hpstat/docs/turnbull.tex

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\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 $\hat{F}(t)$ decreases only on the set of what are now called ‘Turnbull intervals’, or ‘innermost intervals’, $(q_j, p_j]$ for $j = 1, 2, …, m$.
	
	Let $s_j$ be the probability of failure within the interval $(q_j, p_j]$. We seek a maximum likelihood estimator for the vector $\symbf{s} = (s_1, s_2, …, s_m)^\mathrm{T}$.
	
	Take the $i$-th observation, $1 ≤ i ≤ n$ , whose failure time falls in $(L_i, R_i]$. Let $α_{i,j} = \mathrm{I}\left((q_j, p_j] \subseteq (L_i, R_i]\right)$.
	
	As discussed by Turnbull [1], noting that we consider only the case of no truncation, we commence with an arbitrary initial guess for $\hat{\symbf{s}}$, and iteratively apply:
	%
	\begin{align*}
		μ_{ij}(\hat{\symbf{s}}) &= \frac{α_{i,j} \hat{s}_j}{\sum_{k=1}^m α_{i,k} \hat{s}_k} \\
		π_j(\hat{\symbf{s}}) &= \frac{\sum_{i=1}^n μ_{ij}(\hat{\symbf{s}})}{n} \\
		\hat{s}_j &\leftarrow π_j(\hat{\symbf{s}}), \qquad \text{for all $j = 1, 2, …, m$}
	\end{align*}
	%
	This yields the maximum likelihood estimator $\hat{\symbf{s}}$.
	
	Now let $\hat{F}_0 = 0 ≤ \hat{F}_1 ≤ \hat{F}_2 ≤ … ≤ \hat{F}_m = 1$ be the values of $\hat{F}(t)$ outside the Turnbull intervals, such that $\hat{s}_j = \hat{F}_j - \hat{F}_{j-1}$. We seek the standard errors of these $\hat{\symbf{F}} = (\hat{F}_1, \hat{F}_2, …, \hat{F}_{m-1})^\mathrm{T}$.
	%
	Note that the log-likelihood $\mathcal{L}_i$ for the $i$-th observation is:
	%
	\begin{align*}
		\mathcal{L}_i &= \log\left(\sum_{j=1}^m α_{i,j} \hat{s}_j\right) \\
		&= \log\left(\sum_{j=1}^m 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}^m 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}^m 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 [2].
	
	%{\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 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}
	
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