IPWE_Mopt function

Estimate the Mean-optimal Treatment Regime

IPWE_Mopt aims at estimating the treatment regime which maximizes the marginal mean of the potential outcomes.

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

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:

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

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.