\HeaderA{ks.test}{Kolmogorov-Smirnov Tests}{ks.test}
\keyword{htest}{ks.test}
\begin{Description}\relax
Performs one or two sample Kolmogorov-Smirnov tests.
\end{Description}
\begin{Usage}
\begin{verbatim}
ks.test(x, y, ...,
        alternative = c("two.sided", "less", "greater"),
        exact = NULL)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a numeric vector of data values.
\item[\code{y}] either a numeric vector of data values, or a character string
naming a cumulative distribution function or an actual cumulative
distribution function such as \code{pnorm}.
\item[\code{...}] parameters of the distribution specified (as a character
string) by \code{y}.
\item[\code{alternative}] indicates the alternative hypothesis and must be
one of \code{"two.sided"} (default), \code{"less"}, or
\code{"greater"}.  You can specify just the initial letter of the
value, but the argument name must be give in full.
See \sQuote{Details} for the meanings of the possible values.
\item[\code{exact}] \code{NULL} or a logical indicating whether an exact
p-value should be computed.  See \sQuote{Details} for the meaning of
\code{NULL}.  Not used for the one-sided two-sample case.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{y} is numeric, a two-sample test of the null hypothesis
that \code{x} and \code{y} were drawn from the same \emph{continuous}
distribution is performed.

Alternatively, \code{y} can be a character string naming a continuous
(cumulative) distribution function, or such a function.  In this case,
a one-sample test is carried out of the null that the distribution
function which generated \code{x} is distribution \code{y} with
parameters specified by \code{...}.

The presence of ties generates a warning, since continuous
distributions do not generate them.

The possible values \code{"two.sided"}, \code{"less"} and
\code{"greater"} of \code{alternative} specify the null hypothesis
that the true distribution function of \code{x} is equal to, not less
than or not greater than the hypothesized distribution function
(one-sample case) or the distribution function of \code{y} (two-sample
case), respectively.  This is a comparison of cumulative distribution
functions, and the test statistic is the maximum difference in value,
with the statistic in the \code{"greater"} alternative being
\eqn{D^+ = \max_u [ F_x(u) - F_y(u) ]}{}.   Thus in the two-sample case
\code{alternative="greater"} includes distributions for which \code{x}
is stochastically \emph{smaller} than \code{y} (the CDF of \code{x} lies
above and hence to the left of that for \code{y}), in contrast to
\code{\LinkA{t.test}{t.test}} or \code{\LinkA{wilcox.test}{wilcox.test}}.

Exact p-values are not available for the one-sided two-sample case, or
in the case of ties.  If \code{exact = NULL} (the default), an exact
p-value is computed if the sample size is less than 100 in the
one-sample case, and if the product of the sample sizes is less than
10000 in the two-sample case.  Otherwise, asymptotic distributions are
used whose approximations may be inaccurate in small samples.  In the
one-sample two-sided case, exact p-values are obtained as described in
Marsaglia, Tsang \& Wang (2003).  The formula of Birnbaum \& Tingey
(1951) is used for the one-sample one-sided case.

If a single-sample test is used, the parameters specified in
\code{...} must be pre-specified and not estimated from the data.
There is some more refined distribution theory for the KS test with
estimated parameters (see Durbin, 1973), but that is not implemented
in \code{ks.test}.
\end{Details}
\begin{Value}
A list with class \code{"htest"} containing the following components:
\begin{ldescription}
\item[\code{statistic}] the value of the test statistic.
\item[\code{p.value}] the p-value of the test.
\item[\code{alternative}] a character string describing the alternative
hypothesis.
\item[\code{method}] a character string indicating what type of test was
performed.
\item[\code{data.name}] a character string giving the name(s) of the data.
\end{ldescription}
\end{Value}
\begin{References}\relax
Z. W. Birnbaum \& Fred H. Tingey (1951),
One-sided confidence contours for probability distribution functions.
\emph{The Annals of Mathematical Statistics}, \bold{22}/4, 592--596.

William J. Conover (1971),
\emph{Practical Nonparametric Statistics}.
New York: John Wiley \& Sons.
Pages 295--301 (one-sample Kolmogorov test),
309--314 (two-sample Smirnov test).

Durbin, J. (1973)
\emph{Distribution theory for tests based on the sample distribution
function}.  SIAM.

George Marsaglia, Wai Wan Tsang \& Jingbo Wang (2003),
Evaluating Kolmogorov's distribution.
\emph{Journal of Statistical Software}, \bold{8}/18.
\url{http://www.jstatsoft.org/v08/i18/}.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{shapiro.test}{shapiro.test}} which performs the Shapiro-Wilk test for
normality.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(graphics)

x <- rnorm(50)
y <- runif(30)
# Do x and y come from the same distribution?
ks.test(x, y)
# Does x come from a shifted gamma distribution with shape 3 and rate 2?
ks.test(x+2, "pgamma", 3, 2) # two-sided, exact
ks.test(x+2, "pgamma", 3, 2, exact = FALSE)
ks.test(x+2, "pgamma", 3, 2, alternative = "gr")

# test if x is stochastically larger than x2
x2 <- rnorm(50, -1)
plot(ecdf(x), xlim=range(c(x, x2)))
plot(ecdf(x2), add=TRUE, lty="dashed")
t.test(x, x2, alternative="g")
wilcox.test(x, x2, alternative="g")
ks.test(x, x2, alternative="l")
\end{ExampleCode}
\end{Examples}

