# Function to estimate the quantile-optimal treatment regime: the independent censoring Case This function implements the estimation method proposed in Chapter 2 of if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="zhou2018quantile",package="QTOCen",cached_env=.Rdpack.currefs) . It estimates the quantile-optimal treatment regime for a given quantile level of interest from a single-stage clinical randomized experiment or a single-stage observational study under the independent censoring assumption. In other words, we estimate the parameters indexing the quantile-optimal treatment regime. Our assumption of independent censoring means the distribution of the censoring time is the same conditional on baseline covariates, treatment group and the two potential survival times. ```r IPWE_Qopt_IndCen(data, regimeClass, tau, moPropen = "BinaryRandom", Domains = NULL, cluster = FALSE, p_level = 1, s.tol = 1e-04, it.num = 8, pop.size = 6000, sign_beta1 = NULL, Penalty.level = 0) ``` ## Arguments - `data`: a data.frame, containing variables in the `moPropen` and `RegimeClass` and also the response variables, namely `censor_y` as the censored response, and `delta` as the censoring indicator. - `regimeClass`: a formula specifying the class of treatment regimes to search, e.g. if `regimeClass = a~x1+x2`, and then this function will search the class of treatment regimes of the form $$ d(x) = I \left(\beta_0 +\beta_1 x_1 + \beta_2 x_2 > 0\right).d(x)=I(\beta_0 +\beta_1 * x1 + \beta_2 * x2 > 0). $$ Polynomial arguments are also supported. - `tau`: a value between 0 and 1. This is the quantile of interest. - `moPropen`: The propensity score model for the probability of receiving treatment level 1. When `moPropen` equals the string "BinaryRandom", the proportion of observations receiving treatment level 1 in the sample will be plugged in as an estimate of the propensity. Otherwise, this argument should be a formula/string, based on which this function will fit a logistic regression on the treatment level. e.g. `a1~x1`. - `Domains`: default is NULL. Otherwise, the object should be a `nvars *2` matrix used as the space of parameters, which will be supplied to `rgenoud::genoud`. - `cluster`: default is FALSE, meaning do not use parallel computing for the genetic algorithm(GA). - `p_level`: choose between 0,1,2,3 to indicate different levels of output from the genetic function. Specifically, 0 (minimal printing), 1 (normal), 2 (detailed), and 3 (debug). - `s.tol`: tolerance level for the GA algorithm. This is input for parameter `solution.tolerance` in function `rgenoud::genoud`. - `it.num`: the maximum GA iteration number - `pop.size`: an integer with the default set to be 3000. This is roughly the number individuals for the first generation in the genetic algorithm (`rgenoud::genoud`). - `sign_beta1`: logical. Default is NULL. FALSE if the coefficient for the first continuous variable is fixed to be negative one; TRUE if positive one. - `Penalty.level`: 0: stop if the marginal quantiles cannot be further optimized; 1: continue the search among treatment regimes with with same value for the TR with the smallest intended proportion of treatment. ## Details The input argument `data` is the dataframe that contains: 1. `a` observed treatment assignment 2. `censor_y` the censored response variable 3. `delta` the censoring indicator The naming of these three columns should be strict. Note that this function currently only works for scenarios in which treatment is binary. ## Examples ```r GenerateData <- function(n) { x1 <- runif(n, min=-0.5,max=0.5) x2 <- runif(n, min=-0.5,max=0.5) error <- rnorm(n, sd= 1) ph <- exp(-0.5+1*(x1+x2))/(1+exp(-0.5 + 1*(x1+x2))) a <- rbinom(n = n, size = 1, prob=ph) c <- 1 + 1*a + runif(n = n, min=0, max=2) cmplt_y <- pmin(2+x1+x2 + a*(1 - x1 - x2) + (0.2 + a*(1+x1+x2)) * error, 4.4) censor_y <- pmin(cmplt_y, c) delta <- as.numeric(c > cmplt_y) return(data.frame(x1=x1,x2=x2,a=a, censor_y = censor_y, delta=delta)) } n <- 400 data <- GenerateData(n) fit1 <- IPWE_Qopt_IndCen(data = data, regimeClass = a~x1+x2, tau=0.25) # We can used the returned model to visualize the Kaplan-meier # estimate of survival function of the censoring time variable, # justified by the independent censoring assumption. library(survminer) ggsurvplot(fit1$survfitCensorTime, data=fit1$data_aug, risk.table = TRUE) ``` ## References Rdpack::insert_ref(key="zhou2018quantile",package="QTOCen") Rdpack::insert_ref(key="horowitz1992smoothed",package="QTOCen")

IPWE_Qopt_IndCen function