% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/LRT.stat.R
\name{LRT.stat}
\alias{LRT.stat}
\title{Likelihood ratio test statistic for contingency tables}
\usage{
LRT.stat(tab)
}
\arguments{
\item{tab}{A \code{K x C} matrix (contingency table) of counts. See details.}
}
\value{
The calculated value of the LRT statistic.
}
\description{
Calculate the likelihood ratio test statistic for general two-way contingency tables.
}
\details{
Suppose that \code{tab} consists of counts from \eqn{K} populations (rows) in \eqn{C} categories.  The likelihood ratio test statistic is computed as
\deqn{
  2 \sum_{i=1}^K \sum_{j=1}^N O_{ij} \log(p^A_{ij}/p^0_{j}),
}{
  2 \sum_ij O_ij log(p_ij/p_0j),
}
where \eqn{O_{ij}}{O_ij} is the observed number of counts in the \eqn{i}th row and \eqn{j}th column of \code{tab}, \eqn{p^A_{ij} = O_{ij}/\sum_{j=1}^C O_{ij}}{p_ij = O_ij/(\sum_j O_ij)} is the unconstrained estimate of the proportion of category \eqn{j} in population \eqn{i}, and \eqn{p^0_j = \sum_{i=1}^K O_{ij} / \sum_{i=1}^K\sum_{j=1}^C O_{ij}}{p_0j = \sum_i O_ij / \sum_ij O_ij} is the estimate of this proportion under \eqn{H_0} that the populations have indentical proportions in each category.  If any column has only zeros it is removed before calculating the LRT statistic.
}
\examples{
# simple contingency table
ctab <- rbind(pop1 = c(5, 3, 0, 3),
                pop2 = c(4, 10, 2, 5))
colnames(ctab) <- LETTERS[1:4]
ctab
LRT.stat(ctab) # likelihood ratio statistic
}