Estimation of the Average Treatment Effect among Responders for Survival Outcomes

```
RATE.surv(
response,
post.treatment,
treatment,
censoring,
tau,
data,
M = 5,
pr.treatment,
call.response,
args.response = list(),
SL.args.post.treatment = list(family = binomial(), SL.library = c("SL.mean", "SL.glm")),
call.censoring,
args.censoring = list(),
preprocess = NULL,
...
)
```

## Arguments

- response
Response formula (e.g., Surv(time, event) ~ D + W).

- post.treatment
Post treatment marker formula (e.g., D ~ W).

- treatment
Treatment formula (e.g., A ~ 1).

- censoring
Censoring formula (e.g., Surv(time, event == 0) ~ D + A + W)).

- tau
Time-point of interest, see Details.

- data
data.frame.

- M
Number of folds in cross-fitting (M=1 is no cross-fitting).

- pr.treatment
(optional) Randomization probability of treatment.

- call.response
Model call for the response model (e.g. "mets::phreg").

- args.response
Additional arguments to the response model.

- SL.args.post.treatment
Additional arguments to SuperLearner for the post treatment indicator model.

- call.censoring
Similar to call.response.

- args.censoring
Similar to args.response.

- preprocess
(optional) Data pre-processing function.

- ...
Additional arguments to lower level data pre-processing functions.

## Details

Estimation of
$$
\frac{P(T \leq \tau|A=1) - P(T \leq \tau|A=1)}{E[D|A=1]}
$$
under right censoring based on plug-in estimates of \(P(T \leq \tau|A=a)\) and \(E[D|A=1]\).

An efficient one-step estimator of \(P(T \leq \tau|A=a)\) is constructed using
the efficient influence function
$$
\frac{I\{A=a\}}{P(A = a)} \Big(\frac{\Delta}{S^c_{0}(\tilde T|X)} I\{\tilde T \leq \tau\} + \int_0^\tau \frac{S_0(u|X)-S_0(\tau|X)}{S_0(u|X)S^c_0(u|X)} d M^c_0(u|X))\Big)\\
+ \Big(1 - \frac{I\{A=a\}}{P(A = a)}\Big)F_0(\tau|A=a, W) - P(T \leq \tau|A=a).
$$
An efficient one-step estimator of \(E[D|A=1]\) is constructed using the efficient influence function
$$
\frac{A}{P(A = 1)}\left(D-E[D|A=1, W]\right) + E[D|A=1, W] -E[D|A=1].
$$

## Author

Andreas Nordland, Klaus K. Holst