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:
supx : the supremum of the set of distances
Fn(x) : the empirical distribution function for n id observations Xi
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.
ks.text(x, y, …, alternative = c(“two.sided”, “less”, “greater”), exact= NULL, tol= 1e-8,
simulate.p.value = FALSE, B=2000)
x: numeric vector of data values
y: 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.
- 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:
- 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.
- Step 4: Now perform the K-S test on these two samples. For that purpose, use the ks.test() of the dgof package.
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.
Here we are generating both the samples using the rnorm() functions and then plot them.
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