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<h1 class="documentFirstHeading">Lab3.Rnw</h1>
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<p>%
% NOTE -- ONLY EDIT THE .Rnw FILE!!! The .tex file is
% likely to be overwritten.
%
% \VignetteIndexEntry{Seattle Lab 3}
%\VignetteDepends{Biobase. golubEsets, ellipse,lattice, mva, MASS,cluster }
%\VignetteKeywords{Microarray}
\documentclass<a href="#ref12pt">[12pt]</a>{article}</p>
<p>\usepackage{amsmath,pstricks}
\usepackage<a href="#refauthoryear,round">[authoryear,round]</a>{natbib}
\usepackage{hyperref}</p>
<p>\textwidth=6.2in
\textheight=8.5in
%\parskip=.3cm
\oddsidemargin=.1in
\evensidemargin=.1in
\headheight=-.3in</p>
<p>\newcommand{\scscst}{\scriptscriptstyle}
\newcommand{\scst}{\scriptstyle}</p>
<p>\bibliographystyle{plainnat}</p>
<p>\title{Cluster Analysis Using R and Bioconductor}</p>
<p>\begin{document}</p>
<p>\maketitle</p>
<p>We continue our extended example involving the dataset from \citet{Golub99}.
In this lab, we will compute distances between tumor samples and use
multidimensional scaling to represent these distances. We will then
consider different procedures for clustering tumor samples.</p>
<p><<loadlibs>>=
library(Biobase)
library(annotate)
library(golubEsets)
library(genefilter)
library(ellipse)
library(lattice)
library(cluster)
library(mva)
@</p>
<p>We will set up the data from scratch once again.</p>
<p><<setup>>=
X<-exprs(golubTrain)
X[X<100]<-100
X[X>16000]<-16000</p>
<p>mmfilt <- function(r=5, d=500, na.rm=TRUE) {
function(x) {
minval <- min(x, na.rm=na.rm)
maxval <- max(x, na.rm=na.rm)
(maxval/minval > r) && (maxval-minval > d)
}
}</p>
<p>mmfun <- mmfilt()</p>
<p>ffun <- filterfun(mmfun)
sub <- genefilter(X, ffun )
sum(sub) ## Should get 3051</p>
<p>X <- X<a href="#refsub,">[sub,]</a>
X <- log2(X)
golubTrainSub<-golubTrain<a href="#refsub,">[sub,]</a>
golubTrainSub@exprs <- X</p>
<p>Y <- golubTrainSub$ALL.AML</p>
<p>Y<-paste(golubTrain$ALL.AML,golubTrain$T.B.cell)
Y<-sub("NA","",Y)</p>
<p>@</p>
<p>Now that the data are set up, we are ready to compute distances between
tumor samples and apply clustering procedures to these samples.</p>
<p>\section{Distances}</p>
<p>We first compute the correlation distance between the 38
tumor samples from the training set, using all 3,051 genes retained after the above filtering
procedure.</p>
<p>%%FIXME: other distances?</p>
<p><<cordist>>=
r<-cor(X)
dimnames(r)<-list(as.vector(Y),as.vector(Y))
d<-1-r
@</p>
<p>We rely on the \texttt{plotcorr} function from the \textit{ellipse}
package and the \texttt{levelplot} function from \textit{lattice} for displaying this 38 $\times 38$
distance matrix.</p>
<p><<plotcorr1,fig=TRUE>>=
plotcorr(r,
main="Leukemia data: Correlation matrix for 38 mRNA samples\n All 3,051 genes")
@</p>
<p><<plotcorr2,fig=TRUE>>=
plotcorr(r,numbers=TRUE,
main="Leukemia data: Correlation matrix for 38 mRNA samples\n All 3,051 genes")
@</p>
<p><<levelplot>>=
levelplot(r,col.region=heat.colors(50),
main="Leukemia data: Correlation matrix for 38 mRNA samples\n All 3,051 genes")
@</p>
<p>\section{Multidimensional Scaling}</p>
<p>We now turn to {\em multidimensional scaling} (MDS) for representing
the tumor sample distance matrix. MDS is a data reduction method that is appropriate for
general distance matrices. It is much like principal component
analysis (PCA), but it is not the same -- except for Euclidean distances.
Given an $n \times n$ distance
matrix, MDS is concerned with identifying
$n$ point in Euclidean space with a {\em similar} distance structure.
The purpose is to provide a graphical representation of the distances
which conveys information on the relationships between objects, such
as the existence of clusters or one-dimensional structure in the data
(e.g., seriation). </p>
<p><<mds2,fig=TRUE>>=
mds<- cmdscale(d, k=2, eig=TRUE)
plot(mds$points, type="n", xlab="", ylab="",
main="MDS for ALL AML data, correlation matrix, G=3,051 genes, k=2")
text(mds$points<a href="#ref,1">[,1]</a>,mds$points<a href="#ref,2">[,2]</a>,Y, col=codes(factor(Y))+1, cex=0.8)
@</p>
<p><<mds3,fig=TRUE>>=
mds<- cmdscale(d, k=3, eig=TRUE)
pairs(mds$points,
main="MDS for ALL AML data, correlation matrix, G=3,051 genes, k=3",
pch=c("B","T","M")[codes(factor(Y))], col = codes(factor(Y))+1)
@</p>
<p>Note, if we are measuring distances between samples on the basis of
$G=3,051$ probes then we are essentially looking at points in 3,051 dimensional
space.
As with PCA, the quality of the representation will depend on the
magnitude of the first $k$ eigenvalues.
To assess how many components there are, a \textit{scree} plot similar to that
used for principal components can be created.
This plot of the eigenvalues suggests that much of the information is
contained in the first component. One might consider using either three
or four components as well.</p>
<p><<scree,fig=TRUE>>=
mdsScree <- cmdscale(d, k=8, eig=TRUE)
plot(mdsScree$eig, pch=18, col="blue")
@</p>
<p>\section{Hierarchical Clustering}</p>
<p>We can use the \texttt{hclust} function in the \textit{mva} package for
{\em agglomerative hierarchical clustering}. We consider the following
agglomeration methods, i.e., methods for computing {\em between
cluster distances}: </p>
<p>\begin{description}
\item[Single linkage] The distance between two clusters is the minimum
distance between any two objects, one from each cluster.
\item[Average linkage] The distance between two clusters is the
average of all pairwise distances between the members of both
clusters.
\item[Complete linkage] The distance between two clusters is the
maximum distance between two objects, one from each cluster.
\end{description}</p>
<p>The nested sequence of clusters resulting from hierarchical clustering
methods can be represented graphically using a dendrogram. A {\em
dendrogram} is a binary tree diagram in which the terminal nodes, or
leaves, represent individual observations and in which the height of
the nodes reflects the dissimilarity between the two clusters that are
joined.
While dendrograms are quite appealing because of their
apparent ease of interpretation, they can be misleading. </p>
<p>First, the
dendrogram corresponding to a given hierarchical clustering is not
unique, since for each merge one needs to specify which subtree should
go on the left and which on the right. For $n$ observations, there
are $n-1$ merges and hence $2^{(n-1)}$ possible orderings of the
terminal nodes. Therefore, $2^{(n-1)}$ different dendrograms are
consistent with any given sequence of hierarchical clustering
operations. The ordering is of practical importance because it will
result in different genes being placed next to each other in the
heat-maps and possibly different interpretations of the same data. A
number of heuristics have been suggested for ordering the terminal
nodes in a dendrogram. The default in the R function \texttt{hclust}
from the \textit{cluster} package is to order the subtrees so that the
tighter cluster is on the left (i.e., the most recent merge of the
left subtree is at a lower value than the last merge of the right
subtree). </p>
<p>A second, and perhaps less recognized shortcoming of dendrograms, is
that they {\em impose} structure on that data, instead of {\em
revealing} structure in these data. Indeed, dendrograms are often
viewed as graphical summaries of the data, rather than of the results
of the clustering algorithm. Such a
representation will be valid only to the extent that the pairwise
dissimilarities possess the hierarchical structure imposed by the
clustering algorithm.
Note that in particular, the same matrix of pairwise distances between
observations will be represented by a different dendrogram depending
on the distance function that is used to compute between-cluster
distances (e.g., complete or average linkage).
The {\em cophenetic correlation coefficient} can be used to measure
how well the hierarchical structure from the dendrogram represents the
actual distances. This measure is defined as the correlation between
the $n(n-1)/2$ pairwise dissimilarities between observations and their
{\em cophenetic} dissimilarities from the dendrogram, i.e., the
between cluster dissimilarities at which two observations are first
joined together in the same cluster. \\ </p>
<p>Next, we apply three agglomerative hierarchical clustering methods to
cluster tumor samples. For each methods, we plot a dendrogram, compute
the corresponding cophenetic correlation, and compare
the three clusters obtained from cutting the tree into three branches
to the actual tumor labels (ALL B-cell, ALL T-cell, and AML).</p>
<p><<average,fig=TRUE>>=
hc1 <- hclust(as.dist(d), method="average")
coph1<-cor(cophenetic(hc1),as.dist(d)) </p>
<p>plot(hc1,
main=paste("Dendrogram for ALL AML data: Coph = ", round(coph1,2)),
sub="Average linkage, correlation matrix, G=3,051 genes") </p>
<p>cthc1 <- cutree(hc1, 3)
table(Y, cthc1)</p>
<p>@</p>
<p><<single,fig=TRUE>>=
hc2 <- hclust(as.dist(d), method="single")
coph2<-cor(cophenetic(hc2),as.dist(d)) </p>
<p>plot(hc2,
main=paste("Dendrogram for ALL AML data: Coph = ", round(coph2,2)),
sub="Single linkage, correlation matrix, G= 3,051 genes") </p>
<p>cthc2 <- cutree(hc2, 3)
table(Y, cthc2)</p>
<p>@</p>
<p><<complete,fig=TRUE>>=
hc3 <- hclust(as.dist(d), method="complete")
coph3<-cor(cophenetic(hc3),as.dist(d)) </p>
<p>plot(hc3,
main=paste("Dendrogram for ALL AML data: Coph = ", round(coph3,2)),
sub="Complete linkage, correlation matrix, G= 3,051 genes") </p>
<p>cthc3 <- cutree(hc3, 3)
table(Y, cthc3)
@</p>
<p>Divisive hierarchical clustering is available using \texttt{diana} from the
\textit{cluster} package.</p>
<p><<diana,fig=TRUE>>=
di1 <- diana(as.dist(d))
cophdi<- cor(cophenetic(di1), as.dist(d))</p>
<p>plot(di1, which.plots=2, #$
main=paste("Dendrogram for ALL AML data: Coph = ", round(cophdi,2)),
sub="Divisive algorithm, correlation matrix, G= 3,051 genes") </p>
<p>ct.di <- cutree(di1, 3)
table(Y, ct.di)
@</p>
<p>\section{Partitioning Methods}</p>
<p>Here we apply the {\em partitioning around medoids} (PAM) method to cluster
tumor mRNA samples. </p>
<p><<PAM3,fig=TRUE>>=
##PAM
set.seed(12345) </p>
<p>pm3 <- pam(as.dist(d), k=3, diss=TRUE)
table(Y, pm3$clustering)</p>
<p>clusplot(d, pm3$clustering, diss=TRUE, labels=3, #$
col.p=1, col.txt=codes(factor(Y))+1,
main="Bivariate cluster plot for ALL AML data\n Correlation matrix, K=3, G=3,051 genes") </p>
<p>@
<<PAM3plot>>=
plot(pm3,which.plots=1)</p>
<p>@</p>
<p><<PAM3plot2>>=
plot(pm3,which.plots=2)
@</p>
<p>One of the more difficult questions that arises when clustering data is the
assessment of how many clusters there are in the data.
We now consider using PAM with four clusters.</p>
<p><<PAM4,fig=TRUE>>=
##what happens if we try four?
set.seed(12345)
pm4 <- pam(as.dist(d), k=4, diss=TRUE)</p>
<p>clusplot(d, pm4$clustering, diss=TRUE, labels=3, #$
col.p=1, col.txt=codes(factor(Y))+1,
main="Bivariate cluster plot for ALL AML data\n Correlation matrix, K=4, G=3,051 genes")
@</p>
<p>\end{document}
</p>
</div>
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