, such that $\hat{p}_j = \hat{F}_j - \hat{F}_{j-1}$. We seek the standard errors of these $\hat{\symbf{F}} = (\hat{F}_1, \hat{F}_2, …, \hat{F}_{k-1})^\mathrm{T}$.
	%
	Note that the log-likelihood $\mathcal{L}_i$ for the $i$-th observation is:
	%
	\begin{align*}
		\mathcal{L}_i &= \log\left(\sum_{j=1}^k α_{i,j} \hat{p}_j\right) \\
		&= \log\left(\sum_{j=1}^k α_{i,j} (\hat{F}_j - \hat{F}_{j-1})\right)
	\end{align*}
	%
	Note the gradient $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ is the vector whose $h$-th element is:
	%
	\begin{align*}
		\frac{\partial \mathcal{L}_i}{\partial \hat{F}_h} &= \frac{α_{i,h} - α_{i,h+1}}{\sum_{j=1}^k α_{i,j} (\hat{F}_j - \hat{F}_{j-1})}
	\end{align*}
	%
	And so the Hessian $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ has $(h, k)$-th elements:
	%
	\begin{align*}
		\frac{\partial \mathcal{L}_i}{\partial \hat{F}_h \partial \hat{F}_k} &= - \frac{( α_{i,h} - α_{i,h+1} ) ( α_{i,k} - α_{i,k+1} )}{\left( \sum_{j=1}^k α_{i,j} (\hat{F}_j - \hat{F}_{j-1}) \right)^2}
	\end{align*}
	%
	The sum of all $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ yields the Hessian of the log-likelihood $\nablasub{\hat{\symbf{F}}} \mathcal{L}$.
	
	The covariance matrix of $\hat{\symbf{F}}$ is given by the inverse of $-\nablasub{\hat{\symbf{F}}} \mathcal{L}$. The standard errors for each of $\hat{\symbf{F}}$ are the square roots of the diagonal elements of the covariance matrix, as required.
	
	Alternatively, when \textit{--se-method oim-drop-zeros} is passed, columns/rows of $\nablasub{\hat{\symbf{F}}} \mathcal{L}$ corresponding with intervals where $\hat{s}_i = 0$ are dropped before the matrix is inverted, which enables greater numerical stability but whose theoretical justification is not well explored [3].
	
	In the further alternative, when \textit{--se-method likelihood-ratio} is passed, confidence intervals for $\hat{\symbf{F}}$ are computed by inverting a likelihood ratio test at each point, as described by Goodall, Dunn \& Babiker~[3].
	
	{\vspace{0.5cm}\scshape\centering References\par}
	%{\pagebreak\scshape\centering References\par}
	
	\begin{enumerate}
		\item Turnbull BW. The empirical distribution function with arbitrarily grouped, censored and truncated data. \textit{Journal of the Royal Statistical Society, Series B (Methodological)}. 1976;38(3):290–5. \href{https://doi.org/10.1111/j.2517-6161.1976.tb01597.x}{doi: 10.1111\slash j.2517-6161.1976.tb01597.x}
		\item Anderson-Bergman C. An efficient implementation of the EMICM algorithm for the interval censored NPMLE. \textit{Journal of Computational and Graphical Statistics}. 2017;26(2):463–7. \href{https://doi.org/10.1080/10618600.2016.1208616}{doi: 10.1080\slash 10618600.2016.1208616}
		\item Goodall RL, Dunn DT, Babiker AG. Interval-censored survival time data: confidence intervals for the non-parametric survivor function. \textit{Statistics in Medicine}. 2004;23(7):1131–45. \href{https://doi.org/10.1002/sim.1682}{doi: 10.1002\slash sim.1682}
	\end{enumerate}
	
\end{document}