adace (version 1.0.2)

est_S_Star_Plus_MethodB: Estimate the treatment effects for population S_*+ using Method B

Description

The est_S_Star_Plus_MethodB function produces estimation of treatment effects for the population that can adhere to the experimental treatment (S_*+). This method (Method B) is based on the potential outcome under the hypothetical alternative treatment .

Usage

est_S_Star_Plus_MethodB(X, A, Z, Y, TRT)

Value

A list containing the following components:

trt_diff

Estimate of treatment difference for S_*+ using Method B

se

Estimated standard error

res1

Estimated mean for the treatment group

res0

Estimated mean for the control group

se_res1

Estimated standard error for the treatment group

se_res0

Estimated standard error for the control group

#' @details The average treatment difference can be denoted as

$$latex$$

The method B exploits the joint distribution of X, Z, and A to estimate the probability that a patient would adhere to the hypothetical alternative treatment, and then use IPW to estimate treatment different for a given population. The variance estimation for the treatment effect is constructed using the sandwich method. Details can be found in the references.

The intermediate post-baseline measurements for each intermediate time point are estimated by regressing Z on X using subjects with experimental treatment or placebo. The covariance matrix is estimated based on the residuals of the regression.

The probability of adherence is estimated by regressing A on X, Z by using all data. The logistic regression is used in this function.

The indicator of adherence prior to the first intermediate time point is not included in this model since this function assumes no intercurrent events prior to the first time point. Thus, the first element of Z should not have missing values.

Each element of Z contains the variables at each intermediate time point, i.e., the first element of Z contains the intermediate variables at time point 1, the second element contains the intermediate variables at time point 2, etc.

Arguments

X

Matrix of baseline variables. Each row contains the baseline values for each patient across multiple time points.

A

Matrix of indicator for adherence. Each row of A contains the adherence information for each patient. Each column contains the adherence indicator after each intermediate time point. A = 1 means adherence and A=0 means non-adherence. Monotone missing is assumed.

Z

List of matrices. Intermediate efficacy and safety outcomes that can affect the probability of adherence. For each matrix, the structure is the same as variable X.

Y

Numeric vector of the final outcome (E.g., primary endpoint).

TRT

Numeric vector of treatment assignment. TRT=0 for the control group and TRT =1 for the experimental treatment group.

References

Qu, Yongming, et al. "A general framework for treatment effect estimators considering patient adherence." Statistics in Biopharmaceutical Research 12.1 (2020): 1-18.

Zhang, Ying, et al. "Statistical inference on the estimators of the adherer average causal effect." Statistics in Biopharmaceutical Research (2021): 1-4.

Examples

 library(MASS)
 j<- 500
 p_z <- 6 ## dimension of Z at each time point
 n_t <- 4 ## number of time points
 alphas <- list()
 gammas <- list()
 z_para <- c(-1/p_z, -1/p_z, -1/p_z, -1/p_z, -0.5/p_z,-0.5/p_z, -0.5/p_z,
 -0.5/p_z)
 Z <- list()
 beta = c(0.2, -0.3, -0.01, 0.02, 0.03, 0.04, rep(rep(0.02,p_z), n_t))
 beta_T = -0.2
 sd_z_x = 0.4
 X = mvrnorm(j, mu=c(1,5,6,7,8), Sigma=diag(1,5))
 TRT = rbinom(j, size = 1,  prob = 0.5)
 Y_constant <- beta[1]+(X%*%beta[2:6])
 Y0 <- 0
 Y1 <- 0
 A <- A1 <- A0 <- matrix(NA, nrow = j, ncol = n_t)
 gamma <- c(1,-.1,-0.05,0.05,0.05,.05)
 A0[,1] <- rbinom(j, size = 1, prob = 1/(1+exp(-(gamma[1] +
 (X %*% gamma[2:6])))))
 A1[,1] <- rbinom(j, size = 1, prob = 1/(1+exp(-(gamma[1] +
 (X %*% gamma[2:6])))))
 A[,1] <- A1[,1]*TRT + A0[,1]*(1-TRT)
 for(i in 2:n_t){
   alphas[[i]] <- matrix(rep(c(2.3, -0.3, -0.01, 0.02, 0.03, 0.04, -0.4),
   p_z),ncol=p_z)
   gammas[[i]] <- c(1, -0.1, 0.2, 0.2, 0.2, 0.2, rep(z_para[i],p_z))
   Z0 <- alphas[[i]][1,]+(X%*%alphas[[i]][2:6,]) + mvrnorm(j, mu = rep(0,p_z)
   , Sigma = diag(sd_z_x,p_z))
   Z1 <- alphas[[i]][1,]+(X%*%alphas[[i]][2:6,])+alphas[[i]][7,] +
     mvrnorm(j, mu = rep(0,p_z), Sigma = diag(sd_z_x,p_z))
   Z[[i]] <- Z1*TRT + Z0*(1-TRT)
   Y0 <- (Y0 + Z0 %*% matrix(beta[ (7 + (i-1)*p_z):
   (6+p_z*i)],ncol = 1) )[,1]
   Y1 <- (Y1 + Z1 %*% matrix(beta[ (7 + (i-1)*p_z):
   (6+p_z*i)],ncol = 1) )[,1]
   A0[,i] <- rbinom(j, size = 1,
                    prob = 1/(1+exp(-(gammas[[i]][1]+
                    (X%*%gammas[[i]][2:6])+Z0%*%matrix(gammas[[i]][7:
                    (7+p_z-1)], ncol=1))[,1])))*A0[,i-1]
   A1[,i] <- rbinom(j, size = 1,
                    prob = 1/(1+exp(-(gammas[[i]][1]+
                    (X%*%gammas[[i]][2:6])+Z1%*%matrix(gammas[[i]][7:
                    (7+p_z-1)], ncol=1))[,1])))*A1[,i-1]

   A[,i] <- A1[,i]*TRT + A0[,i]*(1-TRT)
 }
 Y0 <- Y0 + rnorm(j, mean = 0, sd = 0.3) + Y_constant
 Y1 <- Y1 + + beta_T + rnorm(j, mean = 0, sd = 0.3) + Y_constant
 Y <- as.vector( Y1*TRT+Y0*(1-TRT))
 for(i in 2:n_t){
   Z[[i]][A[,(i-1)]==0,] <- NA
 }
 Z[[1]] <- matrix(NA, nrow=nrow(Z1),ncol=ncol(Z1))
 Y[A[,n_t] == 0] <- NA
 # estimate the treatment difference
 fit <- est_S_Star_Plus_MethodA(X, A, Z, Y, TRT)
 fit
 # Calculate the true values
 true1 <- mean(Y1[A1[,n_t]==1])
 true1
 true0 <- mean(Y0[A1[,n_t]==1])
 true0
 true_d  =  true1 - true0
 true_d