NAG Library Routine Document

g08cdf (test_ks_2sample)

1
Purpose

g08cdf performs the two sample Kolmogorov–Smirnov distribution test.

2
Specification

Fortran Interface
Subroutine g08cdf ( n1, x, n2, y, ntype, d, z, p, sx, sy, ifail)
Integer, Intent (In):: n1, n2, ntype
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x(n1), y(n2)
Real (Kind=nag_wp), Intent (Out):: d, z, p, sx(n1), sy(n2)
C Header Interface
#include <nagmk26.h>
void  g08cdf_ (const Integer *n1, const double x[], const Integer *n2, const double y[], const Integer *ntype, double *d, double *z, double *p, double sx[], double sy[], Integer *ifail)

3
Description

The data consists of two independent samples, one of size n1, denoted by x1,x2,,xn1, and the other of size n2 denoted by y1,y2,,yn2. Let Fx and Gx represent their respective, unknown, distribution functions. Also let S1x and S2x denote the values of the sample cumulative distribution functions at the point x for the two samples respectively.
The Kolmogorov–Smirnov test provides a test of the null hypothesis H0: Fx=Gx against one of the following alternative hypotheses:
(i) H1: FxGx.
(ii) H2: Fx>Gx. This alternative hypothesis is sometimes stated as, ‘The x's tend to be smaller than the y's’, i.e., it would be demonstrated in practical terms if the values of S1x tended to exceed the corresponding values of S2x.
(iii) H3: Fx<Gx. This alternative hypothesis is sometimes stated as, ‘The x's tend to be larger than the y's’, i.e., it would be demonstrated in practical terms if the values of S2x tended to exceed the corresponding values of S1x.
One of the following test statistics is computed depending on the particular alternative null hypothesis specified (see the description of the argument ntype in Section 5).
For the alternative hypothesis H1.
For the alternative hypothesis H2.
For the alternative hypothesis H3. g08cdf also returns the standardized statistic Z=n1+n2 n1n2 ×D, where D may be Dn1,n2, Dn1,n2+ or Dn1,n2- depending on the choice of the alternative hypothesis. The distribution of this statistic converges asymptotically to a distribution given by Smirnov as n1 and n2 increase; see Feller (1948), Kendall and Stuart (1973), Kim and Jenrich (1973), Smirnov (1933) or Smirnov (1948).
The probability, under the null hypothesis, of obtaining a value of the test statistic as extreme as that observed, is computed. If maxn1,n22500 and n1n210000 then an exact method given by Kim and Jenrich (see Kim and Jenrich (1973)) is used. Otherwise p is computed using the approximations suggested by Kim and Jenrich (1973). Note that the method used is only exact for continuous theoretical distributions. This method computes the two-sided probability. The one-sided probabilities are estimated by halving the two-sided probability. This is a good estimate for small p, that is p0.10, but it becomes very poor for larger p.

4
References

Conover W J (1980) Practical Nonparametric Statistics Wiley
Feller W (1948) On the Kolmogorov–Smirnov limit theorems for empirical distributions Ann. Math. Statist. 19 179–181
Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin
Kim P J and Jenrich R I (1973) Tables of exact sampling distribution of the two sample Kolmogorov–Smirnov criterion Dmnm<n Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
Smirnov N (1933) Estimate of deviation between empirical distribution functions in two independent samples Bull. Moscow Univ. 2(2) 3–16
Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281

5
Arguments

1:     n1 – IntegerInput
On entry: the number of observations in the first sample, n1.
Constraint: n11.
2:     xn1 – Real (Kind=nag_wp) arrayInput
On entry: the observations from the first sample, x1,x2,,xn1.
3:     n2 – IntegerInput
On entry: the number of observations in the second sample, n2.
Constraint: n21.
4:     yn2 – Real (Kind=nag_wp) arrayInput
On entry: the observations from the second sample, y1,y2,,yn2.
5:     ntype – IntegerInput
On entry: the statistic to be computed, i.e., the choice of alternative hypothesis.
ntype=1
Computes Dn1n2, to test against H1.
ntype=2
Computes Dn1n2+, to test against H2.
ntype=3
Computes Dn1n2-, to test against H3.
Constraint: ntype=1, 2 or 3.
6:     d – Real (Kind=nag_wp)Output
On exit: the Kolmogorov–Smirnov test statistic (Dn1n2, Dn1n2+ or Dn1n2- according to the value of ntype).
7:     z – Real (Kind=nag_wp)Output
On exit: a standardized value, Z , of the test statistic, D , without any correction for continuity.
8:     p – Real (Kind=nag_wp)Output
On exit: the tail probability associated with the observed value of D, where D may be Dn1,n2,Dn1,n2+ or Dn1,n2- depending on the value of ntype (see Section 3).
9:     sxn1 – Real (Kind=nag_wp) arrayOutput
On exit: the observations from the first sample sorted in ascending order.
10:   syn2 – Real (Kind=nag_wp) arrayOutput
On exit: the observations from the second sample sorted in ascending order.
11:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, n1=value.
Constraint: n11.
On entry, n2=value.
Constraint: n21.
ifail=2
On entry, ntype=value.
Constraint: ntype=1, 2 or 3.
ifail=3
The iterative process used in the approximation of the probability for large n1 and n2 did not converge. For the two sided test p=1 is returned. For the one-sided test p=0.5 is returned.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.

8
Parallelism and Performance

g08cdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

The time taken by g08cdf increases with n1 and n2, until n1n2>10000 or maxn1,n22500. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with n1 and n2.

10
Example

This example computes the two-sided Kolmogorov–Smirnov test statistic for two independent samples of size 100 and 50 respectively. The first sample is from a uniform distribution U0,2. The second sample is from a uniform distribution U0.25,2.25. The test statistic, Dn1,n2, the standardized test statistic, Z, and the tail probability, p, are computed and printed.

10.1
Program Text

Program Text (g08cdfe.f90)

10.2
Program Data

Program Data (g08cdfe.d)

10.3
Program Results

Program Results (g08cdfe.r)