# Analyzing the Permutation Distribution of K-S Statistic with R Programming

We embark on a captivating journey into the world of statistical analysis. This exploration focuses on two distinct data samples, employing R programming to compute the Kolmogorov-Smirnov (K-S) test statistic. We scrutinize the null hypothesis, which assumes identical distributions for both samples, and employ permutation testing to uncover the K-S statistic's permutation distribution. These statistical techniques empower us to make data-driven inferences regarding the similarity or dissimilarity of the two samples. This comprehensive analysis showcases the power of statistical testing and the practical application of R programming in unraveling hidden insights within data.

## Problem Description

In this challenging R programming assignment, we delve into the fascinating world of statistical analysis to understand the permutation distribution of the Kolmogorov-Smirnov (K-S) statistic. This investigation focuses on two distinct samples, denoted as sample 1 (A) and sample 2 (B), both containing a total of 5 data points. Our primary goal is to determine whether these samples exhibit differences in their distributions, all while assuming no ties among the data points.

Step 1: Building the K-S Test Statistic Function Our journey begins with the creation of an R function named computeKS. This function plays a pivotal role in our assignment. Its purpose is to compute the K-S test statistic for a given permutation. The process commences by transforming the permutation string into a character vector, and subsequently, generating a dummy dataset that retains the order of observations. Finally, the K-S statistic is calculated based on the permutation, resulting in the maximum absolute value of the statistic.

Step 2: Permutation Generation and Test Statistic Computation We are presented with a set of 10 unique permutations, each denoted by a string of "1"s and "2"s, signifying the origin of each data point. Our task is to systematically iterate through these permutations. At each iteration, the computeKS function is applied to compute the K-S statistic, which is then recorded in the D_values vector.

Step 3: Uncovering the Permutation Distribution Since all permutations are considered equally likely under the null hypothesis (which postulates that both samples share the same distribution), we proceed to count the occurrences of each K-S statistic value within the D_values vector. Subsequently, these counts are converted into proportions. The outcome represents the permutation distribution of the K-S statistic, offering insights into its variability.

Step 4: Interpretation of Results To make informed decisions, we set a significance level. By comparing this level to the proportions obtained from the permutation distribution, we can determine whether there is substantial evidence to conclude that the two samples originate from distinct distributions.

## Sample Output:

R- code

``` > print(prop.table(table(D_values))) D_values 0.333333333333333 0.5 0.666666666666667 1 0.1 0.3 0.4 0.2 ```

This sample output illustrates the proportions of K-S statistics in the permutation distribution. These proportions are instrumental in making statistical inferences regarding the similarity or dissimilarity of the two samples' distributions.

Additional Exercises (Chapter 2, Exercise #4): In this supplementary section, we explore diverse statistical hypotheses and their applications using R. These exercises provide hands-on experience with different statistical tests.

1. Testing for Equality of Medians (Exercise 2.4a): We scrutinize two populations, labeled A and B, with the goal of comparing their medians. The null hypothesis asserts that these medians are equal, while the alternative hypothesis suggests their inequality. By employing the rank-sum test, we calculate a p-value and draw meaningful conclusions based on this test.
2. Testing for Equality of Medians (Exercise 2.4b): Similar to the previous exercise, we compare medians, but this time with a distinct dataset. The Wilcoxon rank-sum test is employed to compute a p-value, and we subsequently interpret the results to make informed statistical judgments.
3. Comparing Group Averages (a & b): In these exercises, we embark on a journey to compare the means of three groups, labeled as Group 1, Group 2, and Group 3. Two different approaches are considered - a permutation F-test and a one-way Analysis of Variance (ANOVA). We compute the test statistics and corresponding p-values, enabling us to determine whether there is evidence of a difference in the group means.
4. Kruskal-Wallis Test: To further enrich our statistical toolkit, we apply the Kruskal-Wallis test to compare the distributions of data across the same three groups. The null hypothesis posits that these distributions are identical, while the alternative hypothesis suggests at least one differing distribution. By calculating the test statistic and p-value, we evaluate the available evidence and make decisions based on the chosen significance level.

These exercises offer a comprehensive overview of different statistical tests and their real-world applications. They provide a holistic understanding of data analysis, empowering you to make informed decisions based on rigorous statistical analysis.