The Kolmogorov-Smirnov Test is a type of non-parametric test of the equality of discontinuous and continuous of a 1D probability distribution that is used to compare the sample with the reference probability test (known as one-sample K-S Test) or among two samples (known as two-sample K-S test). A K-S Test quantifies a distance between the cumulative distribution function of the given reference distribution and the empirical distributions of given two samples, or between the empirical distribution of given two samples. In a one-sample K-S test, the distribution that is considered under a null hypothesis can be purely discrete or continuous or mixed. In the two-sample K-S test, the distribution considered under the null hypothesis is generally continuous distribution but it is unrestricted otherwise. The Kolmogorov-Smirnov test can be done very easily in R Programming.

#### Kolmogorov-Smirnov Test Formula

The formula for the Kolmogorov-Smirnov test can be given as:

where,

sup_{x}:the supremum of the set of distances

Fthe empirical distribution function for n id observations X_{n}(x) :_{i}

The empirical distribution function is a distribution function that is associated with the empirical measures of the chosen sample. Being a step function, this cumulative distribution jumps up by a 1/n step at each and every n data points.

#### Implementation in R

The K-S test can be performed using the **ks.test()** function in R.

Syntax:ks.text(x, y, …, alternative = c(“two.sided”, “less”, “greater”), exact= NULL, tol= 1e-8,

simulate.p.value = FALSE, B=2000)

Parameters:

x:numeric vector of data valuesy:numeric vector of data values or a character string which is used to name a cummulative distribution function.…:the parameters which are defined by the y value

alternative:used to indicate the alternate hypothesis.exact:usually NULL or it indicates a logic that an exact p-value should be computed.

tol:an upper bound used for rounding off errors in the data values.simulate.p.value:a logic that checks whether to use Monte Carlo method to compute the p-value.B:an integer value that indicates the number of replicates to be created while using the Monte Carlo method.

Let us understand how to execute a K-S Test step by step using an example of a two-sample K-S test.

**Step 1:**At first**install the required packages**. For performing the K-S test we need to install the “**dgof**” package using the**install.packages()**function from the R console.

install.packages("dgof")

**Step 2:**After a successful installation of the package,**load the required package**in our R Script. for that purpose, use the**library()**function as follows:

## R

`# loading the required package` `library` `(` `"dgof"` `)` |

**Step 3:**Use the**rnorm()**function and the**runif()**function to**generate to samples**say x and y. The**rnorm()**function is used to generate random variates while the**runif()**function is used to generate random deviates.

## R

`# loading the required package` `library` `(dgof) ` ` ` `# generating random variate` `# sample 1` `x <- ` `rnorm` `(50)` ` ` `# generating random deviates` `# sample 2` `y <- ` `runif` `(30)` |

**Step 4:**Now**perform the K-S test**on these two samples. For that purpose, use the**ks.test()**of the**dgof**package.

## R

`# loading the required package` `library` `(dgof) ` ` ` `# generating random variate` `# sample 1` `x <- ` `rnorm` `(50)` ` ` `# generating random deviates` `# sample 2` `y <- ` `runif` `(30)` ` ` `# performing the K-S Test` `# Do x and y come from ` `# the same distribution?` `ks.test` `(x, y)` |

**Output:**

Two-sample Kolmogorov-Smirnov test data: x and y D = 0.84, p-value = 5.151e-14 alternative hypothesis: two-sided

#### Visualization of the Kolmogorov- Smirnov Test in R

Being quite sensitive to the difference of shape and location of the empirical cumulative distribution of the chosen two samples, the two-sample K-S test is efficient, and one of the most general and useful non-parametric test. Hence we will see how the graph represents the difference between the two samples.

**Example:**

Here we are generating both the samples using the **rnorm()** functions and then plot them.

## R

`# loading the required package` `library` `(dgof) ` ` ` `# sample 1` `# generating a random variate` `x <- ` `rnorm` `(50)` ` ` `# sample 2` `# generating a random variate` `x2 <- ` `rnorm` `(50, -1)` ` ` `# plotting the result` `# visualization` `plot` `(` `ecdf` `(x), ` ` ` `xlim = ` `range` `(` `c` `(x, x2)), ` ` ` `col = ` `"blue"` `)` `plot` `(` `ecdf` `(x2), ` ` ` `add = ` `TRUE` `, ` ` ` `lty = ` `"dashed"` `,` ` ` `col = ` `"red"` `)` ` ` `# performing the K-S ` `# Test on x and x2` `ks.test` `(x, x2, alternative = ` `"l"` `)` |

**Output:**

Two-sample Kolmogorov-Smirnov test data: x and x2 D^- = 0.34, p-value = 0.003089 alternative hypothesis: the CDF of x lies below that of y