View source for Kolmogorov-Smirnov Tests

Are the data points in a sample drawn from a specific known distribution? For example, are they drawn from a <code>[[Gamma]](3,2)</code> distribution?  Are the data points in one sample drawn from the same distribution as the data points in a second sample?

The Kolmogorov-Smirnov test(s) can be used to answer either of these questions.
* The data must be univariate and continuous (i.e., each sample point is a single real number).
* It works for any continuous distribution -- there is no assumption that the distribution is [[Normal]] or anything else.

The tests return a p-value, which is the probability of seeing as much distance between the two distributions (in the one-sample case, between the sample and the reference) if there is no difference in distribution. A small p-value, say p-value < 0.05, means a difference was detected, with a false-positive rate equal to 0.05. If the p-value > 0.05, you can't conclude that they are the same distribution -- only that the test and amount of data did not provide enough power to detect any difference if one exists.

== Kolmogorov-Smirnov Tests Library ==

'''Download:''' [[media:Kolmogorov-Smirnov Tests Library.ana]]

To use this library, use '''File / Add Library...''' from the File menu after downloading.

This library provides two functions:
; <code>KS_test_one_dist( x, I, refCdf )</code>
: Tests whether the empirical distribution of data «x», which is indexed by «<code>I</code>», matches a hypothesized distribution whose cumulative probability function is passed as the 3rd parameter.
; <code>KS_test_two_dist( xi, xj, I, J )</code>
:Tests whether the data in «xi» and data in «xj» appear to be from the same distribution.  «xi» is indexed by «<code>I</code>» and «xj» is indexed by «<code>J</code>».

Both functions return the p-value.
;If p-value < 0.05
:Conclude (with statistical significance 0.05) that the distributions differ.
;If p-value > 0.05
:You can't conclude that they have the same or different distributions. Just that there is not enough statistical power in the test, or enough data. to detect a difference in distribution if one exists.

== Using KS_test_one_dist ==

To do a one-sample test, you need to provide a cdf function for your hypothesized distribution. You can do this either by creating a new global [[UDF]], or by using a [[Local functions and local value captures|Local function]].

To test whether the data in <code>X</code> is distributed as <code>Gamma( 5.9, 1.65 )</code>, you can first create a [[UDF]]:
:<code>Function MyHypothCDF(x) ::= CumGamma( x, 5.9, 1.65)</code>
And then pass this to the function using:
:<code>KS_test_one_dist( x, I, MyHypothCDF )</code>

Alternately, you can use a local function like this:
:<code>[[Local functions and local value captures|Function]] hypCdf(x) : CumGamma( x, 5.9, 1.65 );</code>
:<code>KS_test_one_dist( x, I, hypCdf )</code>

It is important to notice that a sole colon is used between the function declaration and its defining expression in the local declaration (i.e., not an assignment operator, <code>:=</code>).

== Using KS_test_two_dist ==
When comparing two data sets, each may have its own index, and each can be a different length. The usage is easy -- just pass each, then pass the index of each:
:<code>KS_test_two_dist( xi, xj, I, J )</code>

== Limitations ==
There are some things that should be cleaned up in this model.
* The identifiers that appear in the '''Examples''' module don't have a namespace prefix. Hence, they are at risk for name collisions. (If name collisions occur, you'll see a dialog when you Add the library to your model).
* You'll get a warning if you open this in a release prior to [[Analytica 6.3]]. You can ignore the warning as long as your release is fairly recent (like 6.x).