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Generalised Smirnov two-sample homogeneity tests

Authorship This software was created by Melvin Brown Maple Primes: MelvinBrown email: anon-uz39rm6p18ug@base.google.com Acknowledgements The author is grateful to Andrey Nikiforov, who in 1994 published the GSMIRN algorithm which is implemented in the MAPLE package described in this worksheet, and who kindly gave the author permission to publish this transciption of GSMIRN from FORTRAN into MAPLE. Thanks also go to Andrey Nikiforov and to Diab Jerius for providing data and details of the tests that were carried out on the GSMIRN algorithm. These tests are implemented in this worksheet. Author disclaimer of liability The author makes no representations or warranties, express or implied, nor assumes any liability for the use of this software. Maintenance of notice In the interest of clarity regarding the origin and status of this software, the author requests that any recipient of it maintains these notices affixed to any distribution by the recipient that contains a copy or derivative of this software. 4 June 2011

The problem addressed by this worksheet is: Given two samples of data, which may contain ties, how may one test the hypothesis that they are drawn from the same distribution? The worksheet demonstrates the use of a MAPLE implementation of an algorithm to perform two-sample homogeneity tests, based on any one of three Kolmogorov-Smirnov (K-S) test statistics. K-S tests are nonparametric tests for the equality of one-dimensional probability distributions. The tests are used to compare a sample with a reference probability distribution (one-sample K-S test), or to compare two samples (two-sample K-S test). In particular, the K-S test statistic and its distribution may be used to perform statistical significance tests on the hypothesis of the equality of two sample distributions or of one sample distribution with a given distribution. Numerous tables and methods for calculating the distribution of K-S statistics have been published in the last 80 years; some of this activity is documented in [3-18]. The MAPLE package KSNstat, which is introduced in this worksheet, contains the MAPLE procedure gsmirn which implements the GSMIRN algorithm given in 1994 by Nikiforov [1] to calculate exact p-values for generalised (conditionally distribution-free) two-sample homogeneity tests based on two-sided and one-sided Kolomogorov-Smirnov statistics. Notably, the Nikiforov algorithm covers the range from discrete to continuous distributions; specifically, it handles tied data data points. On his website [1], Nikiforov provides an update (June 2006) in which he cautions against using the GSMIRN for large samples, advising use instead of his GSMIRN2 for sample sizes over ~10,000. GSMIRN2 is not implemented here, because this MAPLE implementation of GSMIRN display benefits from not having the unexpected behaviour, demonstrated by Diab Jerius, at the large values of K-S statistic and sample sizes to which Nikiforov refers. The Nikiforov algorithm [1] has also been implemented in the statistical package R [2]. JSFH

Given the two samples LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJYRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiYtSSVtc3ViR0YkNiVGMS1GIzYkLUkjbW5HRiQ2JFEiMUYnRj5GPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY7Ni1RIy4uRidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTUSYwLjBlbUYnLUZXNiVGMS1GIzYkLUZXNiUtRiw2JVEiTkYnRjRGNy1GIzYkLUYsNiVRInhGJ0Y0RjdGPkZpbkY+RmluRj5GK0Y+RitGPg== and 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 of independent and identically distributed observations from the distributions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJGRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJHRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg==, the homogeneity hypothesis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg==: 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 is tested against the following alternative hypotheses 1) two-sided: 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 2) one-sided: 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 3) one-sided: 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 for at least one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The tests are based on the Kolomogorov-Smirnov type statistics: 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J:`n\\tN\\NbZOJ:3:\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlcT5GKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQhKSpGKi0lKUNISUxEUkVORzYi in which the starred functions are the empirical distribution functions based on the samples LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The core algorithm (in procedure gsmirn) requires a statistic (of kind 1,2, or 3) for the two samples, and the density function of the pooled observations in all categories. The algorithm then computes the conditional probability LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that a statistic less that the observed statistic would be found for a random assignment of the pooled data to two samples of the same sizes as the input samples. The algorithm actually reports the (so-called) p-value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJRRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiYtSSNtbkdGJDYkUSIxRidGPi1GOzYtUSgmbWludXM7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmhuLUYsNiVRIlBGJ0Y0RjdGPkYrRj5GK0Y+, which is the probability that a random assignment of the pooled data to two sets of the sizes of the input data sets has a KS test statistic at least as great as that of the input data. This p-value is the basis of significance testing. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The KSNstat package comprises three procedures: stcalc: The procedure stcalc computes three kinds of K-S statistic and the pooled density function for a pair of samples. (stcalc is a transcription of Nikiforov's STCALC [1]; it takes advantage of MAPLE's ability to process sets.) gsmirn: The procedure gsmirn computes the p-value for any of the three statistics that may be calculated by stcalc. (gsmirn is a transcription of Nikiforov's GSMIRN [1]; it takes advantage of MAPLE's ability to compute factorials of large numbers.) gstest: The utility procedure gstest calls the latter two procedures, to compute and report all three K-S statistics and their p-values for a given pair of samples. The hierarchy of dependency is: gstest > stcalc > gsmirn.

The MAPLE procedures in the KSNstat package have been used to test gsmirn against the test set provided by Nikiforov [1], and have been found to replicate the results from his FORTRAN implementation of GSMIRN either exactly or to at least 14 decimal places. gsmirn has also been tested against the test set that was used by Diab Jerius to demonstrate unexpected p-values ~1 as produced by GSMRIN at K-S statistic = 0.5 for large (~20,000) sample sizes. The MAPLE procedure gsmirn has been found not to produce these unexpected results; instead it returns either exactly zero or zero to at least 12 decimal places, as expected. The details of the above tests are reported in this worksheet.

After installing the KSNstat package, its help database (.hdb) and archive (.mla) must be included in the MAPLE variable libname, for example: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= libname := "S:\134\134MRBwork\134\134MAPLE\134\134MRBlibrary\134\134KSNstatPackage\134\134KS.hdb",libname,"S:\134\134MRBwork\134\134MAPLE\134\134MRBlibrary\134\134KSNstatPackage\134\134KS.mla";

NiVRUlM6XE1SQndvcmtcTUFQTEVcTVJCbGlicmFyeVxLU05zdGF0UGFja2FnZVxLUy5oZGI2IlE+QzpcUHJvZ3JhbX5GaWxlc1xNYXBsZX4xNS9saWJGJFFSUzpcTVJCd29ya1xNQVBMRVxNUkJsaWJyYXJ5XEtTTnN0YXRQYWNrYWdlXEtTLm1sYUYk

Notice that the help database is place first in the list, and is searched first when a help command is executed. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKEtTTnN0YXRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

NyVJJ2dzbWlybkc2IkknZ3N0ZXN0R0YkSSdzdGNhbGNHRiQ=

Help on the KSNstat procedures may be obtained in the usual way: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEoZ3NtaXJuO0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw== A short description of any of the procedures may be obtained using the Describe command: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpRGVzY3JpYmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSdnc21pcm5GJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

# Original FORTRAN:-Algorithm AS 288: Exact two-sample Smirnov test for # arbitrary distributions. Appl.Stat., vol.43, No. 1. pp.265-270 (1994), by # Andrey Nikiforov; this MAPLE transcription by Melvin Brown, 2011. P-value # calculation for the generalized two-sample Smirnov tests. The tests allow for # ties in the pooled sample. Inputs: nx = number of observations in the first # sample (of F); ny = number of observations in the second sample (of G); kind # ={1:F<>G |2:F>G |3:F<G},i.e. the hypothesis tested; m = number of # observations in each of K (ascending) categories; dstat = the statistic. # Outputs: probability p-value; OR fault indicator =1 if nx<1 or ny<1, = 2 if # kind <> 1,2 or 3, =3 if p-value <= 0, =4 if m inconsistent with nx and ny or # has elements <= 0. gsmirn( nx::integer, ny::integer, kind::integer, m::list, dstat::realcons )

In this section, we test the gsmirn procedure against the test cases provided in Nikiforov's datasets [1]. The following subsections are titled with the names of Nikiforov's test datasets. The results demonstrate that the MAPLE implementation of Nikiforov's algorithm agrees with his FORTRAN implementation to very high precision, where there is not exact agreement. The tests are also examples of the use of gsmirn.

For 1.smp we have lx:=[1,2,3];ly:=[4,5,6,7];

NyUiIiIiIiMiIiQ=

NyYiIiUiIiYiIiciIig=

Compute the statistics of the two samples using stcalc: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIk

IiIl

NykiIiJGI0YjRiNGI0YjRiM=

NyUiIiJGIyIiIQ==

These statistics agree with those provided for the example 1.smp given by Nikiforov. We now use these statistics in gsmirn to compute the p-values for the three KS statistics for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIwciZHOWRHOWQhIzs=

JCIwJ0c5ZEc5ZEchIzs=

IiIi

This agrees with the results in 1.smp to 15 digits.

For 1and9.smp we have lx:=[1];ly:=[2,3,4,5,6,7,8,9,10];

NyMiIiI=

NysiIiMiIiQiIiUiIiYiIiciIigiIikiIioiIzU=

Compute then statistics of the two samples: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIi

IiIq

NywiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

NyUiIiJGIyIiIQ==

These statistics agree with those provided for the example 2.smp given by Nikiforov. We now use these statistics in gsmirn() to compute the p-values for the three KS statistics for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxKysrKysrKz8hIzs=

JCIxKysrKysrKzUhIzs=

IiIi

This precisely agrees with the results in 1and9.smp.

For 1and19A.smp we have lx:=[1];ly:=[2,2.1,2.3,3,3.1,4,4.1,5,5.1,6,6.1,7,7.1,8,8.1,9.1,9,10.1,10];

NyMiIiI=

NzUiIiMkIiNAISIiJCIjQkYmIiIkJCIjSkYmIiIlJCIjVEYmIiImJCIjXkYmIiInJCIjaEYmIiIoJCIjckYmIiIpJCIjIilGJiQiIyIqRiYiIiokIiQsIkYmIiM1

Compute then statistics of the two samples: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIi

IiM+

NzYiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGIw==

NyUiIiJGIyIiIQ==

These statistics agree with those provided for the example 2.smp given by Nikiforov. We now use these statistics in gsmirn to compute the p-values for the three KS statistics for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxKysrKysrKzUhIzs=

JCIwKysrKysrKyYhIzs=

IiIi

This precisely agrees with 1and19A.smp.

For 1and19B.smp we have lx:=[1];ly:=[2,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10];

NyMiIiI=

NzUiIiNGI0YjIiIkRiQiIiVGJSIiJkYmIiInRiciIihGKCIiKUYpIiIqRioiIzVGKw==

Compute then statistics of the two samples: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIi

IiM+

NywiIiIiIiQiIiNGJUYlRiVGJUYlRiVGJQ==

NyUiIiJGIyIiIQ==

These statistics agree with those provided for the example 2.smp given by Nikiforov. We now use these statistics in gsmirn to compute the p-values for the three KS statistics for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIwKysrKysrKyYhIzs=

JCIwKysrKysrKyYhIzs=

IiIi

This precisely agrees with the results in 1and19B.smp.

We test against the example 2.smp in Nikiforov [1]: lx:=[1,1,3,4,4,5];ly:=[2,2,4];

NygiIiJGIyIiJCIiJUYlIiIm

NyUiIiNGIyIiJQ==

Compute then statistics of the two samples: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIn

IiIk

NyciIiNGIyIiIiIiJEYk

NyUjIiIiIiIkRiNGIw==

These statistics agree with those provided for the example 2.smp given by Nikiforov. We now use these statistics in gsmirn to compute the p-values for the three KS statistics for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxJ0c5ZEc5ZEcqISM7

JCIxSDlkRzlkR2shIzs=

JCIxWyE+dy8+dy8lISM7

These results agree (except in the last digit) with those in 2.smp.

For the 3.smp test we have: lx:=[1,2,2];ly:=[1,2];

NyUiIiIiIiNGJA==

NyQiIiIiIiM=

sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIk

IiIj

NyQiIiMiIiQ=

NyUjIiIiIiInIiIhRiM=

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

IiIi

IiIi

JCIxKysrKysrK3EhIzs=

which agree precisely with the results for 3.smp.

Now for 5.smp: lx:=[1];ly:=[1,2];

NyMiIiI=

NyQiIiIiIiM=

sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIi

IiIj

NyQiIiMiIiI=

NyUjIiIiIiIjRiMiIiE=

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

IiIi

JCIxbm1tbW1tbW0hIzs=

IiIi

This agrees precisely with 5.smp results.

For 6.smp we have lx:=[1];ly:=[2];

NyMiIiI=

NyMiIiM=

sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiIi

IiIi

NyQiIiJGIw==

NyUiIiJGIyIiIQ==

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

IiIi

JCIxKysrKysrK10hIzs=

IiIi

This agrees precisely with 6.smp.

Now for 11.smp: lx:=[1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3];ly:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3]; sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

Nz8iIiJGI0YjRiNGI0YjRiNGI0YjRiMiIiNGJEYkRiRGJEYkRiRGJEYkRiQiIiRGJUYlRiVGJUYlRiVGJUYl

N0oiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGIyIiI0YkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiQiIiRGJUYlRiVGJQ==

IiNI

IiNT

NyUiI0kiI0QiIzk=

NyUjIiNWIiRLIyIiIUYj

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxX1AzbFxEUD4hIzs=

IiIi

JCIxQScpUkZLOGI1ISM7

This agrees with 11.smp to 14 significant figures.

Now test against 300_650.smp which is the Example 2 presented on page 5 of Nikiforov paper [1]: lx:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10]: ly:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10]: sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiQrJA==

IiRdJw==

NywiJFMiIiNTIiMhKSIjISoiJCsiRidGJ0YnRidGJw==

NyUjIiIqIiRJIiMiIiQiI2xGIw==

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxTnA0bCU9R0UiISM7

JCIxV0doLlIyY0MhIzs=

JCIwSkYkZSplITNqISM7

These results agree with those given in Example 2, p.6 [1] to all significant figures given there.

For nair5.smp [Nair, 1987, J.Am.Stat.Acc., table 5], we have lx:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5];ly:=[1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5];

N1xwIiIiRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjIiIjRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJCIiJEYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJSIiJUYmRiZGJiIiJkYnRidGJ0YnRidGJ0Yn

N1xwIiIiRiNGI0YjRiNGI0YjRiNGI0YjRiMiIiNGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiQiIiRGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiUiIiVGJkYmRiZGJkYmRiZGJiIiJkYnRidGJ0YnRidGJ0YnRidGJ0YnRidGJ0Yn

sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiMhKQ==

IiMhKQ==

NyciI0YiI1wiI10iIzciI0E=

NyUjIiIiIiImRiMiIiE=

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIwJ3pkYiIpXGY/ISM7

JCIwXUUhcDR2SDUhIzs=

IiIi

This agrees with results for nair4.smp to within 5e-16 and 3e-14 respectively .

Now the new2.smp example which appears in the Nikiforov paper on page 5, as Example 1 of [1]: lx:=[1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4];ly:=[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4];

N2RyIiIiRiNGI0YjRiNGI0YjRiNGI0YjIiIjRiRGJEYkRiRGJEYkRiRGJEYkIiIkRiVGJUYlRiVGJUYlRiVGJUYlRiNGI0YjRiNGI0YjRiNGI0YjRiMiIiVGJkYmRiZGJkYmRiZGJkYmRiZGI0YjRiNGI0YjRiNGI0YjRiNGI0YkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiVGJUYlRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJg==

N2J0IiIiRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGIyIiI0YkRiRGJEYkRiRGJEYkRiRGJCIiJEYlRiVGJUYlRiVGJUYlRiVGJUYjRiNGI0YjRiNGI0YjRiNGI0YjIiIlRiZGJkYmRiZGJkYmRiZGJkYmRiNGI0YjRiNGI0YjRiNGI0YjRiNGJEYkRiRGJEYkRiRGJEYkRiRGJEYmRiZGJkYmRiZGJkYmRiZGJkYmRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYkRiRGJEYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJg==

sstats:=stcalc(lx,ly):nx:=sstats[1];ny:=sstats[2];m:=sstats[3];dstat:=sstats[4];

IiQ/Ig==

IiRdIg==

NyYiIyEpIiNxIiNTRiM=

NyUjIiIiIiM1I0YkIiM3RiM=

for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,ny,kind,m,dstat[kind]);print(qprob); end do:

JCIxcEopeiMpNCZcPCEjOw==

JCIxc1htXSJ6YmEiISM7

JCIwU0c+TWE3VSohIzs=

These results agree with those given in Example 1, p.6 [1] to all significant figures given there.

We are motivated by Nikiforov's remark on his website [1] that for samples sizes >10,000 his routine GSMIRN2 should be used, instead of GSMIRN, in light of unexpected results under particular conditions in tests of GSMIRN recorded by Diab Jerius [1]. Under the same test conditions, we find that gsmirn does not produce unexpected results.

Here, we investigate two different methods of calculating the products and ratios of factorial terms in nx, ny, which are used in gsmirn to determine the the number of ways two samples may be drawn from the pool of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjbnhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkwtRiw2JVEjbnlGJ0YvRjJGOQ==. In his GSMIRN, Nikiforov uses the logarithm of factorials to avoid numerical overflows for large sample sizes, in the final calculation of the p-value. In the MAPLE procedure gsmirn, we compute the factorials directly without the use of logarithms, so that the calculation of the p-value includes the following function of nx and ny: fnxy:=(nx,ny)->(nx!)*(ny!)/(nx+ny)!;

Zio2JEkjbnhHNiJJI255R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqKC1JKmZhY3RvcmlhbEclKnByb3RlY3RlZEc2I0YkIiIiLUYsNiNGJkYvLUYsNiMsJkYkRi9GJkYvISIiRiVGJUYl

An alternative expression of this function is in the product form: fprod:=(nx,ny)->product(k/(nx+k),k=1..ny);

Zio2JEkjbnhHNiJJI255R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSShwcm9kdWN0R0YlNiQqJkkia0dGJSIiIiwmRiRGL0YuRi8hIiIvRi47Ri9GJkYlRiVGJQ==

The two forms are equal: simplify(fnxy(nx,ny)/fprod(nx,ny));

IiIi

We are interested in which form executes faster. We can compute their respective run times using the following test: numit:=10:nx:=50000:ny:=50000: st := time(): for i from 1 by 1 to numit do fnxy(nx,ny): end do: tfact:=(time() - st)/numit; st := time(): for i from 1 by 1 to numit do fprod(nx,ny): end do: tprod:=(time() - st)/numit; tratio:=tprod/tfact; vratio:=fprod(nx,ny)/fnxy(nx,ny);

JCIrKysrdFohIzU=

JCIrKytndjohIio=

JCIrNSZvNUkkISIq

IiIi

This test shows that the full factorial ratio executes between 15 and 2 times faster than the product evaluation. At nx=ny=50,000 execution time is ~ 1s; at nx=ny= 100, execution time is ~1e-5 s. The product version becomes faster than the factorial version when nx>>ny, and very much slower when ny>>nx. The reason for this is easly seen from the above expressions. The ratio of factorials form is symmetric in nx and ny, whereas ny determines the number of terms in the product form. Thus, in the latter, we should always choose the smallest of nx and ny as the limit of the product, then for big differences in sample sizes we may get an advantage using the product form. But in general, the product form is slower. MAPLE is clearly able easily to handle very large factorials, with ease. 10000! executes seemingly instantaneously. We conclude that we may continue to use the ratio factorials form; it is this form that is used in gsmirn.

Here we present code for the critical Jerius test using gsmirn. The results of this testing are reported both to screen (with error reporting from gsmirn) and to a text file (without error reporting from gsmirn) . The author is grateful to Diab Jerius for providing the data files containing the results of the tests he executed on GSMIRN. nx:=20000;ny:=20000;dstall:=0.5:dstat:=[dstall,dstall,dstall];m:=[seq(1,kat=1..nx+ny)]:

IiYrKyM=

IiYrKyM=

NyUkIiImISIiRiNGIw==

fd := fopen("S:\134\134MRBwork\134\134MAPLE\134\134MRBlibrary\134\134KSNstatPackage\134\134Jerius/MapleTest1.txt",APPEND): printf("%76.76s\134n","-----------------------------------------------------------------------------------"): fprintf(fd,"%76.76s\134n","-----------------------------------------------------------------------------------"): printf("%8s %8s %8s %22s %21s\134n",NX,NY,KIND,DSTATS,Q): fprintf(fd,"%8s %8s %8s %22s %21s\134n",NX,NY,KIND,DSTATS,Q): for nny from 11500 by 200 to 12500 do printf("%76.76s\134n","-----------------------------------------------------------------------------------"): fprintf(fd,"%76.76s\134n","-----------------------------------------------------------------------------------"): for kind from 1 by 1 to 3 do qprob:=gsmirn(nx,nny,kind,m,dstat[kind]): printf("%8d %8d %8d %22.7f %21.16f\134n",nx,nny,kind,dstat[kind],qprob): fprintf(fd,"%8d %8d %8d %22.7f %21.16f\134n",nx,nny,kind,dstat[kind],qprob): end do: end do: printf("%76.76s\134n","-----------------------------------------------------------------------------------"): fprintf(fd,"%76.76s\134n","-----------------------------------------------------------------------------------"): fclose(fd):

---------------------------------------------------------------------------- NX NY KIND DSTATS Q ---------------------------------------------------------------------------- ifault = 3 20000 11500 1 0.5000000 0.0000000000000000 ifault = 3 20000 11500 2 0.5000000 0.0000000000000000 ifault = 3 20000 11500 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- 20000 11700 1 0.5000000 0.0000000000025247 20000 11700 2 0.5000000 0.0000000000025247 20000 11700 3 0.5000000 0.0000000000025247 ---------------------------------------------------------------------------- ifault = 3 20000 11900 1 0.5000000 0.0000000000000000 ifault = 3 20000 11900 2 0.5000000 0.0000000000000000 ifault = 3 20000 11900 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- 20000 12100 1 0.5000000 0.0000000000005261 20000 12100 2 0.5000000 0.0000000000005261 20000 12100 3 0.5000000 0.0000000000005261 ---------------------------------------------------------------------------- ifault = 3 20000 12300 1 0.5000000 0.0000000000000000 ifault = 3 20000 12300 2 0.5000000 0.0000000000000000 ifault = 3 20000 12300 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- ifault = 3 20000 12500 1 0.5000000 0.0000000000000000 ifault = 3 20000 12500 2 0.5000000 0.0000000000000000 ifault = 3 20000 12500 3 0.5000000 0.0000000000000000 ----------------------------------------------------------------------------

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Observe that the ifault = 3 print out appears on screen, but it is sent to the results file; gsmirn produces output = 0 under the ifault = 3 condition (i.e. when computed p-value <= 0). The output sent to the results file is as follows: ---------------------------------------------------------------------------- NX NY KIND DSTATS Q ---------------------------------------------------------------------------- 20000 11500 1 0.5000000 0.0000000000000000 20000 11500 2 0.5000000 0.0000000000000000 20000 11500 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- 20000 11700 1 0.5000000 0.0000000000025247 20000 11700 2 0.5000000 0.0000000000025247 20000 11700 3 0.5000000 0.0000000000025247 ---------------------------------------------------------------------------- 20000 11900 1 0.5000000 0.0000000000000000 20000 11900 2 0.5000000 0.0000000000000000 20000 11900 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- 20000 12100 1 0.5000000 0.0000000000005261 20000 12100 2 0.5000000 0.0000000000005261 20000 12100 3 0.5000000 0.0000000000005261 ---------------------------------------------------------------------------- 20000 12300 1 0.5000000 0.0000000000000000 20000 12300 2 0.5000000 0.0000000000000000 20000 12300 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- 20000 12500 1 0.5000000 0.0000000000000000 20000 12500 2 0.5000000 0.0000000000000000 20000 12500 3 0.5000000 0.0000000000000000 ---------------------------------------------------------------------------- in which Q is the p-value. This test demonstrates that gsmirn does not produce the unexpected p-values (ranging from ~0.99999.. ~0.0004 over the range 11500..12500 of ny) for K-S statistic = 0.5 which were obtained by Diab Jerius in his corresponding test of Nikiforov's GSMIRN. For such large samples sizes and K-S statistic = 0.5 (regardless of kind), the p-value is expected to be very close to zero; the results obtained from gsmirn are consistent with this as are those as obtained by Diab Jerius when repeating the same test using GSMIRN2.

Here, we investigate some properties of the p-value distribution. Note that, for convenience, the following examples have pool size n = nx + ny; i.e. the pool size is the sum of the sizes of the two samples. This will not be the case, if the samples have common categories.

We plot the dependence of p-value on the K-S statistic for different sample sizes: nx:=23;ny:=50;m:=[seq(1,kat=1..nx+ny)]:

IiNC

IiNd

gfun:=(x,kind)->gsmirn(nx,ny,kind,m,x): plot(['gfun(x,1)','gfun(x,2)','gfun(x,3)'],x=0..0.6,color=[red,green,blue],labels=["K-S statistic","p-value"], gridlines,legend=["type 1", "type 2", "type 3"],title=cat("p-value as function of KS-statistic \134nfor two samples: nx = ",nx," ny = ",ny," randomly drawn\134n from pooled sample without ties"));

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The p-value distribution is the same function for the two one-sided K-S statistics (blue); the two-sided statistic (red) has a larger value at a given level of signiticance. The discontinuous nature of the p-value distribution is most obvious when nx = ny; for certain values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSNueEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSNueUYnRjRGN0Y+RitGPg== the distribution is much finer grained). This behaviour is related to how the number of ways that two samples can be drawn from the pool varies with (nx, ny). For example, with nx=23, ny=50, the p-value appears to vary relatively 'smoothly' with K-S statistic, compared to nx=25, ny=50, which is very obviously discontinuous. JSFH

We now show (using the case of equal sample sizes) how the p-value distribution depends on the pooled sample size. We choose the two-sided K-S statistic as the independent variable for two samples each of size n/2 and no overlapping categories. m:=[seq(1,kat=1..200)]: gfun:=(x,kind,nz)->gsmirn(nz,nz,kind,m,x): plot(['gfun(x,1,20)','gfun(x,1,50)','gfun(x,1,100)'],x=0..0.5,color=[red,green,blue],labels=["K-S statistic","p-value"],gridlines,legend=["n=40","n=100","n=200"],title="p-value as function of two-sided KS-statistic \134nfor two equal samples of n/2 randomly drawn\134n from pooled sample of n without ties");

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= This plot illustrates how, with increasing pooled sample size, the distribution of the p-value shifts towards the zero K-S statistic, demonstrating how the values of significant K-S statistics depend strongly on sample size. Notice that, with reference to the Jerius test in a previous section, a K-S statistic = 0.5 for n~10000, is expected to have a very small value indeed. JSFH

Next, we explain why the distributions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIzRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= are the same. Referring to the domain (x,y) of the pairs (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) as defined in [1], we see that it has dimension 1 x 1, so two samples of different size each have domain of size 1. In this (x,y) domain, lines of constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIzRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= run parallel and respectively below and above the line y=x. A pair of samples taken from the pool is represented as a trajectory in this domain in the manner described in [1]. Thus, for the case, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIzRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 , there are two lines parallel to the line x=y and equidistant from it. By symmetry, it follows that the number of trajectories exceeding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIzRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= for at least one of their points exactly equals the number exceeding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUScmIzkxNjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo=. Thus, the p-value as a function of K-S statistic is the same for kinds 2 and 3. We can also see the p-value for as a function of K-S statistic for kind 1 is greater than for kinds 2 and 3. Let us investigate the latter conclusions for various values of nx and ny: nx:=50:ny:=10:m:=[seq(1,kat=1..nx+ny)]: gfun:=(x,kind)->gsmirn(nx,ny,kind,m,x): p2:=plot(['gfun(x,1)','gfun(x,2)','gfun(x,3)'],x=0..0.6,color=[red,green,blue],labels=["K-S statistic","p-value"], gridlines,legend=["type 1", "type 2", "type 3"],title=cat("p-value as function of KS-statistic \134nfor two samples: nx = ",nx," ny = ",ny," randomly drawn\134n from pooled sample without ties")):p2;

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Various plots of this type confirm that the p-value distributions of K-S statistics of kind=2,3 are the same, but are different from that of the K-S statistic of kind=1. JSFH

We have presented the KSNstat package, which includes an MAPLE implementation gsmirn of Nikiforov's GSMIRN Fortran subroutine [1] for performing generalised Smirnov two-sample homogeneity tests. With the exception of its handing of the product and division of factorials of (potentially large) integer numbers, gsmirn is a transcription from Fortran to MAPLE. Tests on gsmirn have shown its p-values to be in close agreement with those of GSMIRN in tests provided by Nikiforov, and in the case of a test provided by Jerius, gsmirn does not give the unexpected results that GSMIRN does. The utility procedures gstest and stcalc are provided in the KSNstat package to support the use of gsmirn. Execution times for gsmirn vary widely; for example, for the large (~10000) sample sizes and extreme value of K-S statistic = 0.5 of the Jerius test presented in this worksheet, execution time was ~8,000 seconds for 18 calls to gsmirn on a 2.4 GHz 4 core 64-bit machine (not multi-threaded). However, for more practical samples sizes and statistics, the execution time is ~1s or less, as illustrated in the tests provided by Nikiforov.

The main reference for this worksheet is [1], however, in the course of the developing the KSNstat package, the author created a list of more general references relating to K-S tests; these may be of interest to user of the package. [1] Exact Smirnov two-sample tests for arbitrary distributions, A. Nikiforov, Appl.Stat., vol.43, No. 1. pp.265-270, 1994. (Follow this link to the author's publication page, where the paper publication, the algorithm, the FORTRAN implementation, tests, and comments can be found.) [2] Thomas Waterhouse, "ks.test - The two-sample two-sided Kolmogorov-Smirnov test with tie", R-developers website/forum. 22 July 2009. [3] G08CDF \342\200\223 NAG Fortran Library Routine Document [4] NAG C Library Function Document: nag_prob_2_sample_ks (g01ezc) [5] Marsaglia, G., Tsang, W. W., Wang, J. (2003) "Evaluating Kolmogorov\342\200\231s Distribution", Journal of Statistical Software, 8 (18), 1\342\200\2234. [6] Kolmogorov\342\200\223Smirnov test, Wikipedia. [7] Engineering Statistics Handbook [8] Short explantory introduction [9] JavaScript implementation of one-sided tests [10] ERI Distance Learning Centre: Table 7 K-S distribution (two-sided) [11] Kolmogorov-Smirnov one-sided test [12] Computing the Two-Sided Kolmogorov-Smirnov Distribution, March 2011 [13] Critical Values for the Two-sample Kolmogorov-Smirnov test (2-sided) [14] DEFINITIONS AND FORMULAE WITH STATISTICAL TABLES FOR ELEMENTARY STATISTICS AND QUANTITATIVE METHODS COURSES [15] Nonparametric Statistics, Bendre, S. M., Hyderbad (Note: the scaling of the statistics appears to be missing the square-root) [16] Journal of Statistical Software [17] Two-Sample Kolmogorov-Smirnov Test [18] Kim, P. J. and Jennrich, R. I. (1973). Tables of the exact sampling distribution of the two-sample Kolmogorov-Smirnov criterion, Dmn, m \342\211\244 n. (Page 79 of Selected Tables in Mathematical Statistics (Edited by H. L. Harter and D. B. Owen), Vol. I. American Mathematical Society, 1973 second print with revisions to original print 1970.)

The author would welcome feedback from users of the KSNstat package, particularly of errors that are found or of suggestions for improvements.

Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2009. Maplesoft and Maple are trademarks of Waterloo Maple Inc. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities. 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