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\begin{document}

\title{\bf Using copa}

\author{James W. MacDonald}

\maketitle



\section{Overview}

In certain cancers (lymphoma, sarcoma, leukemia), it is common to have recurrent 
chromosomal rearrangements that may be a causal factor in the progression of the
disease \citet{Rowley:2001}. However, until recently these rearrangements have not 
been commonly found in other carcinomas \citet{Tomlins:2005}. Tomlins et al. describe 
a method they call Cancer Outlier Profile Analysis (COPA) that can be used to 
detect recurrent chromosomal rearrangements using microarray data. Their method 
however is limited to use with the Oncomine website \url{www.oncomine.org}, and
does not at this time either pre-filter the data for likely candidates, nor
compute any sort of inferential statistic.

\section{Introduction}

The idea behind COPA is very simple; it is well known that genetic translocations
occur in cancer cells, and that these translocations can result in the up-regulation
of oncogenes that may affect the progression of the cancer. This happens when the 5' 
activation domain of a constitutively expressed (or up-regulated) gene is fused to the 
3' portion of a given oncogene, thus increasing the expression of the oncogene. This 
translocation can happen between the activating gene and multiple oncogenes, as was
shown by Tomlins \textit{et al.}, as well as others \citet{Fonseca:2004}.

Since a given translocation is only likely to occur once per sample, if there were
multiple partners for a given activating gene, we would expect to see certain cancer samples
with a high expression of say, gene A, whereas other cancer samples might have high expression
of gene B, but these samples would be mutually exclusive. In addition, we would expect
that the normal samples would not have high expression for either gene A nor B. We can use
this idea to both pre-filter genes as well as finding interesting genes that may be 
involved in translocations.

Common methods for detecting differences between  tumor and normal samples will 
not work for finding these genes (e.g., $t$-tests); we need to find those genes where only a subset
of the samples have high expression. To do this, we center and scale the data
(on a row-wise basis) using the median and median average difference (MAD). We can then
select a common value (default is 5) as a cutoff for 'outlier' status and apply this to 
all genes. We then simply look for pairs of genes that have a large number of mutually 
exclusive outlier (cancer) samples, but few or no normal outliers. The candidate gene pairs
will be ranked based on the sum of outlier samples for each pair, as this seems to
be the most reasonable criterion for ranking. 

Since there may be several gene pairs with
the same number of outliers we need to add an additional criterion to rank the ties. We use a
modification of the ranking scheme used by Tomlins \textit{et al.}. They simply ranked the
genes using the 75$^{th}$, 95$^{th}$ and 99$^{th}$ percentiles of the centered and scaled expression values.
Since we are looking at pairs rather than individual genes, we take the difference between the
75$^{th}$ percentile of the tumor and normal samples, and then compute the sum of these differences
for each gene pair. This value quantifies how different the outlier pairs are from their 
corresponding normals. We chose the 75$^{th}$ percentile for ranking rather than say, the 
95$^{th}$ because the values at the higher percentile are what caused the gene pairs to be 
selected in the first place, so we want to use a less extreme percentile to distinguish between
the tied pairs.

\section{Using copa}

To search for gene pairs that may be involved in translocations is very simple. Just 
load the \Rpackage{copa} package, and run the \Rfunction{copa} function using your
microarray expression values. For this vignette, we will be using the \Rpackage{colonCA}
package, which contains an \Robject{ExpressionSet} with normal and tumor colon expression data.

<<>>=
library(copa)
library(colonCA)
data(colonCA)
head(pData(colonCA), 10)
@ 

We will use the third column of the \Robject{phenoData} object as our classlabel, which
tells \Rfunction{copa} which samples are tumor and which are normal. There is no need to 
pre-filter the expression data; \Rfunction{copa} has an internal pre-filtering step that
selects the top \Rfunarg{pct}(percentile) of the data, based on the number of outliers.
The default is to use the $95^{th}$ percentile as a cutoff; if this results in more than
1000 genes, \Rfunction{copa} will give a warning and allow you to abort the run (and 
presumably re-run with a higher value for \Rfunarg{pct}).

One thing to keep in mind is that \Rfunction{copa} is going to be computing all pairwise
sums of outliers, which can get to be a large number of computations really quickly (hence
the warning at n = 1000). Although this portion of the function is written in C, and is 
actually quite fast, a large number of comparisons will tend to slow things down.

<<>>=
rslt <- copa(colonCA, as.numeric(pData(colonCA)[,3]))
@ 

We can now look at a plot showing the number of outliers for each gene.

<<echo=true, eval=false>>=
plotCopa(rslt, idx = 1, col = c("lightgreen","salmon"))
@ 

\begin{figure}
\centering
<<fig=true, width=6, height=6>>=
plotCopa(rslt, idx = 1, col = c("lightgreen","salmon"))
@ 
\caption{Plot of 'Top' Gene Pair}
\label{fig:copa}
\end{figure}

Figure~\ref{fig:copa} shows the outlier status of the 'top' gene pair (based on having 
the most outlier samples). If using an Affymetrix GeneChip for which there is an 
annotation package, one can label 
the plots with the corresponding gene symbol by specifying the \Rfunarg{lib} argument. 
Unfortunately, the colonCA data is based on a Hum6000 Affy chip, for which there is no 
annotation package.

This plot doesn't look that impressive, as there are only a few samples that fulfill 
the criterion for outlier status. We can look at how many gene pairs there are with a
given number of outliers using the \Rfunction{tableCopa} function.

<<>>=
tableCopa(rslt)
@ 

We might then want to know which genes have
\Sexpr{as.numeric(names(tableCopa(rslt)[1]))}. We can list them out
using the \Rfunction{summaryCopa} function.

<<>>=
summaryCopa(rslt, 9)
@ 


We might now want to know how significant this result is. 
We can test this hypothesis by permuting the 
class labels of the samples many times and then checking to see how often we see
pairs of genes with a certain number of outliers. By permuting the class labels
we are mixing up the tumor and normals, so any pair with a large number of outliers
by definition has arisen by chance. If we get many gene pairs with say, 
\Sexpr{as.numeric(names(tableCopa(rslt)[1]))} outliers then it is fairly likely that our
observed results could have arisen by chance as well. However, if the opposite is
true, then we have some reassurance that the observed results are not a chance 
event, and these gene pairs may well be undergoing some sort of recombination.

<<>>=
prm <- copaPerm(colonCA, rslt, 9, 24)
sum(prm >= 9)
@ 

In this instance, there are \Sexpr{sum(prm >= 9)} times that the permuted data
resulted in a number of outliers as large or larger than what we observed. This
indicates that there may well be some recombination going on here, and it might be 
worthwhile to explore further.

A few notes about this function. First, it repeatedly re-runs the \Rfunction{copa}
function after permuting the classlabels, so any caveats that I gave above about the 
number of genes to use above applies a hundred fold here. Note that the percentile
cutoff used to create the \Robject{copa} object will be re-used for the permutations,
so if the cutoff is too lenient, you may repeatedly be queried because of too many genes.

Second, the default for this function is 100 permutations. This is enough to get a basic idea,
but is far too few to calculate a $p$-value or false discovery rate (FDR). For that, one should
use at least 500 - 1000 permutations. Even at 1000 permutations, the smallest $p$-value will 
be 0.001 (actually the smallest will be 0, but the second smallest will be 0.001). Running
\Rfunction{copaPerm} here on approximately \Sexpr{dim(rslt$mat)[1]} genes takes about 90 seconds.
Increasing either the number of genes or the permutations may necessitate an overnight run.





\bibliography{copa}

\end{document}
