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\begin{document}
\title{Identifying interesting SNP interactions with
\texttt{logicFS}}

\author{Holger Schwender\\  holger.schwender@udo.edu}
\date{}

\maketitle

\vspace*{0.5cm}

\section{Introduction}

Logic regression proposed by Ruczinski et al.\ (2003)
is a classification method that attempts to predict the
case-control status based on Boolean combinations of
binary variables. It has already been successfully applied to
SNP data by Kooperberg et al.\ (2001).

This package contains functions that use logic regression
as a wrapper to identify interesting combinations of binary
variables and to measure the importance of these interactions.
A description of the used methods is given in Schwender and Ickstadt (2006).
Even though the intended purpose of this package is the identification of
SNP interactions, it can also be applied to other types
of binary (or categorical) data.   

\texttt{logicFS} also contains a first basic Bagging version
of the logic regression and a fast implementation of the 
Quine-McCluskey algorithm based on matrix algebra. The latter
is described in Schwender (2006) and can be used to identify 
a minimum disjunctive normal form of a given truth table.

\section{SNP Data and their Transformation into Dummy Variables}\label{data}

As always, the package has to be loaded.

<<>>=
library(logicFS)
@

As example data set, we use the data set contained in \texttt{logicFS}.

<<>>=
data(data.logicfs)
@

\texttt{data(data.logicfs)} consists of two objects: A simulated matrix, \texttt{data.logicfs},
containing the data of 15 variables (columns) and 400 observations (rows) and
a vector, \texttt{cl.logicfs}, consisting of the class labels of the 400 observations.

Each of the variables in \texttt{data.logicfs} is categorical and can take the realizations 1, 2 and 3.
The class label of the first 200 observations is 1 (for case) and the class label
of the remaining observations is 0 (for control). If one of the following expression
is \texttt{TRUE}, then the corresponding observation is a case:

\begin{itemize}
\item[] SNP1 == 3
\item[] SNP2 == 1  \&  SNP4 == 3
\item[] SNP3 == 3  \&  SNP5 == 3  \&  SNP6 == 1
\end{itemize}

where SNP1 is in the first column of \texttt{data.logicfs}, SNP2 in the second, and so on.

Even though the variables are called SNPs, these variables have not been generated
by an algorithm for simulating SNP data. So most of the values of a variable can, e.g.,
be 3 even though 3 should actually code the homozygous variant genotype.

Logic regression and hence the functions in \texttt{logicFS} can only handle binary
predictors. So the categorical SNP variables have to be transformed into two binary
dummy variables. If the homozygous reference genotype of the SNPs is coded by 1, the
heterozygous genotype by 2 and the homozygous variant type by 3, then the function
\texttt{make.snp.dummy} can be used to code these SNPs. Thus, a binary 
representation of the variables in \texttt{data.logicfs} can be obtained by

<<>>=
bin.snps <- make.snp.dummy(data.logicfs)
@


\section{Feature Selection Using Logic Regression}\label{fs}

The binary dummy variables in \texttt{bin.snps} can now be used in \texttt{logic.fs}
to identify potentially interesting combinations of these variables and to
measure the importance of these interactions. Since the default version of the logic
regression and thus the current version of \texttt{logic.fs} is based on 
simulated annealing, we set

<<>>=
my.anneal <- logreg.anneal.control(start = 2, end = -2, iter = 10000)
@

and the number \texttt{B} of considered logic regression models to 20
to speed up the computation. The feature selection is then performed by

<<>>=
log.out <- logicFS(bin.snps, cl.logicfs, B = 20, nleaves = 10, 
   rand = 1234, anneal.control = my.anneal)
@

where we allow that a maximum of \texttt{nleaves} $=10$ variables is in the
logic regression models and set \texttt{rand} to 123 to make the results
of this vignette reproducible.

The output of \texttt{logicFS} is given by

<<>>=
log.out
@

Not very surprisingly, only the interactions shown in Section \ref{data}
have a high importance, where, e.g., \texttt{!SNP2\us 1} stands
for NOT SNP2\us 1, i.e.\ SNP2\us $1=0$. Thus, the logic expression 
\texttt{!SNP2\us 1 \& SNP4\us 2} means ``SNP2 is of the homozygous
reference genotype and SNP4 is of the homozygous variant genotype."

As displayed in Figure 1, the variable importance can be plotted by

<<eval=FALSE>>=
plot(log.out)
@

Since the names of SNPs are usually pretty long, coded names are used in
the plot, where \texttt{X1} codes the first variable in \texttt{data},
\texttt{X2} the second, and so on. The original variable names are
displayed in the plot if \texttt{coded} is set to \texttt{FALSE} in
\texttt{plot}.

In the above analysis, the single tree approach of the logic regression is
used. It is, however, also possible to perform a multiple tree analysis
by changing the default of \texttt{ntrees}. Note that if this approach is
applied to \texttt{bin.snps} one will get a lot of warnings (from \texttt{glm})
because of the structure of this data set (cases are unambiguously 
classified by the above logic expressions). With other data sets this usually 
will not happen.

\begin{figure}[!ht]
\vspace{18pt} \small
\centerline{
\includegraphics[width=0.9\textwidth]{logicFSnew}}\vspace{-10pt}
\caption{Variable Importance Plot}
\end{figure}

\section{Bagged Logic Regression}

A first basic Bagging version of the logic regression is implemented
in the function \texttt{logic.bagging}. Similar to the analysis with
\texttt{logic.fs} \texttt{logic.bagging} is applied to \texttt{bin.snps} by

<<>>=
bag.out <- logic.bagging(bin.snps, cl.logicfs, B = 20, nleaves = 10, 
   rand = 1234, anneal.control = my.anneal)
bag.out
@

By default, both the out-of-bag error (\texttt{obb=TRUE}) and the importance 
of the variable combinations (\texttt{importance=TRUE}) are computed. The results
of Section \ref{fs} can also be obtained by

<<>>=
bag.out$vim
@

and the importance can be plotted either by \texttt{plot(bag.out\$vim)} or simply
by \texttt{plot(bag.out)}.

The class of a new observation is predicted by majority voting and can be computed by

<<>>=
cl.preds <- predict(bag.out, bin.snps)
@

where we here assume that \texttt{bin.snps} contains the new observations. The 
misclassification rate is then computed by

<<>>=
mean(cl.preds != cl.logicfs)
@

Warning: There is currently no checking if the data set, \texttt{newbin}, containing
the new observations and \texttt{data} in \texttt{logic.bagging} contain the same
variables in the same order.



\begin{thebibliography}{}
\item[Kooperberg, C.,] Ruczinski, I., LeBlanc, M., and Hsu, L.\ (2001).
Sequence Analysis Using Logic Regression. \textit{Genetic
Epidemiology}, 21, 626--631.

\item[Ruczinski, I.,] Kooperberg, C., and LeBlanc, M.\ (2003).
Logic Regression.
\textit{Journal of the Computational and Graphical Statistics}, 12,
475--511.

\item[Schwender, H.\ (2006).] Minimization of Boolean Expressions Using Matrix Algebra.
\textit{Technical Report}, SFB 475, University of Dortmund, Germany.

\item[Schwender, H.,] and Ickstadt, K.\ (2006).
Identification of SNP Interactions Using Logic Regression.
\textit{Technical Report}, SFB 475, University of Dortmund, Germany.


\end{thebibliography}


\end{document}
