# overlapping

    library(OPL)

### Introduction

The overlapping function performs an overlap analysis between two datasets, typically used to compare populations before and after a policy intervention. The function applies Principal Component Analysis (PCA) to the covariates of both datasets (train and test) and then calculates the Kolmogorov-Smirnov (KS) test to assess the statistical overlap between the two distributions.

### Usage

overlapping(train_data, test_data, x)

### Arguments

train_data: A data frame representing the training dataset, typically indicating the old policy population. test_data: A data frame representing the test dataset, typically indicating the new policy population. x: A vector of predictor variable names (column names of train_data and test_data) to be used in the analysis.

### Output

The function generates the following outputs:

A superposition graph comparing the distributions of the first principal component (PC1) for both the training and test datasets. The results of the Kolmogorov-Smirnov test, which quantifies the overlap between the two datasets.

### Details

The function follows these steps:

Principal Component Analysis (PCA): PCA is applied to the covariates of both the training and test datasets to reduce the dimensionality and focus on the most significant variation. Kolmogorov-Smirnov Test: The Kolmogorov-Smirnov test is performed on the first principal components to test the hypothesis that the distributions of the two datasets are the same.

### Example

    # Load example data
    set.seed(123)
    train_data <- data.frame(
      var1 = rnorm(100),
      var2 = rnorm(100),
      var3 = rnorm(100)
    )
    test_data <- data.frame(
      var1 = rnorm(100),
      var2 = rnorm(100),
      var3 = rnorm(100)
    )

    # Perform overlap analysis
    overlapping(train_data, test_data, x = c("var1", "var2", "var3"))

### Interpretation of Results

The graph shows the density distributions of the first principal component for both the training and test datasets. Overlap is indicated by how much the density curves intersect. The Kolmogorov-Smirnov test provides a p-value indicating whether the distributions of the two datasets are statistically different. A high p-value suggests that the distributions are similar, while a low p-value indicates significant differences.

This vignette provides an overview of the `overlapping` function and demonstrates its usage for comparing the overlap between two datasets using PCA and the Kolmogorov-Smirnov test. For further details, consult the package documentation.

## Acknowledgment

The development of this software was supported by FOSSR (Fostering Open Science in Social Science Research), a project funded by the European Union - NextGenerationEU under the NPRR Grant agreement n. MURIR0000008.