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% File src/library/stats/man/ks.test.Rd
% Part of the R package, http://www.R-project.org
% Copyright 1995-2008 R Core Development Team
% Distributed under GPL 2 or later

\name{ks.test}
\alias{ks.test}
\title{Kolmogorov-Smirnov Tests}
\description{
  Performs one or two sample Kolmogorov-Smirnov tests.
}
\usage{
ks.test(x, y, \dots,
        alternative = c("two.sided", "less", "greater"),
        exact = NULL)
}
\arguments{
  \item{x}{a numeric vector of data values.}
  \item{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{\dots}{parameters of the distribution specified (as a character
    string) by \code{y}.}
  \item{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{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.}

}
\details{
  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{\dots}.

  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{\link{t.test}} or \code{\link{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{\dots} 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}.
}
\value{
  A list with class \code{"htest"} containing the following components:
  \item{statistic}{the value of the test statistic.}
  \item{p.value}{the p-value of the test.}
  \item{alternative}{a character string describing the alternative
    hypothesis.}
  \item{method}{a character string indicating what type of test was
    performed.}
  \item{data.name}{a character string giving the name(s) of the data.}
}
\references{
  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/}.

}
\seealso{
  \code{\link{shapiro.test}} which performs the Shapiro-Wilk test for
  normality.
}
\examples{
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")
}
\keyword{htest}