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\begin{document}

\title{MACAT - MicroArray Chromosome Analysis Tool }
\author{Joern Toedling, Sebastian Schmeier,Matthias Heinig,\\
        Benjamin Georgi, and Stefan Roepcke}
\date{}
\maketitle
\tableofcontents
\pagebreak

\section{Introduction}
This project aims at linking the term \emph{differential gene expression} to the chromosomal localization of genes. \\
Microarray data analysts have defined tumor subtypes by specific gene expression profiles, consisting of genes that show differential expression between subtypes (see, e.g., \cite{Yeoh2002}). 
However, tumor subtypes have also been characterized by phenomena involving
large chromosomal regions. For instance, Christiansen et al.
\cite{Christiansen2004} report on
a subtype of acute myeloid leukemia, showing mutations
in the \emph{AML1} gene on chromosome 21 along with deletions or loss of
chromosome arm 7q.
A natural approach to bridging the gap between these two paradigms is to
link scoring for differential gene expression to the chromosomal localization of genes. Tumor subtypes can be defined by differential expression patterns affecting sizeable regions of certain chromosomes.\\
In the following, we propose a statistical approach for identifying significantly differentially 
expressed regions on the chromosomes based on a regularized t-statistic 
(see section \ref{sec:scoringttest}). We address the problem of interpolating the scores between genes
by applying kernel functions (see section \ref{smoothing}). In order to evaluate the significance of 
scores we conduct permutation experiments (section \ref{StatEval}). An integral part of the project is
the visualization of the results in order to provide convenient access to the statistical analysis
without requiring a profound mathematical background (section \ref{sec:viz}).  \\

The package is implemented in the R statistical programming language (\emph{www.r-project.org}).
It requires functionalities provided by the Bioconductor package \cite{bioconductor}, 
which is a collection of R-libraries dealing with
various aspects of the analysis of biological data including normalization, assessing background information
on genes, and visualization of data. \\
We apply our method on a publicly available data set of acute lymphoblastic leukemia (ALL), described
in \cite{Yeoh2002}. This data set consists of 327 tumor samples subdivided into 10 classes. In order to investigate chromosomal phenomena defining one tumor class, we consider the expression levels of
that class versus all the other subclasses.

\section{Methods}
\subsection{Data Preprocessing}
We assume normalized expression data, which can be provided already as a matrix in R or in form of a delimited text file. In the preprocessing step the expression data is integrated with gene location data for the given microarray into one common data format. 
For this \Rpackage{macat} provides the \Rfunction{preprocessedLoader} or its 
convenience-wrapper \Rfunction{buildMACAT}, which employ various Bioconductor 
\cite{bioconductor} functions.\\
The resulting data format is a list containing:
\begin{itemize}
\item Gene identifier
\item Gene location (chromosome, strand, coordinate)
\item Sample labels, denoting for instance tumor (sub)types
\item Expression levels as a matrix
\item An identifier for the type of microarray, used in the experiments
\end{itemize}
Currently \Rpackage{macat} is limited to commercial Affymetrix$^\textup{\textregistered}$  oligonucleotide microarrays.

\subsection{Scoring of Differential Gene Expression}
\label{sec:scoringttest}
For each gene we compute a statistic denoting the degree of differential
expression between two given groups of samples. 
In the context of the leukemia data set
the two groups of samples are given by one tumor class
in the first group and the remaining nine classes forming the second.
The statistic is the regularized t-score introduced in \cite{Tusher2001}. This so-called
\emph{relative difference} is defined as
$$ d(i) = \frac{ \overline{x}_A(i) - \overline{x}_B(i) }
{ s(i) + s_0 } $$
where $\overline{x}_A(i)$ and $\overline{x}_B(i)$ are the mean expression levels
of gene $i$ in group $A$ and $B$ respectively, $s(i)$ is the pooled
standard deviation of the expression values of gene $i$, and $s_0$ is constant for
all genes.
In essence, $d(i)$ is Student's t-statistic augmented by a
fudge factor $s_0$ in the denominator, which is introduced to prevent a high
statistic for genes with a very low standard deviation $s(i)$. 
We set $s_0$ to
the median over all gene standard deviations $s(i)$, analogous
to \cite{Tibshirani2002}. \\
Since the null-distribution of these regularized t-scores is conceptually
different from a known t-distribution, we have to approximate it by
random permutations (for details see section \ref{StatEval}).\\
However, \Rpackage{macat} also allows to compute Student's classical t-statistic
for each gene and assess significance based on the underlying t-distribution.

\subsection{Smoothing Kernels}
\label{smoothing}
On each chromosome only a limited number of genes can be measured with
a microarray. The distance in base pairs between measured genes also varies
greatly.
Since we want to compute differential expression statistics for larger
chromosomal regions, we need a method to interpolate scores between 
measured values.
This interpolation, however, does not aim to assign statistics to
non-coding regions, but to provide a
smooth estimate of differential expression over large chromosomal regions.\\
Formally this can be seen as a regression problem, 
$y = f(x)$, with $y$ being the t-score and $x$ being one
base pair position on the chromosome. 
(In the following, $x$ will often be called \emph{chromosomal coordinate}.)
Score $y_j$ is to be estimated for a large number of coordinates $x_j$, when
$f(x)$ is known only for few observations $x_i$.
Kernel methods achieve \emph{smooth} estimates of the function $f(x)$ by 
fitting different models at each query point $x_0$ using only observations
close to $x_0$ \cite{HastieElementsChapter6}.\\
Three different kernel approaches will be discussed here:

\label{kernel}
\begin{itemize}
\item k-Nearest Neighbor: For every chromosomal coordinate compute the average of the k nearest genes.
\item Radial basis function (rbf): For every chromosomal coordinate compute the average over all genes weighted by distance as explained in detail below.
\item Base-Pair-Distance Kernel: Similar to the k-Nearest-Neighbors, but
using this kernel the average is taken over all genes within a certain radius
of the position, whose value has to be determined.
\end{itemize}

\label{math}
The kernel approaches presented above can be expressed as weighted sums of 
scores. Thus, we use a matrix multiplication as the basic
framework for the computations, where we multiply our score matrix with 
the so called kernel matrix. 

Let $E$ be the score matrix, 
with rows corresponding to $n$ genes and columns corresponding
to $m$ samples. 
For each gene $g$ the chromosomal coordinate $geneLocation_g$ is known.
Further, let $K$ be the kernel matrix. 
Let us assume that we want to interpolate the
scores at $s$ genomic locations that are stored in a
vector we call $\vec{s}$. So the kernel matrix has the dimensions
($n \times s$). One entry of $K(g, l)$ represents the weight
that the gene $g$ has at the location $l$. So the product
$$ E^TK = S  $$
gives the smoothed matrix $S$ of dimension ($m \times s$), where one
entry $S(i, l)$ represents the interpolation of
the score of sample $i$ at location $\vec{s}_{l}$.

The smoothed matrix $S$ is the product of the score matrix
$E$ times the kernel matrix $K$.
The weights in $K$ depend only on the location of the genes relative
to the steps, for which interpolated values are to be computed.\\
For the three kernels listed above the kernel functions are:
\begin{itemize}
\item k-nearest neighbors
\[
 K_{kNN}(k, g, l) = \left\{ \begin{array}{ll}
1 & \mbox{if $g$ is one of the $k$ nearest neighbors of $l$} \\
0 & \mbox{else} \\
\end{array} \right. \]

\item radial basis function
\[
 K_{rbf}(\gamma, g, l) = \left. \begin{array}{l}
\exp (-\gamma \cdot \| geneLocation_{g} - \vec{s}_{l} \|^2) \\
\end{array} \right.
\]

\item base-pair distance
\[
 K_{bpd}(d, g, l) = \left\{ \begin{array}{ll}
1 & \mbox{if $geneLocation_{g}$ is within radius $d$ of $l$} \\
0 & \mbox{else}
\end{array} \right. \]
\end{itemize}

\label{params}
The free parameters ($k,~\gamma,~d$) of the kernel functions
determine the degree of smoothing. 
Take for instance the kNN kernel: the smaller $k$ gets, the fewer
genes are averaged and in the extreme case interpolations take only
the value of the next gene ($k = 1$). In the case of the base-pair distance
kernel with a very small distance $d$, 
you will see spikes at the locations, where
genes are located, and zero elsewhere. 
For very large distances the smoothing will remove
all spikes and the value for each position is roughly the same. \\
More formally, kernel parameters define the width of the
window, over which values are used to estimate the regression function. 
Changing the window width is a
\emph{bias-variance tradeoff} situation. If the window around $x_0$ is narrow, 
only a small number of observations will
be used to estimate $f(x_0)$. In this case the estimate will have a relatively
large variance but only a small bias. However, if the window is wide, a large
number of observations will be used for the estimate. This estimate
tends to show a low variance but comes with a high bias, since now
observations $x_i$ far from $x_0$ are used as well and we still assume that
$f(x_i)$ is close to $f(x_0)$ \cite{HastieElementsChapter6}.\\
This shows that
a good choice of the kernel parameters is very important. For the three kernel
functions presented above, we fit the parameters from the data:
\begin{itemize}
\item kNN: The number of genes on the chromosome is determined and $k$ is set
to cover approximately 10\% of the genes.
\item rbf: The width of the window is controlled by the parameter $\gamma$.
We require that for each position $l$, for which we interpolate the score,
there should be at least two genes within the window for averaging.
If both of these genes are located equally far from $l$, 
both will receive a weight of 1/2, thus  resulting in a simple average over 
these two genes' scores. To allow for these weights, 
the maximal gene distance ($max$) between any two neighboring genes
on the chromosome has to be determined. Then $\gamma$ can be computed as:
$$\gamma = \frac{\ln{2}}{(max/2)^2}$$
\item base-pair distance: as with the rbf kernel, we require that the average
is computed at least over two genes. So the parameter $d$, which describes the
radius, within which genes are to be used to compute the average,
is set to $max$, the maximal gene distance between any two genes
on the chromosome.
\end{itemize}
These biologically motivated heuristics for choosing the kernel parameters
provide good smoothing results on test data without any overfitting of
estimated scores to the observed values.\\
Nevertheless, we recommend to determine the optimal parameter settings by
cross-validation \cite{HastieElementsChapter7}, for which 
\Rpackage{macat} provides the function \Rfunction{evaluateParameters}. Hereby, one part of
the genes is left out, the interpolation is computed based on the remaining
genes only, and the quality of interpolation is assessed by the residual
sum of squares between the scores of the left-out genes and their interpolated
values. The above mentioned biologically heuristics, by default, provide a
starting point for a grid-search over multiples of these parameter settings.
The optimal parameter setting is the one with the minimal residual 
sum of squares. For instance, in Figure \ref{fig:evalparam}, the
result of an evaluation of the parameter  $k$ from the 
$k$-nearest-neighbor kernel is depicted. From this evaluation, 
the optimal $k$ seems to be approximately 30.\\

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[width = 0.9 \columnwidth]{evalkNN6}
    \caption{Result plot for evaluating kNN parameter settings}
    \label{fig:evalparam}
  \end{center}
\end{figure}

\subsection{Statistical Evaluation}
\label{StatEval}
To judge the significance of differential gene expression, we propose to
investigate random permutations of the class labels. To obtain a reliable
simulation of the empirical distribution, we suggest to choose at least
$B \geq 1000$
permutations, preferably more. For each of these permutations the (regularized)
t-statistic is recomputed for each gene. Thus, for each gene we obtain
$B$ permutation statistics and consequently an \emph{empirical p-value},
denoting the proportion of the $B$ permutation statistics being greater
or equal than the actual statistic that is based on the true class labels.
The permutation statistics also provide upper and lower significance
borders, which are smoothed using the same kernel
function as for the original statistics.

Optionally,  to judge the significance of differential expression over
whole chromosomal regions, one could instead investigate permutations of
the ordering of genes on chromosomes. For each of these permutations
the smoothing of gene-specific scores is recalculated.
This way, one obtains a null-distribution for scores over chromosomal regions.
Given this null-distribution, it is possible to define confidence intervals
for scores from random sample groupings and assess significance of scores
observed in a relevant sample grouping.\\
However, the standard procedure in \Rpackage{macat} is to permute the class labels,
which is considerably faster and yields results that are 
easier to validate and interpret statistically.\\
This approach is implemented in the \Rpackage{macat}-function
\Rfunction{evalScoring}, which can be seen as the core function of \Rpackage{macat}. See
appendix \ref{ExampleSession} for an exemplary use of the function and its arguments.\\
One important issue is that, when analyzing many chromosomes and classes, one
might obtain statistically significant results by chance (\emph{multiple-
testing problem}). In the given setting, classical procedures
correcting for multiple testing, such as the \emph{Bonferroni} correction, cannot be applied.
We advise the user to be aware of the problem and to validate results
by alternative methods.

\subsection{Visualization}
In order to facilitate a better understanding of both the data and the statistical analysis, 
it is helpful to employ meaningful and concise visualizations.
\Rpackage{macat} includes plotting functionality for several questions of interest.
\label{sec:viz}

\begin{itemize}
\item Plotting raw and kernelized expression levels versus coordinates of genes on one chromosome.
\item Visualize raw and kernelized statistical scores versus coordinates of genes on one chromosome.
\item Emphasis of interesting chromosomal regions with listing of relevant genes.
\end{itemize}

Chromosomal regions showing significant differential expression can be visualized
by plotting the result of the \Rfunction{evalScoring} function (see figure
\ref{fig:exampleplot}). Hereby scores for genes (black
dots), the sliding average of the 0.025 and 0.975 quantiles of the
permuted scores (grey lines), the sliding average permuted scores
(red line), and highlighted regions (yellow dotted), where the score
exceeds the quantiles, are plotted along one chromosome. 
The yellow regions are deemed interesting, showing significant over- or under-expression 
according to the underlying statistic. 
The plot region ranges from zero to the length of the respective chromosome.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[width = 0.9 \columnwidth]{chrom6TkNN}
    \caption{Example plot of plot.MACATevalScoring}
    \label{fig:exampleplot}
  \end{center}
\end{figure}

One can generate an HTML-page (see figure \ref{fig:html}) on-the-fly by setting the argument 
\Rfunarg{output} in the function \Rfunction{plot.MACATevalScoring} to ``html''. 
The generated HTML-page provides information about genes located within the highlighted chromosomal regions. 
For each gene some annotation, a click-able LocusID linked to the NCBI web site, and the empirical p-value is provided.

\begin{figure}[htbp]
  \begin{center}
    %\epsfig{file=chrom6T.eps, width = 0.9 \columnwidth}
    \includegraphics[ width = 0.9 \columnwidth]{chrom6T}
    \caption{Example for a generated HTML-page}
    \label{fig:html}
  \end{center}
\end{figure}



\section{Results}

This section describes the results, we obtain from an exemplary analysis 
on ``T- vs. B-lymphocyte ALL''. \\
First the regularized t-score \cite{Tibshirani2002} is computed as described in section 
\ref{sec:scoringttest}. To include information about the distances between the 
genes, we used the radial basis kernel for smoothing 
with the default parameters (see section \ref{smoothing}).
We have investigated regions of chromosome six for significant differential expression.
Figure \ref{fig:html} shows that there is a region on the p-arm of chromosome six
that is significantly under-expressed. The genes within that region comprise also the
well known MHC class II genes that are known to be expressed by B-lymphocytes, but not
by T-lymphocytes.
Other genes in this region remain to be investigated in more detail.\\
Apart from this region the data reveals no other significant differences in 
gene expression between T- and B-lymphocyte ALL on chromosome six.

In table \ref{examplegenes}, we show a list of genes found to be located in
significantly differentially expressed regions on different chromosomes when
analyzing different subtypes of leukemia versus all other subtypes.

\begin{table}[htbp]

  \begin{center}
    \begin{tabular}{llllp{7.5cm}}
\bf{Class} & \bf{Chrom} & \bf{Cytoband} & \bf{LocusID} & \bf{OMIM Annotation} \\

MLL & 8 & 8q24.12 & 4609 & ''Alitalo et al. \cite{Alitalo} found that the MYC gene, which is involved by \emph{translocation} 
in the generation of \emph{Burkitt lymphoma}, is amplified, resulting in homogeneously staining chromosomal regions (HSRs) 
in a human neuroendocrine tumor cell line derived from a \emph{colon cancer}. The HSR resided on a distorted X chromosome; 
amplification of MYC had been accompanied by translocation of the gene from its normal position on 8q24.''\\

MLL & 9 & 9p22.3 & 6595 &  ''Mammalian SWI/SNF complexes are ATP-dependent chromatin remodeling enzymes 
that have been implicated in the regulation of gene expression, cell cycle control, 
and \emph{oncogenesis}.''\\

MLL & 9 & 9p24.3 & 23189 & '' The suggested role for this protein is in tumorogenesis of renal cell
\emph{carcinoma}.''\\

MLL & 11 & 11p13 & 7490 & ''Mutations in this gene can be associated with the development of \emph{Wilms tumors} 
in the kidney or with abnormalities of the genito-urinary tract.'' \\
E2A & 1 & 1q23 & 5087 &  ''Wiemels et al.\cite{wiemels} sequenced the genomic fusion between the PBX1 and E2A genes in 22 pre-B acute lymphoblastic leukemias and 2 cell lines. Kamps et al. \cite{kamps}  discussed the chimeric genes created by the human t(1;19) translocation in pre-B-cell acute lymphoblastic leukemias. The authors cloned 2 different E2A-PBX1 fusion transcripts and showed that NIH-3T3 cells transfected with cDNAs encoding the fusion proteins were able to cause malignant tumors in nude mice.''\\

E2A & 9 & 9p24.3 & 23189 & ''By RT-PCR and Western blot analysis, Sarkar et al. \cite{sarkar} demonstrated that KANK expression was suppressed in most renal tumors and in kidney tumor cell lines due to methylation at CpG sites in the gene.''

    \end{tabular}
    \caption{Example genes with OMIM annotation \cite{omim} detected by MACAT to be located in differentially expressed regions. 'Class' denotes the analyzed tumor subtype.}
  \label{examplegenes}    
  \end{center}
\end{table}


\section{Discussion and Outlook}

As described in the previous section, applying our method, we detected a chromosomal region as
significantly differentially expressed, which is in agreement with biological
knowledge of the two sample classes involved. This gives an indication that the chromosomal regions
annotated as being significant by the method are indeed biologically meaningful.
Many of the genes found by MACAT (see table \ref{examplegenes}) are known oncogenes or are at least
associated with oncogenesis.  

This fits well with our expectation, since it appears reasonable that different tumor subtypes can be
characterized on the molecular level by different expression levels for genes relevant to oncogenesis. 

One point of interest for future research would be the application of 
\Rpackage{macat} on other publicly
available data sets and contrasting the results with relevant biological expert knowledge to evaluate
performance. This way, one could get an clearer impression of the extend of possible applications.

Another point would be the exploration of different approaches for obtaining
relative gene expression values for groups of samples. For instance one could use another data set as a 
reference for samples with ''normal'' gene expression levels (``normal'' denoting samples from
patients with a tumor). Of course, due to the noisy nature of microarray data, this approach would 
have to overcome some structural and technical difficulties to make the measured expression levels
comparable. Given that a suitable data set could be found and the compatibility issues resolved, one
could examine the characteristic patterns of chromosomal phenomena in tumor subclasses as opposed to
normal tissue.

The method, which we have described, can detect significant differential expression for chromosomal regions. However, the reason for the differential expression, be it a mutation, translocation, hypermethylation, loss of heterozygosity, or another event, remains to be investigated. 

Thus, results obtained by our method should be verified by suitable experiments. One could think of a customized 
cDNA or oligo chip that contains all known genes, including fusion-genes, stemming from translocation events, 
in the regions that were found to be differentially expressed. To validate
the borders of found regions, it would be useful to also incorporate genes that neighbor found regions.

Other possibly useful experiments include Real-time PCR to measure the amount of mRNA for one (or few) specific genes,
comparing the results to measured expression levels.
The nature of the chromosomal event leading to the differential expression can be investigated by
methylation array experiments or by fluorescence in situ hybridization.

Nevertheless, we have shown that our method provides a solid baseline for gene-wise experiments.

\pagebreak
\appendix

\section{Example Session}
\label{ExampleSession}

\subsection{The Beginning}
In this section, we show an example \Rpackage{macat} session. First of
all, one has to include the library and to load the data set provided in the data
package \Rpackage{stjudem}.
\begin{Schunk}
\begin{Sinput}
> library(macat)
\end{Sinput}
\begin{Soutput}
Loading required package: Biobase
Loading required package: annotate
Loading MicroArray Chromosome Analysis Tool...
Loading required packages...
You need package 'stjudem', if you want to see the demo and examples.
Type 'loaddatapkg("stjudem")' for automatic install!

Type 'demo(macatdemo)' for a quick tour...
\end{Soutput}
\begin{Sinput}
> loaddatapkg("stjudem")
\end{Sinput}
\begin{Soutput}
Note: Using download type: Win32 
[1] "Attempting to download stjudem from http://compdiag.molgen.mpg.de/software/Rrepos/"
[1] "Download complete."
[1] "Installing stjudem"
Note: Running R version 2.3.0 and package stjudem was built for R version 2.0.0
 , installing anyway.
 
[1] "Installation complete"
Loading data set 'stjude' ...
Type 'summary(stjude)' for an overview of the data structure.
See package 'macat' for examplary analyses.
\end{Soutput}
\end{Schunk}
We now have a data object called \Robject{stjude}.

\subsection{First Investigation}
Let us have a closer look on the data. 
The data is already in the right format. It has been pre-processed by the function \Rfunction{preprocessedLoader}.
\begin{Schunk}
\begin{Sinput}
> summary(stjude)
\end{Sinput}
\begin{Soutput}
             Length  Class  Mode     
geneName       12637 -none- character
geneLocation   12637 -none- numeric  
chromosome     12637 -none- character
expr         4128375 -none- numeric  
labels           327 -none- character
chip               1 -none- character
\end{Soutput}
\end{Schunk}
We can for example access the first 10 gene names by typing
\begin{Schunk}
\begin{Sinput}
> stjude$geneName[1:10]
\end{Sinput}
\begin{Soutput}
 [1] "34916_s_at" "34917_at"   "34462_at"   "163_at"     "163_at"    
 [6] "35219_at"   "31641_s_at" "33300_at"   "33301_g_at" "38950_r_at"
\end{Soutput}
\end{Schunk}
The different labels in the data set can be accessed by
\begin{Schunk}
\begin{Sinput}
> unique(stjude$labels)
\end{Sinput}
\begin{Soutput}
 [1] "BCR"        "E2A"        "Hyperdip"   "Hyperdip47" "Hypodip"   
 [6] "MLL"        "Normal"     "Pseudodip"  "T"          "TEL"       
\end{Soutput}
\begin{Sinput}
> table(stjude$labels)
\end{Sinput}
\begin{Soutput}
       BCR        E2A   Hyperdip Hyperdip47    Hypodip        MLL     Normal 
        15         27         64         23          9         20         18 
 Pseudodip          T        TEL 
        29         43         79 
\end{Soutput}
\end{Schunk}
There are ten different classes of tumor patients. The next
question is how many probe sets are mapped to chromosome 1.
\begin{Schunk}
\begin{Sinput}
> sum(stjude$chromosome == 1)
\end{Sinput}
\begin{Soutput}
[1] 1255
\end{Soutput}
\end{Schunk}
Now for some visualization. We want to plot the sliding average 
of the expression values from sample 3 along chromosome 6 with the default rbf-kernel (for details see section \ref{sec:viz}).
\begin{Schunk}
\begin{Sinput}
> plotSliding(stjude, 6, sample = 3, kernel = rbf)
\end{Sinput}
\end{Schunk}
\begin{figure}[htbp]
 \begin{center}
%<<fig=TRUE,echo=FALSE>>=
%print(plotSliding(stjude, chrom=6, sample=3, kernel=rbf)) 
%@
 \includegraphics[width = 0.9 \columnwidth]{Slidingchrom6s3} 
 \caption{Sliding Average of the expression values of chromosome 6 with rbf-kernel.}
 \label{fig:sliding}
 \end{center}
\end{figure}
See the result in figure \ref{fig:sliding}.


\subsection{Deeper Investigation}
Next we investigate the data for chromosomal regions showing differential expression. Again, we search chromosome 6 
for some specific differences between T-lymphocyte ALL (class "T") and B-lymphocyte ALL (all other classes). 
Take a look on section \ref{sec:scoringttest} (Scoring differential expression) for details on the score. \\
We want to use a kNN-kernel for interpolation between the scores.
First, we determine the optimal parameter settings, using the function
\Rfunction{evaluateParameters}.
\begin{Schunk}
\begin{Sinput}
> evalkNN6 = evaluateParameters(stjude, class = "T", chromosome = 6, 
+     kernel = kNN, paramMultipliers = c(0.01, seq(0.1, 2, 0.1), 
+         2.5))
\end{Sinput}
\end{Schunk}
Let's have a look at the result:
\begin{verbatim}
> evalkNN6$best

$k
[1] 30

> plot(evalkNN6)
\end{verbatim}
The result should look similar to Figure \ref{fig:evalparam}.
According to this result,
the optimal parameter setting for the kNN kernel with our data seems to be
$k=30$.
We use this $k$ for the search for significantly differentially expressed
chromosome regions. The \Rfunction{evalScoring} is used 
to build a \Rclass{MACATevalScoring} object. 
This may take some time due to the number of permutations. 

\begin{verbatim}
> e1 = evalScoring(stjude, class = "T", chromosome = 6, nperms = 1000,
       kernel = kNN, kernelparams = evalkNN6$best, cross.validate= FALSE)

Investigating 43 samples of class T ...
Compute observed test statistics...
Building permutation matrix...
Compute 1000 permutation test statistics...
250 ...500 ...750 ...1000 ...
Compute empirical p-values...
Compute quantiles of empirical distributions...
Computing sliding values for scores...
Compute sliding values for permutations...
All done.
\end{verbatim}


\subsection{Collecting Results}
Next, we use the plot function \Rfunction{plot.MACATevalScoring} to generate an HTML-page. 
Therefore, we set the parameter \Rfunarg{output} to "html" (see section \ref{sec:viz}). 
\begin{verbatim}
> plot(e1, output="html")
\end{verbatim}
See the result in your web browser; it should look similar to
Figure \ref{fig:html}. Some annotation of genes, which lie in the highlighted regions, is provided.\\[0.3cm]
This ends our short example session. Have fun using \Rpackage{macat}!

\newpage

\section*{Acknowledgments}
The authors would like to thank Stefanie Scheid and Florian Markowetz for
helpful discussions and proof-reading, Claudio Lottaz for countless hints
on building an R-package, and Alexander Schliep, Rainer Spang, Eike Staub,
and Martin Vingron for setting up our  knowledge and motivation for this
project. 
Our special thanks go to two reviewers, whose comments have led to great
improvements of \Rpackage{macat}.

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\end{document}
