Responder Average Treatment Effect — RATE.surv • targeted

Estimation of the Average Treatment Effect among Responders for Survival Outcomes

Usage

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: Arguments to SuperLearner for the post treatment indicator
call.censoring: Similar to call.response.
args.censoring: Similar to args.response.
preprocess: (optional) Data preprocessing function
...: Additional arguments to lower level functions

Value

estimate object

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