# Introduction to drcarlate

This introduction has three purposes:

- Describes the functions and structures included with `drcarlate`.

- Show how to use the functions in `drcarlate` for numerical simulation to get results similar to those in Jiang et al.(2022).

- Provides an example of research using `drcarlate`.

# Section 1

There are six families of functions in drcarlate:

- Data generation: these functions provide a way to generate random data according to the three DGPs in Jiang et al.(2022).

- Statistics calculation: these functions are used to calculate the statistics and the LATE used in Jiang et al.(2022).

- Estimation strategy: these functions provide a method for estimating treatment effects according to the three estimation strategies including L, NL and R presented in Jiang et al.(2022).

- Output function: this function integrates the functions of the above three types of functions and provides a means of output results.

- JLTZ function: this function calls all of the above functions and helps the user reproduce the results of the data simulation section of Jiang et al.(2022).

- ATE functions: these functions are based on some of the above functions, focusing on calculating average treatment effect(ATE) under the full compliance condition, one characteristic of these functions is that they begin with the ATE prefix.

## Data generation

This section contains two functions, `FuncDGP` and `CovAdptRnd`:

- `FuncDGP` allows users to generate the corresponding data using the three algorithms declared in the data generation process in Jiang et al.(2022).

- `CovadptRnd` provides four CAR schemes proposed in Jiang et al.(2022).

- Combined with the above two functions, users can generate 3 x 4 = 12 sets of data.

    # Parameter dgpflag declares three ways to generate random data.
    # FuncDGP will generate random data according to the method specified in (i), (ii) or (iii) when dgpflag = 1, 2 or 3 respectively.

    # Parameter rndflag declares four ways to randomly assign treatment effects.
    # rndflag = 1 - SRS; rndflag = 2 - WEI; rndflag = 3 - BCD; rndflag = 4 - SBR
    # Note that CovadpRnd is built into FuncDGP, so it is not necessary to use CovadpRnd alone in the actual operation of generating random data

    # Let's take dgpflag=1 and rndflag=2 for example
    random_dgp <- FuncDGP(dgptype = 1, rndflag = 2, n = 100, g = 4, pi = c(0.5, 0.5, 0.5, 0.5))
    # We can see that the return value of FuncDGP is a list of nine matrices, We can easily extract what we need from it
    Y <- random_dgp$Y
    X <- random_dgp$X
    S <- random_dgp$S
    A <- random_dgp$A
    Y1 <- random_dgp$Y1
    Y0 <- random_dgp$Y0
    D1 <- random_dgp$D1
    D0 <- random_dgp$D0
    D <- random_dgp$D

## Calculated statistics

This section contains four functions including `TrueValue`, `pihat`, `tau` and `stanE`.

- `pihat`, `tau` and `stanE` computes estimated treatment assignment probabilities, LATE and standard deviation respectively, see Jiang et al.(2022) for more details.

    # compute estimated LATE
    tauhat <- tau(muY1 = Y1, muY0 = Y0, muD1 = D1, muD0 = D0, A = A, S = S, Y = Y, D = D)

    #compute estimated treatment assignment probabilities
    pihat(A = A, S = S)
    #>             [,1]
    #>   [1,] 0.4615385
    #>   [2,] 0.5600000
    #>   [3,] 0.4117647
    #>   [4,] 0.5600000
    #>   [5,] 0.5000000
    #>   [6,] 0.5000000
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    #>  [10,] 0.4615385
    #>  [11,] 0.4615385
    #>  [12,] 0.5600000
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    #>  [19,] 0.4117647
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    #>  [69,] 0.4615385
    #>  [70,] 0.5000000
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    #>  [73,] 0.4117647
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    #>  [80,] 0.4615385
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    #>  [84,] 0.4117647
    #>  [85,] 0.5600000
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    #>  [87,] 0.4615385
    #>  [88,] 0.4117647
    #>  [89,] 0.5000000
    #>  [90,] 0.4615385
    #>  [91,] 0.5600000
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    #>  [94,] 0.4615385
    #>  [95,] 0.4117647
    #>  [96,] 0.5000000
    #>  [97,] 0.5000000
    #>  [98,] 0.4615385
    #>  [99,] 0.4117647
    #> [100,] 0.5600000

    # compute estimated standard deviation
    stanE(muY1 = Y1, muY0 = Y0, muD1 = D1, muD0 = D0, A = A, S = S, Y = Y, D = D, tauhat = tauhat)
    #> [1] 4.418164

- `TrueValue` differs a little from the above three functions，it calculates the LATE of all four kinds of CAR schemes (rndflag = 1, 2, 3 and 4) under the specified DGP (dgpflag = 1, 2, or 3). So the user who runs `Truevalue` is supposed to get four values at once.

    # let's take dgpflag = 1 for example.
    true_value <- TrueValue(dgptype = 1, vIdx = 1:4, n = 100, g = 4, pi = c(0.5, 0.5, 0.5, 0.5))
    true_tau <- true_value$tau

    # SRS - WEI - BCD - SBR
    true_tau
    #> [1] 0.9252152 0.9237937 0.9192154 0.9215521

## Estimation strategy

This section contains three functions including `LogisticReg`, `feasiblePostLassoMatTool` and `LinearLogit`.

- `LinearLogit` provides users with three regression estimation combination strategies: L (modelflag = 1), NP (modelflag = 2), and R (modelflag = 3), see Jiang et al.(2022) for more details.

- `feasiblePostLassoMatTool` is the computing engine for function `LinearLogit` when modelflag = 3.

- `LogisticReg` provides a simple way to use logistic CDF.

    # remember that we set dgpflag = 1 and rndflag = 2 before
    LinearLogit(Y = Y, D = D, A = A, X = X, S = S, s = 4, modelflag = 1, iridge = 0.001)
    #> $theta_0s
    #>           [,1]
    #> [1,] 0.4104139
    #> [2,] 2.1014655
    #>
    #> $theta_1s
    #>            [,1]
    #> [1,] -0.4238703
    #> [2,]  0.7981997
    #>
    #> $beta_0s
    #>            [,1]
    #> [1,] 0.10671410
    #> [2,] 0.01534334
    #>
    #> $beta_1s
    #>            [,1]
    #> [1,] -0.2452247
    #> [2,]  0.2143404

## Output function

- `Output` is the most important integration function, which has the functions of generating random data, estimating LATE by regression analysis based on different estimation strategies: (1) NA (2) LP (3) LG (4) F (5) NP (6) R (when dgp = 3) (7) TSLS (8) R (when dgp = 1 or 2) and summarizing analysis results. This function is key to generating the simulation results in Jiang et al.(2022). See the paper for more details about all the estimation strategies.

    # set random seed
    set.seed(1)

    # get true tau
    true_tau <- TrueValue(dgptype = 1, vIdx = 1:4, n = 1000, g = 4, pi = c(0.5, 0.5, 0.5, 0.5))
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    # see the output: size, iPert = 0
    Output(ii = 1, tau = tauhat[1], dgptype = 1, rndflag = 1, n = 1000, g = 4,
           pi = c(0.5, 0.5, 0.5, 0.5),
           iPert = 0, iq = 0.05, iridge = 0.001)
    #> Currently at 1 th sample!
    #> $vtauhat
    #>          [,1]     [,2]     [,3]     [,4]     [,5] [,6]      [,7]     [,8] [,9]
    #> [1,] 1.236257 1.011659 1.014877 1.034401 1.070561  NaN 0.9797705 1.116384  NaN
    #>      [,10] [,11] [,12]
    #> [1,]   NaN   NaN   NaN
    #>
    #> $vsighat
    #>          [,1]     [,2]     [,3]     [,4]     [,5] [,6]     [,7]     [,8] [,9]
    #> [1,] 14.71728 10.93291 10.97075 10.81983 9.691528  NaN 11.10584 9.792131  NaN
    #>      [,10] [,11] [,12]
    #> [1,]   NaN   NaN   NaN
    #>
    #> $vstat
    #>           [,1]      [,2]      [,3]      [,4]      [,5] [,6]      [,7]      [,8]
    #> [1,] 0.1822901 0.4042457 0.3935777 0.3420046 0.2638352  NaN 0.4887518 0.1131412
    #>      [,9] [,10] [,11] [,12]
    #> [1,]  NaN   NaN   NaN   NaN
    #>
    #> $vdeci
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
    #> [1,]    0    0    0    0    0  NaN    0    0  NaN   NaN   NaN   NaN

    # see the output: power, iPert = 1
    Output(ii = 1, tau = tauhat[1], dgptype = 1, rndflag = 1, n = 1000, g = 4,
           pi = c(0.5, 0.5, 0.5, 0.5),
           iPert = 1, iq = 0.05, iridge = 0.001)
    #> Currently at 1 th sample!
    #> $vtauhat
    #>          [,1]     [,2]     [,3]     [,4]     [,5] [,6]     [,7]     [,8] [,9]
    #> [1,] 1.848939 1.270918 1.271093 1.313234 1.036465  NaN 1.266284 1.060324  NaN
    #>      [,10] [,11] [,12]
    #> [1,]   NaN   NaN   NaN
    #>
    #> $vsighat
    #>          [,1]     [,2]     [,3]     [,4]     [,5] [,6]     [,7]     [,8] [,9]
    #> [1,] 14.14772 10.91436 10.91949 10.57321 9.554213  NaN 10.48309 9.306626  NaN
    #>      [,10] [,11] [,12]
    #> [1,]   NaN   NaN   NaN
    #>
    #> $vstat
    #>           [,1]     [,2]     [,3]     [,4]     [,5] [,6]     [,7]     [,8] [,9]
    #> [1,] 0.6760993 2.551125 2.549418 2.506875 3.690303  NaN 2.670054 3.707407  NaN
    #>      [,10] [,11] [,12]
    #> [1,]   NaN   NaN   NaN
    #>
    #> $vdeci
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
    #> [1,]    0    1    1    1    1  NaN    1    1  NaN   NaN   NaN   NaN

## JLTZ function

- `JLTZ` provides users with a convenient way to reproduce the simulation results in Jiang et al.(2022), The user only needs to set parameters according to the corresponding DGP, sample size and Monte Carlo simulation times in the paper, and the results consistent with Jiang et al.(2022) can be obtained under the condition that the random number seed is set as 1. It should be noted that because the results in the original text are implemented in MATLAB, the results obtained by using JLTZ are not numerically the same as those in the paper, but they are very close.

    # For example, if we wanted to get the results shown in Panel A in Table 1, we could run the following two functions separately.

    # First of all, There are four strata data (g = 4), the probability of data being treated at each strata is equal to 0.5 (pi = c(0.5, 0.5, 0.5, 0.5)), the random data generation process follows the data generation process 1 (DGP = 1), the total sample size is 200 (n = 200), run 10,000 Monte Carlo simulations (iMonte = 10000), the confidence level for hypothesis testing is 5%, and finally，get the size of the Monte Carlo simulation (iPert = 0).
    # Time Consumeing. This command will take about 1.4 hours to execute, so do not run it unless necessary.

    # JLTZ(iMonte = 10000, dgptype = 1, n = 200, g = 4, pi = c(0.5, 0.5, 0.5, 0.5), iPert = 0, iq = 0.05, iridge = 0.001)

    # Second of all, There are four strata data (g = 4), the probability of data being treated at each strata is equal to 0.5 (pi = c(0.5, 0.5, 0.5, 0.5)), the random data generation process follows the digital generation process 1 (DGP = 1), the total sample size was 200 (n = 200), run 10,000 Monte Carlo simulations (iMonte = 10000), the confidence level for hypothesis testing is 5%, and finally，get the power of the Monte Carlo simulation (iPert = 1).
    # Time Consumeing. This command will take about 1.4 hours to execute, so do not run it unless necessary.

    # JLTZ(iMonte = 10000, dgptype = 1, n = 200, g = 4, pi = c(0.5, 0.5, 0.5, 0.5), iPert = 1, iq = 0.05, iridge = 0.001)

## ATE functions

- `ATEDGP` is a version of `FuncDGP` under full compliance condition.
- `ATETrueValue` is a version of `TrueValue` under full compliance condition.
- `ATEOutput` is a version of `Output` under full compliance condition.
- `ATEJLTZ` is a version of `JLTZ` under full compliance condition.
- The arguments and results of the ATE- functions and the non-ATE functions are basically identical. It should be noted that the results of `ATEOutput` is a list containing four 1x4 vectors rather than a list containing four 1x12 vectors, which is the result of `Output`. For details, please refer to the help documentation for related functions.

# Section 2

In this section we will show the user how to reproduce the results of the numerical simulation section of Jiang et al.(2022) by using the `JLTZ` function. Since it may take a long time to get the results of all three kinds of GDPS in Jiang et al. (2022), we will focus on only one case as an example.

Let’s consider a case like this:

- First, the total sample was set at 200, which means `size = 200`.

- And second, the number of Monte Carlo simulations was set at 10,000, which means `iMonte = 10000`.

- Third, calculate the size of the hypothesis test instead of the power and consider a five percent confidence interval in hypothesis testing,, which means `iPert = 0` and `iq = 0.05`.

- Fourth, the data generation process follows DGP-1 in Jiang et al.(2022), which means `dgptype = 1`.

- Fifth, the samples are divided into four strata, and the probability of the samples being processed in each strata is 0.5, which means `g = 4` and `pi = c(0.5, 0.5, 0.5, 0.5)`.

- Sixth, the penalization parameter in ridge regression is 0.001, which means `iridge = 0.001`.

    # With the above Settings in mind, we get the following JLTZ function.

    # JLTZ(iMonte = 10000, dgptype = 1, n = 200, g = 4, pi = c(0.5,0.5,0.5,0.5), iPert = 0,iq = 0.05, iridge = 0.001)

    # Since it takes about 1.4 hours to run this function, we'll give you the results directly that users can check them out themselves.

    #        vProb_d1   vProb_d2   vProb_d3   vProb_d4
    # [1,] 0.03139898 0.03548546 0.03071509 0.03139898
    # [2,] 0.04356403 0.04189256 0.03951368 0.04356403
    # [3,] 0.04307085 0.04238541 0.03919373 0.04307085
    # [4,] 0.05622226 0.04632824 0.05327148 0.05622226
    # [5,] 0.10833470 0.08854937 0.09486482 0.10833470
    # [6,]        NaN        NaN        NaN        NaN
    # [7,] 0.03435805 0.03334976 0.03455447 0.03435805
    # [8,] 0.04586553 0.05076392 0.05087186 0.04586553

# Section 3

In this section we further show an example of using the `drcarlate` package to analyze real data. For detailed information on this case, see Section 7 and Table 5 of Jiang et al.(2022).

    # Set up ------------------------------------------------------------------
    library(drcarlate)
    library(pracma)

    # Load data ---------------------------------------------------------------
    # data_for_final.csv add covariates b_total_income log_b_total_income b_exp_total_30days
    # edu_index b_asset_index b_health_index to data_for_tab4.csv

    data_table <- drcarlate::data_table

    data1 <- as.matrix(data_table)

    colnames(data1) <- NULL

    # create S for wave 1: number of stratnum, total 41 stratnum
    n <- size(data1,1)
    S <- zeros(n,1)
    for (i in 1:41) {
      S[data1[, size(data1,2)-i+1] == 1] <- 41 - i + 1
    }

    data1 <- cbind(data1, S)

    # create Y, X, S, D: number of outcomes considered is 9:
    # save the results for NA, TSLS, L, NL, F
    vtauhat <- NaN * ones(8,5)
    vsighat <- NaN * ones(8,5)
    n_vec <- NaN * ones(8,1)

    iridge = 0.01 #tuning parameter for in ridge regression

    for (i in 1:8) {
      if (i <= 4) {
      # data_used: 1st col- Y, 2nd col- A, 3rd col- D, 4~(end-1)th col- X,
      # last col is strata number
        data_used <- cbind(data1[, 4+(i-1)*3], data1[, 2:3], data1[, 5:6],
                           data1[, size(data1,2)-1-41], data1[, size(data1,2)])
      } else {
        data_used <- cbind(data1[, 4+(i-1)*3], data1[,2:3], data1[, (4+(i-1)*3+1):(4+(i-1)*3+2)],
                           data1[, size(data1,2)-1-41], data1[, size(data1,2)])
      }

      # delete the outcome variables with NaN
      data_used <- data_used[!is.nan(data_used[,1]), ]

      # check whether a strata has obs less than 10
      for (s in 1:41) {
        if (length(data_used[data_used[, size(data_used,2)] == s,1]) < 10) {
          # delete strata has obs less than 10
         data_used[data_used[, size(data_used,2)] ==s, ] <- matrix()
         data_used <- data_used[!is.na(data_used[,1]),]
        }
      }

      # update n
      n <- size(data_used,1)
      n_vec[i] <- n

      # find the unique value for stratnum
      stratnum <-  sort(base::unique(data_used[, size(data_used,2)]))

      # create A
      A <-  matrix(data_used[, 2])

      # create D
      D <- matrix(data_used[, 3])

      # create Y
      Y <- matrix(data_used[, 1])

      # create S
      S <- matrix(data_used[, size(data_used,2)])

      # create X
      # use baseline total income
      X <- data_used[,4:6]

      # make a index
      print(stringr::str_c("Now i equals to ", i, " !"))

      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      #  %% No adjustment (z) NA
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      muY0_z <- zeros(n,1)
      muY1_z <- zeros(n,1)
      muD0_z <- zeros(n,1)
      muD1_z <- zeros(n,1)

      vtauhat[i,1] <- tau(muY1 = muY1_z, muY0 = muY0_z, muD1 = muD1_z, muD0 = muD0_z,
                          A = A, S = S, Y = Y, D = D, stratnum = stratnum)
      vsighat[i,1] <- stanE(muY1 = muY1_z, muY0 = muY0_z, muD1 = muD1_z, muD0 = muD0_z,
                            A = A, S = S, Y = Y, D = D, tauhat = vtauhat[i,1], stratnum = stratnum)/sqrt(n)

      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      #  %% TSLS
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      mS_iv <- zeros(n, length(stratnum))

      for (j in 1:length(stratnum)) {
        s <- stratnum[j]
        mS_iv[S==s, j] <- 1
      }

      mX_iv <- cbind(D, X, mS_iv)
      mZ_iv <- cbind(A, X, mS_iv)
      K <- size(mX_iv,2)

      vPara_iv <- inv(t(mZ_iv) %*% mX_iv) %*% t(mZ_iv) %*% Y
      mE_iv2 <- diag(((Y - mX_iv %*% vPara_iv)^2)[,])
      m0_iv <- inv(t(mZ_iv) %*% mX_iv/n) %*% (t(mZ_iv) %*% mE_iv2 %*% mZ_iv/n) %*% inv(t(mX_iv) %*% mZ_iv/n)
      vtauhat[i,2] <- vPara_iv[1]
      vsighat[i,2] <- sqrt(m0_iv[1,1])/sqrt(n)

      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      #  %% Linear+linear model (L)
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      muY0_a <- NaN*ones(n,1)
      muY1_a <- NaN*ones(n,1)
      muD0_a <- NaN*ones(n,1)
      muD1_a <- NaN*ones(n,1)
      for (j in 1:length(stratnum)) {
        s = stratnum[j]

        result <- LinearLogit(Y = Y, D = D, A = A, X = X, S = S, s = s, modelflag = 1, iridge = iridge)

        theta_0s_a <- result[["theta_0s"]]
        theta_1s_a <- result[["theta_1s"]]
        beta_0s_a <- result[["beta_0s"]]
        beta_1s_a <- result[["beta_1s"]]

        muY0_a[S==s] <- X[S==s,] %*% theta_0s_a
        muY1_a[S==s] <- X[S==s,] %*% theta_1s_a
        muD0_a[S==s] <- X[S==s,] %*% beta_0s_a
        muD1_a[S==s] <- X[S==s,] %*% beta_1s_a
      }

      vtauhat[i,3] <- tau(muY1 = muY1_a, muY0 = muY0_a, muD1 = muD1_a, muD0 = muD0_a,
                          A = A, S = S, Y = Y, D = D, stratnum = stratnum)
      vsighat[i,3] <- stanE(muY1 = muY1_a, muY0 = muY0_a, muD1 = muD1_a, muD0 = muD0_a,
                            A = A, S = S, Y = Y, D = D, tauhat = vtauhat[i,3], stratnum = stratnum)/sqrt(n)
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      #  %% Linear+logistic model (NL)
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      muY0_b <- NaN*ones(n,1)
      muY1_b <- NaN*ones(n,1)
      muD0_b <- NaN*ones(n,1)
      muD1_b <- NaN*ones(n,1)

      for (j in 1:length(stratnum)) {
        s = stratnum[j]

        result <- LinearLogit(Y = Y, D = D, A = A, X = X, S = S, s = s, modelflag = 2, iridge = iridge)

        theta_0s_b <- result[["theta_0s"]]
        theta_1s_b <- result[["theta_1s"]]
        beta_0s_b <- result[["beta_0s"]]
        beta_1s_b <- result[["beta_1s"]]

        muY0_b[S==s] <- X[S==s,] %*% matrix(theta_0s_b[2:length(theta_0s_b)]) + theta_0s_b[1]
        muY1_b[S==s] <- X[S==s,] %*% matrix(theta_1s_b[2:length(theta_0s_b)]) + theta_1s_b[1]
        muD0_b[S==s] <- LogisticReg(x = (X[S==s,] %*% matrix(beta_0s_b[-1]) + beta_0s_b[1]))
        muD1_b[S==s] <- LogisticReg(x = (X[S==s,] %*% matrix(beta_1s_b[-1]) + beta_1s_b[1]))
      }

      vtauhat[i,4] <- tau(muY1 = muY1_b, muY0 = muY0_b, muD1 = muD1_b, muD0 = muD0_b,
                          A = A, S = S, Y = Y, D = D, stratnum = stratnum)
      vsighat[i,4] <- stanE(muY1 = muY1_b, muY0 = muY0_b, muD1 = muD1_b, muD0 = muD0_b,
                            A = A, S = S, Y = Y, D = D,tauhat = vtauhat[i,4], stratnum = stratnum)/sqrt(n)

      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      #  %% Further Efficiency Improvement
      #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      X_c <- cbind(X, muD1_b, muD0_b)
      muY0_c <- NaN*ones(n,1)
      muY1_c <- NaN*ones(n,1)
      muD0_c <- NaN*ones(n,1)
      muD1_c <- NaN*ones(n,1)

      for (j in 1:length(stratnum)) {
        s = stratnum[j]

        result <- LinearLogit(Y = Y, D = D, A = A, X = X_c, S = S, s = s, modelflag = 1, iridge = iridge)

        theta_0s_c <- result[["theta_0s"]]
        theta_1s_c <- result[["theta_1s"]]
        beta_0s_c <- result[["beta_0s"]]
        beta_1s_c <- result[["beta_1s"]]

        muY0_c[S==s] <- X_c[S==s,] %*% theta_0s_c
        muY1_c[S==s] <- X_c[S==s,] %*% theta_1s_c
        muD0_c[S==s] <- X_c[S==s,] %*% beta_0s_c
        muD1_c[S==s] <- X_c[S==s,] %*%beta_1s_c
      }

      vtauhat[i,5] <- tau(muY1 = muY1_c, muY0 = muY0_c, muD1 = muD1_c, muD0 = muD0_c,
                          A = A, S = S, Y = Y, D = D, stratnum = stratnum)
      vsighat[i,5] <- stanE(muY1 = muY1_c, muY0 = muY0_c, muD1 = muD1_c, muD0 = muD0_c,
                            A = A, S = S, Y = Y, D = D, tauhat = vtauhat[i,5], stratnum = stratnum)/sqrt(n)
      }
    #> [1] "Now i equals to 1 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 2 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 3 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 4 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 5 !"
    #> Warning: glm.fit: algorithm did not converge

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 6 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> Warning: glm.fit: algorithm did not converge
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 7 !"
    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
    #> [1] "Now i equals to 8 !"
    #> Warning: glm.fit: algorithm did not converge

    #> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred

    # show the LATE estimates and standard errors in Table 5 of Jiang et al.(2022).
    # LATE Estimates
    vtauhat
    #>             [,1]        [,2]        [,3]        [,4]        [,5]
    #> [1,]   2.7866925   7.1528747   7.1694104   7.2542016   8.4805342
    #> [2,]  20.5577389  21.1544145  22.1604884  22.3940102  22.7366686
    #> [3,]  -0.2076838  -0.1735085  -0.2912842  -0.2947078  -0.3099807
    #> [4,]  20.3993940  21.0965957  21.9244498  22.1835123  22.3253690
    #> [5,] -10.8261902  -7.4558010  -9.0040607  -8.9405514  -8.6914634
    #> [6,]  -1.9332078  -2.3327993  -1.2417154  -1.2579026  -1.5505684
    #> [7,]  -3.6211274  -3.3462369  -1.4278123  -1.5513955  -1.4265200
    #> [8,] -17.6431819 -14.3166215 -15.6652067 -15.8118165 -15.5119289

    # Standard Errors
    vsighat
    #>           [,1]      [,2]      [,3]      [,4]      [,5]
    #> [1,] 7.2902879 6.3684935 6.1971428 6.2712205 6.1394941
    #> [2,] 3.0667814 3.0151296 2.9646743 3.0037782 2.9512801
    #> [3,] 0.2233596 0.2241912 0.2119157 0.2144341 0.2094491
    #> [4,] 3.0890583 3.0344251 2.9792857 3.0221352 2.9639732
    #> [5,] 5.0034112 4.4034599 4.4007221 4.3728318 4.4028981
    #> [6,] 1.9711334 1.8580997 1.7936724 1.8155761 1.8076464
    #> [7,] 2.0402640 1.9989033 1.8654540 2.0344097 1.8689752
    #> [8,] 6.1995757 5.3509212 5.1849623 5.2346367 5.1130911

## Reference

Jiang L, Linton O B, Tang H, Zhang Y. Improving estimation efficiency via regression-adjustment in covariate-adaptive randomizations with imperfect compliance [J]. 2022.