# Computational formulas in cifcurve()

## Formulas for the Kaplan–Meier estimator

Let \(t_1 < t_2 < \cdots < t_D\) represent the distinct event times. For each event time \(j=1,\ldots,D\), let \(Y_j\) be the size of the risk set (the number of surviving observations) just prior to \(t_j\).

Survival data is assumed when `outcome.type = "survival"`. Let \(d_j\) be the number of failures at \(t_j\). The Kaplan–Meier estimator of the survival function \(S(t)\) is as below.

Kaplan–Meier estimator

\[ \hat{S}(t) = \prod_{t_j \le t} \left(1 - \frac{d_j}{Y_j} \right) \] Note that the estimator is defined to be right-continuous, so the events at \(t_j\) are included in the estimate of \(\hat{S}(t_j)\).

The variance or standard error of the Kaplan–Meier estimator is often calculated with the Greenwood formula. This formula is derived from a binomial argument, so extension to the weighted case is ad hoc. Alternatively, Tsiatis (1981) proposes a slightly different formula based on a counting process argument which includes the weighted case.

Greenwood variance \[ \textrm{Var}\!\left\{\hat{S}(t)\right \} = \hat{S}(t)^2\sum_{t_j \le t} \frac{d_j}{Y_j\,(Y_j - d_j)} \]

Tsiatis variance \[ \textrm{Var}\!\left\{\hat{S}(t)\right \} = \hat{S}(t)^2\sum_{t_j \le t} \frac{d_j}{Y_j^2} \]

Xie and Liu (2005) considered Greenwood-type variance for the weighted Kaplan–Meier estimator. Suppose that the survival data consists of \(\{T_i, d_i, w_i\}\) (\(i=1,...,N\)), an independent sample of right-censored survival data with weights. Let \(d_j^w\) and \(Y_j^w\) be the weighted number of failures and the weighted number at risk, respectively, at time \(t_j\). The weighted Kaplan–Meier estimator of the survival function \(S(t)\) is

\[ \hat{S}(t) = \prod_{t_j \le t} \left( 1 - \frac{d_j^w}{Y_j^w} \right). \] The variance proposed by Xie and Liu (2005) is expressed as an adjusted Greenwood formula,

Greenwood variance for weighted Kaplan–Meier \[ \textrm{Var}\!\left\{\hat{S}(t)\right \} = \hat{S}(t)^2\sum_{t_j \le t} \frac{d_j^w Y_j}{M_j Y_j^w(Y_j^w - d_j^w)}, \] where \(M_j\) is an adjustment factor defined as \[ M_j = \frac{\left(\sum_{t_i \geq t_j} w_i \right)^2}{\sum_{t_i \geq t_j} w_i^2}. \] The Tsiatis-type variance is calculated as follows in the same spirit.

Tsiatis variance for weighted Kaplan–Meier \[ \textrm{Var}\!\left\{\hat{S}(t)\right \} = \hat{S}(t)^2\sum_{t_j \le t} \frac{d_j^w Y_j}{M_j (Y_j^w)^2} \]

## Formulas for the Aalen-Johansen estimator

Competing risks (`outcome.type = "competing-risk"`) arise in studies in which individuals are exposed to two or more mutually exclusive failure events. When a failure occurs, we observe the time to event \(T\) and the cause of failure \(\epsilon\). Suppose that \(\epsilon=1\) and \(\epsilon=2\) represent the event of interest and the competing risk, respectively. Let \(d_{jk}\) be the number of failures of cause \(k\) at time \(t_j\), and now the total number of failures at \(t_j\) is \(d_j = d_{j1} + d_{j2}\).

The Aalen-Johansen estimator of CIF for cause \(\epsilon=k\) is as below.

Aalen-Johansen estimator

\[ \hat{F}_k(t) = \sum_{t_j \le t} \frac{d_{jk}}{Y_j}\,\hat{S}(t_{j-1}). \]

where \(\hat{S}(t)\) is the overall survival function.

Two variance estimators of the Aalen-Johansen estimator are commonly used: one based on counting process theory (Aalen, 1978) and the other based on the delta method.

Aalen variance \[ \begin{aligned} \textrm{Var}\!\left\{\hat{F}_k(t)\right \} &= \sum_{t_j \le t} \bigl[\hat{F}_k(t) - \hat{F}_k(t_j)\bigr]^2 \frac{d_j}{(Y_j-1)(Y_j - d_j)} \\[2pt] &\quad + \sum_{t_j \le t} \hat{S}^2(t_{j-1}) \frac{d_{jk}\,(Y_j - d_{jk})}{Y_j^2\,(Y_j-1)} \\[2pt] &\quad - 2 \sum_{t_j \le t} \bigl[\hat{F}_k(t) - \hat{F}_k(t_j)\bigr]\, \hat{S}(t_{j-1}) \frac{d_{jk}\,(Y_j - d_{jk})}{Y_j\,(Y_j - d_j)\,(Y_j-1)}. \end{aligned} \]

Delta method variance \[ \begin{aligned} \textrm{Var}\!\left\{\hat{F}_k(t)\right \} &= \sum_{t_j \le t} \bigl[\hat{F}_k(t) - \hat{F}_k(t_j)\bigr]^2 \frac{d_j}{Y_j\,(Y_j - d_j)} \\[2pt] &\quad + \sum_{t_j \le t} \hat{S}^2(t_{i-1}) \frac{d_{jk}\,(Y_j - d_{jk})}{Y_j^3} \\[2pt] &\quad - 2 \sum_{t_j \le t} \bigl[\hat{F}_k(t) - \hat{F}_k(t_j)\bigr]\, \hat{S}(t_{j-1}) \frac{d_{jk}}{Y_j^2}. \end{aligned} \]

Variance based on influence functions

It is known that the Aalen-Johansen estimator can be expanded under regularity conditions as
\[ n^{1/2}\{\hat F_k(t) - F_k(t)\} = n^{-1/2} \sum_{i=1}^n IF_{ik}(t) + o_p(1) \] and the process \(n^{1/2}\{\hat F_{ik}(t) - F_{ik}(t)\}\) converges weakly to a tight Gaussian process. Here \(IF_{ik}(t)\) is the influence function, the contribution of \(i\)-th observation to the Aalen-Johansen estimator, and may be written as \[ IF_{ik}(t) = \int_0^t \frac{n S(u^-)}{Y(u)}\,dM_i(u) - \int_0^t \frac{n F_k(u^-)}{Y(u)}\,dM_{ik}(u), \]

where \(M_i(t)\) and \(M_{ik}(t)\) is the Martingale process of the total count and the count of cause \(k\) of \(i\)-th observation, respectively, and \(Y(t)\) is the at-risk process. A consistent variance estimator for \(n^{1/2}\{\hat F_k(t) - F_k(t)\}\) is \(n^{-1} \sum_{i=1}^n \{\widehat{IF}_{ik}(t)\}^2\).

## Confidence interval options

Standard errors The default in `cifcurve()` with `weights=NULL` is the Greenwood SE when `outcome.type="survival"` and the delta SE when `outcome.type="competing-risk"`. The default in `cifcurve()` with weights is the SE based on influence functions. By default `cifcurve()` rescales the Greenwood/Tsiatis quantities so that `std.err` is reported on the probability scale; set `report.survfit.std.err = TRUE` to return the conventional log-survival SEs from `survival::survfit()`.

Confidence intervals `cifcurve()` constructs intervals on the probability scale using the requested transformation: `"arcsine-square root"`/`"arcsin"`/`"a"` (default), `"plain"`, `"log"`, `"log-log"`, or `"logit"`. Passing `"none"`/`"n"` skips interval computation entirely. The function exponentiates back to the probability scale, clips bounds to [0, 1], and replaces undefined values with `NA` so that interval endpoints remain well behaved in plots and summaries.