Computationally efficient estimation of the influence function for Kaplan-Meier

In riskRegression: Risk Regression Models and Prediction Scores for Survival Analysis with Competing Risks

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We assume that the event times are sorted and possibly tied such that $\tilde{T}{1} < \ldots < \tilde{T}{i} = \ldots = \tilde{T}{i+k} < \tilde{T}{i+k+1} < \ldots < \tilde{T}n$. We use the following algorithm to preserve memory and the number of iterations for say $\mu^{(i)} = \int 1{{t \leqslant \tau, d=1}} \frac{f_i(t-)}{G(t-)}P(dz)$. The idea is to split the sum into two terms: \begin{gather} \frac{1}{n} \sum_{j=1}^n \frac{\hat{f_i}(\tilde{T}j - )1{ { \tilde{T}j \leq \tau, \Delta_j =1 }}}{\hat{G}(\tilde{T_j}-)} =\ \frac{1}{n} \left(\sum{j=2}^{i+k} \frac{g(j) 1_{ { \tilde{T}j \leq \tau, \Delta_j =1 }}}{\hat{G}(\tilde{T_j}-)} + h(i) \sum{j=i+k+1}^n \frac{1_{ { \tilde{T}_j \leq \tau, \Delta_j =1 }}}{\hat{G}(\tilde{T_j}-)} \right) \end{gather}

since $\hat{f_i}(\tilde{T}j - )$ only depends on $i$ for $i+k > j$ and only depends on $j$ for $i+k\leq j$, so these values are calculated a priori. Also the first term will always be zero, since we are looking at the value of the integral before any observed event (hence the sum starts at $j=2$). One can check in the estimation of the Influence Curve for the censoring, which does not depend on the covariates that we need to calculate $2n$ values (i.e. $n$ values for $g(i)$ and $n$ for $h(j)$). This is how we can avoid memory issues. The algorithm is: \begin{algorithm}[H] \DontPrintSemicolon \BlankLine $t := 1$ \; $\hat{\mu}{2}:= \sum_{j=1}^n \frac{1_{ { \tilde{T}j \leq \tau, \Delta_j =1 }}}{\hat{G}(\tilde{T_j}-)}$ \; \While{$\tilde{T}_1 = \tilde{T}_t $ and $t \leq n$ } { \If{$\tilde{T}_t \leq \tau$ and $\Delta_t = 1$}{ $\hat{\mu}{2} = \hat{\mu}{2} - \frac{1}{G(\tilde{T}_t-)}$ \; } $t = t + 1$ \; } $tieEnd := t-1$ \; $\hat{\mu}{1}:=0$ \; \For{$i = 1$ to $n$}{ $\hat{\mu}^{(i)}=\frac{1}{n} \left(\hat{\mu}{1}+ h(i) \hat{\mu}{2} \right) $\; \If{$tieEnd \leq i$}{ $t = i+1$ \; \While{$\tilde{T}1 = \tilde{T}_t $ and $t \leq n$ } { \If{$\tilde{T}_t \leq \tau$ and $\Delta_t = 1$}{ $\hat{\mu}{2} = \hat{\mu}{2} - \frac{1}{G(\tilde{T}_t-)}$ \; $\hat{\mu}{1} = \hat{\mu}{1} + \frac{g(t) 1{ { \tilde{T}{t} \leq \tau, \Delta{t} =1 }}}{\hat{G}(\tilde{T}_{t}-)}$\; } Let $t = t + 1$ \; } } Let $tieEnd = t-1$ \; } return $\hat{\mu}^{(i)}$ for each $i = 1, \ldots, n$ \; \end{algorithm}

The idea is that but we keep on adding and subtracting the terms with tied values in the event times. Then we do not need to calculate a sum for each $i$.

riskRegression documentation built on Sept. 8, 2023, 6:12 p.m.