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\name{contingency2xt.comb}
\alias{contingency2xt.comb}
\title{
Combined Kruskal-Wallis Tests for the 2 x t Contingency Tables
}
\description{
This function uses the Kruskal-Wallis criterion to test
the hypothesis of no association between the counts
for two responses
"A" and "B" across t categories and across \eqn{M} blocks.
}
\usage{
contingency2xt.comb(\dots,
method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
}
\arguments{
\item{\dots}{
Either several lists \eqn{L_1,\ldots,L_M}, each
of two equal length vectors \eqn{A_i} and
\eqn{B_i}, \eqn{i=1,\ldots,M}, of counts \eqn{\ge 0},
where the common length \eqn{t_i} of \eqn{A_i} and
\eqn{B_i} may vary from list to list
or a list of \code{M} such lists
}
\item{method}{= \code{c("asymptotic","simulated","exact")}, where
\code{"asymptotic"} uses only an asymptotic chi-square approximation
with \eqn{(t_1-1)+\ldots+(t_M-1)} degrees of freedom
to approximate the \eqn{P}-value, This calculation is always done.
\code{"simulated"} uses \code{Nsim} simulated counts for the two vectors
\eqn{A_i} and \eqn{B_i} in list \eqn{L_i},
with the observed marginal totals, \eqn{m_i=\sum A_i},
\eqn{n_i = \sum B_i}, \eqn{d_i = A_i+B_i}.
It does this independently from list to list using the same \code{Nsim} each time,
adding the resulting Kruskal-Wallis criteria across lists
to get \code{Nsim} such summed values to estimate the \eqn{P}-value.
\code{"exact"} enumerates all counts for \eqn{A_i} and \eqn{B_i} with
the respective observed marginal totals to get an exact distribution for each list.
These distributions are then convolved to obtain the \eqn{P}-value.
It is used only when \code{Nsim} is at least as large as the product across blocks
of the number \code{choose(m+t-1,t-1)} of full enumerations per block, where
\eqn{t = t_1,\ldots, t_M}.
Otherwise, \code{method} reverts to \code{"simulated"} using the given \code{Nsim}.
}
\item{dist}{\code{FALSE} (default) or \code{TRUE}. If \code{TRUE}, the
simulated or fully enumerated null distribution \code{null.dist} is returned
for the Kruskal-Wallis test statistic. Otherwise \code{null.dist = NULL} is returned.
}
\item{Nsim}{\code{=10000} (default), number of simulated \eqn{A_i} splits to use per block.
It is only used when \code{method = "simulated"},
or when \code{method = "exact"} reverts to \code{method = "simulated"}, as previously explained.
}
}
\details{
For details on the calculation of the Kruskal-Wallis criterion and its exact or simulated
distribution for each block see \code{\link{contingency2xt}}.
}
\value{
A list of class \code{kSamples} with components
\item{test.name}{\code{"Combined 2 x t Contingency Tables"}}
\item{t}{vector giving the number of classification categories per block}
\item{M}{number of blocked tables}
\item{kw.list}{a list of the \code{KW.cont} output componenents from
\code{\link{contingency2xt}} for each of the blocks}
\item{null.dist}{simulated or enumerated null distribution
of the combined test statistic. It is given as an \code{L} by 2 matrix,
where the first column (named \code{KW}) gives the \code{L} unique ordered
values of the combined Kruskal-Wallis
statistic and the second column (named \code{prob}) gives the corresponding (simulated or exact)
probabilities.
\code{null.dist = NULL} is returned when \code{dist = FALSE} or when
\code{method =} \code{"asymptotic"}.}
\item{method}{the \code{method} used.}
\item{Nsim}{the number of simulations.}
}
\note{
The required level for \code{Nsim} in order for \code{method = "exact"}
to be carried out, is conservative, but there is no transparent way to get a
better estimate. The actual dimension \code{L} of the realized \code{null.dist}
will typically be much smaller, since the distribution is compacted to
its unique support values.
}
\section{warning}{\code{method = "exact"} should only be used with caution.
Computation time is proportional to the number of enumerations. In most cases
\code{dist = TRUE} should not be used, i.e.,
when the returned distribution objects
become too large for R's work space.}
\examples{
out <- contingency2xt.comb(list(c(25,15,20),c(16,6,18)),
list(c(12,4,5),c(13,8,9)),method = "simulated", dist=FALSE, Nsim=1e3)
}
\keyword{nonparametric}
\keyword{htest}
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