# Estimate the Mean-optimal Treatment Regime `IPWE_Mopt` aims at estimating the treatment regime which maximizes the marginal mean of the potential outcomes. ```r IPWE_Mopt(data, regimeClass, moPropen = "BinaryRandom", max = TRUE, s.tol = 1e-04, cl.setup = 1, p_level = 1, it.num = 10, hard_limit = FALSE, pop.size = 3000) ``` ## Arguments - `data`: a data frame, containing variables in the `moPropen` and `RegimeClass` and a component `y` as the response. - `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. See also 'Details'. - `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 employed as a good estimate of the probability for each observation. 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`. - `max`: logical. If `max=TRUE`, it indicates we wish to maximize the marginal mean; If `max=FALSE`, we wish to minimize the marginal mean. The default is `TRUE`. - `s.tol`: This is the tolerance level used by `genoud`. Default is $10^{-5}$ times the difference between the largest and the smallest value in the observed responses. This is particularly important when it comes to evaluating `it.num`. - `cl.setup`: the number of nodes. >1 indicates choosing parallel computing option in `rgenoud::genoud`. Default is 1. - `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.) - `it.num`: integer > 1. This argument will be used in `rgeound::geound` function. If there is no improvement in the objective function in this number of generations, `rgenoud::genoud` will think that it has found the optimum. - `hard_limit`: logical. When it is true the maximum number of generations in `rgeound::geound` cannot exceed 100. Otherwise, in this function, only `it.num` softly controls when `genoud` stops. Default is `FALSE`. - `pop.size`: an integer with the default set to be 3000. This is the population number for the first generation in the genetic algorithm (`rgenoud::genoud`). ## Returns This function returns an object with 6 objects. Both `coefficients` and `coef.orgn.scale` were normalized to have unit euclidean norm. - **`coefficients`**: the parameters indexing the estimated mean-optimal treatment regime for standardized covariates. - **`coef.orgn.scale`**: the parameter indexing the estimated mean-optimal treatment regime for the original input covariates. - **`hatM`**: the estimated marginal mean when a treatment regime indexed by `coef.orgn.scale` is applied on everyone. See the 'details' for connection between `coef.orgn.scale` and `coefficient`. - **`call`**: the user's call. - **`moPropen`**: the user specified propensity score model - **`regimeClass`**: the user specified class of treatment regimes ## Details Note that all estimation functions in this package use the same type of standardization on covariates. Doing so would allow us to provide a bounded domain of parameters for searching in the genetic algorithm. This functions returns the estimated parameters indexing the mean-optimal treatment regime under two scales. The returned `coefficients` is the set of parameters when covariates are all standardized to be in the interval [0, 1] by subtracting the smallest observed value and divided by the difference between the largest and the smallest value. While the returned `coef.orgn.scale` corresponds to the original covariates, so the associated decision rule can be applied directly to novel observations. In other words, let $\beta$ denote the estimated parameter in the original scale, then the estimated treatment regime is: $$ d(x)= I\{\hat{\beta}_0 + \hat{\beta}_1 x_1 + ... + \hat{\beta}_k x_k > 0\}.d(x)= I{\beta_0 + \beta_1*x_1 + ... + \beta_k*x_k > 0}. $$ The estimated $\beta$ is returned as `coef.orgn.scale`. If, for every input covariate, the smallest observed value is exactly 0 and the range (i.e. the largest number minus the smallest number) is exactly 1, then the estimated `coefficients` and `coef.orgn.scale` will render identical. ## Examples ```r GenerateData.test.IPWE_Mopt <- function(n) { x1 <- runif(n) x2 <- runif(n) tp <- exp(-1+1*(x1+x2))/(1+exp(-1+1*(x1+x2))) error <- rnorm(length(x1), sd=0.5) a <- rbinom(n = n, size = 1, prob=tp) y <- 1+x1+x2 + a*(3 - 2.5*x1 - 2.5*x2) + (0.5 + a*(1+x1+x2)) * error return(data.frame(x1=x1,x2=x2,a=a,y=y)) } n <- 500 testData <- GenerateData.test.IPWE_Mopt(n) fit <- IPWE_Mopt(data=testData, regimeClass = a~x1+x2, moPropen=a~x1+x2, pop.size=1000) fit ``` ## References Rdpack::insert_ref(key="zhang2012robust",package="quantoptr") ## Author(s) Yu Zhou, zhou0269@umn.edu , with substantial contribution from Ben Sherwood.

IPWE_Mopt function