The Doubly Robust Estimator of the Quantile-Optimal Treatment Regime

Description

DR_Qopt implements the doubly robust estimation method to estimate the quantile-optimal treatment regime. The double robustness property means that it is consistent when either the propensity score model is correctly specified, or the conditional quantile function is correctly specified. Both linear and nonlinear conditional quantile models are considered. See 'Examples' for an illustrative example.

Usage

DR_Qopt(data, regimeClass, tau, moPropen = "BinaryRandom",
  nlCondQuant_0 = FALSE, nlCondQuant_1 = FALSE, moCondQuant_0,
  moCondQuant_1, max = TRUE, length.out = 200, s.tol, it.num = 8,
  cl.setup = 1, p_level = 1, pop.size = 3000, hard_limit = FALSE,
  start_0 = NULL, start_1 = NULL)

Arguments

Details

• Standardization on covariates AND explanation on the differences between the two returned regime parameters.

  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 quantile-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 2016 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)= I\{\hat{\beta}_0 + \hat{\beta}_1 x_1 + ... + \hat{\beta}_k x_k > 0\}.

     The estimated \bm{\hat{\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 each 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.

• Property of the doubly robust(DR) estimator. The DR estimator DR_Qopt is consistent if either the propensity score model or the conditional quantile regression model is correctly specified. (Wang et. al. 2016)

Value

This function returns an object with 9 objects. Both coefficients and coef.orgn.scale were normalized to have unit euclidean norm.

coefficients

the parameters indexing the estimated quantile-optimal treatment regime for standardized covariates.

coef.orgn.scale

the parameter indexing the estimated quantile-optimal treatment regime for the original input covariates.

tau

the quantile of interest

hatQ

the estimated marginal tau-th quantile when the 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

moCondQuant_0

the user specified conditional quantile model for treatment 0

moCondQuant_1

the user specified conditional quantile model for treatment 1

Author(s)

Yu Zhou, zhou0269@umn.edu

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

See Also

dr_quant_est, augX

Examples

ilogit <- function(x) exp(x)/(1 + exp(x))
GenerateData.DR <- function(n)
{
 x1 <- runif(n,min=-1.5,max=1.5)
 x2 <- runif(n,min=-1.5,max=1.5)
 tp <- ilogit( 1 - 1*x1^2 - 1* x2^2)
 a <-rbinom(n,1,tp)
 y <- a * exp(0.11 - x1- x2) + x1^2 + x2^2 +  a*rgamma(n, shape=2*x1+3, scale = 1) +
 (1-a)*rnorm(n, mean = 2*x1 + 3, sd = 0.5)
 return(data.frame(x1=x1,x2=x2,a=a,y=y))
}

regimeClass <- as.formula(a ~ x1+x2)
moCondQuant_0 <- as.formula(y ~ x1+x2+I(x1^2)+I(x2^2))
moCondQuant_1 <- as.formula(y ~ exp( 0.11 - x1 - x2)+ x1^2 + p0 + p1*x1
                           + p2*x1^2 + p3*x1^3 +p4*x1^4 )
start_1 = list(p0=0, p1=1.5, p2=1, p3 =0,p4=0)



n <- 400
testdata <- GenerateData.DR(n)

## Examples below correctly specified both the propensity model and 
##  the conditional quantile model.
  
 system.time(
 fit1 <- DR_Qopt(data=testdata, regimeClass = regimeClass, 
                 tau = 0.25,
                 moPropen = a~I(x1^2)+I(x2^2),
                 moCondQuant_0 = moCondQuant_0,
                 moCondQuant_1 = moCondQuant_1,
                 nlCondQuant_1 = TRUE,  start_1=start_1,
                 pop.size = 1000))
 fit1
 ## Go parallel for the same fit. It would save a lot of time.
 ### Could even change the cl.setup to larger values 
 ### if more cores are available.
  
 system.time(fit2 <- DR_Qopt(data=testdata, regimeClass = regimeClass, 
                 tau = 0.25,
                 moPropen = a~I(x1^2)+I(x2^2),
                 moCondQuant_0 = moCondQuant_0,
                 moCondQuant_1 = moCondQuant_1,
                 nlCondQuant_1 = TRUE,  start_1=start_1,
                 pop.size = 1000, cl.setup=2))
 fit2

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

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) |).

Usage

IPWE_MADopt(data, regimeClass, moPropen = "BinaryRandom", s.tol, it.num = 8,
  hard_limit = FALSE, cl.setup = 1, p_level = 1, pop.size = 3000)

Arguments

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\}.

   The estimated \bm{\hat{\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.

Value

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

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

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

Estimate the Mean-optimal Treatment Regime

Description

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

Usage

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

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\}.

The estimated \bm{\hat{\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.

Value

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

Author(s)

Yu Zhou, zhou0269@umn.edu, with substantial contribution from Ben Sherwood.

References

Zhang B, Tsiatis AA, Laber EB and Davidian M (2012). “A robust method for estimating optimal treatment regimes.” Biometrics, 68(4), pp. 1010–1018.

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

Estimate the Quantile-optimal Treatment Regime

Description

Estimate the Quantile-optimal Treatment Regime by inverse probability of weighting

Usage

IPWE_Qopt(data, regimeClass, tau, moPropen = "BinaryRandom", max = TRUE,
  s.tol, it.num = 8, hard_limit = FALSE, cl.setup = 1, p_level = 1,
  pop.size = 3000)

Arguments

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 quantile-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)= I\{\hat{\beta}_0 + \hat{\beta}_1 x_1 + ... + \hat{\beta}_k x_k > 0\}.

   The estimated \bm{\hat{\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.

Value

This function returns an object with 7 objects. Both coefficients and coef.orgn.scale were normalized to have unit euclidean norm.

coefficients

the parameters indexing the estimated quantile-optimal treatment regime for standardized covariates.

coef.orgn.scale

the parameter indexing the estimated quantile-optimal treatment regime for the original input covariates.

tau

the quantile of interest

hatQ

the estimated marginal tau-th quantile when the 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

Author(s)

Yu Zhou, zhou0269@umn.edu with substantial contribution from Ben Sherwood.

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

Examples

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= 0.5)
  tp <- exp(-1+1*(x1+x2))/(1+exp(-1+1*(x1+x2)))
  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 <- 300
testData <- GenerateData(n)

# 1. Estimate the 0.25th-quantile optimal treatment regime. ###

fit1 <- IPWE_Qopt(data = testData, regimeClass = "a~x1+x2",
           tau = 0.25, moPropen="a~x1+x2")
fit1


# 2. Go parallel. This saves time in calculation. ###

fit2 <- IPWE_Qopt(data = testData, regimeClass = "a~x1+x2",
           tau = 0.25, moPropen="a~x1+x2", cl.setup=2)
fit2




# 3. Set a quardratic term in the class #######################

fit3 <- IPWE_Qopt(data = testData, regimeClass = "a~x1+x2+I(x1^2)",
                  tau = 0.25, moPropen="a~x1+x2", pop.size=1000)
fit3


# 4. Set screen prints level. #######################
# Set the p_level to be 0, 
# then all screen prints from the genetic algorithm will be suppressed.

fit4 <- IPWE_Qopt(data = testData, regimeClass = "a~x1+x2",
           tau = 0.25, moPropen="a~x1+x2", cl.setup=2, p_level=0)
fit4

Estimate the Two-stage Mean-Optimal Treatment Regime

Description

This function implements the estimator of two-stage mean-optimal treatment regime by inverse probability of weighting proposed by Baqun Zhang. As there are more than one stage, the second stage treatment regime could take into account the evolving status of an individual after the first stage and the treatment level received in the first stage. We assume the options at the two stages are both binary and take the form:

d_1(x_{stage1})=I\left(\beta_{10} +\beta_{11} x_{11} +...+ \beta_{1k} x_{1k} > 0\right),

d_2(x_{stage2})=I\left(\beta_{20} +\beta_{21} x_{21} +...+ \beta_{2j} x_{2j} > 0\right)

Usage

TwoStg_Mopt(data, regimeClass.stg1, regimeClass.stg2,
  moPropen1 = "BinaryRandom", moPropen2 = "BinaryRandom", max = TRUE,
  s.tol, cl.setup = 1, p_level = 1, it.num = 10, pop.size = 3000,
  hard_limit = FALSE)

Arguments

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.

For every stage k, k=1,2, this estimated parameters indexing the two-stage mean-optimal treatment regime are returned in two scales:

1. , the returned coef.k is the set of parameters that we estimated after standardizing every covariate available for decision-making at stage k 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 2016 to get the estimated regime parameters, coef.k, based on the standardized data. For the identifiability issue, we force the Euclidean norm of coef.k to be 1.

2. The difference between coef.k and coef.orgn.scale.k is that the latter set of parameters correspond 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\{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k > 0\},

   where the \beta values are returned as coef.orgn.scale.k, and the the vector (1, x_1,...,x_k) corresponds to the specified class of treatment regimes in the kth stage.

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 coef.k and coef.orgn.scale.k will render identical.

Value

This function returns an object with 6 objects. Both coef.1, coef.2 and coef.orgn.scale.1, coef.orgn.scale.2 were normalized to have unit euclidean norm.

coef.1, coef.2

the set of parameters indexing the estimated mean-optimal treatment regime for standardized covariates.

coef.orgn.scale.1, coef.orgn.scale.2

the set of parameter indexing the estimated mean-optimal treatment regime for the original input covariates.

hatM

the estimated marginal mean when the treatment regime indexed by coef.orgn.scale.1 and coef.orgn.scale.2 is applied on the entire population. See the 'details' for connection between coef.orgn.scale.k and coef.k.

call

the user's call.

moPropen1, moPropen2

the user specified propensity score models for the first and the second stage respectively

regimeClass.stg1, regimeClass.stg2

the user specified class of treatment regimes for the first and the second stage respectively

Author(s)

Yu Zhou, zhou0269@umn.edu

References

Zhang B, Tsiatis AA, Laber EB and Davidian M (2013). “Robust estimation of optimal dynamic treatment regimes for sequential treatment decisions.” Biometrika, 100(3).

Examples

ilogit <- function(x) exp(x)/(1 + exp(x))
GenerateData.2stg <- function(n){
 x1 <- runif(n)
 p1 <- ilogit(-0.5+x1)
 a1 <- rbinom(n, size=1, prob=p1)
 
 x2 <- runif(n, x1, x1+1)
 p2 <- ilogit(-1 + x2)
 a2 <- rbinom(n, size=1, prob=p2)
 
 mean <- 1+x1+a1*(1-3*(x1-0.2)^2) +x2 + a2*(1-x2-x1)
 y <- mean + (1+a1*(x1-0.5)+0.5*a2*(x2-1))*rnorm(n,0,sd = 1)
 return(data.frame(x1,a1,x2,a2,y))
}

n <- 400
testdata <- GenerateData.2stg(n)

fit <- TwoStg_Mopt(data=testdata, 
                   regimeClass.stg1="a1~x1", regimeClass.stg2="a2~x1+a1+x2",
                   moPropen1="a1~x1", moPropen2="a2~x2",
                   cl.setup=2)
fit

fit2 <- TwoStg_Mopt(data=testdata, 
                   regimeClass.stg1="a1~x1", regimeClass.stg2="a2~a1+x1*x2",
                   moPropen1="a1~x1", moPropen2="a2~x2",
                   cl.setup=2)
fit2

Estimate the Two-stage Quantile-optimal Treatment Regime

Description

This function implements the estimator of two-stage quantile-optimal treatment regime by inverse probability of weighting proposed by Lan Wang, et al. As there are more than one stage, the second stage treatment regime could take into account the evolving status of an individual after the first stage and the treatment level received in the first stage. We assume the options at the two stages are both binary and take the form:

d_1(x)=I\left(\beta_{10} +\beta_{11} x_{11} +...+ \beta_{1k} x_{1k} > 0\right),

d_2(x)=I\left(\beta_{20} +\beta_{21} x_{21} +...+ \beta_{2p} x_{2p} > 0\right)

Usage

TwoStg_Qopt(data, tau, regimeClass.stg1, regimeClass.stg2,
  moPropen1 = "BinaryRandom", moPropen2 = "BinaryRandom", s.tol = 1e-04,
  it.num = 8, max = TRUE, cl.setup = 1, p_level = 1, pop.size = 1000,
  hard_limit = FALSE)

Arguments

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.

For every stage k, k=1,2, this estimated parameters indexing the two-stage quantile-optimal treatment regime are returned in two scales:

1. , the returned coef.k is the set of parameters that we estimated after standardizing every covariate available for decision-making at stage k 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, coef.k, based on the standardized data. For the identifiability issue, we force the Euclidean norm of coef.k to be 1.

2. The difference between coef.k and coef.orgn.scale.k is that the latter set of parameters correspond 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\{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k > 0\},

   where the \beta values are returned as coef.orgn.scale.k, and the the vector (1, x_1,...,x_k) corresponds to the specified class of treatment regimes in the kth stage.

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 coef.k and coef.orgn.scale.k will render identical.

Value

This function returns an object with 7 objects. Both coefficients and coef.orgn.scale were normalized to have unit euclidean norm.

coef.1, coef.2

the set of parameters indexing the estimated quantile-optimal treatment regime for standardized covariates.

coef.orgn.scale.1, coef.orgn.scale.2

the set of parameter indexing the estimated quantile-optimal treatment regime for the original input covariates.

tau

the quantile of interest

hatQ

the estimated marginal quantile when the treatment regime indexed by coef.orgn.scale.1 and coef.orgn.scale.2 is applied on the entire population. See the 'details' for connection between coef.orgn.scale.k and coef.k.

call

the user's call.

moPropen1, moPropen2

the user specified propensity score models for the first and the second stage respectively

regimeClass.stg1, regimeClass.stg2

the user specified class of treatment regimes for the first and the second stage respectively

Author(s)

Yu Zhou, zhou0269@umn.edu

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

Examples

ilogit <- function(x) exp(x)/(1 + exp(x))
GenerateData.2stg <- function(n){
 x1 <- runif(n)
 p1 <- ilogit(-0.5+x1)
 a1 <- rbinom(n, size=1, prob=p1)
 
 x2 <- runif(n,x1,x1+1)
 p2 <- ilogit(-1 + x2)
 a2 <- rbinom(n, size=1, prob=p2)
 
 mean <- 1+x1+a1*(1-3*(x1-0.2)^2) +x2 + a2*(1-x2-x1)
 y <- mean + (1+a1*(x1-0.5)+0.5*a2*(x2-1))*rnorm(n,0,sd = 1)
 return(data.frame(x1,a1,x2,a2,y))
}

n <- 400
testdata <- GenerateData.2stg(n)
fit <- TwoStg_Qopt(data=testdata, tau=0.2,
                   regimeClass.stg1=a1~x1, regimeClass.stg2=a2~x1+a1+x2,
                   moPropen1=a1~x1, moPropen2=a2 ~ x2,
                   cl.setup=2)
fit

Estimate the Gini's mean difference/mean absolute difference(MAD) for a Given Treatment Regime

Description

Estimate the MAD if the entire population follows a treatment regime indexed by the given parameters. This function supports the IPWE_MADopt function.

Usage

abso_diff_est(beta, x, y, a, prob, Cnobs)

Arguments

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

See Also

The function IPWE_MADopt is based on this function.

Examples

library(stats)
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*0.3*(-1+x1+x2<0) +  a*-0.3*(-1+x1+x2>0)) * error
  return(data.frame(x1=x1,x2=x2,a=a,y=y))
}



n <- 500
testData <- GenerateData.MAD(n)
logistic.model.tx <- glm(formula = a~x1+x2, data = testData, family=binomial)
ph <- as.vector(logistic.model.tx$fit)
Cnobs <- combn(1:n, 2)
abso_diff_est(beta=c(1,2,-1), 
              x=model.matrix(a~x1+x2, testData),
              y=testData$y,
              a=testData$a,
              prob=ph,
              Cnobs = Cnobs)

Generate Pseudo-Responses Based on Conditional Quantile Regression Models

Description

This function supports the DR_Qopt function. For every observation, we generate pseudo-observations corresponding to treatment 0 and 1 respectively based on working conditional quantile models.

Usage

augX(raw.data, length.out = 200, txVec, moCondQuant_0, moCondQuant_1,
  nlCondQuant_0 = FALSE, nlCondQuant_1 = FALSE, start_0 = NULL,
  start_1 = NULL, clnodes)

Arguments

Details

This function implements the algorithm to generate individual level pseudo responses for two treatment levels respectively.

For each observation, two independent random variables from {unif}[0,1] are generated. Denote them by u_0 and u_1. Approximately, this function then estimates the u_0th quantile of this observation were treatment level 0 is applied via the conditional u_0th quantile regression. This estimated quantile will be the pseudo-response for treatment 0. Similarly, this function the pseudo-response for treatment 1 will be estimated and returned.

See the reference paper for a more formal explanation.

Value

It returns a list object, consisting of the following elements:

1. y.a.0, the vector of estimated individual level pseudo outcomes, given the treatment is 0;

2. y.a.1, the vector of estimated individual level pseudo outcomes, given the treatment is 1;

3. nlCondQuant_0, logical, indicating whether the y.a.0 is generated based on a nonlinear conditional quantile model.

4. nlCondQuant_1, logical, indicating whether the y.a.1 is generated based on a nonlinear conditional quantile model.

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

Examples

ilogit <- function(x) exp(x)/(1 + exp(x))
GenerateData.DR <- function(n)
{
  x1 <- runif(n,min=-1.5,max=1.5)
  x2 <- runif(n,min=-1.5,max=1.5)
  tp <- ilogit( 1 - 1*x1^2 - 1* x2^2)
  a <-rbinom(n,1,tp)
  y <- a * exp(0.11 - x1- x2) + x1^2 + x2^2 +  a*rgamma(n, shape=2*x1+3, scale = 1) +
       (1-a)*rnorm(n, mean = 2*x1 + 3, sd = 0.5)
  return(data.frame(x1=x1,x2=x2,a=a,y=y))
}
regimeClass = as.formula(a ~ x1+x2)
moCondQuant_0 = as.formula(y ~ x1+x2+I(x1^2)+I(x2^2))
moCondQuant_1 = as.formula(y ~ exp( 0.11 - x1 - x2)+ x1^2 + p0 + p1*x1
+ p2*x1^2 + p3*x1^3 +p4*x1^4 )
start_1 = list(p0=0, p1=1.5, p2=1, p3 =0,p4=0)

## Not run: 
n<-200
testdata <- GenerateData.DR(n)
fit1 <- augX(raw.data=testdata, txVec = testdata$a,
             moCondQuant_0=moCondQuant_0, moCondQuant_1=moCondQuant_1,
             nlCondQuant_0=FALSE,   nlCondQuant_1=TRUE,
             start_1=start_1, 
             clnodes=NULL)  
 
# How to use parallel computing in AugX(): ##
 
# on Mac OSX/linux
 clnodes <- parallel::makeForkCluster(nnodes =getOption("mc.cores",2))
 fit2 <- augX(raw.data=testdata, length.out = 5, txVec = testdata$a,
             moCondQuant_0=moCondQuant_0, moCondQuant_1=moCondQuant_1,
             nlCondQuant_0=FALSE,   nlCondQuant_1=TRUE,
             start_1=start_1, 
             clnodes=clnodes)  
  
# on Windows
 clnodes <- parallel::makeCluster(2, type="PSOCK")
 fit3 <- augX(raw.data=testdata, length.out = 5, txVec = testdata$a,
             moCondQuant_0=moCondQuant_0, moCondQuant_1=moCondQuant_1,
             nlCondQuant_0=FALSE,   nlCondQuant_1=TRUE,
             start_1=start_1, 
             clnodes=clnodes)  
 
## End(Not run)
 
 
 

The Doubly Robust Quantile Estimator for a Given Treatment Regime

Description

Given a fixed treatment regime, this doubly robust estimator estimates the marginal quantile of responses when it is followed by every unit in the target population. It took advantages of conditional quantile functions for different treatment levels when they are available.

Usage

dr_quant_est(beta, x, y, a, prob, tau, y.a.0, y.a.1, num_min = FALSE)

Arguments

Details

The double robustness property means that it can consistently estimate the marginal quantile when either the propensity score model is correctly specified, or the conditional quantile function is correctly specified.

See Also

augX

Get the OS from R

Description

Get the type of the operating system. The returned value is used in configuring parallel computation for the implemented algorithms.

Usage

get_os()

References

This function is adapted from https://www.r-bloggers.com/identifying-the-os-from-r/

The Inverse Probability Weighted Estimator of the Marginal Mean Given a Specific Treatment Regime

Description

Estimate the marginal mean of the response when the entire population follows a treatment regime. This function implements the inverse probability weighted estimator proposed by Baqun Zhang et. al..

This function supports the mestimate function.

Usage

mean_est(beta, x, a, y, prob)

Arguments

References

Zhang B, Tsiatis AA, Laber EB and Davidian M (2012). “A robust method for estimating optimal treatment regimes.” Biometrics, 68(4), pp. 1010–1018.

The Mean-Optimal Treatment Regime Wrapper Function

Description

The wrapper function for mean-optimal treatment regime that calls a genetic algorithm. This function supports the IPWE_Mopt function.

Usage

mestimate(x, y, a, prob, p_level, nvars, hard_limit = FALSE, max = TRUE,
  cl.setup = 1, s.tol = 1e-04, it.num = 8, pop.size = 3000)

Arguments

References

Zhang B, Tsiatis AA, Laber EB and Davidian M (2012). “A robust method for estimating optimal treatment regimes.” Biometrics, 68(4), pp. 1010–1018.

See Also

The function IPWE_Mopt is based on this function.

The Quantile-Optimal Treatment Regime Wrapper Function

Description

The wrapper function for quantile-optimal treatment regime that calls a genetic algorithm. This function supports the IPWE_Qopt function.

Usage

qestimate(tau, x, y, a, prob, p_level, nvars, hard_limit, max = TRUE,
  cl.setup = 1, s.tol = 1e-04, it.num = 8, pop.size = 3000)

Arguments

References

Wang L, Zhou Y, Song R and Sherwood B (2017). “Quantile-Optimal Treatment Regimes.” Journal of the American Statistical Association.

See Also

The function IPWE_Qopt is based on this function.

Estimate the Marginal Quantile Given a Specific Treatment Regime

Description

Estimate the marginal quantile if the entire population follows a treatment regime indexed by the given parameters. This function supports the qestimate function.

Usage

quant_est(beta, x, y, a, prob, tau)

Arguments