TwoSampleKolmogorovSmirnovTest Class

The Kolmogorov-Smirnov test tries to determine if two samples have been drawn from the same probability distribution. The Kolmogorov-Smirnov test has an interesting advantage in which it does not requires any assumptions about the data. The distribution of the K-S test statistic does not depend on which distribution is being tested.

The K-S test has also the advantage of being an exact test (other tests, such as the chi-square goodness-of-fit test depends on an adequate sample size). One disadvantage is that it requires a fully defined distribution which should not have been estimated from the data. If the parameters of the theoretical distribution have been estimated from the data, the critical region of the K-S test will be no longer valid.

The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.

This class uses an efficient and high-accuracy algorithm based on work by Richard Simard (2010). Please see KolmogorovSmirnovDistribution for more details.

References:

• Wikipedia, The Free Encyclopedia. Kolmogorov-Smirnov Test. Available at: http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
• NIST/SEMATECH e-Handbook of Statistical Methods. Kolmogorov-Smirnov Goodness-of-Fit Test. Available at: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
• Richard Simard, Pierre L’Ecuyer. Computing the Two-Sided Kolmogorov-Smirnov Distribution. Journal of Statistical Software. Volume VV, Issue II. Available at: http://www.iro.umontreal.ca/~lecuyer/myftp/papers/ksdist.pdf
• Kirkman, T.W. (1996) Statistics to Use. Available at: http://www.physics.csbsju.edu/stats/

In the following example, we will be creating a K-S test to verify if two samples have been drawn from different populations. In this example, we will first generate a number of samples from two different distributions and then check if the K-S can indeed see the difference between them:

// Generate 15 points from a Normal distribution with mean 5 and sigma 2
double[] sample1 = new NormalDistribution(mean: 5, stdDev: 1).Generate(25);

// Generate 15 points from an uniform distribution from 0 to 10
double[] sample2 = new UniformContinuousDistribution(a: 0, b: 10).Generate(25);

// Now we can create a K-S test and test the unequal hypothesis:
var test = new TwoSampleKolmogorovSmirnovTest(sample1, sample2,
    TwoSampleKolmogorovSmirnovTestHypothesis.SamplesDistributionsAreUnequal);

bool significant = test.Significant; // outputs true

The following example comes from the stats page of the College of Saint Benedict and Saint John's University (Kirkman, 1996). It is a very interesting example as it shows a case in which a t-test fails to see a difference between the samples because of the non-normality of the sample's distributions. The Kolmogorov-Smirnov nonparametric test, on the other hand, succeeds.

The example deals with the preference of bees between two nearby blooming trees in an empty field. The experimenter has collected data measuring how much time does a bee spent near a particular tree. The time starts to be measured when a bee first touches the tree, and is stopped when the bee moves more than 1 meter far from it. The samples below represents the measured time, in seconds, of the observed bees for each of the trees.

double[] redwell = 
{
    23.4, 30.9, 18.8, 23.0, 21.4, 1, 24.6, 23.8, 24.1, 18.7, 16.3, 20.3,
    14.9, 35.4, 21.6, 21.2, 21.0, 15.0, 15.6, 24.0, 34.6, 40.9, 30.7, 
    24.5, 16.6, 1, 21.7, 1, 23.6, 1, 25.7, 19.3, 46.9, 23.3, 21.8, 33.3, 
    24.9, 24.4, 1, 19.8, 17.2, 21.5, 25.5, 23.3, 18.6, 22.0, 29.8, 33.3,
    1, 21.3, 18.6, 26.8, 19.4, 21.1, 21.2, 20.5, 19.8, 26.3, 39.3, 21.4, 
    22.6, 1, 35.3, 7.0, 19.3, 21.3, 10.1, 20.2, 1, 36.2, 16.7, 21.1, 39.1,
    19.9, 32.1, 23.1, 21.8, 30.4, 19.62, 15.5 
};

double[] whitney = 
{
    16.5, 1, 22.6, 25.3, 23.7, 1, 23.3, 23.9, 16.2, 23.0, 21.6, 10.8, 12.2,
    23.6, 10.1, 24.4, 16.4, 11.7, 17.7, 34.3, 24.3, 18.7, 27.5, 25.8, 22.5,
    14.2, 21.7, 1, 31.2, 13.8, 29.7, 23.1, 26.1, 25.1, 23.4, 21.7, 24.4, 13.2,
    22.1, 26.7, 22.7, 1, 18.2, 28.7, 29.1, 27.4, 22.3, 13.2, 22.5, 25.0, 1,
    6.6, 23.7, 23.5, 17.3, 24.6, 27.8, 29.7, 25.3, 19.9, 18.2, 26.2, 20.4,
    23.3, 26.7, 26.0, 1, 25.1, 33.1, 35.0, 25.3, 23.6, 23.2, 20.2, 24.7, 22.6,
    39.1, 26.5, 22.7
};

// Create a t-test as a first attempt.
var t = new TwoSampleTTest(redwell, whitney);

Console.WriteLine("T-Test");
Console.WriteLine("Test p-value: " + t.PValue);    // ~0.837
Console.WriteLine("Significant? " + t.Significant); // false

// Create a non-parametric Kolmogorov-Smirnov test
var ks = new TwoSampleKolmogorovSmirnovTest(redwell, whitney);

Console.WriteLine("KS-Test");
Console.WriteLine("Test p-value: " + ks.PValue);    // ~0.038
Console.WriteLine("Significant? " + ks.Significant); // true

Reference