%\VignetteIndexEntry{R2OpenBUGS}
\documentclass{Z}
\usepackage{RdRW,thumbpdf}
\newcommand{\WinBUGS}{{\proglang{WinBUGS}}{}}
\newcommand{\OpenBUGS}{{\proglang{OpenBUGS}}{}}
\newcommand{\RW}{{\pkg{R2WinBUGS}}{}}
\newcommand{\RO}{{\pkg{R2OpenBUGS}}{}}
\renewcommand{\R}{{\proglang{R}}{}}
\setlength{\textheight}{23.5cm}

\title{\RO{}:\protect\linebreak A Package for Running \OpenBUGS{} from \R{}}
\author{Sibylle Sturtz\thanks{\email{sturtz@statistik.tu-dortmund.de}}\\Fachbereich Statistik\\Universit\"at Dortmund\\Germany
    \And
        Uwe Ligges\thanks{\email{ligges@statistik.tu-dortmund.de}}\\Fachbereich Statistik\\Universit\"at Dortmund\\Germany
    \And
        Andrew Gelman\thanks{\email{gelman@stat.columbia.edu}}\\Department of Statistics\\Columbia University\\USA
}
\Plainauthor{Sibylle Sturtz, Uwe Ligges, Andrew Gelman}
\Plaintitle{R2OpenBUGS: A Package for Running OpenBUGS from R}

\Abstract{
The \RO{} package provides convenient functions to call \OpenBUGS{} from \R{}. It
automatically writes the data and scripts in a format readable by \OpenBUGS{} for processing in batch mode, which
is possible since version 1.4. After the \OpenBUGS{} process has finished, it is possible either to read the
resulting data into \R{} by the package itself---which gives a compact graphical summary of inference and
convergence diagnostics---or to use the facilities of the \pkg{coda} package for further analyses of the
output. Examples are given to demonstrate the usage of this package.
}

\Keywords{\R{}, \OpenBUGS{}, interface, MCMC}
\Plainkeywords{R, OpenBUGS, interface, MCMC}

\begin{document}
An earlier version of this vignette has been published by the Journal of Statistical Software:\\
Sturtz S, Ligges U, Gelman A (2005):
``\RW{}: A Package for Running \WinBUGS{} from \R{}.''
{\em Journal of Statistical Software}, 12(3), 1--16.  
\RO{} was adapted from \RW{} by Neal Thomas.


\section{Introduction}\label{Introduction}
The usage of Markov chain Monte Carlo (MCMC) methods became very popular within the last decade. \OpenBUGS{}
\citep[\textbf{B}ayesian inference \textbf{U}sing \textbf{G}ibbs \textbf{S}ampling, ][]
{Spiegelhalter;Thomas;Best:2003} is a popular software for analyzing complex statistical models using MCMC
methods. This software uses Gibbs sampling \citep{Geman;Geman:1984,Gelfand;Smith:1990,Casella;George:1992} and the
Metropolis algorithm \citep{Metropolis;Rosenbluth;Rosenbluth;Teller;Teller:1953}
to generate a Markov chain by sampling from full conditional
distributions.
The \OpenBUGS{} software is available for free
at \url{http://www.OpenBUGS.info}.
An introduction to MCMC methods is given in \cite{Gilks;Richardson;Spiegelhalter:1996}.

Using \OpenBUGS{}, the user must specify the model to run, and to load data and initial values for a specified
number of Markov chains. Then it is possible to run the Markov chain(s) and to save the results for the parameters
the user is interested in. Summary statistics of these data, convergence diagnostics, kernel estimates
etc.\ are available as well.
Nevertheless, some users of this software might be interested in saving the output and reading it into \R{}
\citep{RCore:2004} for further analyses.
\OpenBUGS{} comes with the ability to run the software in batch mode using scripts.

The \RO{} package makes use of this feature and provides the tools to call \OpenBUGS{}
directly after data manipulation in \R{}. Furthermore, it is possible to
work with the results after importing them back into \R{} again, for example to create posterior predictive simulations or, more generally, graphical displays of data and posterior simulations \citep{gelman:2004}.
Embedding in R can also be useful for frequently changed data or processing a bunch of data sets,
because it is much more convenient to use some \R{} functions (possibly within a loop)
rather than using ``copy \& paste'' to update data in \OpenBUGS{} each time; however difficulties have been encountered in this area because both \R{} and \OpenBUGS{} can lock up RAM in the Windows operating system.

\R{} is a ``language for data analysis and graphics'' and an open source and
freely available statistical software package implementing that language, see \url{http://www.R-project.org/}.
Historically, \R{} is an implementation of the award-winning \proglang{S}
language and system \citep{becker:1984r,becker:1988r,chambers:1992,chambers:1998}.
\R{} and \RO{} are available from \emph{CRAN} (Comprehensive \R{} Archive Network),
i.e., \url{http://CRAN.R-Project.org} or one of its mirrors.
\RO{} could be ported to the commercial \proglang{S} implementation
\textsc{S-Plus}.  Minor adaptions would be needed since \textsc{S-Plus}
lacks some of \R{}'s functions and capabilities.
If an internet connection is available, \RO{} can be installed by typing
\verb+install.packages("R2OpenBUGS")+ at the \R{} command prompt.
Do not forget to load the package with \verb+library("R2OpenBUGS")+.

The package \pkg{coda} by \cite{Plummer;Best;Cowles;Vines:2004} is very useful for the analysis of \OpenBUGS{}'
output, the reader might want to install this package as well. The CRAN package \pkg{boa} (Bayesian Output
Analysis Program) by \cite{Smith:2004} has similar aims.
\proglang{JAGS} (Just Another Gibbs Sampler)
by \cite{Plummer:2003} is a program for analysis of Bayesian hierarchical models using Gibbs sampling that aims for
the same functionality as classic \proglang{BUGS}. \proglang{JAGS} is developed to work closely together with \R{} and
the \pkg{coda} package.

%% Revision 2005-01-10 end

In this paper, we give two examples, involving educational testing experiments in schools (cf.~Section~\ref{School}),
and incidence of childhood leukaemia depending on benzene emissions (cf.~Section~\ref{Leukaemia}). Details on the
functions of \RO{} are given in Section~\ref{Programming}. These functions automatically write the data and a
script in a format readable by \OpenBUGS{} for processing in batch mode, and call \OpenBUGS{} from \R{}. After the
\OpenBUGS{} process has finished, it is possible either to read the resulting data into \R{} by the package itself
or to use the facilities of the \pkg{coda} package for further analyses of the output. In Section~\ref{Example2},
we demonstrate how to apply the functions provided by \RO{} on the examples' data, and how to analyze the output
both with package \pkg{coda} and with \RO{}'s methods to \verb+plot()+ and \verb+print()+ the output.


\section{Examples}\label{Examples}
In this Section, we introduce two examples which will be continued in Section~\ref{Example2}.

\subsection{Schools data}\label{School}
The Scholastic Aptitude Test (SAT) measures the aptitude of high-schoolers in order to help colleges to make
admissions decisions. It is divided into two parts, verbal (SAT-V) and mathematical (SAT-M).
Our data comes from the SAT-V (Scholastic Aptitude Test-Verbal) on eight different high schools, from an experiment conducted in the late 1970s.
SAT-V is a standard multiple choice test administered by the Educational Testing Service.
This Service was interested in the effects of coaching programs for each of the selected schools.

The study included coached and uncoached pupils, about sixty in each of the eight different schools; see
\cite{Rubin:1981}. All of them had already taken the PSAT (Preliminary SAT) which results were used as covariates.
%Even if the test is constructed to be resistant to short-term
%efforts directed specifically toward improving test performance, each of the schools is successful at increasing
%SAT scores.
For each school, the estimated treatment effect and the standard error of the effect estimate are given.
These are calculated by an analysis of covariance adjustment appropriate for a completely randomized experiment
\citep{Rubin:1981}. This example was analyzed using a hierarchical normal model in \cite{Rubin:1981} and
\citeauthor{Gelman:2003} (\citeyear{Gelman:2003}, Section 5.5).

\subsection{Leukaemia registration data}\label{Leukaemia}
Spatial data usually arises on different, non-nesting spatial scales. One example is childhood leukaemia
registration data analyzed by \cite{Best;Cockings;Bennett;Wakefield;Elliot:2001}
using ecologic regression. Data are given for Greater London bounded by the M25 orbital motorway.
%% Revision 2005-01-10 start
The data are not available as an example in \RO{} but we use the example here
to illustrate
alternative calls to the \verb+bugs()+ function and output analysis using the \pkg{coda} package.
%% Revision 2005-01-10 end

The observed number of leukaemia cases among children under 15 years old is given at ward level. Census wards are
administrative areas containing approximately 5000 to 10\,000 people. Central London is divided into 873 wards. The
number of incident cases of leukaemia in children is available from 1985 until 1996 from the Office of National
Statistics and the Thames Cancer Registry. A plot of these numbers is given in Figure~\ref{observed}.

\begin{figure}
\begin{center}
\includegraphics[width=0.85\textwidth]{countssw}
\caption{\label{observed}Observed number of cases of childhood leukaemia in 1985--1996}
\end{center}
\end{figure}

Additionally, the number of expected cases (cf.~Fig.~\ref{expected}) is calculated on the same
resolution using population numbers for different age-sex-strata and the national leukaemia
rate for the corresponding strata, for details see \cite{Best;Cockings;Bennett;Wakefield;Elliot:2001}.

\begin{figure}
\begin{center}
\includegraphics[width=0.85\textwidth]{expectedsw}
\caption{\label{expected}Expected number of cases of childhood leukaemia in 1985--1996}%, as obtained from a simple demographic model}
\end{center}
\end{figure}

It is assumed that benzene emissions have an effect on the incidence rate of leukaemia. Benzene emission rates are
available in tonnes per year from an atmospheric emissions inventory for London
\citep{Buckingham;Clewley;Hutchinson;Sadler;Shah:1997} produced by the London Research Centre. They are provided
at 1km $\times$ 1km grid cells, giving 2132 grid cells in total.
Their spatial distribution is shown in Figure~\ref{benzene}.

\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{benzolsw}
\caption{\label{benzene}Benzene emissions in tonnes per year}
\end{center}
\end{figure}

For further details on the data see \cite{Best;Cockings;Bennett;Wakefield;Elliot:2001}.

We model these data by Poisson-Gamma models introduced by \cite{Best;Ickstadt;Wolpert:2000} using \OpenBUGS{}.
A linking matrix containing information which grid cell belongs to which ward
and to which amount is required. This matrix is calculated using \R{}.
Unfortunately, \OpenBUGS{} does not support a list format such as directly produced by \R{}.
Therefore, the data must be provided as a matrix with 2132 rows and 873 columns (or vice versa).
Most of the entries of this matrix are zeroes, but using \verb+dump()+ to export it from \R{}
yields in a file size of 14.2~MB.
Unfortunately, opening a file of such size really slows \OpenBUGS{} down,
and it was not even possible on some of our PCs.
Importing data written by our \RO{} package does not make any problems using the batch mode,
probably due to memory management issues in \OpenBUGS{}.


\section{Implementation}\label{Programming}
The implementation of the \RO{} package is straightforward.
The ``main'' function \verb+bugs()+ is intended to be called by the user.
In principle, it is a wrapper for several other functions called therein step by step as follows:
\begin{enumerate}
    \item \verb+bugs.data.inits()+ writes the data files \file{data.txt}, and \file{inits1.txt}, \file{inits2.txt}, ...
        into the working directory. These files will be used by \OpenBUGS{} during batch processing.

        In particular, input for \OpenBUGS{} must not exceed a certain number of digits.
        Moreover, it needs an \verb+E+ instead of an \verb+e+ in scientific notation.
        Scientific notation is particularly desirable because of the ``number of digits'' limitation.
        The default (\verb+digits = 5+) is to, e.g., reformat the number \verb+123456.789+ to \verb@1.23457E+05@.
    \item \verb+bugs.script()+ writes the file \file{script.txt} that is used by \OpenBUGS{} for batch processing.
    \item \verb+bugs.run()+ updates the lengths of the adaptive phases in the \WinBUGS{} registry
        (using a function \verb+bugs.update.settings()+),
        calls \WinBUGS{}, and runs it in batch mode with \file{script.txt}.
    \item \verb+bugs.sims()+ is only called if the argument \verb+codaPkg+ has been set to \verb+FALSE+ (the default).
        Otherwise \verb+bugs()+ returns the filenames of stored data. These can, for example,
        be imported by package \pkg{coda} (see the example in Section~\ref{Leukaemia2}, page~\pageref{codaexample}),
        which provides functions for convergence diagnostics,
        calculation of Monte Carlo estimates, trace plots, and so forth.

        The function \verb+bugs.sims()+ reads simulations from \OpenBUGS{} into \R{} (not necessarily called by \verb+bugs()+
        itself), formats them,
        monitors convergence, performs convergence checks, and computes medians and quantiles.
        It also prepares the output for \verb+bugs()+ itself.
\end{enumerate}
These functions are not intended to be called by the user directly.
Arguments are passed from \verb+bugs()+ to the other functions, if appropriate.
A shortened help file of \verb+bugs()+ listing all arguments is given in Appendix~\ref{Doc};
for the full version type \verb+?bugs+ in \R{} after having installed and loaded the package \RO{}
(see Section~\ref{Introduction}).

%\newpage
As known from \OpenBUGS{}, one must specify the \verb+data+ in form of a list, with list names equal to the names
of data in the corresponding \OpenBUGS{} model. Alternatively, it is possible to specify a vector or list of names
(of mode \verb+character+). In that case objects of that names are looked for in the environment in which
\verb+bugs()+ has been called (usually that is the user's Workspace, \verb+.GlobalEnv+).
%% Revision 2005-01-10 start
If data have already been written in a file called \file{data.txt} to the working directory,
it is possible to specify \verb+data = "data.txt"+.
%% Revision 2005-01-10 end
One will usually want to supply initial values.
This can be done either in the form of a function \verb+inits()+ that creates these values, so that
different chains can be automatically initialized at different points (see Section \ref{School2}), or by specifying them
directly (see Section \ref{Leukaemia2}).
If \verb+inits()+ is not specified,
\verb+bugs()+ just uses the starting values created by \OpenBUGS{}; but in practice \OpenBUGS{} can crash when
reasonable initial values are not specified, and so we recommend constructing a simple \verb+inits()+ function to
simulate reasonable starting points \cite[Section C.2]{Gelman:2003}. It is also necessary to specify which
parameters should be saved for monitoring by specifying \verb+parameters.to.save+.

The user might also want to change the defaults for the length of the burn-in (\verb+n.burnin+, which defaults to half the length of the chain)
period for every MCMC run and the number of iterations (\verb+n.iter+, default value 3)
that are used to calculate Monte Carlo estimates.
%SS: Achtung: n.iter=n.burn.in + length of stored chain!
The specification of a thinning parameter (\verb+n.thin+) is possible as well; this is useful when the number of parameters is large, to keep the saved output to a reasonably-sized R object.  

By setting the argument \verb+debug = TRUE+,
\OpenBUGS{} remains open after the run.
This way it is possible to find errors in the code or the data structure,
or even to work with that software as in a usual run.  This feature is not
available with Linux execution.

It is possible to run one or more Markov chains.
The number of chains (\verb+n.chains+) must be specified together with the chains' initial values (\verb+inits+).
If more than one Markov chain is requested and \verb+codaPkg+ is set to \verb+FALSE+, the convergence diagnostic
$\hat{R}$ \citep{Brooks;Gelman:1998} is calculated by \verb+bugs.sims()+ for each of the saved parameters.

%% Revision 2005-01-10 start
Since the communication between \OpenBUGS{} and \R{} is based on files,
rather huge files will be saved in the working directory by the \verb+bugs()+ call,
either files to be read in by \verb+bugs()+ itself, or by the \pkg{coda} package.
The user might want to delete those files after the desired contents has been imported into \R{},
and save those objects, e.g., as compressed \R{} data files.
%% Revision 2005-01-10 end

The function \verb+bugs()+ returns a rather complex object of class \verb+bugs+,
if called with argument \verb+codaPkg = FALSE+.
In order to look at the structure of such an object, type \verb+str(objectname)+.
For convenience, \RO{} provides methods corresponding to class \verb+bugs+ for
the generic functions \verb+print()+ and \verb+plot()+.

So that user will not be overwhelmed with information; summaries of the output are provided by the
\verb+print()+ method. That is, some parameters of the \verb+bugs()+ call are summarized, and mean,
standard deviation, several quantiles of the parameters and convergence diagnostics based on \cite{Gelman;Rubin:1992}
are printed. See the example in
Section~\ref{School2}, page~\pageref{schoolssim}, for a typical output. As with \cite{Spiegelhalter;Best;Carlin;vdLinde:2002},
%\pagebreak
 the DIC computed by \verb+bugs.sims()+ is defined as the posterior mean
of the deviance plus $p_D$, the estimated effective number of parameters in the posterior distribution. We define
$p_D$ as half the posterior variance of the deviance and estimate it as half the average of the within-chain
variances of the deviance.\footnote{In contrast, \cite{Spiegelhalter;Best;Carlin;vdLinde:2002},
 and OpenBUGS, define $p_D$ as the
posterior mean of the deviance evaluated at the posterior mean of the parameter values.  We cannot use that
definition because the deviance function is not available to our program, which calls \OpenBUGS{} from the
``outside''.  Both definitions of $p_D$---ours and that introduced by \cite{Spiegelhalter;Best;Carlin;vdLinde:2002}---can be
derived from the asymptotic $\chi^2$ distribution of the deviance relative to its minimum \citep[Section 6.7]{Gelman:2003}.
We make no claim that our measure of $p_D$ is superior to that of \cite{Spiegelhalter;Best;Carlin;vdLinde:2002};
we choose this measure purely because it is computationally possible given what is available to us from the \OpenBUGS{} output.}


The \verb+plot()+ for objects of class \verb+bugs+
provides information condensed in some plots conveniently arranged within the same graphics device.
For an example, see Figure~\ref{plot} in Section~\ref{School2}.  It is intended to adapt this function to work with MCMC output in general, even if obtained from software other than OpenBUGS.



\section{Examples continued}\label{Example2}
The Examples introduced in Section~\ref{Example2} are continued in this Section.
We apply the functions provided by \RO{}
to the examples' data and analyze the output.

\subsection{Schools data}\label{School2}
Schools example data (see Section~\ref{School}) are available with the \RO{} package:
\begin{Code}
    > data(schools)
    > schools
      school estimate    sd
    1      A    28.39  14.9
    2      B     7.94  10.2
    3      C    -2.75  16.3
    4      D     6.82  11.0
    5      E    -0.64   9.4
    6      F     0.63  11.4
    7      G    18.01  10.4
    8      H    12.16  17.6
\end{Code}
For modeling these data, we use a hierarchical model as proposed by \citeauthor{Gelman:2003} (\citeyear{Gelman:2003}, Section 5.5).
We assume a normal distribution for the observed estimate for each school with
mean \verb+theta+ and inverse-variance \verb+tau.y+.
The inverse-variance is given as $1/\verb+sigma.y+^{2}$ and its prior distribution is uniform on (0,1000). For the mean
\verb+theta+, we employ another normal distribution with mean \verb+mu.theta+ and inverse-variance \verb+tau.theta+. For
their prior distributions, see the following \OpenBUGS{} code:


\begin{Code}
 model {
       for (j in 1:J)
         {
         y[j] ~ dnorm (theta[j], tau.y[j])
         theta[j] ~ dnorm (mu.theta, tau.theta)
         tau.y[j] <- pow(sigma.y[j], -2)
         }
       mu.theta ~ dnorm (0.0, 1.0E-6)
       tau.theta <- pow(sigma.theta, -2)
       sigma.theta ~ dunif (0, 1000)
     }
\end{Code}
This model must be stored in a separate file, e.g.~\file{schools.bug}\footnote{Emacs Speaks Statistics (ESS)
by \cite{rossini04}, a package available with Gnu Emacs \citep{stallmann99},
recognizes and properly formats Bugs model files that have the .bug extension.},
in an appropriate directory, say \path{c:/schools/}.
In \R{} the user must prepare the data inputs the \verb+bugs()+ function needs.
This can be a list containing the name of each data vector, e.g.
\begin{Code}
    > J <- nrow(schools)
    > y <- schools$estimate
    > sigma.y <- schools$sd
    > data <- list ("J", "y", "sigma.y")
\end{Code}
Using these data and the model file, we can run an MCMC simulation to get estimates for \verb+theta+,
\verb+mu.theta+ and \verb+sigma.theta+. Before running, the user must decide how many chains to be run
(\verb+n.chain = 3+) for how many iterations (\verb+n.iter = 1000+).
If the length of burn-in is not specified, \verb+n.burnin = floor(n.iter/2)+ is used, that is, 500 in this example.
Additionally, the user must specify initial values for the chains, for example by writing a function. This can be done by
\begin{Code}
    > inits <- function(){
    +     list(theta = rnorm(J, 0, 100), mu.theta = rnorm(1, 0, 100),
    +          sigma.theta = runif(1, 0, 100))
    + }
\end{Code}
Now, the user can start the MCMC simulation by typing
\begin{Code}
    > schools.sim <- bugs(data, inits, model.file = "c:/schools/schools.bug",
    +                     parameters = c("theta", "mu.theta", "sigma.theta"),
    +                     n.chains = 3, n.iter = 1000)
\end{Code}
in \R{}.\label{schoolssim} 
For other available arguments, see Appendix \ref{Doc}.

The results in objects \verb+schools.sim+ can conveniently be printed by \verb+print(schools.sim)+.
The generic function \verb+print()+ calls the print method for an object of class \verb+bugs+
provided by \RO{}.
For this example, you will get something like

\small
\begin{Code}
    > print(schools.sim)
    Inference for Bugs model at "c:/schools/schools.bug"
     3 chains, each with 1000 iterations (first 500 discarded)
     n.sims = 1500 iterations saved
                mean  sd  2.5%  25%  50%  75% 97.5% Rhat n.eff
    theta[1]    11.1 9.1  -3.0  5.0 10.0 16.0  31.8  1.1    39
    theta[2]     7.6 6.6  -4.7  3.3  7.8 11.6  21.1  1.1    42
    theta[3]     5.7 8.4 -12.5  0.6  6.1 10.8  21.8  1.0   150
    theta[4]     7.1 7.0  -6.6  2.7  7.2 11.5  21.0  1.1    42
    theta[5]     5.1 6.8  -9.5  0.7  5.2  9.7  18.1  1.0    83
    theta[6]     5.7 7.3  -9.7  1.0  6.2 10.2  20.0  1.0    56
    theta[7]    10.4 7.3  -2.1  5.3  9.8 15.3  25.5  1.1    27
    theta[8]     8.3 8.4  -6.6  2.8  8.1 12.7  26.2  1.0    64
    mu.theta     7.6 5.9  -3.0  3.7  8.0 11.0  19.5  1.1    35
    sigma.theta  6.7 5.6   0.3  2.8  5.1  9.2  21.2  1.1    46
    deviance    60.8 2.5  57.0 59.1 60.2 62.1  66.6  1.0   170
     pD = 3 and DIC = 63.8 (using the rule, pD = var(deviance)/2)

     For each parameter, n.eff is a crude measure of effective sample size,
     and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
    DIC is an estimate of expected predictive error (lower deviance is better).
\end{Code}
\normalsize
Additionally, the user can generate a plot of the results by typing \verb+plot(schools.sim)+.
The resulting plot is given in Figure~\ref{plot}.
\begin{figure}[t]
\begin{center}
\fbox{
\includegraphics[width=0.9\textwidth]{plot}
} \caption{\label{plot}Plot produced by \RO{} package for the schools example. }
\end{center}
\end{figure}
In this plot, the left column shows a quick
summary of inference and convergence ($\widehat{R}$ is close to 1.0 for all parameters, indicating good mixing of
the three chains and thus approximate convergence); and the right column shows inferences for each set of
parameters.  As can be seen in the right column, \RO{} uses the parameter names in \OpenBUGS{} to structure the
output into scalar, vector, and arrays of parameters, in addition to storing the parameters as a long vector.

For the interpretation of these results see \citeauthor{Gelman:2003} (\citeyear{Gelman:2003}, Section 5.5).

\subsection{Leukaemia registration data}\label{Leukaemia2}
The leukaemia registration data (see Section \ref{Leukaemia}) are used to show data modeling and output reading
into \R{} using the \pkg{coda} package. A simple model for these data looks as follows:
\begin{Code}
model{
    beta.0 ~ dgamma(a.0, tau.0)
    beta.benz ~ dgamma(a.benz, tau.benz)
    a.0    <- 0.575
    tau.0  <- a.0*2
    a.benz <- 0.575
    tau.benz <- a.benz*2
\end{Code}
\clearpage
\begin{Code}
    for (i in 1:I)
        {
        count[i] ~ dpois(lambda[i])
        lambda[i] <- p[i]*expect[i]
        for (j in 1:J)
            {
            prop[j,i] <- gamma[j,i]*(benz[j] - benzbar)
            }
        p[i]<- beta.0 + beta.benz*sum(prop[,i])
        }
    }
\end{Code}
Here \verb+count+ denotes the number of observed incidences of childhood leukaemia in ward~\verb+i+. These are
assumed to be Poisson distributed with mean \verb+lambda+ depending on the number of expected cases \verb+expect+
in ward \verb+i+ and an area-specific risk rate \verb+p+. For calculation of this area specific risk rate we use
an intercept \verb+beta.0+ and a term depending on the weighted sum of benzene emissions \verb+benz+ in each grid
cell \verb+j+. The weights are chosen proportional to the amount of area that ward \verb+i+ and grid cell \verb+j+ have
in common.

In \R{} we can define all these data and then initialize the model. The data needed for this example are
\begin{description}
\item[\tt{benzbar}:]  arithmetic mean of all benzene values,
\item[\tt{benz}:]     a vector containing benzene emissions of all 2132 grid cells,
\item[\tt{expect}:]   expected number of cases of childhood leukaemia in each of the 873 wards,
\item[\tt{count}:]    observed number of childhood leukaemia in these wards,
\item[\tt{gamma}:]    a $2132\times 873$ matrix containing the amount of area each grid cell and each ward have in common,
\item[\tt{J}:]        total number of grid cells, i.e.~2132, and
\item[\tt{I}:]        total number of ward cells, i.e.~873.
\end{description}
The parameters we want to store are regression coefficients \verb+beta.0+ and \verb+beta.benz+ as well as
\verb+p+, the area specific relative risk compared to the reference rate. This reference rate was used to
calculate the expected number of cases in each ward.

Since we want to use the \pkg{coda} package for reading the data into \OpenBUGS{}, we specify \verb+codaPkg = TRUE+
in the \verb+bugs()+ call:
\begin{Code}
    > data <- list(benzbar = mean(benz), benz = benz, expect = expect,
    +     count = count, gamma = gamma, J = J, I = I)
    > parameters <- c("beta.0", "beta.benz", "p")
    > inits1 <- list(beta.0 = 1, beta.benz = 1)
    > inits2 <- list(beta.0 = 0.5, beta.benz = 0.5)
    > inits <- list(inits1, inits2)
    > model <- bugs(data, inits, parameters, model.file = "c:/model.bug",
    +     n.chains = 2, n.iter = 8000, n.burnin = 5000, n.thin = 1,
    +     codaPkg = TRUE )
\end{Code}

\clearpage
Starting with, e.g.,\label{codaexample}
\begin{Code}
    > library("coda")
    > codaobject <- read.bugs(model)
    > plot(codaobject)
\end{Code}
it is now possible to use the \pkg{coda} package for output analyses.


\section*{Acknowledgments}
The work of Uwe Ligges has been supported by the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 475.
The work of Andrew Gelman has been supported by the U.S. National Science Foundation.

\bibliography{literatur}

\clearpage
\begin{appendix}
\section[Help page for the function bugs()]{Help page for the function \code{bugs()}\label{Doc}}
\small
This help page has been shortened.
\input{bugs}
\end{appendix}
\end{document}
