NAG Library Function Document
nag_2_sample_ks_test (g08cdc)
1 Purpose
nag_2_sample_ks_test (g08cdc) performs the two sample Kolmogorov–Smirnov distribution test.
2 Specification
#include <nag.h> 
#include <nagg08.h> 
void 
nag_2_sample_ks_test (Integer n1,
const double x[],
Integer n2,
const double y[],
Nag_TestStatistics dtype,
double *d,
double *z,
double *p,
NagError *fail) 

3 Description
The data consist of two independent samples, one of size ${n}_{1}$, denoted by ${x}_{1},{x}_{2},\dots ,{x}_{{n}_{1}}$, and the other of size ${n}_{2}$ denoted by ${y}_{1},{y}_{2},\dots ,{y}_{{n}_{2}}$. Let $F\left(x\right)$ and $G\left(x\right)$ represent their respective, unknown, distribution functions. Also let ${S}_{1}\left(x\right)$ and ${S}_{2}\left(x\right)$ 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
${H}_{0}$ :
$F\left(x\right)=G\left(x\right)$ against one of the following alternative hypotheses:
(i) 
${H}_{1}$ : $F\left(x\right)\ne G\left(x\right)$. 
(ii) 
${H}_{2}$ : $F\left(x\right)>G\left(x\right)$. 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 ${S}_{1}\left(x\right)$ tended to exceed the corresponding values of ${S}_{2}\left(x\right)$. 
(iii) 
${H}_{3}$ : $F\left(x\right)<G\left(x\right)$. 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 ${S}_{2}\left(x\right)$ tended to exceed the corresponding values of ${S}_{1}\left(x\right)$. 
One of the following test statistics is computed depending on the particular alternative null hypothesis specified (see the description of the argument
dtype in
Section 5).
For the alternative hypothesis
${H}_{1}$.
 ${D}_{{n}_{1},{n}_{2}}$ – the largest absolute deviation between the two sample cumulative distribution functions.
For the alternative hypothesis
${H}_{2}$.
 ${D}_{{n}_{1},{n}_{2}}^{+}$ – the largest positive deviation between the sample cumulative distribution function of the first sample, ${S}_{1}\left(x\right)$, and the sample cumulative distribution function of the second sample, ${S}_{2}\left(x\right)$. Formally ${D}_{{n}_{1},{n}_{2}}^{+}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{S}_{1}\left(x\right){S}_{2}\left(x\right),0\right\}$.
For the alternative hypothesis
${H}_{3}$.
 ${D}_{{n}_{1},{n}_{2}}^{}$ – the largest positive deviation between the sample cumulative distribution function of the second sample, ${S}_{2}\left(x\right)$, and the sample cumulative distribution function of the first sample, ${S}_{1}\left(x\right)$. Formally ${D}_{{n}_{1},{n}_{2}}^{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{S}_{2}\left(x\right){S}_{1}\left(x\right),0\right\}$.
nag_2_sample_ks_test (g08cdc) also returns the standardized statistic
$Z=\sqrt{\left({n}_{1}+{n}_{2}/{n}_{1}{n}_{2}\right)}\times D$ where
$D$ may be
${D}_{{n}_{1},{n}_{2}}$,
${D}_{{n}_{1},{n}_{2}}^{+}$ or
${D}_{{n}_{1},{n}_{2}}^{}$ depending on the choice of the alternative hypothesis. The distribution of this statistic converges asymptotically to a distribution given by Smirnov as
${n}_{1}$ and
${n}_{2}$ 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
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\le 2500$ and
${n}_{1}{n}_{2}\le 10000$ then an exact method given by Kim and Jenrich is used. Otherwise
$p$ is computed using the approximations suggested by Kim and Jenrich (see
Kim and Jenrich (1973)). Note that the method used is only exact for continuous theoretical distributions. This method computes the twosided probability. The onesided probabilities are estimated by halving the twosided probability. This is a good estimate for small
$p$, that is
$p\le 0.10$, 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 ${D}_{mn}\left(m<n\right)$ Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society
Siegel S (1956) Nonparametric 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:
$\mathbf{n1}$ – IntegerInput

On entry: the number of observations in the first sample, ${n}_{1}$.
Constraint:
${\mathbf{n1}}\ge 1$.
 2:
$\mathbf{x}\left[{\mathbf{n1}}\right]$ – const doubleInput

On entry: the observations from the first sample, ${x}_{1},{x}_{2},\dots ,{x}_{{n}_{1}}$.
 3:
$\mathbf{n2}$ – IntegerInput

On entry: the number of observations in the second sample, ${n}_{2}$.
Constraint:
${\mathbf{n2}}\ge 1$.
 4:
$\mathbf{y}\left[{\mathbf{n2}}\right]$ – const doubleInput

On entry: the observations from the second sample, ${y}_{1},{y}_{2},\dots ,{y}_{{n}_{2}}$.
 5:
$\mathbf{dtype}$ – Nag_TestStatisticsInput

On entry: the statistic to be computed, i.e., the choice of alternative hypothesis.
 ${\mathbf{dtype}}=\mathrm{Nag\_TestStatisticsDAbs}$
 Computes ${D}_{{n}_{1}{n}_{2}}$, to test against ${H}_{1}$.
 ${\mathbf{dtype}}=\mathrm{Nag\_TestStatisticsDPos}$
 Computes ${D}_{{n}_{1}{n}_{2}}^{+}$, to test against ${H}_{2}$.
 ${\mathbf{dtype}}=\mathrm{Nag\_TestStatisticsDNeg}$
 Computes ${D}_{{n}_{1}{n}_{2}}^{}$, to test against ${H}_{3}$.
Constraint:
${\mathbf{dtype}}=\mathrm{Nag\_TestStatisticsDAbs}$, $\mathrm{Nag\_TestStatisticsDPos}$ or $\mathrm{Nag\_TestStatisticsDNeg}$.
 6:
$\mathbf{d}$ – double *Output

On exit: the Kolmogorov–Smirnov test statistic (
${D}_{{n}_{1}{n}_{2}}$,
${D}_{{n}_{1}{n}_{2}}^{+}$ or
${D}_{{n}_{1}{n}_{2}}^{}$ according to the value of
dtype).
 7:
$\mathbf{z}$ – double *Output

On exit: a standardized value, $Z$, of the test statistic, $D$, without any correction for continuity.
 8:
$\mathbf{p}$ – double *Output

On exit: the tail probability associated with the observed value of
$D$, where
$D$ may be
${D}_{{n}_{1},{n}_{2}},{D}_{{n}_{1},{n}_{2}}^{+}$ or
${D}_{{n}_{1},{n}_{2}}^{}$ depending on the value of
dtype (see
Section 3).
 9:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
 NE_BAD_PARAM

On entry, argument
dtype had an illegal value.
 NE_G08CD_CONV

The iterative procedure used in the approximation of the probability for large
n1 and
n2 did not converge. For the twosided test,
${\mathbf{p}}=1.0$ is returned. For the onesided test,
${\mathbf{p}}=0.5$ is returned.
 NE_INT_ARG_LT

On entry,
n1 must not be less than 1:
${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$.
On entry,
n2 must not be less than 1:
${\mathbf{n2}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
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
nag_2_sample_ks_test (g08cdc) is not threaded in any implementation.
The time taken by nag_2_sample_ks_test (g08cdc) increases with ${n}_{1}$ and ${n}_{2}$, until ${n}_{1}{n}_{2}>10000$ or $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\ge 2500$. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with ${n}_{1}$ and ${n}_{2}$.
10 Example
The following example computes the twosided Kolmogorov–Smirnov test statistic for two independent samples of size 100 and 50 respectively. The first sample is from a uniform distribution $U\left(0,2\right)$. The second sample is from a uniform distribution $U\left(0.25,2.25\right)$. The test statistic, ${D}_{{n}_{1},{n}_{2}}$, the standardized test statistic, $Z$, and the tail probability, $p$, are computed and printed.
10.1 Program Text
Program Text (g08cdce.c)
10.2 Program Data
Program Data (g08cdce.d)
10.3 Program Results
Program Results (g08cdce.r)