## Introduction

This vignette demonstrates the usage of the ATE.ERROR.Y function in
the ATE.ERROR package. The ATE.ERROR.Y function provides a method for
estimating the Average Treatment Effect (ATE) considering both naive and
true estimates, and bootstrapping to assess variability.

## Generating Simulated Data

First, we generate our simulated data using the data(Simulated_data)
syntax.

```
library(ATE.ERROR)
set.seed(1)
data(Simulated_data)
Y_star <- Simulated_data$Y_star
A <- Simulated_data$T
Z <- Simulated_data$Z
X_star <- Simulated_data$X_star
X <- Simulated_data$X
Y <- Simulated_data$Y
p11 <- 0.8
p10 <- 0.2
bootstrap_number <- 1000
```

In this section, we load the required libraries and set the seed for
reproducibility. The simulated data is loaded from the Simulated_data
dataset. The variables Y_star, A, Z, X_star, X, and Y are extracted for
further analysis. The probabilities p11 and p10 are set to 0.8 and 0.2,
respectively. The number of bootstrap samples is set to 1000.

## Applying the ATE.ERROR.Y Function:

The ATE.ERROR.Y function is applied to estimate the ATE using the
generated data and specified parameter values, where we use 1000
bootstrap samples to obtain a standard error and the resulting 95%
confidence interval:

```
set.seed(1)
result <- ATE.ERROR.Y(Y_star, A, Z, X, p11, p10, bootstrap_number)
```

## Adding True ATE to the Result Summary:

The True ATE is added to the result summary, and the columns are
reordered to report the true ATE and the naive estimate for ATE:

```
True_ATE <- True_Estimation(Y, A, Z, X)
result_summary <- result$summary
result_summary <- data.frame(True_ATE = round(True_ATE, 3), result_summary)
print(result_summary)
#> True_ATE Naive_ATE_Y ATE SE CI
#> 1 0.162 0.093 0.155 0.033 (0.087, 0.217)
```

## Visualizing the Distribution of ATE Estimates Using a Boxplot

```
boxplot_with_true <- result$boxplot +
geom_hline(aes(yintercept = True_ATE, color = "true estimate"),
linetype = "dashed") +
scale_color_manual(name = NULL, values = c("naive estimate" = "red",
"true estimate" = "blue")) +
labs(title = "ATE Estimates from ATE.ERROR.Y Method", y = "ATE Estimate") +
theme_minimal() +
theme(legend.position = "right") +
guides(fill = guide_legend(title = NULL, order = 1),
color = guide_legend(title = NULL, override.aes = list(linetype = "dashed")
, order = 2))
Scale for colour is already present.
Adding another scale for colour, which will replace the existing scale.
print(boxplot_with_true)
```

The boxplot illustrates the distribution of the ATE estimates using
the ATE.ERROR.Y method. The blue dashed line represents the true
estimate of ATE, and the red dashed line represents the naive estimate
of ATE. The median of the estimated ATEs obtained from the ATE.ERROR.Y
method is closer to the true estimate than the naive estimate of ATE, as
expected.