\HeaderA{loglin}{Fitting Log-Linear Models}{loglin}
\keyword{category}{loglin}
\keyword{models}{loglin}
\begin{Description}\relax
\code{loglin} is used to fit log-linear models to multidimensional
contingency tables by Iterative Proportional Fitting.
\end{Description}
\begin{Usage}
\begin{verbatim}
loglin(table, margin, start = rep(1, length(table)), fit = FALSE,
       eps = 0.1, iter = 20, param = FALSE, print = TRUE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{table}] a contingency table to be fit, typically the output from
\code{table}.
\item[\code{margin}] a list of vectors with the marginal totals to be fit.

(Hierarchical) log-linear models can be specified in terms of these
marginal totals which give the \sQuote{maximal} factor subsets contained
in the model.  For example, in a three-factor model,
\code{list(c(1, 2), c(1, 3))} specifies a model which contains
parameters for the grand mean, each factor, and the 1-2 and 1-3
interactions, respectively (but no 2-3 or 1-2-3 interaction), i.e.,
a model where factors 2 and 3 are independent conditional on factor
1 (sometimes represented as \sQuote{[12][13]}).

The names of factors (i.e., \code{names(dimnames(table))}) may be
used rather than numeric indices.

\item[\code{start}] a starting estimate for the fitted table.  This optional
argument is important for incomplete tables with structural zeros
in \code{table} which should be preserved in the fit.  In this
case, the corresponding entries in \code{start} should be zero and
the others can be taken as one.
\item[\code{fit}] a logical indicating whether the fitted values should be
returned.
\item[\code{eps}] maximum deviation allowed between observed and fitted
margins.
\item[\code{iter}] maximum number of iterations.
\item[\code{param}] a logical indicating whether the parameter values should
be returned.
\item[\code{print}] a logical.  If \code{TRUE}, the number of iterations and
the final deviation are printed.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The Iterative Proportional Fitting algorithm as presented in
Haberman (1972) is used for fitting the model.  At most \code{iter}
iterations are performed, convergence is taken to occur when the
maximum deviation between observed and fitted margins is less than
\code{eps}.  All internal computations are done in double precision;
there is no limit on the number of factors (the dimension of the
table) in the model.

Assuming that there are no structural zeros, both the Likelihood
Ratio Test and Pearson test statistics have an asymptotic chi-squared
distribution with \code{df} degrees of freedom.

Note that the IPF steps are applied to the factors in the order given
in \code{margin}.  Hence if the model is decomposable and the order
given in \code{margin} is a running intersection property ordering
then IPF will converge in one iteration.

Package \pkg{MASS} contains \code{loglm}, a front-end to \code{loglin}
which allows the log-linear model to be specified and fitted in a
formula-based manner similar to that of other fitting functions such
as \code{lm} or \code{glm}.
\end{Details}
\begin{Value}
A list with the following components.
\begin{ldescription}
\item[\code{lrt}] the Likelihood Ratio Test statistic.
\item[\code{pearson}] the Pearson test statistic (X-squared).
\item[\code{df}] the degrees of freedom for the fitted model.  There is no
adjustment for structural zeros.
\item[\code{margin}] list of the margins that were fit.  Basically the same
as the input \code{margin}, but with numbers replaced by names
where possible.
\item[\code{fit}] An array like \code{table} containing the fitted values.
Only returned if \code{fit} is \code{TRUE}.
\item[\code{param}] A list containing the estimated parameters of the
model.  The \sQuote{standard} constraints of zero marginal sums
(e.g., zero row and column sums for a two factor parameter) are
employed.  Only returned if \code{param} is \code{TRUE}.
\end{ldescription}
\end{Value}
\begin{Author}\relax
Kurt Hornik
\end{Author}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Haberman, S. J. (1972)
Log-linear fit for contingency tables---Algorithm AS51.
\emph{Applied Statistics}, \bold{21}, 218--225.

Agresti, A. (1990)
\emph{Categorical data analysis}.
New York: Wiley.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{table}{table}}.

\code{\LinkA{loglm}{loglm}} in package \pkg{MASS} for a user-friendly wrapper.

\code{\LinkA{glm}{glm}} for another way to fit log-linear models.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## Model of joint independence of sex from hair and eye color.
fm <- loglin(HairEyeColor, list(c(1, 2), c(1, 3), c(2, 3)))
fm
1 - pchisq(fm$lrt, fm$df)
## Model with no three-factor interactions fits well.
\end{ExampleCode}
\end{Examples}

