IPWE_MADopt function

Estimation of the Optimal Treatment Regime defined as Minimizing Gini's Mean Differences

IPWE_MADopt seeks to estimated the treatment regime which minimizes

the Gini's Mean difference defined below.

Besides mean and quantile criterion, in some applications people seek minimization of dispersion in the outcome, which, for example, can be described by Gini's mean difference. Formally, it is defined as the absolute differences of two random variables $Y_1$ and $Y_2$ drawn independently from the same distribution: [REMOVE_ME]$MAD:= E(|Y_1-Y_2|). [REMOVE_ME_2]$

Given a treatment regime $d$, define the potential outcome of a subject following the treatment recommended by d as $Y^{*}(d)$. When $d$ is followed by everyone in the target population, the Gini's mean absolute difference is [REMOVE_ME]$MAD(d):= E(| Y_1^{*}(d)-Y_2^{*}(d) |). [REMOVE_ME_2]$

IPWE_MADopt(data, regimeClass, moPropen = "BinaryRandom", s.tol, it.num = 8,
  hard_limit = FALSE, cl.setup = 1, p_level = 1, 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.
• 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.
• 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.
• 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.)
• 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 MAD-optimal treatment regime for standardized covariates.
• coef.orgn.scale: the parameter indexing the estimated MAD-optimal treatment regime for the original input covariates.
• hat_MAD: the estimated MAD 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

Description

IPWE_MADopt seeks to estimated the treatment regime which minimizes

the Gini's Mean difference defined below.

Besides mean and quantile criterion, in some applications people seek minimization of dispersion in the outcome, which, for example, can be described by Gini's mean difference. Formally, it is defined as the absolute differences of two random variables $Y_1$ and $Y_2$ drawn independently from the same distribution:

$$MAD:= E(|Y_1-Y_2|).$$

Given a treatment regime $d$, define the potential outcome of a subject following the treatment recommended by d as $Y^{*}(d)$. When $d$ is followed by everyone in the target population, the Gini's mean absolute difference is

$$MAD(d):= E(| Y_1^{*}(d)-Y_2^{*}(d) |).$$

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 estimated parameters indexing the MAD-optimal treatment regime are returned in two scales:

1. The returned coefficients is the set of parameters after covariates $X$

   are standardized to be in the interval [0, 1]. To be exact, every covariate is subtracted by the smallest observed value and divided by the difference between the largest and the smallest value. Next, we carried out the algorithm in Wang et al. 2017 to get the estimated regime parameters, coefficients, based on the standardized data. For the identifiability issue, we force the Euclidean norm of coefficients

   to be 1.

2. In contrast, 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:

$$d(x;\bm{\hat{\beta}})= I\{\hat{\beta}_0 + \hat{\beta}_1 x_1 + ... + \hat{\beta}_k x_k \> 0\}.d(x)= I{\beta_0 + \beta_1*x_1 + ... + \beta_k*x_k \> 0}.$$

The estimated $\beta$ is returned as coef.orgn.scale. The same as coefficients, we force the Euclidean norm of coef.orgn.scale

to be 1.

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.MAD <- function(n)
{
  x1 <- runif(n)
  x2 <- runif(n)
  tp <- exp(-1+1*(x1+x2))/(1+exp(-1+1*(x1+x2)))
  a<-rbinom(n = n, size = 1, prob=tp)
  error <- rnorm(length(x1))
  y <- (1 + a*0.6*(-1+x1+x2<0) +  a*-0.6*(-1+x1+x2>0)) * error
  return(data.frame(x1=x1,x2=x2,a=a,y=y))
}
# The true MAD optimal treatment regime for this generative model
# can be deduced trivially, and it is:  c( -0.5773503,  0.5773503,  0.5773503).

# With correctly specified propensity model   ####

n <- 400
testData <- GenerateData.MAD(n)
fit1 <- IPWE_MADopt(data = testData, regimeClass = a~x1+x2,
                    moPropen=a~x1+x2, cl.setup=2)
fit1

             

# With incorrectly specified propensity model ####

fit2 <- IPWE_MADopt(data = testData, regimeClass = a~x1+x2,
                    moPropen="BinaryRandom", cl.setup=2)
fit2

References

Rdpack::insert_ref(key="wang2017quantile",package="quantoptr")