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@ 

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\newcommand{\myincheatmap}[5]{%
  \begin{figure}[tbp]
  \begin{center}
  \includegraphics[width=#2\textwidth]{\prefix{#1}}
  \caption[Heatmap of #3.]{\label{#1}\textbf{Heatmap of #3}. Data from #4 features with evidence of 
    differential expression between the two clusters are shown. #5.}
  \end{center}
  \end{figure}
}
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\begin{document}

\bioctitle[Analysis report: Ohnishi et al., 2014]{{\Large Executable Analysis Document Supporting:}\\
  \textbf{Cell-to-cell expression variability followed by signal reinforcement progressively segregates early mouse lineages}\\
  {\Large Yusuke Ohnishi, Wolfgang Huber, Akiko Tsumura, Minjung Kang, Panagiotis Xenopoulos, Kazuki Kurimoto, Andrzej K.~Ole\'s, Marcos J.~Ara\'uzo-Bravo, Mitinori Saitou, Anna-Katerina Hadjantonakis and Takashi Hiiragi\\
  Nature Cell Biology 16(1), 27-37 (2014)\\\href{http://dx.doi.org/10.1038/ncb2881}{doi:10.1038/ncb2881}}}

\author{Authors of this document: Andrzej K.~Ole\'s, Wolfgang Huber}

\maketitle
\tableofcontents

\newpage
\listoffigures

\newpage
%--------------------------------------------------
\section{Data import and preparations}
%--------------------------------------------------
We first load the required R package and set the random seed.
%
<<packages, results=hide>>=
library("Hiiragi2013")
set.seed(2013)
@
%
The array data consist of a set of CEL files (the output from the Affymetrix scanner
/ image analysis software), whose annotation is provided in an Excel table.
The CEL files are deposited at Array Express under the accession code E-MTAB-1681. The import of
these data and metadata is performed by the script \texttt{readdata.R}, whose code is shown on
page~\pageref{readdata} and following. This script also performs
data preprocessing (``normalisation'') using the RMA
method~\cite{IrizarryAffy} and arranges the metadata to support the
analyses presented in the following. Let us load the result of
\texttt{readdata.R}.
%
<<loadx>>=
data("x")
x
@
%
\texttt{x} is an \emph{\Sexpr{class(x)}} object containing the
normalized data for the \Sexpr{ncol(x)} arrays. These include \Sexpr{nrow(subset(pData(x), genotype=="WT"))} wild type (WT) samples
<<showWT>>=
with(subset(pData(x), genotype=="WT"),
     addmargins(table(Embryonic.day, Total.number.of.cells), 2))
@
and \Sexpr{nrow(subset(pData(x), genotype=="FGF4-KO"))} FGF4-KO mutants
<<showFGF4KO>>=
with(subset(pData(x), genotype=="FGF4-KO"), table(Embryonic.day))
@
%
%--------------------------------------------------
\subsection{Grouping of samples}
%--------------------------------------------------

The preprocessed data object defines the grouping of the samples and an associated colour map, which will be used in the plots throughout this report.
%
<<sampleColours>>=
groups = split(seq_len(ncol(x)), pData(x)$sampleGroup)
sapply(groups, length)
@
%
Each sample has assigned a colour which will be used in the subsequent plots.
%
<<sampleColours>>=
sampleColourMap = setNames(unique(pData(x)$sampleColour), unique(pData(x)$sampleGroup))
sampleColourMap
@
%
For some analyses, we need to specifically address the four probes mapping to FGF4.
<<FGF4probes>>=
FGF4probes = (fData(x)$symbol == "Fgf4")
stopifnot(sum(FGF4probes)==4)
@ 


%--------------------------------------------------
\clearpage
\section{How many genes are expressed?}
%--------------------------------------------------
In this section, we aim to determine how many distinct mRNAs were detected by the arrays
over the background level in the 66 WT samples.
%
<<expressed1,cache=TRUE>>=
selectedSamples = with(pData(x), genotype=="WT") 
xe = x[, selectedSamples]
stopifnot(ncol(xe)==66)
@ 
%
Because of the presence of background signal (stray light, cross-hybridisation), the
answer to the question whether a transcript is present in a sample, based on Affymetrix
GeneChip data, is not straightforward. In addition, the problem is complicated by the fact
that the background signal is probe-sequence dependent. We perform two approaches that are
used in the literature.

%----------------------------------------------------------
\subsection{By variability}
%----------------------------------------------------------
The first approach is based on the notion that the existence of \emph{variability} of
signal across samples is a more specific indicator of a transcript's presence, than the
absolute signal intensity~\cite{Bourgon:2010:PNAS}. Let us plot the histogram of the
standard deviation of each probe set's signal, across the \Sexpr{ncol(xe)} samples
(Figure~\ref{figHistSds}).
%
<<figHistSds,fig=TRUE,width=6,height=4>>=
sdxe = rowSds(exprs(xe))
thresh = 0.5
hist(sdxe, 100, col = "skyblue")
abline(v = thresh)
@ 
%
\incfig{figHistSds}{0.6\textwidth}{Histogram of
standard deviation of each probe set's signal}{across the \Sexpr{ncol(xe)} samples.}
Based on the (visually) chosen threshold\footnote{One could also come up with a more
  automated, ``statistical'' rule, but the downstream result would be very similar.} \Robject{thresh},
we find the following numbers of probe sets and unique target gene identifiers.
<<anaHistSds>>=
table(sdxe>=thresh)
length(unique(fData(xe)$ensembl[ sdxe>=thresh ]))
@ 

%----------------------------------------------------------
\subsection{By Affymetrix present/absent calls}
%----------------------------------------------------------
The second approach uses the Wilcoxon signed rank-based gene expression presence/absence
detection algorithm first implemented in the Affymetrix Microarray Suite version 5.
%
<<mas5calls,cache=TRUE,results=hide>>=
data("a")
stopifnot(nrow(pData(a))==ncol(x))
mas5c = mas5calls(a[, selectedSamples])
@
%
where \Robject{a} is an \emph{\Sexpr{class(a)}} object containing the unprocessed raw intensity values.
Some bookkeeping and shuffling around is needed to compute the number of genes, as defined by
unique Ensembl identifiers, for the classes \emph{present} (P), \emph{marginal} (M) and
\emph{absent} (A). If multiple probe sets map to one gene (which happens frequently on this array
type), then P trumps M trumps A.
%
<<numCalls1,cache=TRUE>>=
myUnique = function(x) setdiff(unique(x), "")
allEnsemblIDs = myUnique(fData(xe)$ensembl)

callsPerGenePerArray = matrix(0, nrow = length(allEnsemblIDs), ncol = ncol(mas5c)+1,
                              dimnames = list(allEnsemblIDs, NULL))

for(j in seq_len(ncol(mas5c))) {
  for(k in 1:2) {
    ids = myUnique(fData(xe)$ensembl[ exprs(mas5c)[, j]==c("M","P")[k] ])
    callsPerGenePerArray[ids, j] = k
  }
}
 
fractionOfArrays = 0.1
for(k in 1:2) {
  ids = myUnique(fData(xe)$ensembl[ apply(exprs(mas5c)==c("M","P")[k], 1, 
                                          function(v) (mean(v)>fractionOfArrays)) ])
  callsPerGenePerArray[ids, ncol(mas5c)+1] = k
}

numCalls = apply(callsPerGenePerArray, 2, table)
numCalls = numCalls[rev(seq_len(nrow(numCalls))), ]
<<numCalls2>>=
numCalls[, 67]
<<figmas5calls,fig=TRUE,width=12,height=5>>=
barplot2(numCalls, names.arg = paste(seq_len(ncol(callsPerGenePerArray))),
         col = c(brewer.pal(8, "Paired")[2:1], "#e8e8e8"), ylab = "number of genes")
@ 
%
See Figure~\ref{figmas5calls}.
\incfig{figmas5calls}{\textwidth}{Present/absent calls.}{The barplot shows, for each of the
  \Sexpr{ncol(mas5c)} arrays, the number of genes targeted by probe sets with "A" (light
  grey), "M" (light blue) and "P" (blue) calls. 
  The \Sexpr{ncol(mas5c)+1}-th bar at the very right corresponds to detection in at least
  \Sexpr{fractionOfArrays*100}\% of arrays.}


%----------------------------------------------------------
\clearpage
\section{Cluster stability analysis}
\subsection{E3.25 and E3.5 WT samples}\label{sec:cluewt}
%-----------------------------------------------------------
In this section, we investigate the hypothesis that the data for E3.5 fall
'naturally' into two clusters (associated with PE and EPI), while the data for E3.25 do not.
For this, we use the framework of the \Rpackage{clue}
package~\cite{clue}. Briefly, the below function
\Rfunction{clusterResampling} performs the following steps:
\begin{enumerate}
\item \label{clue1} Draw a random subset of the full data (the full data are either
  all E3.25 or all E3.5 samples) by selecting 67\% of the
  samples.
\item Select the top \Robject{ngenes} (see below) features by overall variance
  (in the subset).
\item \label{clue2} Apply $k$-means clustering, and predict the
  cluster memberships of the samples that were not in the subset with the
  \Rfunction{cl\_predict} method, through their proximity to the
  cluster centres.
\item Repeat steps \ref{clue1}-\ref{clue2} for \Robject{B} $=250$ times.
\item Apply consensus clustering (\Rfunction{cl\_consensus}).
\item \label{clueagree} For each of the \Robject{B} $=250$ clusterings, measure
  the agreement with the consensus (\Rfunction{cl\_agreement}); here,
  good agreement is indicated by a value of 1, and less agreement by
  smaller values. If the agreement is generally high, then the clustering into $k$ classes
  can be considered stable and reproducible; inversely, if it is low, then no stable
  partition of the samples into $k$ clusters is evident. 
\end{enumerate}

As a measure of between-cluster distance for the consensus clustering,
the \emph{Euclidean} dissimilarity of the memberships is used, i.~e.,
the square root of the minimal sum of the squared differences of $\mathbf{u}$
and all column permutations of $\mathbf{v}$, where $\mathbf{u}$ and $\mathbf{v}$ are the cluster membership matrices.
As agreement measure for step~\ref{clueagree}, the quantity $1 - d /
m$ is used, where $d$ is the Euclidean dissimilarity,
and $m$ is an upper bound for the maximal Euclidean dissimilarity~\cite{Dimitriadou:2002}.
%
<<clusterResampling>>=
clusterResampling = function(x, ngenes, k = 2, B = 250, prob = 0.67) {
  mat = exprs(x)
  ce = cl_ensemble(list = lapply(seq_len(B), function(b) {
    selSamps = sample(ncol(mat), size = round(prob*ncol(mat)), replace = FALSE)
    submat = mat[, selSamps, drop = FALSE]
    selFeats = order(rowVars(submat), decreasing = TRUE)[seq_len(ngenes)]
    submat = submat[selFeats,, drop = FALSE]
    pamres = pam(t(submat), k = k, metric = "euclidean")
    pred = cl_predict(pamres, t(mat[selFeats, ]), "memberships")
    as.cl_partition(pred)
  }))
  cons = cl_consensus(ce)
  ag = sapply(ce, cl_agreement, y = cons)
  return(list(agreements = ag, consensus = cons))
}
@

<<ce1,cache=TRUE>>=
ce = list(
  "E3.25" = clusterResampling(x[, unlist(groups[c("E3.25")])], ngenes = 20),
  "E3.5"  = clusterResampling(x[, unlist(groups[c("E3.5 (EPI)", "E3.5 (PE)")])],
                              ngenes = 20))
@
%
The results are shown in Figure~\ref{figClue1}. They confirm the hypothesis stated at
the beginning of this section.
%
<<figClue1,fig=TRUE,width=9,height=5>>=
par(mfrow = c(1,2))
colours  = c(sampleColourMap["E3.25"], brewer.pal(9,"Set1")[9])

boxplot(lapply(ce, `[[`, "agreements"), ylab = "agreement probabilities", col = colours)

mems = lapply(ce, function(x) sort(cl_membership(x$consensus)[, 1]))
mgrp = lapply(seq(along = mems), function(i) rep(i, times = length(mems[[i]])))
myjitter = function(x) x+seq(-.4, +.4, length.out = length(x))

plot(unlist(lapply(mgrp, myjitter)), unlist(mems),
     col = colours[unlist(mgrp)], ylab = "membership probabilities",
     xlab = "consensus clustering", xaxt = "n", pch = 16)
text(x = 1:2, y = par("usr")[3], labels = c("E3.25","E3.5"), adj = c(0.5, 1.4), xpd = NA)
@
%
\incfig{figClue1}{0.8\textwidth}{Cluster stability analysis with E3.25 and E3.5 WT samples.}{Left:
  boxplot of the cluster agreements  with the consensus, for the
  \Robject{B}=\Sexpr{formals(clusterResampling)$B}
  clusterings; 1 indicates perfect agreement, and
  the value decreases with worse agreement. The statistical significance of the difference
  is confirmed by a Wilcoxon test in the main text. Right: membership
  probabilities of the consensus clustering; colours are as in the left panel.
  For E3.25, the probabilities are diffuse, indicating that the individual
  (resampled) clusterings disagree a lot, whereas for E3.5, the
  distribution is bimodal, with only one ambiguous sample.}

We can compute a $p$-value for the statistical significance of the two distributions shown
in the boxplot of Figure~\ref{figClue1}.
<<test>>=
wilcox.test(ce$E3.25$agreements, ce$E3.5$agreements)
@

%-----------------------------------------------------------
\subsection{E3.5/E4.5 WT and E4.5 FGF4-KO samples}\label{sec:clueko}
%-----------------------------------------------------------
From the PCA plot in Figure~\ref{figmds-2}, one might derive the impression that the FGF4
KO E4.5 cells cluster together with the EPI E3.5 cells.  Here, we perform a cluster
stability analysis analogous to that in Section~\ref{sec:cluewt}, using the FGF4-KO E4.5
cells together with 
\begin{enumerate}
  \item the WT EPI E3.5 and WT PE E3.5 cells,
  \item the WT EPI E4.5 and WT PE E4.5 cells.
\end{enumerate}
As we will see in the following, in each of the above cases, three distinct clusters
exist, and the above mentioned impression is not substantiated; in addition, we can also
clearly distinguish the WT and KO samples at E4.5.
%
<<ce2,cache=TRUE>>=
sampleSets = list(
  `E3.5` = unlist(groups[c("E3.5 (EPI)", "E3.5 (PE)", "E4.5 (FGF4-KO)")]),
  `E4.5` = unlist(groups[c("E4.5 (EPI)", "E4.5 (PE)", "E4.5 (FGF4-KO)")]))
k = 3
csa = lapply(sampleSets, function(samps) list(
  colours = x$sampleColour[samps], 
  r = clusterResampling(x[!FGF4probes, samps], ngenes = 20, k = k)))
<<figClue2,fig=TRUE,width=8,height=6>>=
par(mfrow = c(2,3))
for(i in seq(along = csa))
  for(j in seq_len(k))
    plot(cl_membership(csa[[i]]$r$consensus)[, j], 
         ylab = paste0("P(cluster ", j, ")"), xlab = "samples",
         main = names(sampleSets)[i], col = csa[[i]]$colours, pch = 16, cex = 1.5)
@ 
%
\incfig{figClue2}{\textwidth}{Cluster stability analysis with E3.5/E4.5 WT and E4.5 FGF4-KO samples.}{Top 
  row: Results of the cluster stability analysis with the E3.5 (EPI), E3.5 (PE), E4.5
  (FGF4-KO) samples. Shown on the $y$-axis is the membership probability P
  in the three clusters. The actual group
  membership of the samples (known to us but not used by the clustering algorithm) is indicated by the
  colours. Bottom row: Similarly for E4.5 (EPI), E4.5 (PE), E4.5 (FGF4-KO). These analyses indicate that the FGF4-KO
  very clearly separate from the WT samples, which between themselves tend to form the 
  clusters already described in Section~\ref{sec:cluewt}.}
%
The results of the above code are shown in Figure~\ref{figClue2} and indicate that
indeed FGF4-KO E4.5 cells are distinct from either of the WT groups.

%-----------------------------------------------------------
\subsection{E3.25 WT and E3.5 FGF4-KO samples}\label{sec:clueko}
%-----------------------------------------------------------

Although in the 2-dimensional PCA projection in Figure~\ref{figmds-2} dots representing FGF4-KO E3.5 cells appear to overlap with ones representing WT E3.25 samples, cluster stability analysis analogous to that from the previous section indicates that they form a distinct population (Figure~\ref{figClue3}).
%
<<ce3,cache=TRUE>>=
sampleSets = unlist(groups[c("E3.25", "E3.5 (FGF4-KO)")])
selSamples = x[!FGF4probes, sampleSets]
resampledSampleSet = clusterResampling(selSamples, ngenes = 20)
@
%
<<figClue3,fig=TRUE,width=8,height=4.5>>=
par(mfrow = c(1,2))
  for(j in seq_len(2))
    plot(cl_membership(resampledSampleSet$consensus)[, j], 
         ylab = paste0("P(cluster ", j, ")"), xlab = "Samples",
         col = x$sampleColour[sampleSets], pch = 16, cex = 1.5)
@ 
%
\incfig{figClue3}{.66\textwidth}{Cluster stability analysis with E3.25 WT and E3.5 FGF4-KO samples.}{Results of the cluster stability analysis with the E3.25 WT and E3.5 FGF4-KO samples. The $y$-axis shows the membership probability in the two clusters. The actual group membership of the samples (known but not used by the clustering algorithm) is indicated by the colours.}

The cluster stability analysis of the two clusters reveals that the FGF4-KO samples at E3.5 form a single, tight cluster, and are consistently together throughout all of the resamplings. The E3.25 WT cells, on the other hand, are much more diffuse and in the course of the resampling, each cell does not necassaryly cluster together with the same cluster all the time.
Therefore, the difference is prevalently one of variability---the E3.25 WT are quite variable and cover a large "expression space", whereas the FGF4-KO are stuck on one particular, narrowly defined expresion profile. This is also evident from the heatmap showed in Figure~\ref{figHeatmapClue}. 

<<figHeatmapClue,fig=TRUE,width=10,height=13>>=
ngenes = 100
selFeats = order(rowVars(exprs(selSamples)), decreasing = TRUE)[seq_len(ngenes)]
myHeatmap(selSamples[selFeats, ], collapseDuplicateFeatures = TRUE, haveColDend = TRUE)
@
\incfig{figHeatmapClue}{\textwidth}{Heatmap of the E3.25 WT and E3.5 FGF4-KO samples.}{Data of the 100 genes with the highest variance. Even though a few individual cells from E3.25 WT do indeed seem to have similar expression profiles as ones from E3.5 FGF4-KO, which all cluster together, the populations are not the same.}

%----------------------------------------------------------
\clearpage
\section{Lineage Markers}
\label{sec:unsuper}
%-----------------------------------------------------------

In this section, the following steps are performed:
\begin{enumerate}
\item Section~\ref{sec:cluster}: Cluster the arrays from E3.5 into two clusters (using $k$-means
  clustering with $k=2$ on the overall expression profiles).
\item Section~\ref{sec:deCluster}: Determine the genes that are differentially expressed between
  these two clusters, according to a $t$-test with a nominal cutoff of
  false discovery rate (FDR) of 10\%.  These genes are reported in the
  table \texttt{differentially-expressed-features-3.5.csv}, which can be
  imported into Excel etc.
\item Section~\ref{sec:deHeatmaps}: Present a heatmap of these genes.
\item Section~\ref{sec:de45}: Produce an analogous table \texttt{differentially-expressed-features-4.5.csv} for
  the E4.5 samples.
\end{enumerate}

%-----------------------------------------------------------------
\subsection{Clustering of E3.5 WT samples}\label{sec:cluster}
%-----------------------------------------------------------------
The function \Rfunction{pamCluster}
selects the \Robject{ngenes} most variable genes in
the data matrix \Robject{x} and seeks for two clusters using the
\emph{partitioning around medoids} method.
To assess the influence of the parameter \Robject{ngenes}, we call the
algorithm for multiple choices, from 10 to 10000.
<<doCluster>>=
ngenes = c(10, 25, 50, 100, 250, 500, 1000, 2500, 5000, 10000)
<<cluster, cache=TRUE>>=
xForClustering = x[, x$Embryonic.day=="E3.5" & x$genotype=="WT"]
clusters = sapply(ngenes, pamCluster, x = xForClustering)
@
The result is displayed in Figure~\ref{figClusters}.
%
<<figClusters, fig=TRUE, width=5, height=5>>=
image(x = seq_len(nrow(clusters)), y = seq_len(ncol(clusters)), z = clusters,
      col = c("#f0f0f0", "#000000"), ylab = "ngenes", xlab = "samples", yaxt = "n")
text(x = 0, y = seq_len(ncol(clusters)), paste(ngenes), xpd = NA, adj = c(1, 0.5))
@
%
\incfig{figClusters}{0.5\textwidth}{The influence of the
  parameter \Robject{ngenes} on the clustering result.}{Each of the \Sexpr{nrow(clusters)}
  samples corresponds to a column of the matrix, the rows of the
  matrix correspond to different choices for
  \Robject{ngenes}. Cluster membership is indicated by the colour code
  (light gray \emph{vs} black). For most samples, cluster membership
  is consistent throughout the range of \Robject{ngenes} from 10 to 1000.
  \Robject{ngenes} = 1000 is used for subsequent analyses.}
%
From Figure~\ref{figClusters} we can conclude that we can proceed
with the choice of
%
<<choosengenes>>=
i = which(ngenes==1000); stopifnot(length(i)==1)
ngenes = ngenes[i]
clusters = factor(clusters[, i])
@
Now we can check how the microarray-data driven clustering compares with
the annotation of the cells that was provided by Yusuke in the Excel table:
<<compareClusteringAnnotation>>=
table(clusters, pData(x)[names(clusters), "lineage"])
@
<<compareClusteringAnnotationCheck,echo=FALSE,results=hide>>=
stopifnot(all(
 table(clusters, pData(x)[names(clusters), "lineage"])==cbind(c(0,11),c(11,0))
))
@
%
As we can see, the clustering perfectly agrees with Yusuke's labeling. \emph{Note}: the
\texttt{lineage} annotation was used here only to assess the clustering output.
It is not used as input for any of the data analyses shown in this document.
%
<<print>>=
cbind(unlist(groups[c("E3.5 (EPI)", "E3.5 (PE)")]), ce[[2]]$consensus$.Data[, 1])
@

%-----------------------------------------------------------------
\subsection{Differentially expressed genes in E3.5 samples}
\label{sec:deCluster}
%-----------------------------------------------------------------
<<deBetweenClusters>>=
deCluster = rowttests(xForClustering, fac = clusters)
@
%
The code below, which produces Figure~\ref{figIndepFiltSetup},
is used to set the parameters for the independent filtering step~\cite{Bourgon:2010:PNAS}.
%
<<figIndepFiltSetup, fig=TRUE, pdf=FALSE, png=TRUE, width=4.5, height=4.5>>=
varianceRank = rank(-rowVars(exprs(xForClustering)))
plot(varianceRank, deCluster$p.value, pch = ".", log = "y",
     main = "Parameters for the independent filtering",
     xlab = "variance rank", ylab = "p-value")
nFilt = 20000
smallpValue = 1e-4
abline(v = nFilt, col = "blue")
abline(h = smallpValue, col = "orange")
@
\incfig{figIndepFiltSetup}{0.5\textwidth}{Determination of the cutoff for independent
  filtering on E3.5 WT samples.}{Each dot corresponds to one of the \Sexpr{nrow(xForClustering)} features on the array, the $x$-axis
  shows the rank of the features overall variance across the  \Sexpr{ncol(xForClustering)} arrays in
  \Robject{xForClustering}, the $y$-axis the $p$-value from the $t$-test on a logarithmic scale.
  The horizontal orange line corresponds to a $p$-value of \Sexpr{smallpValue}.
  The vertical blue line indicates the cutoff implied by taking only the \Sexpr{nFilt} features with the highest
  overall variance. We can see that if we apply this cutoff, we in fact do not miss any of the features with
  $p$-value smaller than \Sexpr{smallpValue}.}
<<checkFilterClaim,echo=FALSE,results=hide>>=
stopifnot(!any((deCluster$p.value<smallpValue)&(varianceRank>nFilt)))
@
%
Let's perform false discovery rate (FDR) $p$-value adjustment using the
Benjamini-Hochberg method~\cite{BH:1995}.
<<BH>>=
passfilter = which(varianceRank<=nFilt)
adjp = rep(NA_real_, nrow(x))
adjp[passfilter] = p.adjust(deCluster$p.value[passfilter], method = "BH")
ord = order(adjp)
numFeaturesReport = 200
differentially = ord[seq_len(numFeaturesReport)]
length(unique(fData(x)$symbol[differentially]))
@
% We identify \Sexpr{length(differentially)} features at an FDR of 10\%,
The FDR of the selected set of \Robject{numFeaturesReport} = \Sexpr{numFeaturesReport} features
is \Sexpr{signif( adjp[tail(differentially,1)]*100, 3)}\%,
and these correspond to \Sexpr{length(unique(fData(x)$symbol[differentially]))} unique genes.
In the following code chunk, we write out the results table into a CSV file.
%
<<write.csv>>=
deFeat35 = cbind(deCluster[differentially,], `FDR-adjusted p-value` = adjp[differentially], 
                 fData(x)[differentially,])
write.csv(deFeat35, file = "differentially-expressed-features-3.5.csv")
@

%-----------------------------------------------------------------
\subsubsection{Heatmaps}\label{sec:deHeatmaps}
%-----------------------------------------------------------------
To visualise the data, we use the helper function \Rfunction{myHeatmap}.
Heatmaps produced by the function calls below are shown in
Figures~\ref{figHeatmapAll}--\ref{figHeatmap35Coll}.
%
<<figHeatmapAll, fig=TRUE, width=16, height=21>>=
myHeatmap(x[differentially, x$genotype=="WT"])
<<figHeatmapAllColl, fig=TRUE, width=14, height=19>>=
myHeatmap(x[differentially, x$genotype=="WT"], collapseDuplicateFeatures = TRUE)
<<figHeatmap35, fig=TRUE, width=10, height=22>>=
myHeatmap(xForClustering[differentially, ])
<<figHeatmap35Coll, fig=TRUE, width=10, height=22>>=
myHeatmap(xForClustering[differentially, ], collapseDuplicateFeatures = TRUE)
@

\myincheatmap{figHeatmapAll}{1}{all WT arrays}{\Sexpr{length(differentially)}}{For some genes, multiple features are shown}
\myincheatmap{figHeatmapAllColl}{1}{all WT arrays with duplicate features collapsed}{\Sexpr{length(differentially)}}{Duplicate features per gene were averaged}
\myincheatmap{figHeatmap35}{0.6}{only the WT arrays from E3.5}{\Sexpr{length(differentially)}}{For some genes, multiple features are shown}
\myincheatmap{figHeatmap35Coll}{0.6}{only the WT arrays from E3.5 with duplicate features collapsed}{\Sexpr{length(differentially)}}{Duplicate features per gene were averaged}

%-----------------------------------------------------------------
\subsection{Differentially expressed genes in E4.5 samples}
\label{sec:de45}
%-----------------------------------------------------------------
The following code is analogous to Section~\ref{sec:deCluster}.
<<de45>>=
x45 = x[, x$Embryonic.day=="E4.5" & x$genotype=="WT"]
de45 = rowttests(x45, fac = "lineage")
<<figIF45, fig=TRUE, pdf=FALSE, png=TRUE, width=4.5, height=4.5>>=
varianceRank = rank(-rowVars(exprs(x45)))
plot(varianceRank, de45$p.value, pch = ".", log = "y",
     main = "Parameters for the independent filtering",
     xlab = "variance rank", ylab = "p-value")
abline(v = nFilt, col = "blue")
abline(h = smallpValue, col = "orange")
@
See Figure~\ref{figIF45}.
\incfig{figIF45}{0.5\textwidth}{Determination of the cutoff for independent
  filtering on E4.5 WT samples.}{Analogous to Figure~\ref{figIndepFiltSetup}.}
<<BH45>>=
passfilter = which(varianceRank<=nFilt)
adjp = rep(NA_real_, nrow(x))
adjp[passfilter] = p.adjust(de45$p.value[passfilter], method = "BH")
ord = order(adjp)
differentially = ord[seq_len(numFeaturesReport)]
length(unique(fData(x)$symbol[differentially]))
@
% We identify \Sexpr{length(differentially)} features at an FDR of 10\%,
The FDR of the selected set of \Robject{numFeaturesReport} = \Sexpr{numFeaturesReport} features
is \Sexpr{signif( adjp[tail(differentially,1)]*100, 3)}\%,
and these correspond to \Sexpr{length(unique(fData(x)$symbol[differentially]))} unique genes.
The following code chunk writes out the results table into a CSV file.
%
<<write.csv>>=
deFeat45 = cbind(de45[differentially, ], `FDR-adjusted p-value` = adjp[differentially], 
                 fData(x)[differentially, ])
write.csv(deFeat45, file = "differentially-expressed-features-4.5.csv")
@


%--------------------------------------------------
\clearpage
\section{Differentially expressed genes from E3.25 to E3.5}
\label{sec:E325toE35}
%--------------------------------------------------
To understand the transitions shown in the MDS plots in
molecular terms, a look at the genes that are differentially expressed
along the transitions from E3.25 to E3.5 (EPI) and from E3.25 to E3.5
(PE) might be instructive. To this end, we
determine the differentially expressed genes for each of the two
comparisons, build the union set, and display their data in the
heatmap shown in Figure~\ref{figdifferentiallyE325toE35}.
It is interesting that some genes are specifying for either EPI or PE,
whereas other genes show the same trend from E3.25 to E3.5 for both
lineages (i.~e.\ they only reflect time).
%
<<differentiallyE325toE35>>=
samples = unlist(groups[c("E3.25", "E3.5 (EPI)", "E3.5 (PE)")])
deE325toE35 = union(
  getDifferentialExpressedGenes(x, groups, "E3.25", "E3.5 (EPI)"),
  getDifferentialExpressedGenes(x, groups, "E3.25", "E3.5 (PE)"))
<<figdifferentiallyE325toE35, width=9, height=24, fig=TRUE>>=
myHeatmap(x[deE325toE35, samples], collapseDuplicateFeatures = TRUE)
@
%
\incfig{figdifferentiallyE325toE35}{0.5\textwidth}{Heatmap of
  differentially expressed genes from E3.25 to E3.5 (EPI), and from
  E3.25 to E3.5 (PE).}{A
  standalone PDF file with this figure is also available, under the name
  \texttt{Hiiragi2013-figdifferentiallyE325toE35.pdf}.}

%-----------------------------------------------------------------
\clearpage
\section{Principal Component Analysis}
\label{sec:E325}
%-----------------------------------------------------------------
As a first attempt at getting an overview over the data, see Figure~\ref{figProjection}.
%
<<E325samples>>=
projection = deCluster$dm[differentially] %*% exprs(x)[differentially, ]
<<figProjection, fig=TRUE, width=7, height=15>>=
plotProjection(projection, label = sampleNames(x),
               col = x$sampleColour, colourMap = sampleColourMap)
@
%
\incfig{figProjection}{8cm}{Projection of sample expression profiles on
the differential expression signature from Section~\ref{sec:unsuper}.}{Expression
profiles and signature are each vectors of length \Sexpr{length(differentially)},
and shown is the scalar product between each sample's expression profile for
these \Sexpr{length(differentially)} features and the differential expression signature.}

%--------------------------------------------------------------
\subsection{On the WT samples}\label{sec:mdswt}
%--------------------------------------------------------------
Select the WT samples.
<<safeSelect-1>>=
safeSelect = function(grpnames){
  stopifnot(all(grpnames %in% names(groups)))
  unlist(groups[grpnames])
}
g = safeSelect(c("E3.25",
                 "E3.5 (EPI)", "E3.5 (PE)",
                 "E4.5 (EPI)", "E4.5 (PE)"))
@
%
Note: we do not use the data from all the \Sexpr{nrow(x)} features on the microarrays,
since most of these are dominated by noise. Rather, we use the top 100 features according
to overall variance across the WT samples, excluding, however, the FGF4 probes (to avoid
any possible concerns about ``trivial'' effects of the KOs).
%
<<varianceOrder,cache=TRUE>>=
nfeatures = 100
varianceOrder = order(rowVars(exprs(x[, g])), decreasing = TRUE)
varianceOrder = setdiff(varianceOrder, which(FGF4probes))
selectedFeatures = varianceOrder[seq_len(nfeatures)]
xwt = x[selectedFeatures, g]
@
Before embarking on the PCA computation, construct a new data matrix with equal group sizes.
<<doPCA1,cache=TRUE>>=
tab = table(xwt$sampleGroup)
sp = split(seq_len(ncol(xwt)), xwt$sampleGroup)
siz = max(listLen(sp))
sp = lapply(sp, sample, size = siz, replace = (siz>length(x)))
xwte = xwt[, unlist(sp)]
@
Now we are ready to do it.
<<doPCA2,cache=TRUE>>=
thepca = prcomp(t(exprs(xwte)), center = TRUE)
pcatrsf = function(x) scale(t(exprs(x)), center = TRUE, scale = FALSE) %*% thepca$rotation
stopifnot(all( abs(pcatrsf(xwte) - thepca$x) < 1e-6 ))
<<myPCA>>=
myPCAplot = function(x, labels, ...) {
  xy = pcatrsf(x)[, 1:2]
  plot(xy, pch = 16, col = x$sampleColour, cex = 1, xlab = "PC1", ylab = "PC2", ...)
  if(!missing(labels))
    text(xy, labels, cex = 0.35, adj = c(0.5, 0.5))
}
<<figmds-1, fig=TRUE, width=5, height=5, results=hide>>=
myPCAplot(xwt)
@
%
See Figure~\ref{figmds-1}. We also provide an overview over the distributions of
loadings of the PCA components (Figure~\ref{figpcaloadings}) and the 20 most
important genes (10 with highest positive coefficients and 10 with the highest negative
coefficients) in the R output below.
%
\incfig{figmds-1}{.65\textwidth}{PCA plot, using WT samples.}{The colour code is as in
  Figure~\ref{figProjection}.}
\incfig{figpcaloadings}{\textwidth}{Sorted loadings (coefficients) of the first two
  PCA vectors.}{The most important genes are shown in the text.}
%
<<figpcaloadings,fig=TRUE, width=8, height=4>>=
par(mfrow = c(1,2))
for(v in c("PC1", "PC2")) {
  loading = thepca$rotation[, v]
  plot(sort(loading), main = v, ylab = "")
  sel = order(loading)[c(1:10, (-9:0)+length(loading))]
  print(data.frame(
   symbol = fData(x)$symbol[selectedFeatures[sel]],
   probe = names(loading)[sel],
   loading = loading[sel], stringsAsFactors = FALSE
  ))
}
@
%



%--------------------------------------------------------------
\subsection{WT and FGF4-KO samples}\label{sec:mdsko}
%--------------------------------------------------------------
In the following we perform PCA analysis using both WT and FGF4-KO samples.
<<figmds-2, fig=TRUE, width=5, height=5, results=hide>>=
myPCAplot(x[selectedFeatures, ])
@
See Figure~\ref{figmds-2}.
\incfig{figmds-2}{.65\textwidth}{PCA
  plot for WT and FGF4-KO samples.}{The
  colour code is as in Figure~\ref{figProjection}.}
%
<<figmds-3, fig=TRUE, width=5, height=5, results=hide>>=
myPCAplot(x[selectedFeatures, ], labels = paste(seq_len(ncol(x))))
@
See Figure~\ref{figmds-3}.
\incfig{figmds-3}{.65\textwidth}{Same as Figure~\ref{figmds-2}, with labels
  indicating the array (sample) number.}{This may be useful to detect outlier arrays.}
%
<<figmds-4, fig=TRUE, width=5, height=5, results=hide>>=
myPCAplot(x[selectedFeatures, ], labels = paste(x$Total.number.of.cells))
@
See Figure~\ref{figmds-4}.
\incfig{figmds-4}{.65\textwidth}{Same as Figure~\ref{figmds-2}, with labels
  indicating \Robject{Total.number.of.cells}.}{This may be useful to detect ``batch effects''.}

%--------------------------------------------------------------
\subsection{Heatmap of all WT and FGF4-KO samples}
%--------------------------------------------------------------
<<hmprep>>=
mat = exprs(x[selectedFeatures, ])
rownames(mat) = fData(x)[selectedFeatures, "symbol"]
<<fighmko,width=14,height=17,fig=TRUE>>=
heatmap.2(mat, trace = "none", dendrogram = "none", scale = "row",
          col = bluered(100), keysize = 0.9,
          ColSideColors = x$sampleColour, margins = c(7,5))
@

See Figure~\ref{fighmko}.
\incfig{fighmko}{\textwidth}{Heatmap of all arrays.}{Data from the \Sexpr{nfeatures} with the highest variance across the WT samples, excluding the FGF4 probes. The colour code of the bar at the top is as in Figure~\ref{figProjection}.}

%----------------------------------------------------------------
\clearpage
\section{Further analyses of FGF4-KO}\label{sec:fg4ko}
%----------------------------------------------------------------
\subsection{FGF4's expression pattern in E3.25 samples}\label{sec:fgf4:exprpat}
%----------------------------------------------------------------
In Figure~\ref{figFGF4expression}, produced by the code below, we
visualise the expression pattern of FGF4 in the E3.25 samples. The
striking result is that there is a lot of natural variation in FGF4
expression even in the WT samples, and some of the lowest levels in
the WT samples approach the background signal level seen for the KOs.
%
<<figFGF4expression, fig=TRUE, width=10, height=5>>=
x325  = x[, with(pData(x), Embryonic.day=="E3.25")]
rv325 = rowVars(exprs(x325))
featureColours = brewer.pal(sum(FGF4probes), "Dark2")
py = t(exprs(x325)[FGF4probes, ])
matplot(py, type = "p", pch = 15, col = featureColours,
        xlab = "arrays", ylab = expression(log[2] ~ signal),
        ylim = range(py) + c(-0.7, 0))
legend("topright", legend = rownames(fData(x325))[FGF4probes], fill = featureColours)
points(seq_len(nrow(py)), rep(par("usr")[3]+0.2, nrow(py)),
       pch = 16, col = x325$sampleColour)
@
%
\incfig{figFGF4expression}{\textwidth}{FGF4 expression.}{The
  plot shows data from the
  \Sexpr{sum(FGF4probes)} features on the microarray annotated to
  FGF4. From these data, we conclude that \texttt{1420085\_at} and
  \texttt{1420086\_x\_at} essentially report the same, and are likely
  to be good reporters for the FGF4 isoform that we are interested in,
  whereas the other two features measure something else. The circle symbols
  at the bottom of the plot indicate the samples' genotypes.}
%
<<checkClaim1, echo=FALSE>>=
stopifnot(all(c("1420085_at", "1420086_x_at") %in% rownames(fData(x325))[FGF4probes]),
          all(x325$genotype[1:36]=="WT"),
          all(x325$genotype[37:44]=="FGF4-KO"))
@
%
For presentation, we also produce another visualisation, this time only
showing one value per array, which we obtain by averaging over the two
"good" features (Figure~\ref{figFGF4jitter}).
%
<<FGF4expression2>>=
fgf4Expression = colMeans(exprs(x325)[c("1420085_at", "1420086_x_at"), ])
fgf4Genotype = factor(x325$genotype, 
                      levels = sort(unique(x325$genotype),decreasing = TRUE))
@
<<figFGF4jitter, fig=TRUE, width=5, height=4.5>>=
plot(x = jitter(as.integer(fgf4Genotype)),
     y = fgf4Expression,
     col = x325$sampleColour, xlim = c(0.5, 2.5), pch = 16,
     ylab = expression(FGF4~expression~(log[2]~units)),
     xlab = "genotype", xaxt = "n")
cm = sampleColourMap[sampleColourMap %in% x325$sampleColour]
legend("topright", legend = names(cm), fill = cm)
@
\incfig{figFGF4jitter}{0.6\textwidth}{FGF4 expression (microarray signal) in WT and KO samples.}{}

%----------------------------------------------------------------
\subsection{Are the E3.25 WT samples with low FGF4 expression more
  similar to the FGF4-KO samples than those with high FGF4?}
%----------------------------------------------------------------
This question is addressed by the code below, whose result is shown in
Figure~\ref{figFgf4MDS}.
%
<<Fgf4MDS>>=
zero2one = function(x) (x-min(x))/diff(range(x))
rgb2col = function(x) {x = x/255; rgb(x[, 1], x[, 2], x[, 3])}

colours = x325$sampleColour
wt325 = x325$genotype=="WT"
colourBar = function(x) rgb2col(colorRamp(c("yellow", "blue"))(zero2one(x)))
colours[wt325] = colourBar(fgf4Expression)[wt325]

selMDS = order(rv325, decreasing = TRUE)[seq_len(100)]
<<figFgf4MDS,fig=TRUE,width=6,height=6,results=hide>>=
MDSplot(x325[selMDS, ], col = colours)
<<figFgf4MDSColourBar,fig=TRUE,width=1.5,height=4>>=
atColour = seq(min(fgf4Expression), max(fgf4Expression), length = 20)
image(z = rbind(seq(along = atColour)), col = colourBar(atColour),
      xaxt = "n", y = atColour, ylab = "")
@
%
\begin{figure}\begin{center}
\includegraphics[width=.5\textwidth]{\prefix{figFgf4MDS}}
\includegraphics[width=2.5cm]{\prefix{figFgf4MDSColourBar}}
\caption[MDS plot of the E3.25 wild type and FGF4-KO samples.]{\label{figFgf4MDS}
  \textbf{MDS plot of the E3.25 wild type
  (yellow--blue) and FGF4-KO (orange) samples.}  The yellow--blue colour
  scale represents the FGF4 expression level, as indicated in the
  colour bar.}
\end{center}\end{figure}
%
Figure~\ref{figFgf4MDS} indicates that
\begin{enumerate}
  \item there is a relationship between FGF4 expression and overall global expression
    patterns in the WT samples;
  \item WT samples with low FGF4 are more similar to the FGF4 KOs than
    than WT samples with high FGF4.
\end{enumerate}
To more formally explore statement 2, we compute the Euclidean distance between
each WT sample and the mean of the KOs, plot this against the FGF4
expression level (Figure~\ref{figCorFgf4Dist}) and test for correlation:
%
<<corFgf4Dist>>=
KOmean = rowMeans(exprs(x325)[selMDS, x325$genotype=="FGF4-KO"])
dists = colSums((exprs(x325)[selMDS, wt325] - KOmean)^2)^0.5
ct = cor.test(fgf4Expression[wt325], dists, method = "spearman")
ct
<<figCorFgf4Dist,fig=TRUE,width=4.5,height=4.5>>=
plot(fgf4Expression[wt325], dists, pch = 16, main = "E3.25 WT samples",
     xlab = "FGF4 expression", ylab = "Distance to FGF4-KO", col = colours)
@
%
\incfig{figCorFgf4Dist}{0.6\textwidth}{Relationship between
  FGF4 expression and similarity of the transcription profile to the KO.}{FGF4
  expression of each sample is shown along the $x$-axis, the
  $y$-axis corresponds to the distance to the mean FGF4-KO expression profile
  computed over the \Sexpr{length(selMDS)} most variable features as
  selected in \Robject{selMDS}.}
%
There is a significant correlation
between FGF4 expression in WT and similarity of the overall expression
profile with that of the FGF4 KOs.
<<echo=FALSE>>=
stopifnot(ct$p.value<0.01)
@

%-------------------------------------------------------------------
\subsection{Variability of the FGF4-KO samples compared to WT samples}
%-------------------------------------------------------------------
To address this question, let us take some precaution
against possible batch effects. Therefore, we split the samples first by the
\Robject{sampleGroup} classification defined above, but then also into
groups according to the value of \Robject{Total.number.of.cells},
assuming that the samples within each such group have been processed
together. See below.
%
<<FGF4variab1>>=
varGroups = split(seq_len(ncol(x)), f = list(x$sampleGroup,x$Total.number.of.cells),
                  sep = ":", drop = TRUE)
@ 
%
We can see how many samples are in each of these groups, and what the value of
\Robject{ScanDate} is.
<<ScanDate>>=
data.frame(
  `number arrays` = listLen(varGroups),
  `ScanDates` = sapply(varGroups, function(v) 
    paste(as.character(unique(x$ScanDate[v])), collapse = ", ")),
  stringsAsFactors = FALSE)
@
%
Again select the top \Sexpr{nfeatures} genes according to \Robject{varianceOrder}.
%
<<FGF4variab1>>=
sel   = varianceOrder[seq_len(nfeatures)]
myfun = function(x) median(apply(exprs(x), 1, mad))

sds = lapply(varGroups, function(j) myfun(x[sel, j]))
names(sds) = sprintf("%s (n=%d)", names(sds), listLen(varGroups))

varGroupX = factor(sapply(strsplit(names(varGroups), split = ":"),`[`, 1))
<<figFGF4variab,fig=TRUE,width=5,height=5>>=
op = par(mai = c(2,0.7,0.1,0.1))
plot(jitter(as.integer(varGroupX)), sds, xaxt = "n", xlab = "", ylab = "")
axis(1, las = 2, tick = FALSE, at = unique(varGroupX), labels = unique(varGroupX))
par(op)
@
%
\incfig{figFGF4variab}{0.6\textwidth}{Variability of different groups of
  samples.}{The groups were defined by \Robject{sampleGroup} and \Robject{ScanDate} (see
  definition of \Robject{varGroups} above). Variability was measured by the \emph{median}
  (across the \Sexpr{length(sel)} top variable genes) of the \emph{median absolute
    deviation} across samples.}
%
See Figure~\ref{figFGF4variab}. Due to the small numbers of samples, it is difficult
to decide whether or not batch effects play an important role for estimating
variability. This view is corroborated by the diagnostic plots in the \emph{quality
  assessment report} from the \Rpackage{arrayQualityMetrics} package.

So let us set aside the batch effect worries, and just compute and compare distributions
of standard deviations\footnote{Since these standard deviations are computed on the
  logarithmic scale, they correspond to coefficient of variation on the
  not-log-transformed scale.}.
<<sds>>=
gps = split(seq_len(ncol(x)), f = x$sampleGroup)[c("E3.25", "E3.25 (FGF4-KO)")]
sds = sapply(gps, function(j) apply(exprs(x)[sel, j], 1, mad))
summary(sds)
apply(sds, 2, function(x) c(`mean` = mean(x), `sd` = sd(x)))
@ 

%--------------------------------------------------
\subsection{Do the FGF4-KO samples correspond to a particularly early substage
  within E3.25 (as indicated by the number of cells)?}
%--------------------------------------------------
See Figures~\ref{figNumberOfCells-100} and
\ref{figNumberOfCells-1000}, which are produced by the code below.
%
<<numberOfCells, results=hide>>=
for(n in c(100, 1000)) {
  sel = order(rv325, decreasing = TRUE)[seq_len(n)]
  KOmean = rowMeans(exprs(x325)[sel, x325$genotype=="FGF4-KO"])
  dists = colSums((exprs(x325)[sel, wt325] - KOmean)^2)^0.5

  pdf(file = sprintf("Hiiragi2013-figNumberOfCells-%d.pdf", n), width = 5, height = 10)
  par(mfrow = c(2,1))
  plot(x325$Total.number.of.cells[wt325], dists, pch = 16, main = "",
       xlab = "Total number of cells", ylab = "Distance to FGF4-KO")
  MDSplot(x325[sel, ], pointlabel = ifelse(x325$genotype=="WT",
                                           paste(x325$Total.number.of.cells), "KO"), cex = 1)
  dev.off()
}
@
%
\incfig{figNumberOfCells-100}{.66\textwidth}{Distance of the E3.25 WT samples to the mean profile of FGF4-KO.}
  {The 100 features with highest overall variance were used.}
\incfig{figNumberOfCells-1000}{.66\textwidth}{Distance of the E3.25 WT samples to the mean profile of FGF4-KO.}
  {The 1000 features with highest overall variance were used.}

%--------------------------------------------------
\subsection{Heatmap of E3.25 WT and E3.25 FGF-KO samples}
%--------------------------------------------------
For data visualisation, we produce a heatmap (Figure~\ref{figHeatmapKO}) that
shows the data from the following groups
<<selectedGroups>>=
selectedGroups = c("E3.25", "E3.25 (FGF4-KO)")
xKO = x[, safeSelect(selectedGroups)]
selectedFeatures = order(rowVars(exprs(xKO)), decreasing = TRUE)[seq_len(100)]
@
%
<<figHeatmapKO, fig=TRUE, width=12, height=16>>=
myHeatmap(xKO[selectedFeatures, ], collapseDuplicateFeatures = TRUE, haveColDend = TRUE)
@
<<checkClaim2, echo=FALSE>>=
stopifnot("Fgf4" %in% fData(xKO)$symbol[selectedFeatures])
@
%
\incfig{figHeatmapKO}{\textwidth}{Heatmap of all E3.25 WT and E3.25 FGF-KO samples.}{The
  \Sexpr{length(selectedFeatures)} features with highest overall
  variance were used. One of them shows Fgf4.}

%-----------------------------------------------------------------
\subsection{Differentially expressed genes between FGF4-KO and WT (PE, EPI) at E3.5}\label{sec:deWTFGF4KO}
%-----------------------------------------------------------------
Let us compute the differentially expressed genes between WT and FGF4-KO,
<<dewtko,cache=TRUE>>=
x35 = x[, safeSelect(c("E3.5 (FGF4-KO)", "E3.5 (EPI)", "E3.5 (PE)"))]
f1 = f2 = x35$sampleGroup
f1[f1=="E3.5 (PE)" ] = NA
f2[f2=="E3.5 (EPI)"] = NA
x35$EPI = factor(f1, levels = c("E3.5 (FGF4-KO)", "E3.5 (EPI)"))
x35$PE  = factor(f2, levels = c("E3.5 (FGF4-KO)", "E3.5 (PE)"))
de = list(`EPI` = rowttests(x35, "EPI"),
          `PE`  = rowttests(x35, "PE"))
for(i in seq(along = de))
  de[[i]]$p.adj = p.adjust(de[[i]]$p.value, method = "BH")
@
<<figIndepFiltDeWtKo, fig=TRUE, pdf=FALSE, png=TRUE, width=7.3, height=9>>=
par(mfcol = c(3,2))
rkv = rank(-rowVars(exprs(x35)))
fdrthresh = 0.05
fcthresh =  1
for(i in seq(along = de)) {
  hist(de[[i]]$p.value, breaks = 100, col = "lightblue", main = names(de)[i], xlab = "p")
  plot(rkv, -log10(de[[i]]$p.value), pch = 16, cex = .25, main = "",
       xlab = "rank of overall variance", ylab = expression(-log[10]~p))
  plot(de[[i]]$dm, -log10(de[[i]]$p.value), pch = 16, cex = .25, main = "",
      xlab = "average log fold change", ylab = expression(-log[10]~p))
  abline(v = c(-1,1)*fcthresh[i], col = "red")
}
@
%
\incfig{figIndepFiltDeWtKo}{\textwidth}{Differentially expressed genes between
  FGF4-KO and WT (\Sexpr{paste(names(de), collapse=", ")}) at E3.5.}{The upper panels show
  the histograms of $p$-values, middle panels the scatter plots of rank of overall
  variance (\Robject{rkv}) versus $-\log_{10} p$, lower panels $log_2$ fold-change versus
  $-\log_{10} p$ (``volcano plots''). The red lines indicate possibly useful fold change thresholds.}
%
The plots are shown in Figure~\ref{figIndepFiltDeWtKo}. In contrast to
Figure~\ref{figIndepFiltSetup}, this plot indicates no obvious choice of threshold
from overall-variance (i.~e., independent) filtering. In fact, there seem to be many
probes with apparently significant changes but very low average fold changes. These could
be caused by ``batch effects'', and we will apply the fold change threshold
\Robject{fcthresh} to remove these.
%
<<fig2DWTKO,fig=TRUE,png=TRUE,pdf=FALSE,width=6,height=6>>=
isSig = ((pmin(de$PE$p.adj, de$EPI$p.adj) < fdrthresh) &
         (pmax(abs(de$PE$dm), abs(de$EPI$dm)) > fcthresh))
table(isSig)
plot(de$PE$dm, de$EPI$dm, pch = 16, asp = 1,
     xlab = expression(log[2]~FGF4-KO / PE),
     ylab = expression(log[2]~FGF4-KO / EPI),
     cex  = ifelse(isSig, 0.6, 0.1), col = ifelse(isSig, "red", "grey"))
@ 
%
See Figure~\ref{fig2DWTKO}.
\incfig{fig2DWTKO}{0.6\textwidth}{Differentially expressed genes between
  FGF4-KO and WT (\Sexpr{paste(names(de), collapse=", ")}) at E3.5.}{Shown is a
  scatterplot of the average fold changes for both comparisons. Large values along the
  $x$-axis indicate a relatively higher level of expression in E3.5 (FGF4-KO) compared to
  E3.5 (PE), small (i.~e.\ negative) values indicate higher expression in E3.5 (PE);
  analogously for the $y$-axis.}
%
<<writetab35>>=
df = cbind(fData(x35)[isSig, ], `log2FC KO/PE` = de$PE$dm[isSig],
           `log2FC KO/EPI` = de$EPI$dm[isSig])
write.csv(df, file = "differentially_expressed_E3.5_vs_FGF4-KO.csv")
ctrls = grep("^AFFX", rownames(df), value = TRUE)
@ 
 
\subsubsection{The probes for FGF4}
Let us look at the data for the four probes annotated to FGF4.
<<figFGF4ProbesInKO, fig=TRUE, width=8, height=7>>=
par(mfrow = c(2,2))
for(p in which(FGF4probes))
  plot(exprs(x35)[p, ], type = "p", pch = 16, col = x35$sampleColour, 
       main = rownames(fData(x325))[p], ylab = expression(log[2]~expression)) 
<<checkClaim3>>=
stopifnot(sum(FGF4probes)==4)
@ 
%
See Figure~\ref{figFGF4ProbesInKO}.
\incfig{figFGF4ProbesInKO}{\textwidth}{The probes for FGF4.}{As expected, the signal
  from these probes in the KO samples is consistent with background.}

\subsubsection{Behaviour of the control probes} 
A notable problem (probably associated with the "batch effects" discussed above) is that
\Robject{length(ctrls)=}\Sexpr{length(ctrls)} control probes appear as significant in this
analysis. Let us plot the first 8 of them (see Figure~\ref{figCtrls}).
<<figCtrls, echo=FALSE>>=
stopifnot(length(ctrls)>=8)
<<figCtrls, fig=TRUE, width=8, height=10>>=
par(mfcol = c(4, 2))
for(p in ctrls[1:8])
  plot(exprs(x35)[p, ], type = "p", pch = 16, col = x35$sampleColour, 
       main = p, ylab = expression(log[2]~expression)) 
@ 
%
\incfig{figCtrls}{\textwidth}{Behaviour of some control probe sets.}{}

%--------------------------------------------------
\subsubsection{Gene set enrichment analysis}
%------------------------------------------------------------
%Fisher tests against KEGG pathways.
<<kegg,cache=TRUE,eval=FALSE,echo=FALSE>>=
allGenes = unique(fData(x35)$symbol)
sigGenes = unique(df$symbol)
ntot = length(allGenes)
n1   = length(sigGenes)
keggpw = as.list(mouse4302PATH2PROBE)
fts = lapply(keggpw, function(ps) {
   pwGenes = unique(as.character(mouse4302SYMBOL[ps]))
   ovlGenes = intersect(pwGenes, sigGenes)
   n2  = length(pwGenes)
   nov = length(ovlGenes)
   mat = matrix(c(ntot-n1-n2+nov, n2-nov, n1-nov, nov), ncol = 2, nrow = 2,
                dimnames = list(`pathway` = c("no","yes"), `diff. expressed` = c("no","yes")))
   ft = fisher.test(mat)
   list(mat = mat, p.value = ft$p.value, estimate = ft$estimate, 
        genes = paste(sort(ovlGenes), collapse = ", ")) 
   })
@ 
Kolmogorov-Smirnov tests against KEGG pathways.
<<kegg,cache=TRUE>>=
keggpw = as.list(mouse4302PATH2PROBE)
dat  = de$PE$dm+de$EPI$dm
fts = lapply(keggpw, function(ps) {
  inpw = rownames(fData(x35)) %in% ps
  ft   = ks.test( dat[inpw], dat[!inpw] )
  list(p.value = ft$p.value, statistic = ft$statistic, n = sum(inpw)) 
})
@ 
%
First, let us select some prominent signalling pathways.
<<pws1>>=
pws = c("04010", "04070", "04150", "04310", "04330", "04340", "04350", "04370", "04630")
@ 
%
In addition, let us add those pathways with the strongest enrichment signal: 
% less than 40 genes, $p$-value less than 0.01 and odds ratio greater than 2.
% Table~\ref{tab_KEGG}, which we compute below, indicates that these pathways primarily
% represent various parts of metabolism.
<<pws2,eval=FALSE,echo=FALSE>>=
pws = c(pws, names(fts)[ sapply(fts, function(x) 
    with(x, (sum(mat[2, ])<=40) && (p.value<0.01) && (estimate>=2))) ] )
@ 
<<pws2>>=
pws = unique( c(pws, names(fts)[ sapply(fts, function(x) 
    with(x, (n<=100) && (p.value<0.01))) ] ) )
@ 
%
Make an online query at the KEGG database.
%
<<pwInfo,cache=TRUE>>=
query  = paste0("mmu", pws)
query  = split(query, seq(along = query) %/% 10)
pwInfo = unlist(lapply(query, keggGet), recursive = FALSE)
@ 
<<assertion,echo=FALSE>>=
stopifnot(all(pws %in% names(fts)))
stopifnot(length(pwInfo)==length(pws))
<<pwImgs,cache=TRUE,eval=FALSE,echo=FALSE>>=
pwImgs = lapply(query, keggGet, option = "image")
stopifnot(length(pwImgs)==length(pws))
@ 
<<xtable,eval=FALSE,echo=FALSE>>=
df = data.frame(
  `id`   = pws,
  `name` = sub(" - Mus musculus (mouse)", "", sapply(pwInfo, `[[`, "NAME"), fixed = TRUE),
  `n`    = sapply(pws, function(x) sum(fts[[x]]$mat[2, ])),
  `differentially expressed genes` = sapply(pws, function(x) fts[[x]]$genes),
  `p`          = as.character(signif(sapply(pws, function(x) fts[[x]]$p.value),  2)),
  `odds-ratio` = as.character(signif(sapply(pws, function(x) fts[[x]]$estimate), 2)),
  stringsAsFactors = FALSE, check.names = FALSE)
<<xtable>>=
df = data.frame(
  `id`   = pws,
  `name` = sub(" - Mus musculus (mouse)", "", sapply(pwInfo, `[[`, "NAME"), fixed = TRUE),
  `n`         =                     sapply(pws, function(x) fts[[x]]$n),
  `p`         = as.character(signif(sapply(pws, function(x) fts[[x]]$p.value),  2)),
  `statistic` = as.character(signif(sapply(pws, function(x) fts[[x]]$statistic), 2)),
  stringsAsFactors = FALSE, check.names = FALSE)
<<printxtable>>=
print(xtable(df,
  caption = paste("Gene set enrichment analysis of selected KEGG pathways, for the",
                   "differentially expressed genes between E3.5 FGF-4 KO and WT samples.",
                   "n: number of microarray features annotated to genes in the pathway."),
  label = "tab_KEGG", align = c("l", "l","p{4cm}","r","r","r")),
  type = "latex", file = "tab_KEGG.tex")
@
\input{tab_KEGG.tex}
Table~\ref{tab_KEGG} shows 
\begin{enumerate}
\item first, the data for the manually selected signalling pathways,
\item then, the pathways with the most prominent changes.
\end{enumerate}
We visualize their data in Figure~\ref{figShowPws}, see code below.
<<figShowPws,fig=TRUE,png=TRUE,pdf=FALSE,width=16,height=20>>=
par(mfrow = c(7,4)); stopifnot(length(pws)<=42)
for(i in seq(along = pws)) {
  inpw = factor(ifelse(rownames(fData(x35)) %in% keggpw[[pws[i]]],"in pathway","outside"))
  ord  = order(inpw)
  enr  = paste0("p=", df$p[i], if(as.numeric(df$p[i])<0.05)
                                  paste(" D=", df$statistic[i]) else "")
  multiecdf( (de$PE$dm+de$EPI$dm) ~ inpw, xlim = c(-1,1)*3,
             main = paste(df$name[i], enr, sep = "\n"),
             xlab = "mean difference between FGF4-KO and wildtype samples" )
}
@ 
\incfig{figShowPws}{\textwidth}{Differentially expressed genes between
  FGF4-KO and WT (\Sexpr{paste(names(de), collapse=", ")}) at E3.5.}{}


%--------------------------------------------------
\clearpage
\section{Jensen-Shannon Divergence analysis}
\label{sec:JSD}
%--------------------------------------------------
In the following, we will explore whether and how we can detect changes in the degree of
"lack of correlation" ("heterogeneity") between cells. \cite{Buganim:2012} considered a
measure of Jensen-Shannon Divergence
(Figures 2C, D in their paper) with a similar idea in mind. 
From their Supplement:
\begin{quote}
  JSD was calculated to assess within-group similarity of gene
  expression within each cell line according to Lin (1991). Expression
  values of genes were transformed so that they sum up to 1 in each
  cell. Each cell $i$ is thus represented as a vector of probabilities
  $P_i$. Cells from the same line were grouped together and for each
  group, the JSD was calculated from the probability vectors
  $(P_1,P_2,\ldots,P_n)$ of cells in each group.
\begin{equation}
  JSD(P_1,\ldots,P_n)=H\left(\frac{1}{n}\sum_{i=1}^n P_i\right)-\frac{1}{n}\sum_{i=1}^n H(P_i)
\end{equation}
where $H(P)$ is the Shannon entropy given by
\begin{equation}
  H(P)=-\sum_{i=1}^k P(x_i)\log_2P(x_i).
\end{equation}
Confidence intervals (CIs) were estimated by bootstrapping (sampling with replacement). The 95\% CIs were shown as error bars.
\end{quote}

Let's define R functions for this.
<<defJSD>>=
H = function(p) -sum(p*log2(p))
JSD = function(m, normalize = TRUE) {
  stopifnot(is.matrix(m), all(dim(m)>1))
  if(normalize)
    m = m/rowSums(m)
  H(colMeans(m)) - mean(apply(m, 2, H))
}
@
%
For comparison, we consider below also the ordinary within-group standard deviation:
%
<<defSDV>>=
SDV = function(m) sqrt(mean(rowVars(m)))
@
%
This measure of divergence computes for each gene (feature on the
array) the variance across arrays, then computes the average of these
and takes the square root.

Since the majority of the \Sexpr{nrow(x)} features on the array are
dominated ``noise'' and/or potential technical drifts, we want to use
only the most highly variable (i.~e.\ most informative) features,
%
<<numberFeatures>>=
numberFeatures = c(50, 200, 1000, 4000)
@
%
as selected by overall variance. In the following, we
will use the function \Rfunction{computeDivergences} to compute a
measure of divergence, such as \Rfunction{JSD}, for each of the groups
defined by \Robject{strata} and for the different choices of \Robject{numberFeatures}.
%
<<computeDivergences>>=
computeDivergences = function(y, indices, fun) {
  y = y[indices, ]
  exprVals = y[, -1]
  strata = y[, 1]
  numStrata = max(strata)
  stopifnot(setequal(strata, seq_len(numStrata)))

  orderedFeatures = order(rowVars(t(exprVals)), decreasing = TRUE)
  sapply(numberFeatures, function(n) {
    selFeat = orderedFeatures[seq_len(n)]
    sapply(seq_len(numStrata), function(s)
             fun(exprVals[strata==s, selFeat, drop = FALSE]))
  })
}
<<strata>>=
x$sampleGroup = factor(x$sampleGroup)
x$strata = as.numeric(x$sampleGroup)
<<check,echo=FALSE>>=
stopifnot(!any(is.na(x$strata)))
@

Now we are ready to go, we use the bootstrap to assess variability in our divergene estimates:
<<doJSD,cache=TRUE>>=
bootJSD = boot(data = cbind(x$strata, t(exprs(x))),
               statistic = function(...) computeDivergences(..., fun = JSD),
               R = 100, strata = x$strata)
dim(bootJSD$t) = c(dim(bootJSD$t)[1], dim(bootJSD$t0))
<<doSD,cache=TRUE>>=
bootSDV = boot(data = cbind(x$strata, t(exprs(x))),
               statistic = function(...) computeDivergences(..., fun = SDV),
               R = 100, strata = x$strata)
dim(bootSDV$t) = c(dim(bootSDV$t)[1], dim(bootSDV$t0))
@
%
The resulting values (mean and distribution indicated by boxplot) are shown in Figure~\ref{fig-JSD}.
%
<<fig-JSD,fig=TRUE,width=10,height=14>>=
par(mfrow = c(length(numberFeatures),2))
colores = sampleColourMap[levels(x$sampleGroup)]
for(i in seq(along = numberFeatures)) {
  for(what in c("JSD", "SDV")){
    obj = get(paste("boot", what, sep = ""))
    boxplot(obj$t[, , i], main = sprintf("numberFeatures=%d", numberFeatures[i]),
            col = colores, border = colores, ylab = what, xaxtb = "n")
    px = seq_len(ncol(obj$t))
    text(x = px, y = par("usr")[3], labels = levels(x$sampleGroup),
         xpd = NA, srt = 45, adj = c(1,0.5), col = colores)
    points(px, obj$t0[, i], pch = 16)
  }
} 
@
%
\incfig{fig-JSD}{\textwidth}{Jensen-Shannon divergences.}{JSD is
  shown on the left; on the right, for comparison we performed the same
  analysis with \emph{within-group standard deviation} as another
  measure of group spread.}

%--------------------------------------------------
\clearpage
\section{Classification of temporal profiles}
\label{sec:temp}
%--------------------------------------------------
\begin{figure}[tbp]
  \begin{center}
  \includegraphics[width=\textwidth]{FigTemporalChangeLineageMarkerExp}
  \caption[Temporal change of the lineage marker expression]{\label{FigTemporalChangeLineageMarkerExp}%
\textbf{Temporal change of the lineage marker expression} Example figure provided by Takashi in Email of 9 Oct 2012.  
  Red genes are known lineage markers.
% we wanted to place novel markers into this context of sequential activation of different markers.
% \cite{Pina:2012} did similar with the whole transcriptome (Figure 2b). 
% Can we also classify from our microarry data the potential markers into these classes, have something 
% like Figure 2b and identify those genes included in each class?
}
  \end{center}
\end{figure}

\subsection{Comparison of microarray data with qPCR results}
First, let us plot the microarray data for all probesets annotated to the genes shown in
Figure~\ref{FigTemporalChangeLineageMarkerExp}. We write the output into an extra PDF file,
\texttt{exemplaryGenes.pdf}.
%
<<xs>>=
xs = x[, pData(x)$sampleGroup %in%
   c("E3.25", "E3.5 (PE)", "E4.5 (PE)", "E3.5 (EPI)", "E4.5 (EPI)")]
xs$sampleGroup = factor(xs$sampleGroup)
@
<<myBoxplot>>=
myBoxplot = function(ps) {
  fac = factor(pData(xs)$sampleGroup)
  boxplot(exprs(xs)[ps, ] ~ fac, col = sampleColourMap[levels(fac)], lim = c(2,14),
          main = sprintf("%s (%s)", fData(x)[ps, "symbol"], ps), show.names = FALSE)
  text(seq(along = levels(fac)), par("usr")[3] - diff(par("usr")[3:4])*0.02,
       levels(fac), xpd = NA, srt = 90, adj = c(1,0.5), cex = 0.8)
}
<<showGenes,results=hide>>=
exemplaryGenes = read.table(header = TRUE, stringsAsFactors = FALSE, text = "
symbol   thclass  probeset
Gata6      C-1 1425464_at
Tom1l1     C-1 1439337_at
Sox2       C-1 1416967_at
Pdgfra    PE-2 1438946_at
Sox17     PE-2 1429177_x_at
P4ha2     PE-2 1417149_at
Gata4     PE-3 1418863_at
Aldh18a1  PE-3 1415836_at
Col4a1    PE-3 1452035_at
Col4a2    PE-3 1424051_at
Cubn      PE-3 1426990_at
Lamb1     PE-3 1424113_at
Dab2      PE-4 1423805_at
Lrp2      PE-4 1427133_s_at
Amn       PE-4 1417920_at
Fgf4     EPI-2 1420085_at
Nanog    EPI-2 1429388_at
Tdgf1    EPI-2 1450989_at
Cldn4    EPI-4 1418283_at
Enox1    EPI-4 1436799_at")

pdf(file = "exemplaryGenes.pdf", width = 8, height = 11)
par(mfrow = c(5,3))
for(i in seq_len(nrow(exemplaryGenes))) {
  wh = featureNames(x)[fData(x)[, "symbol"]==exemplaryGenes[i, "symbol"]]
  stopifnot(length(wh)>=1)
  sapply(wh, myBoxplot)
}
dev.off()
@
%
Based on these plots, we assigned one ``trustworthy'' probeset to each
gene, indicated in the above table.
%
<<fig-ExemplaryGenes,fig=TRUE,width=13,height=18,results=hide>>=
layout(rbind(c(1, 4,   7, 13),
             c(2, 5,   8, 14),
            c(21, 6,   9, 15),
             c(3, 16, 10, 19),
            c(22, 17, 11, 20),
            c(23, 18, 12, 24)))

xs = x
xs$sampleGroup = factor(xs$sampleGroup)
xs$sampleGroup = relevel(xs$sampleGroup, "E4.5 (PE)")
xs$sampleGroup = relevel(xs$sampleGroup, "E4.5 (EPI)")
xs$sampleGroup = relevel(xs$sampleGroup, "E3.5 (PE)")
xs$sampleGroup = relevel(xs$sampleGroup, "E3.5 (EPI)")
xs$sampleGroup = relevel(xs$sampleGroup, "E3.25")

for(i in seq_len(nrow(exemplaryGenes))) 
  myBoxplot(exemplaryGenes[i, "probeset"])
@
%
The output is shown in Figure~\ref{fig-ExemplaryGenes}, which we
can compare to Figure~\ref{FigTemporalChangeLineageMarkerExp}.
\incfig{fig-ExemplaryGenes}{\textwidth}{Boxplots of the
  microarray expression data for exemplary genes.}{Compare this to
  Figure~\ref{FigTemporalChangeLineageMarkerExp}. The agreement is
  satisfactory.}

%--------------------------------------------------
\subsection{Rule-based classification}
%--------------------------------------------------
For EPI and PE separately, we define 4 classes as indicated in
Figure~\ref{FigTemporalChangeLineageMarkerExp}:
%
\begin{itemize}
  \item class 1: highest in E3.25
  \item class 2: highest in E3.5 and off in the other lineage (at E3.5, E4.5)
  \item class 3: E3.25 < E3.5 < E4.5 and off in the other lineage (at E4.5)
  \item class 4: off in E3.25 and E3.5, on in E4.5, and off in the other lineage (at E4.5) 
\end{itemize}
%
To implement these rules, let us define the helper function \Rfunction{greater}, which
determines if a gene has an average log fold change larger than \Robject{thresh} between
each of the conditions in \Robject{cond1} versus each of the conditions in
\Robject{cond2}.
%
<<avglfc>>=
greater = function(cond1, cond2, thresh = 2) {
  stopifnot(all(cond1 %in% names(groups)), all(cond2 %in% names(groups)))
  rm1 = lapply(groups[cond1], function(j) rowMeans(exprs(x)[, j]))
  rm2 = lapply(groups[cond2], function(j) rowMeans(exprs(x)[, j]))
  res = rep(TRUE, nrow(x))
  for(v1 in rm1) for(v2 in rm2) 
      res = res & ((v1-v2) > thresh)
  return(res)
}
@
%
Next, we compute \Robject{xsc}, a version of the data matrix \Robject{x} that has been
scaled to the range $[0,1]$, such that \emph{for each gene}, the lowest value across
arrays corresponds to 0, and the highest value to 1. (In the code below, we actually use
0.025\% and 97.5\% quantiles instead of lowest and highest values, that is just to ensure
some statistical robustness.)
%
<<minLevel,cache=TRUE>>=
xquantiles = apply(exprs(x), 1, quantile, probs = c(0.025, 0.975))
minLevel = xquantiles[1, ]
maxLevel = xquantiles[2, ]
trsf2Zero2One = function(x) {
  x = (x-minLevel)/(maxLevel-minLevel)
  x[x<0] = 0
  x[x>1] = 1
  return(x)
}
xsc = x
exprs(xsc) = trsf2Zero2One(exprs(x))
@ 
%
We will use \Robject{xsc} for the heatmap (Figure~\ref{figHeatmapthclasses}).
The function \Rfunction{isOff} determines whether a gene is off by
checking whether more than 90\% of the arrays in the condition(s) specified by
\Robject{cond} have values below \Robject{minLevel} + \Robject{thresh}.
%
<<isOff>>=
isOff = function(cond, thresh = 2) {
  stopifnot(all(cond %in% names(groups)))
  samps = unlist(groups[cond])
  rowSums(exprs(x)[, samps] > minLevel+thresh) <= ceiling(length(samps) * 0.1)
}
@ 
%
Now we can do the classification.
%
<<classification>>=
  `C-1` =  greater("E3.25", c("E3.5 (EPI)", "E4.5 (EPI)", "E3.5 (PE)", "E4.5 (PE)"))
`EPI-2` =  greater("E3.5 (EPI)", c("E3.25")) &
           greater("E3.5 (EPI)", "E4.5 (EPI)", thresh = 0.5) &
           isOff("E3.5 (PE)")  & isOff("E4.5 (PE)" )
 `PE-2` =  greater("E3.5 (PE)",  c("E3.25")) &
           greater("E3.5 (PE)",  "E4.5 (PE)", thresh = 0.5)  & 
           isOff("E3.5 (EPI)") & isOff("E4.5 (EPI)")
`EPI-4` = (greater("E4.5 (EPI)", c("E3.25", "E3.5 (EPI)")) &
           isOff(c("E3.25", "E3.5 (EPI)")) & 
           isOff("E4.5 (PE)" ))
 `PE-4` = (greater("E4.5 (PE)", c("E3.25", "E3.5 (PE)")) &
           isOff(c("E3.25", "E3.5 (PE)")) & 
           isOff("E4.5 (EPI)"))
`EPI-3` = (greater("E3.5 (EPI)", "E3.25", thresh = 0.5) &
           greater("E4.5 (EPI)", "E3.5 (EPI)", thresh = 0.5) &
           greater("E4.5 (EPI)", "E3.25", thresh = 3)   & 
           isOff("E4.5 (PE)") & !`EPI-4`)
 `PE-3` = (greater("E3.5 (PE)",  "E3.25", thresh = 0.5) &
           greater("E4.5 (PE)",  "E3.5 (PE)",  thresh = 0.5) &
           greater("E4.5 (PE)",  "E3.25", thresh = 3) & 
           isOff("E4.5 (EPI)") & !`PE-4`)
thclasses = cbind(`C-1`, `EPI-2`, `EPI-3`, `EPI-4`, `PE-2`,  `PE-3`,  `PE-4`)
@ 
We want each feature to be in exactly one group:
<<onegroup>>=
multiclass = thclasses[rowSums(thclasses)>1,, drop = FALSE]
stopifnot(nrow(multiclass)==0)
@ 
The group sizes:
<<cs>>=
colSums(thclasses)
@
<<figHeatmapthclasses, fig=TRUE, width=28, height=66>>=
agr = groups[c("E3.25", "E3.5 (EPI)", "E4.5 (EPI)", "E3.5 (PE)", "E4.5 (PE)")]
fgr = apply(thclasses, 2, which)
xsub = xsc[unlist(fgr), unlist(agr)]
mat = exprs(xsub)
rownames(mat) = fData(xsub)[, "symbol"]
myHeatmap2(mat, keeprownames = TRUE,
           rowGroups = factor(rep(seq(along = fgr), listLen(fgr))),
           colGroups = factor(rep(seq(along = agr), listLen(agr))))
@ 
%
\incfig{figHeatmapthclasses}{9.5cm}{Heatmap of all classes}{of
  the [0,1]-scaled data matrix \Robject{xsc}. From left
  to right, the \Sexpr{length(agr)} blocks correspond to the
  conditions \Sexpr{paste(names(agr), collapse=', ')}. From bottom to top, 
  the \Sexpr{length(fgr)} blocks correspond to the feature
  (gene) classes \Sexpr{paste(names(fgr), collapse=', ')}. Within
  classes, genes are sorted by dendrogram clustering.}
%
The heatmap is shown in Figure~\ref{figHeatmapthclasses}. 

\subsubsection{Table export}

<<writecsv>>=
out = do.call(rbind, lapply(seq(along = fgr), function(i) 
    cbind(`class` = names(fgr)[i], fData(xsc)[fgr[[i]], ])))
write.csv(out, file = "featureclassification.csv")
@ 

\subsubsection{Comparison with manual classification} 

Figure 2b in the paper shows qPCR data for these seven genes:
%
<<fig2bGenes>>=
fig2bGenes = c("Aldh18a1", "Col4a1", "Cubn", "Foxq1", "Gata4", "Lamb1", "Serpinh1")
@
The genes classified as PE-3 in the above microarray data rule-based classification are
<<rbGenes>>=
rbGenes = sort(unique(fData(x)$symbol[thclasses[, "PE-3"]]))
rbGenes
@
Difference:
<<setdiff>>=
setdiff(fig2bGenes, rbGenes)
@


%--------------------------------------------------
\clearpage
\section{qPCR data analysis}
%--------------------------------------------------
\subsection{Heatmaps for all data in \Robject{xq}}
%--------------------------------------------------
We first load the data
<<load>>=
data("xq")
@
%
<<hmprep>>=
xq$sampleColours = sampleColourMap[sub("[[:digit:]]+ ", "", sampleNames(xq))]
stopifnot(!any(is.na(xq$sampleColours)))
@ 
%
Next, we define a function \Rfunction{myHeatmap3} to create heatmaps
<<myHeatmap3>>=
myHeatmap3 = function(x, log = TRUE, 
  col = colorRampPalette(brewer.pal(9, "Blues"))(ncol), ncol = 100, ...) {
  mat = exprs(x)
  if(log){
    mat[mat<1] = 1
    mat = log10(mat)
  }
  rownames(mat) = fData(x)[, "symbol"]
  heatmap.2(mat, trace = "none", dendrogram = "none", scale = "none", col = col,
            keysize = 0.9, ColSideColors = x$sampleColour, margins = c(5,7), ...)
}
@ 
and use it to visualize data in \Robject{xq}.
<<fighm1,width=14,height=10,fig=TRUE>>=
myHeatmap3(xq)
@ 
%
See Figure~\ref{fighm1}.
\incfig{fighm1}{\textwidth}{Heatmap of the qPCR data set.}{
  The data are shown on the logarithmic (base 10) scale. Before transformation, values $<1$ were set to 1.
  The colour code of the bar at the top is as in Figure~\ref{figProjection}.}


\subsection{Heatmaps for the seven selected genes and selected samples}
<<fighm2,width=14,height=7,fig=TRUE>>=
selectedGenes = c("Col4a1", "Lamab1", "Cubn", "Gata4", "Serpinh1", "Foxq1", "Aldh18a1")
myHeatmap3(xq[selectedGenes, ])
<<checkAssertion,echo=FALSE>>=
stopifnot(length(selectedGenes)==7)
@ 
%
See Figure~\ref{fighm2}.
\incfig{fighm2}{\textwidth}{Heatmap of the qPCR data for the 7 selected genes.}{}
 
<<selectedPE>>=
selectedSamples = (xq$Cell.type %in% c("ICM", "PE"))
table(xq$sampleGroup[selectedSamples])
sxq = xq[selectedGenes, selectedSamples]
groups = c("E3.25", "E3.5 (PE)", "E4.5 (PE)")
sxq$fsampleGroup = factor(sxq$sampleGroup, levels = groups)
stopifnot(!any(is.na(sxq$fsampleGroup)))
@ 
<<fighm3,width=14,height=7,fig=TRUE>>=
myHeatmap3(sxq)
@ 
See Figure~\ref{fighm3}.
\incfig{fighm3}{\textwidth}{Heatmap for the 7 selected genes and the
  E3.25/E3.5/E4.5 PE samples.}{Rows and columns are ordered by clustering dendrogram.}

\subsection{Distribution of the data and discretisation}\label{sec:disc}
\incfig{figthr}{\textwidth}{Visualisation of the qPCR data for the seven
  genes.}{Shown are also the within-group means (horizontal dotted light grey lines) and the
  discretisation thresholds (horizontal solid dark grey lines).}
The qPCR expression levels in \Robject{sxq} are compared, separately for each gene,
against the midpoint of the means of successive groups (\Sexpr{paste(groups, collapse = ', ')}).
This is illustrated in Figure~\ref{figthr}.
%
<<discr,cache=TRUE>>=
groupmedians = t(apply(exprs(sxq), 1, function(v) tapply(v, sxq$sampleGroup, median)))
stopifnot(identical(colnames(groupmedians), groups), length(groups)==3)
discrthreshs = (groupmedians[, 2:3] + groupmedians[, 1:2]) / 2
stst = sapply(split(seq(along = sxq$sampleGroup), sxq$sampleGroup), 
    function(v) {i1 = head(v,1); i2 = tail(v,1); stopifnot(identical(v, i1:i2)); c(i1,i2)})
<<show>>=
groupmedians
discrthreshs
<<figthr,width=6,height=7,fig=TRUE>>=
op = par(mfrow = c(nrow(sxq), 1), mai = c(0.1, 0.7, 0.01, 0.01))
for(j in seq_len(nrow(sxq))) {
  plot(exprs(sxq)[j, ], type = "n", xaxt = "n", ylab = fData(sxq)$symbol[j])
  segments(x0 = stst[1, ], x1 = stst[2, ], y0 = groupmedians[j, ], col = "#808080", lty=3)
  segments(x0 = stst[1,1:2], x1 = stst[2,2:3], y0 = discrthreshs[j, ], col = "#404040")
  points(exprs(sxq)[j, ], pch = 16, col = sxq$sampleColour)
}
par(op)
@ 
%

In this way, the data are assigned to three discrete levels: 
\begin{enumerate} 
  \item less than the midpoint of the means of \Sexpr{groups[1]} and 
        \Sexpr{groups[2]},
  \item in between 1.~and 3.,
   \item higher than the midpoint of the means of \Sexpr{groups[2]} and 
        \Sexpr{groups[3]},
\end{enumerate} 
The result of this is shown in Figure~\ref{fighmd}, where these three levels
are represented by white, light blue, dark blue.

<<discretisation1>>=
discretize = function(x) {
  exprs(x) = t(sapply(seq_len(nrow(exprs(x))), function(r) {
      as.integer(cut(exprs(x)[r, ], breaks = c(-Inf, discrthreshs[r, ],+Inf)))
  }))
  return(x)
}
<<discretisation2,cache=FALSE>>=
sxqd = discretize(sxq)
<<fighmd,width=12,height=5,fig=TRUE>>=
myHeatmap3(sxqd, log = FALSE, Colv = FALSE, Rowv = FALSE, key = FALSE, ncol = 3)
@ 
%
\incfig{fighmd}{\textwidth}{Heatmap for the discretized values.}{Rows and
  colums are in the original order.}

\subsection{Temporal order - hierarchy}
\begin{figure}[htbp]
\input{fighmd.tex}
\begin{center}
\caption[Heatmaps, sorted]{\label{fighmd-mult}Heatmaps, ordered to show
  potential hierarchy of successive activation. This was done separately for the E3.25 --
  E3.5 (PE) and the E3.5 (PE) -- E4.5 (PE) comparisons (as indicated by the horizontal
  colour bar). Note: if multiple equally optimally solutions were found, all are displayed.}          
\end{center}
\end{figure}
Now we look for a (putatively: temporal) ordering of genes, so that the number of cases
(samples $\times$ genes) in which a lower level follows a higher one is minimised. More
specifically, the function \Rfunction{costfun1} below adds a penalty of 1 for every
instance that in a sample, white follows \emph{after} blue in an E3.5 sample (for the
E3.25$\to$E3.5 transition), and \Rfunction{costfun2} adds a penalty of 1 when white/light
blue follow \emph{after} dark blue in an E4.5 sample (for E3.5$\to$E4.5) .
%
<<bruteForceOptimisation1,cache=TRUE>>=
stopifnot(all(exprs(sxqd)%in%(1:3)))

costfun1 = function(x, sg) {
  k = (sg=="E3.5 (PE)")
  mean( (x[-1,k]==1) & (x[-nrow(x),k]>1 ) )
}
costfun2 = function(x, sg) {
  k = (sg=="E4.5 (PE)")
  mean( (x[-1,k]<3 ) & (x[-nrow(x),k]==3) )
}

perms = permutations(nrow(sxq), nrow(sxq))
bruteForceOptimisation = function(fun, samps) {
  if (missing(samps)) samps = rep(TRUE, ncol(sxqd))
  apply(perms, 1, function(i) fun(exprs(sxqd)[i, samps], sxq$sampleGroup[samps]))
}

costs = list(
 `E3.25 -> E3.5` = bruteForceOptimisation(costfun1),
  `E3.5 -> E4.5` = bruteForceOptimisation(costfun2))
whopt  = lapply(costs, function(v) which(v==min(v)))
@ 

<<fighmd>>=
outf = file("fighmd.tex", open="w")
for(i in seq(along = costs)) {
  sxqdsub = switch(i,
     { 
       rv = sxqd[, sxqd$sampleGroup %in% groups[2] ]
       exprs(rv)[ exprs(rv)>2 ] = 2
       rv
     }, {
       rv = sxqd[, sxqd$sampleGroup %in% groups[3] ]
       exprs(rv)[ exprs(rv)<2 ] = 2
       exprs(rv) = exprs(rv)-1
       rv
     })
  columnOrder = order(sxqdsub$sampleGroup, colSums(exprs(sxqdsub)))
  for(w in whopt[[i]]) {
    fn = sprintf("fighmd-%d-%d.pdf", i, w)
    pdf(fn, width = 7, height = 3)
    myHeatmap3(sxqdsub[perms[w, ], columnOrder], log = FALSE, Colv = FALSE, Rowv = FALSE,
               key = FALSE, ncol = 2, breaks = seq(0.5, 2.5, by = 1))
    dev.off()
    cat("\\includegraphics[width=0.49\\textwidth]{", fn, "}\n", file = outf, sep = "")
  }
}
close(outf)
@ 
<<printcosts>>=
for(i in seq(along = costs)) {
  k = unique(costs[[i]][whopt[[i]]])
  stopifnot(length(k)==1)
  cat(sprintf("%14s: cost %g\n", names(costs)[[i]], k))
}
@ 
%
See Figure~\ref{fighmd-mult}.

%------------------------------------------------------------
\subsubsection{How significant is this?}
%------------------------------------------------------------
Repeat the above with bootstrapping.
<<boot-takes-a-long-time,cache=TRUE>>=
bopt = lapply(list(costfun1, costfun2), function(fun) {
  boot(data = seq_len(ncol(sxqd)), 
       statistic = function(dummy, idx) min(bruteForceOptimisation(fun, idx)),
       R = 250) 
})
@ 
% Note: 'length(idx)' above will always be equal to 'ncol(x)', since 'boot' samples with duplicates


<<figboot,width=7,height=5,fig=TRUE>>=
names(bopt) = names(costs)
lapply(bopt, `[[`, "t0")
multiecdf(lapply(bopt, `[[`, "t"))
t.test(x = bopt[[1]]$t, y = bopt[[2]]$t)
@ 
%
See Figure~\ref{figboot}.
\incfig{figboot}{.6\textwidth}{Distribution of bootstrap-resampled optimal
  costs ("disorder penalties").}{Shown are the empirical distribution function of the
  bootstrap-sampled cost functions for the two situations. The values for
  \Sexpr{names(costs)[2]} are significantly smaller (see also $t$-test result in the main text).}

\appendix

\clearpage
\section{Influence of cell position on gene expression}

To examine how cell position within the embryo influences its gene expression, cells lying on the surface of the ICM facing the blastocyst cavity were fluorescently labelled and expression profiled versus those located deeper within the ICM. We then used multi-dimensional scaling to compare the labelled and non-labelled cells at E3.5 and E4.5, based on the expression of 10 highly variable genes, as identified from the E3.5 and 4.5 microarray data, and quantified by single-cell qPCR measurements. The result of this analysis is show in Figure~\ref{mdsplot}.

<<mdsplot,width=5,height=5,fig=TRUE,results=hide>>=
data("xql")

labeledSampleColourMap = c(brewer.pal(5, "RdGy")[c(1,2)], brewer.pal(5, "BrBG")[c(4,5)])
names(labeledSampleColourMap) = c("E4.5_high", "E3.5_high", "E3.5_low",  "E4.5_low")
labeledGroups = with(pData(xql), list(
  `E3.5_high` = which(Label=="High" & Embryonic.day=="E3.5"),
  `E3.5_low`  = which(Label=="Low"  & Embryonic.day=="E3.5"),
  `E4.5_high`	= which(Label=="High" & Embryonic.day=="E4.5"),
  `E4.5_low`  = which(Label=="Low"  & Embryonic.day=="E4.5")))

labeledSampleColours = rep(NA_character_, ncol(xql))
for(i in seq(along = labeledGroups))
 labeledSampleColours[labeledGroups[[i]]] = labeledSampleColourMap[names(labeledGroups)[i]]
xql$sampleColours = labeledSampleColours

md = isoMDS( dist(t(log(exprs(xql),2))))$points
plot(md, col = xql$sampleColours, pch = 16, asp = 1, xlab = "",ylab = "")
@
\incfig{mdsplot}{.6\textwidth}{Multi-dimensional scaling plot.}{Light red and light green dots represent E3.5 labeled and unlabeled cells, while red and green ones represent E4.5 labeled and unlabeled cells, respectively.}

\section{Correlation between Fgf ligands and Fgf receptors}

Several Fgf ligands (Fgf3, 4 and 13) and all Fgf receptors (Fgfr1-4) were found to be differentially expressed within the ICM, thus possibly contributing to the EPI versus PrE lineage segregation. We have found that a statistically significant correlation (positive or negative) in gene expression levels is discernible at single cell level for Fgf4 against Fgfr2, Fgf4 against Fgfr3, Fgf3 with Fgfr3, and Fgf3 with Fgfr4 at E3.5 and E4.5 (Figure \ref{figure5b}).

<<figure5b,width=10,height=3,fig=TRUE>>=
xs = x[, (x$genotype %in% "WT") & (x$Embryonic.day %in% c("E3.25","E3.5","E4.5"))]

Fgf3 = c("1441914_x_at")
Fgf4 = c("1420086_x_at")
Fgfr2 = c("1433489_s_at")
Fgfr3 = c("1421841_at")
Fgfr4 = c("1418596_at")

correlationPlot = function(ID){
  xsl = xs[ID, ]
  mat = exprs(xsl)
  rownames(mat) = fData(xsl)[, "symbol"]
  plot(t(mat), pch = 16, asp = 1, cex = 1.25, cex.lab = 1.2, col = xs$sampleColour,
       xlim = c(2,12), ylim = c(3,11))
}

par(mfrow = c(1,4))
correlationPlot(c(Fgf4,Fgfr2))
correlationPlot(c(Fgf4,Fgfr3))
correlationPlot(c(Fgf3,Fgfr3))
correlationPlot(c(Fgf3,Fgfr4))
@ 
%
\incfig{figure5b}{\textwidth}{Correlation between Fgf ligands and Fgf receptors.}{Scatter plots with each dot representing the mRNA expression levels of specific Fgf ligand and receptor pairs in one blastomere. The colour code is as in Figure~\ref{figProjection}, with pink representing E3.25 cells, light blue and light green E3.5 EPI and PrE cells, and blue and green E4.5 EPI and PrE cells, respectively.}


%---------------------------------------------------------
% SessionInfo
%---------------------------------------------------------
\section{Session info}
Below, the output is shown
of \Rfunction{sessionInfo} on the system on which this document was compiled.
<<sessionInfo, results=tex, print=TRUE>>=
toLatex(sessionInfo())
@

%---------------------------------------------------------
% readdata.R
%---------------------------------------------------------
\section{The data import script \texttt{readdata.R}}\label{sec:readdata}
\lstinputlisting[language=R,
label=readdata,breaklines=true,caption={The script \texttt{readdata.R}}]{../inst/scripts/readdata.R}

%------------------------------------------------------------
\bibliographystyle{unsrt}
\bibliography{Hiiragi2013}
%------------------------------------------------------------

\end{document}