\documentclass[a4paper,10pt]{article}
\usepackage{longtable}
\usepackage[noams]{SIunits}
\setlongtables

\newcommand{\code}[1]{{\tt #1}}
\newcommand{\Rfunc}[1]{{\tt #1}}

\newcommand{\myincfig}[3]{%
  \begin{figure}[htbp]
    \begin{center}
      \includegraphics[width=#2]{#1}
      \caption{\label{#1}#3}
    \end{center}
  \end{figure}
}


\title{Microarray segmentation methods significantly influence data
precision}
\author{Ahmed Ahmed \and James D. Brenton}
\begin{document}
\maketitle



\section{Data description}

Expression microarrays containing 6528 pairs of duplicate cDNA spots
were used (Cancer Research UK DNA Microarray Facility at the Institute
of Cancer Research; CR-UK DMF Human 6.5k genome-wide array).  All
microarrays used were from the same printing batch.  Total RNA was
obtained from the cell line HCT116 and an isogenic daughter line with
a targeted disruption of the \emph{EP300} gene derived by homologous
recombination~(Iyer~\emph{et al}, submitted).  Total RNA was used for
reverse transcription and indirect labeling with Cy3 and Cy5 dyes
(Amersham) using random hexamers.  Measurements of the amount of
purified cDNA and Cy3/Cy5 incorporation were made before hybridization
using the Nanodrop ND-1000 spectrophotometer (Nanodrop Technologies,
Inc.).  Two sets of experiments (\emph{A} and \emph{B}) were carried
out, using 6 slides in each with a balanced dye-swap design (three
slides for each dye).  Experiment~\emph{A} and \emph{B} were identical
but used \unit{10}{\micro\gram} and \unit{15}{\micro\gram} total RNA
for labelling for each hybridization.  The names of the slides and the
dyes used for each slide are as follows:


\begin{longtable}{rrrr}

Experiment &  Slide ID & p300 knock-out & HCT116\\
\hline
\\
A&  3084 & Cy3 & Cy5\\

& 3089 & Cy3 & Cy5\\


&  3097 & Cy3 & Cy5\\

&  3092 & Cy5 & Cy3\\

&  3093 & Cy5 & Cy3\\

&  3069 & Cy5 & Cy3\\


\\

B&  3025 & Cy3 & Cy5\\

&  3033 & Cy3 & Cy5\\


&  3054 & Cy3 & Cy5\\

&  3034 & Cy5 & Cy3\\

&  3036 & Cy5 & Cy3\\

&  3037 & Cy5 & Cy3\\

\end{longtable}


Segmentation was performed using QuantArray (Perkin Elmer) and GenePix
Pro 4.1 (Axon Instruments, Inc.) software.  All three methods of
segmentation available within the QuantArray package were evaluated.
When using the segmentation software, channel 1 was always assigned to
cy3 regardless of the sample, knock-out or wild type, used.  The
suffixes A, F, H and G were used, following the slide ID, to name the
raw data files generated by the adaptive, fixed circle, histogram and
genepix segmentation methods respectively.

All statistical analysis was conducted using the R
environment~\cite{Ihaka1996} and the R package ``Statistics for
Microarray Analysis''~\cite{Dudoit2002}. We start the analysis by
startting the sma library then loading the data files into R.


<<>>=
library(sma)

for(i in 1:length(dir(pattern="....[AFHG]"))){
data <- read.table(dir(pattern="....[AFHG]")[i], sep='\t', header= TRUE)
assign(dir(pattern="....[AFHG]")[i], data)
}
@


We generate objects according to the experiment used, A or B, and the
segmentation method.  For experiment A we build \Rfunc{expA.A},
\Rfunc{expA.F}, \Rfunc{expA.H}, \Rfunc{expA.G} for adaptive,
fixedcircle, histogram and genepix respectively:

<<>>=
expA.A <- c( "3084A", "3089A", "3097A",
"3092A", "3093A", "3069A")
expA.F <- c( "3084F", "3089F", "3097F",
"3092F", "3093F", "3069F")
expA.H <- c( "3084H", "3089H", "3097H",
"3092H", "3093H", "3069H")
expA.G <- c( "3084G" , "3089G", "3097G", "3092G", "3093G",
"3069G" )
@

We do the same for experiment B:

<<>>=
expB.A <- c( "3025A", "3033A", "3054A",
"3034A", "3036A", "3037A")
expB.F <- c( "3025F", "3033F", "3054F",
"3034F", "3036F", "3037F")
expB.H <- c( "3025H", "3033H", "3054H",
"3034H", "3036H", "3037H")
expB.G <- c( "3025G", "3033G", "3054G", "3034G", "3036G",
"3037G")
@


Note that the first half of each of the vector of names represents
slides where the test in the experiment, in this case the knock-out
clone, was labeled by cy3 and the the control, in this case the wild
type clone, labeled by cy5.


\section{Statistical methods}

Log intensity ratios for each spot were obtained, as a measure of the
differential expression between the samples, with and without
background subtraction.  All spots from each microarray were included
in the analysis.  Data precision was initially assessed by using
correlation coefficients.

\subsection{The \Rfunc{get.cor} function}

To allow for iterating over the entire data set and to investigate the
effect of various normalization methods and background subtraction we
use a function \Rfunc{get.cor}.

\subsubsection{Description}


This function extracts the log2 ratios for all the slides used as well
as correlations between replicates within each slide and between
different slides.

\subsubsection{Usage}

get.cor(x, BG.subtract=TRUE, normalization)

\subsubsection{Arguments}


\begin{longtable}{rp{8cm}}



x: &  a vector the names of the raw data objects.\\\\
  
BG.subtract: &  a logical argument meaning;
`should the back ground values be subtracted?'. \\\\
 
normalization: & the
method of normalization to be used; `n' for no normalization, `s' for
scaled normalization, `m' for median normalization, `p' for print-tip
group lowess normalization and `l' for global `lowess` normalization.\\
\end{longtable}



\subsubsection{Details}

In the function we build a list containing the raw data R, red=Channel
2, Rb, background for channel 2, G ,green= Channel 1, and Gb,
background for channel 1. Note that, for consistency, we have always
put cy3, G, in Channel 1 regardless of the sample used.  Also note
that by not specifying the background data, Rb and Gb, no background
subtraction is performed.

We then specify the grid information for the images as follows;
nspot.r= the number rows in a block, nspot.c= the number of columns
within a block, ngrid.r= the number of rows of blocks and ngrid.c= the
number of columns of blocks.

We use the normalization function in the sma package.

We then correct the sign for the ratios for the first half of the
elements in (x) to account for the dye-swapping.


We assign the ratios of all the slides to an object called M. Note
that the number of columns in M represents the number of elements in
(x). In other words the number of slides tested.

We define index objects rep.a and rep.b for the first and second
replicates in each slide.

To obtain the within-slide correlations we compute the correlation
coefficients using the \Rfunc{cor} function in R.


For between-slidecorrelations we obtain the correlations for M using
the the \Rfunc{cor} function.


\subsubsection{Value}


\begin{longtable}{rp{8cm}}

within: &  a vector containing the within-slide correlations.\\\\

between:&  a vector containing the between-slidecorrelations.\\\\

ratios: &  a data frame containing the log2 ratios of knock-out over wild-type for each spot
in a slide. The number of columns correspond to the number of slides
and the number of rows to the total number of spots in a slide.\\

\end{longtable}


\subsubsection{Script}


<<>>=
get.cor <-  function(x, BG.subtract=TRUE, normalization){
  correlations <- list (within=NULL, between= NULL, ratios= NULL)
  res <- list(R = NULL, G = NULL, Rb = NULL, Gb = NULL)
    if (BG.subtract==TRUE)
      for(i in (1:length(x))){
        y<-get(x[i])
        res$R <- cbind(res$R,y$ch2.Intensity)
        res$G <- cbind(res$G,y$ch1.Intensity)
        res$Rb <- cbind(res$Rb,y$ch2.Background)
        res$Gb <- cbind(res$Gb,y$ch1.Background)
      }
      else for(i in (1:length(x))){
        y<-get(x[i])
        res$R <- cbind(res$R,y$ch2.Intensity)
        res$G <- cbind(res$G,y$ch1.Intensity)
      }
    
  

data.grid<-list(nspot.r=16,nspot.c=34,ngrid.r=12,ngrid.c=2)


data.ma<-stat.ma(res,data.grid,norm=normalization)


halfn= length(x)/2
data.ma$M[,c(1:halfn)]<- (data.ma$M[,c(1:halfn)])*-1


M<-data.ma$M


rep.a <- rep(c(TRUE,FALSE),12,each=544)
rep.b <- rep(c(FALSE,TRUE),12,each=544)





cors<-NULL


for(i in 1:ncol(M)){
  y<-cor(M[rep.a,i],M[rep.b,i],use='pairwise.complete.obs')
  cors<-c(cors,y)
  }


cors.between <- cor(M,use='pairwise.complete.obs')


upper.diagonal <- col(cors.between)>row(cors.between)
cors.between <- cors.between[upper.diagonal]

correlations$within <- c( correlations$within, cors)
correlations$between <- c( correlations$between, cors.between)
correlations$ratios <- cbind( correlations$ratio, M)
correlations
}
@

\subsection{Obtaining the data}

We then obtain the data for the different methods of segmentation for
experiment A:

First, with background subtraction and no normalization:
<<>>=
dataA.A.BGS.n <- get.cor ( expA.A, BG.subtract=TRUE, normalization='n')
dataA.F.BGS.n <- get.cor ( expA.F, BG.subtract=TRUE, normalization='n')
dataA.H.BGS.n <- get.cor ( expA.H, BG.subtract=TRUE, normalization='n')
dataB.A.BGS.n <- get.cor ( expB.A, BG.subtract=TRUE, normalization='n')
dataB.F.BGS.n <- get.cor ( expB.F, BG.subtract=TRUE, normalization='n')
dataB.H.BGS.n <- get.cor ( expB.H, BG.subtract=TRUE, normalization='n')
@
Then without background subtraction
<<>>=
dataA.A.NoBGS.n <- get.cor ( expA.A, BG.subtract=FALSE, normalization='n')
dataA.F.NoBGS.n <- get.cor ( expA.F, BG.subtract=FALSE, normalization='n')
dataA.H.NoBGS.n <- get.cor ( expA.H, BG.subtract=FALSE, normalization='n')
dataB.A.NoBGS.n <- get.cor ( expB.A, BG.subtract=FALSE, normalization='n')
dataB.F.NoBGS.n <- get.cor ( expB.F, BG.subtract=FALSE, normalization='n')
dataB.H.NoBGS.n <- get.cor ( expB.H, BG.subtract=FALSE, normalization='n')
@
Then we repeat all the process with scaled normalization:
<<>>=
dataA.A.BGS.s <- get.cor ( expA.A, BG.subtract=TRUE, normalization='s')
dataA.F.BGS.s <- get.cor ( expA.F, BG.subtract=TRUE, normalization='s')
dataA.H.BGS.s <- get.cor ( expA.H, BG.subtract=TRUE, normalization='s')
dataB.A.BGS.s <- get.cor ( expB.A, BG.subtract=TRUE, normalization='s')
dataB.F.BGS.s <- get.cor ( expB.F, BG.subtract=TRUE, normalization='s')
dataB.H.BGS.s <- get.cor ( expB.H, BG.subtract=TRUE, normalization='s')

dataA.A.NoBGS.s <- get.cor ( expA.A, BG.subtract=FALSE, normalization='s')
dataA.F.NoBGS.s <- get.cor ( expA.F, BG.subtract=FALSE, normalization='s')
dataA.H.NoBGS.s <- get.cor ( expA.H, BG.subtract=FALSE, normalization='s')
dataB.A.NoBGS.s <- get.cor ( expB.A, BG.subtract=FALSE, normalization='s')
dataB.F.NoBGS.s <- get.cor ( expB.F, BG.subtract=FALSE, normalization='s')
dataB.H.NoBGS.s <- get.cor ( expB.H, BG.subtract=FALSE, normalization='s')


@


\subsection{Extracting the correlations to data frames}


We then build a data frame of the results of within-slide correlations
and the identifiers as follows; the first column contains the
correlation values, the second contains the segmentation method and
the third contains the DNA content whether Low, in experiment A, or
High, in experiment B:
<<>>=
Results.within.BGS <- data.frame( c(dataA.A.BGS.n$within, dataA.F.BGS.n$within,
dataA.H.BGS.n$within, dataB.A.BGS.n$within, dataB.F.BGS.n$within,
dataB.H.BGS.n$within), factor(rep(c('A','F','H'), each=6, 2)),
factor(rep(c('Low', 'High'), each=18) ))

names(Results.within.BGS) <- c('Correlations', 'Method', 'DNA')
@

And with no background subtraction:
<<>>=
Results.within.NoBGS <- data.frame( c(dataA.A.NoBGS.n$within,
dataA.F.NoBGS.n$within, dataA.H.NoBGS.n$within, dataB.A.NoBGS.n$within,
dataB.F.NoBGS.n$within, dataB.H.NoBGS.n$within), factor(rep(c('A','F','H'),
each=6, 2)), factor(rep(c('Low', 'High'), each=18) ))

names(Results.within.NoBGS) <- c('Correlations', 'Method', 'DNA')
@



We do the same for between-slidecorrelations and we add a fourth
column to identify whether the dyes used for the two slide where the
same, S, or different, D. Note that for between-slidecomparisons we
used normalized data:
<<>>=
Results.between.BGS <- data.frame( c(dataA.A.BGS.s$between,
dataA.F.BGS.s$between, dataA.H.BGS.s$between, dataB.A.BGS.s$between,
dataB.F.BGS.s$between, dataB.H.BGS.s$between), factor(rep(c('A','F','H'),
each=15, 2)), factor(rep(c('Low', 'High'), each=45)),
factor(rep(c('S','D','S','D','S'), c(3,6,1,3,2))) )

names(Results.between.BGS) <- c('Correlations', 'Method', 'DNA', 'Dye')






Results.between.NoBGS <- data.frame(  c(dataA.A.NoBGS.s$between,
dataA.F.NoBGS.s$between, dataA.H.NoBGS.s$between, dataB.A.NoBGS.s$between,
dataB.F.NoBGS.s$between, dataB.H.NoBGS.s$between), factor(rep(c('A','F','H'),
each=15, 2)), factor(rep(c('Low', 'High'), each=45)),
factor(rep(c('S','D','S','D','S'), c(3,6,1,3,2))) )

names(Results.between.NoBGS) <- c('Correlations', 'Method', 'DNA', 'Dye')
@





\section{Segmentation method significantly influences within-slide correlations}



To investigate whether the segmentation method was an important
determinant of within-slide correlations, we first plot the
within-slide correlations data categorized by the segmentation
method. We use a function to plot the medians of the different
categories on the dot plots as follows.





<<>>=
stripchart.plot.median.by.factor <-   # plot median bars with "jitter"
  function(x, y, z){
    for (i in 1:length(x)) {
      lines(c(x[i] - z, x[i] + z), c(y[i], y[i]))
    }
  }
@




\begin{center}
<<fig=TRUE,echo=FALSE>>=


op<-par(mfrow=c(1,2), las=1)
stripchart(Results.within.BGS$Correlations[1:18] ~ Results.within.BGS$Method[1:18], ylab="Correlation coefficient",
ylim=c(0,0.8), xlim=c(0.5,3.5), pch=1, vertical=T, method="stack", offset=1,
main="With background subtracted")
stripchart.plot.median.by.factor(1:3, as.vector(by(Results.within.BGS$Correlations[1:18],
Results.within.BGS$Method[1:18], median)), 0.25)


stripchart(Results.within.NoBGS$Correlations[1:18] ~ Results.within.NoBGS$Method[1:18], ylab="Correlation coefficient",
ylim=c(0,0.8), xlim=c(0.5,3.5), pch=1, vertical=T, method="stack", offset=1,
main="With no background subtracted")
stripchart.plot.median.by.factor(1:3, as.vector(by(Results.within.NoBGS$Correlations[1:18],
Results.within.NoBGS$Method[1:18], median, na.rm=F)), 0.25)


@
\end{center}

We then turn the graphics window off. 


<<>>=
dev.off()
@

The dot plot shows differences in the correlations between the three
methods of segmentation, A, F and H. To investigate whether these
differences were significant we used a one-way ANOVA which is suitable
for investigating data grouped into more than two categories. Applying
a simple t test would not account for the problem of multiple testing
of differences between the means of categories (A$-$F, A$-$H and
F$-$H).  First, one-way ANOVA analysis was performed on the background
subtracted within-slide correlations for experiment A (first 18 rows):


<<>>=

ANOVA.within.one.BGS <- lm (Correlations[1:18] ~ Method[1:18],

Results.within.BGS)

summary(ANOVA.within.one.BGS)
@


We found a significant difference ($p<0.001$) between the three
methods of segmentation. Note that the R-squared test equals $0.84$
indicating a good fit.  We also perform a diagnostic QQ plot to
examine the fitness of the model to the data. We find a reasonable fit
of the quantiles.




\begin{center}
<<fig=TRUE,echo=FALSE>>=
qqnorm(ANOVA.within.one.BGS$res)
@
\end{center}

We then use the Tukey HSD test to investigate the significance level at each level of comparison. 



<<>>=

TukeyHSD(aov(Correlations[1:18] ~ Method[1:18], Results.within.BGS))
@


We found that the confidence intervals at each level of comparison do
not include zero indicating that the Null hypothesis of equal category
means is rejected. We then do the same analysis on the data with no
background subtraction and do the diagnostic plot. We find a
reasonable fit for the quantiles.


<<>>=
ANOVA.within.one.NoBGS <- lm (Correlations[1:18] ~ Method[1:18],
Results.within.NoBGS)
summary(ANOVA.within.one.NoBGS)
@

\begin{center}
<<fig=TRUE,echo=FALSE>>=
qqnorm(ANOVA.within.one.NoBGS$res)
@
\end{center}


<<>>=
TukeyHSD(aov(Correlations[1:18] ~ Method[1:18], Results.within.NoBGS))
@

The overall difference between the methods remained ($p<0.001$)
although the difference between the histogram and fixed circle method
was no longer significant.


\section{Histogram segmentation gives lower pixel-to-pixel variability}

We hypothesized that the better precision for the histogram method was
because of less variability in pixel intensity, as fluctuations in
pixel values have been shown to increase noise~\cite{Brown2001}.  The
histogram method summarizes centiles of pixel intensities obtained
from a square centered around the true spot.  From this it follows
that a narrow window of centiles will reduce the within-spot
variability as compared to the other methods of segmentation used
here. We therefore calculated the coefficient of variability (CV) for
foreground and background pixels for each feature in
experiments~\emph{A} and \emph{B} in both Cy3 and Cy5 channels.


\subsection{Categorizing the data}

In order to obtain the CV values we created a data object containing
all the names of the raw data objects created by the three different
methods of quantarray segmentation:

<<>>=
All.slides.methods <- c( expA.A, expA.F, expA.H, expB.A,  expB.F, expB.H)
@

We build the identifier for the interaction between the method of
segmentation and the DNA content:

<<>>=
Method.content <- factor(rep(c('LowA', 'LowF', 'LowH', 'HighA', 'HighF',
'HighH'), each=6))
@

\subsection{The \Rfunc{get.cv} function}

\subsubsection{Description}

This is a function to compute the spot CV values. 

\subsubsection{Usage}

get.cv(x)

\subsubsection{Arguments}

\begin{longtable}{rp{8cm}}

x: & an
object that has the names of the raw data objects.\\

\end{longtable}

\subsubsection{Details}

We first get the median coefficient of variability (CV) values for
each slide.  First the median CV for cy3, channel 1, by dividing the
standard deviation (Std.Dev) over the mean intensity derived from the
total number of pixels per spot.  We then get the median of the CVs
for all the spots on the slide.  We do the same for the background and
for cy5.

\subsubsection{Value}

A list containing the following:

\begin{longtable}{rp{8cm}}


Cy3.foreground: & the medians for all the CVs of cy3 foreground of all spots in a slide.\\\\
Cy3.background: & the medians for all the CVs of cy3 background of all spots in a slide.\\\\
Cy5.foreground: & the medians for all the CVs of cy5 foreground of all spots in a slide.\\\\
Cy5.background: & the medians for all the CVs of cy5 background of all spots in a slide.\\\\

\end{longtable}

\subsubsection{Script}

<<>>=
get.cv <- function(x){
  Median.CV.Values <- list ( Cy3.foreground= NULL, Cy3.background= NULL,
Cy5.foreground=NULL, Cy5.Background=NULL)
  for (i in 1:length(x)){
    data <- get(x[i])
    CV.cy3.fore <- data$ch1.Intensity.Std.Dev/data$ch1.Intensity
    median.cvs.cy3.fore <- median(CV.cy3.fore, na.rm=TRUE)
    CV.cy3.back <- data$ch1.Background.Std.Dev/data$ch1.Background
    median.cvs.cy3.back <- median(CV.cy3.back, na.rm=TRUE)
    CV.cy5.fore <- data$ch2.Intensity.Std.Dev/data$ch2.Intensity
    median.cvs.cy5.fore <- median(CV.cy5.fore, na.rm=TRUE)
    CV.cy5.back <- data$ch2.Background.Std.Dev/data$ch2.Background
    median.cvs.cy5.back <- median(CV.cy5.back, na.rm=TRUE)
    Median.CV.Values$Cy3.foreground <- c(Median.CV.Values$Cy3.foreground,
median.cvs.cy3.fore)
    Median.CV.Values$Cy3.background <- c(Median.CV.Values$Cy3.background,
median.cvs.cy3.back)
    Median.CV.Values$Cy5.foreground <- c(Median.CV.Values$Cy5.foreground,
median.cvs.cy5.fore)
    Median.CV.Values$Cy5.background <- c(Median.CV.Values$Cy5.background,
median.cvs.cy5.back)
    Median.CV.Values
    }
}
@

\section{Results of analysis of CV values}

We get the median CV values for all the data set and plot the results as follows:

<<>>=
Median.CVs <- get.cv( All.slides.methods)
attach(Median.CVs)
@




\begin{center}
<<fig=TRUE, echo=FALSE>>=
op<-par(mfrow=c(2,2), las=1)#, cex=0.7)
stripchart(Cy3.foreground~Method.content , xlab="", ylab="Median CV",
xlim=c(0.5,6.5), ylim=c(0,1), vertical=T, pch=1, method="stack", offset=1,
main="Cy3 foreground")
stripchart.plot.median.by.factor(1:6, as.vector(by(Cy3.foreground,
Method.content , median)), 0.25)
stripchart(Cy3.background~Method.content , xlab="", ylab="Median CV",
xlim=c(0.5,6.5), ylim=c(0,1), vertical=T, pch=1,method="stack", offset=1,
main="Cy3 background")
stripchart.plot.median.by.factor(1:6, as.vector(by(Cy3.background,
Method.content , median)), 0.25)
stripchart(Cy5.foreground~Method.content , xlab="", ylab="Median CV",
xlim=c(0.5,6.5), ylim=c(0,1), vertical=T, pch=1,method="stack", offset=1,
main="Cy5 foreground")
stripchart.plot.median.by.factor(1:6, as.vector(by(Cy5.foreground,
Method.content , median)), 0.25)
stripchart(Cy5.background~Method.content , xlab="", ylab="Median CV",
xlim=c(0.5,6.5), ylim=c(0,1), vertical=T, pch=1,method="stack", offset=1,
main="Cy5 background")
stripchart.plot.median.by.factor(1:6, as.vector(by(Cy5.background,
Method.content , median)), 0.25)

@
\end{center}





<<>>=
detach(Median.CVs)
@

The histogram method had the lowest CV values in both foreground and
background.



\section{Dye-swapping confounds the precision of between-slide comparisons}

We next studied the effect of segmentation on between-slide
variability by deriving a matrix of correlations for all possible
pair-wise comparisons between the slides for each method (fifteen
comparisons for each of three segmentation methods).  A one-way ANOVA
was conducted as above.



<<>>=
ANOVA.between.one.BGS <- lm (Correlations[1:45] ~ Method[1:45],
Results.between.BGS)
summary(ANOVA.between.one.BGS)
@


In contrast to our results from within-slide comparisons, no
significant differences in the correlations were found. The
correlation coefficients between slides with dye swapping were mostly
negative, indicating low overall repeatability of the data. As dye
swapping would be expected to alter correlations between slides, we
reanalyzed the data by restricting the comparisons to those between
replicates in which cDNA probes had been labelled with the same
fluors. We plot the effect of dye swapping on correlations using the
interaction plot:


<<>>=
low <- Results.between.BGS[1:45,]

@





\begin{center}
<<fig=TRUE,echo=FALSE>>=

op <- par(mfrow=c(1,2), las=1)
stripchart(low$Correlations~ low$Dye, xlim=c(0.5,2.5), ylim=c(-0.5, 0.5),
ylab="Correlation coefficient", xlab="", pch=1, vertical=TRUE, method="stack",
offset=1)
tmpFD <- low$Correlations[low$Method=="F" & low$Dye=="D"]
tmpHD <- low$Correlations[low$Method=="H" & low$Dye=="D"]
tmpFS <- low$Correlations[low$Method=="F" & low$Dye=="S"]
tmpHS <- low$Correlations[low$Method=="H" & low$Dye=="S"]
points(x=rep(c(1),9), y=tmpFD, col="magenta")
points(x=rep(c(1),9), y=tmpHD, col="green")
points(x=rep(c(2),6), y=tmpFS, col="magenta")
points(x=rep(c(2),6), y=tmpHS, col="green")
mtext("a", at=0, cex=2, font=2)

interaction.plot(low$Dye,low$Method,low$Correlations, xlab="", ylab="Mean correlation",
lty=1, col=c("black","magenta", "green"),legend=FALSE, ylim=c(-0.5, 0.5),
xtick=TRUE)
mtext("b", at=0.5, cex=2, font=2)

@
\end{center}

<<>>=
dev.off()

@





As expected, comparisons between replicates with the same dyes had
higher correlations than between slides with swapped dyes.  However,
the beneficial effect of histogram segmentation on correlation was
observed for slides with the same dye.






\section{Precision is determined by choice of segmentation method
  and amount of labelled probe}


\subsection{The effect of segmentation method
  and amount of labelled probe on within-slide correlations}

The impact of the quantity of RNA used on the overall precision of
microarray data has been previously reported~\cite{Yue2001}.  In
experiment~\emph{A}, we labelled \unit{10}{\micro\gram} of total RNA
for each slide which yielded a median of \unit{2.1}{\micro\gram} (IQR
1.1--3.1) of cDNA probe after purification, and incorporated a median
of \unit{151}{\pico\mole} (IQR 104--206) of each fluor.  In order to
identify whether the low between-slide correlations were caused by
inadequate specific activity of our samples, we repeated the
experiment using starting material of \unit{15}{\micro\gram} of total
RNA for each sample (experiment~\emph{B}; median labelled cDNA
\unit{5.5}{\micro\gram}, IQR 4.6--6.6, median incorporation of each
fluor \unit{463}{\pico\mole}, IQR 384--534).


First, we look at the effect of the two variables on the within-slide
correlations using an interaction plot.


<<>>=
attach(Results.within.BGS)
@

\begin{center}
<<fig=TRUE,echo=FALSE>>=
#op <- par(mfrow=c(1,2), las=1)
#interaction.plot(DNA,Method,Correlations, ylab="Mean correlation",
#xlab="",lty=1, col=c("black","magenta", "green"), main="Within-slide
#comparison", xtick=TRUE, legend=FALSE)
interaction.plot(DNA,Method,Correlations)
@
\end{center}

<<>>=
detach(Results.within.BGS)
@



The figure indicates a lack of interaction between the two variables
and suggests that differences exist between the categories within each
variable, low and high for the DNA content and A, F, H for the
segmentation method.

In order to investigate whether these differences were significant, we
performed a two-way ANOVA for within-slide correlations to examine the
effect of DNA content and the method of segmentation.  Note that we
fit the interaction term in the model to examine for interactions.

<<>>=
ANOVA.within.two.BGS <- lm (Correlations ~ Method * DNA,  Results.within.BGS)
anova( ANOVA.within.two.BGS)
@

The amount of labelled sample and method of segmentation independently
and significantly ($p<0.001$) influenced correlations.  There was no
significant interaction between the two variables. We then perform a
diagnostic QQ plot to examine how the models fits the data and find a
good fit.


\begin{center}
<<fig=TRUE,echo=FALSE>>=
qqnorm(ANOVA.within.two.BGS$res)
@
\end{center}




We repeat the same process with no background subtraction

<<>>=
ANOVA.within.two.NoBGS <- lm (Correlations ~ Method * DNA,
Results.within.NoBGS)
anova( ANOVA.within.two.NoBGS)


@





\begin{center}
<<fig=TRUE,echo=FALSE>>=
qqnorm(ANOVA.within.two.NoBGS$res)
@
\end{center}



\subsection{The effect of segmentation method
  and amount of labelled probe on between-slide correlations}

We then plot the effect of the two variables on between-slide correlations



<<>>=
attach(Results.between.BGS)
@


\begin{center}
<<fig=TRUE,echo=FALSE>>=
#interaction.plot(DNA,Method,Correlations, ylab="Mean correlation",
#xlab="",lty=1, col=c("black","magenta", "green"), main="Between-slide
#comparison", xtick=TRUE, legend=FALSE)
#legend(x=1.75, y=0.5, c("H","F","A"), lty=1, bty="n", pch="-", col=c("green",
#"magenta", "black"))
interaction.plot(DNA,Method,Correlations)
@
\end{center}

<<>>=

detach(Results.between.BGS)
dev.off()
@

The plot shows obvious interaction between the two variables at the
low DNA concentration level.  In order to formally assess the apparent
differences between categories within each variable we perform a
two-way ANOVA.


<<>>=
ANOVA.between.two.BGS <- lm (Correlations ~ Method * DNA,  Results.between.BGS)
anova( ANOVA.between.two.BGS)
@

Again we perform a diagnostic QQ plot and find a good quantile fit.  






\begin{center}
<<fig=TRUE,echo=FALSE>>=

qqnorm(ANOVA.between.two.BGS$res)
@
\end{center}




For between-slide comparisons, using a larger amount of labelled
sample significantly ($p<0.001$) improved the correlations
independently of the segmentation method used.  However significant
interaction was observed between the amount of labelled sample and the
segmentation method.  Therefore, while there was no advantage for any
segmentation method when low amounts of labelled sample were used,
there were marked differences for the methods when using higher
amounts.
  
We examine the effects with no background subtraction as follows:

<<>>=

ANOVA.between.two.NoBGS <- lm (Correlations ~ Method * DNA,
Results.between.NoBGS)
anova( ANOVA.between.two.NoBGS)

@


\begin{center}
<<fig=TRUE,echo=FALSE>>=

qqnorm(ANOVA.between.two.NoBGS$res)
@
\end{center}





\section{Coefficient of repeatability confirms higher precision}


A low value for the correlation coefficient does not necessarily mean
low repeatability as the correlation coefficient is not a measure of
sameness~\cite{Bland1986}.  Previous reports have shown discrepancies
between correlation coefficients and repeatability
coefficients~\cite{Bland1986, Jenssen2002}.  In order to confirm our
findings, we repeated the analysis using the coefficient of
repeatability (CR) values to compare between the three different
methods of segmentation and included a fourth proprietary method
encoded within the GenePix software package.


\subsection{Data processing for the Genepix segmentation}



We focus the attention on the data from experiment B to avoid the
effect of DNA content. We only examine the data with no background
subtraction to examine the effect independent of background
estimation.

\subsection{The \Rfunc{Get.ratios.genepix} function}

\subsubsection{Description}

A function to automate the process of obtaining the normalized log2
ratios for the genepix method.

\subsubsection{Usage}

Get.ratios.genepix(x)

\subsubsection{Arguments}

\begin{longtable}{rp{8cm}}

x: & an object containing the names of the genepix raw data objects.\\

\end{longtable}

\subsubsection{Details}

For each raw data object we get the intensity values for cy5, Red=
F635.Mean, and for cy3, Green= F532.Mean. We apply the grid
information.  We use scaled normalization as above.


\subsubsection{Value}

A data frame containing the normalized log2 ratios of each spot. The
number of columns corresponds to the number of elements in x.

\subsubsection{Script}


<<>>=
Get.ratios.genepix <- function(x){
res <- list(R = NULL, G = NULL, Rb = NULL, Gb = NULL)
for(i in (1:length(x))){
y<-get(x[i])
res$R <- cbind(res$R,y$F635.Mean)
res$G <- cbind(res$G,y$F532.Mean)
res
}
data.grid<-list(nspot.r=16,nspot.c=34,ngrid.r=12,ngrid.c=2)
data.ma<-stat.ma(res,data.grid,norm="s")
halfn= length(x)/2

data.ma$M[,c(1:halfn)]<- (data.ma$M[,c(1:halfn)])*-1
data.ma$M
}
@

\subsubsection{Obtaining the ratios}

<<>>=
Res.genepix <- Get.ratios.genepix(expB.G)
@





\subsection{The get.sigma function}

\subsubsection{Description}


A function that calculates the coeffecient of repeatability (CR)
values as described by ~\cite{Jenssen2002}.

\subsubsection{Usage}

get.sigma(x,arrays)

\newpage
\subsubsection{Arguments}

\begin{longtable}{rp{8cm}}

x: & a three-dimensional array having 2 columns,
by 6528 rows, by the number of slides in an experiment as a third
dimension.\\\\

arrays: & the number of slides in an experiment\\

\end{longtable}


\subsubsection{Details}


The function computes the CR value for each spot by getting the square
root of the mean of sum of squares for replicates across the
slides. First the mean of the ratios of any two replicates on a slide:

<<>>=
spot.mean <-
function(x){
 y<-mean(x,na.rm=T)}
@

Where x is a vector containing two elements representing the log2
ratios of a duplicate spot on a slide.

Then the sum of squares for a duplicate spot pair on a slide:

<<>>=
sum.sq.spot <-
function(x){
square.difference.spot<-NULL
 for(i in 1:length(x)){
 y<-x[i]-(spot.mean(x))
 y<-y^2
 square.difference.spot<-c(square.difference.spot,y)
 }
 sum.square.difference.spot<-sum(square.difference.spot)
 sum.square.difference.spot
}

@

where x is a vector containing two elements representing the log2 ratios of a
duplicate spot on a slide.

First we get the difference between the ratio of each spot and the
mean of the two spot ratios.

Then we get the sum of the differences

We repeat the same process for each duplicate on a slide: 


<<>>=
sum.sq.allspots <- function(x){
  ss.allspots<-NULL
  for( i in 1:nrow(x)){
    y<-x[i,]
    y<-sum.sq.spot(y)
    ss.allspots<-c(ss.allspots,y)
    ss.allspots
  }
}
@






Where x is a matrix of two columns, representing the two replicates in
a slide, and 6528 rows, representing the different clones on the
slide.

Finally, we obtain the CR values by computing the square root of the
mean of sum of squares for replicates across the slides.


\subsubsection{Value}

An object containing the CR values for each spot in the object x.



\subsubsection{Script}

<<>>=
get.sigma <- function(x,arrays){
 CR.values <- NULL
 for (i in 1:arrays){
   y<-x[,,i]
   y<-sum.sq.allspots(y)
   CR.values<-cbind(CR.values,y)
   CR.values
 }
 res<-apply(CR.values,1,mean,na.rm=T)
 res<-sqrt(res)
 res
}
@


\subsection{Obtaining the CR values}



We convert the data to a three dimensional matrix for the different segmentation
method. 

\subsubsection{The adaptive method}

<<>>=
rep.a <- rep(c(TRUE,FALSE),12,each=544)
rep.b <- rep(c(FALSE,TRUE),12,each=544)

Ratio.adaptive <- NULL
for (i in 1:6){
  y <- cbind(dataB.A.NoBGS.s$ratios[rep.a,i], dataB.A.NoBGS.s$ratios[rep.b,i])
  Ratio.adaptive <- cbind ( Ratio.adaptive, y)
}
dim(Ratio.adaptive) <- c(6528, 2, 6)
@

\subsubsection{The fixed circle method}



<<>>=
Ratio.fixed <- NULL
for (i in 1:6){
  y <- cbind(dataB.F.NoBGS.s$ratios[rep.a,i], dataB.F.NoBGS.s$ratios[rep.b,i])
  Ratio.fixed <- cbind ( Ratio.fixed, y)
}
dim(Ratio.fixed) <- c(6528, 2, 6)
@


\subsubsection{The histogram method}

<<>>=
Ratio.histogram <- NULL
for (i in 1:6){
  y <- cbind(dataB.H.NoBGS.s$ratios[rep.a,i], dataB.H.NoBGS.s$ratios[rep.b,i])
  Ratio.histogram <- cbind ( Ratio.histogram, y)
}
dim(Ratio.histogram) <- c(6528, 2, 6)
@




\subsubsection{The genepix method}
<<>>=
Ratio.genepix <- NULL
for (i in 1:6){
  y <- cbind(Res.genepix[rep.a,i], Res.genepix[rep.b,i])
  Ratio.genepix <- cbind ( Ratio.genepix, y)
}
dim(Ratio.genepix) <- c(6528, 2, 6)
@




\subsubsection{We then obtain the CR values for each method of segmentation.}

<<>>=
CR.adaptive <- get.sigma( Ratio.adaptive, 6)
CR.fixed <- get.sigma( Ratio.fixed, 6)
CR.histogram <- get.sigma( Ratio.histogram, 6)
CR.genepix <- get.sigma( Ratio.genepix, 6)
@


\subsubsection{We then calculate the median values for each method and plot the data.}

<<>>=
  apply(cbind(CR.adaptive, CR.fixed, CR.histogram, CR.genepix), 2, median,
na.rm=T)
@

\begin{center}
<<fig=TRUE,echo=FALSE>>=

boxplot(data.frame(cbind(CR.adaptive, CR.fixed, CR.histogram, CR.genepix)),
ylab="Coefficient of repeatability", xlab= "")
@
\end{center}

 The box plots for the sigma factors obtained for each feature from
the slides from experiment~\emph{B} (no background subtraction) showed
that the histogram methods had the lowest median CR value.





\section{Estimating the differentially expressed genes}

The probability of a gene to be differentially expressed is dependent
on the variability of the data for that gene across the replicates of
an experiment~\cite{Lee2000}.  It follows from our findings that the
segmentation method could have a direct effect on the number of
differentially expressed genes identified.  In order to test this
assumption we used a Bayesian method to estimate the number of
differentially expressed genes at a $p$ value of $0.01$ from the data
set of experiment~\emph{B}~\cite{Lonnstedt2002}.

We take account of each the duplicate clones on a slide in estimating
significance. To achieve this we reorder the clones so that duplicates
are sequentially in pairs as this is required by the stat.bayes
function in sma.

\subsection{The order.clones function}

\subsubsection{Description}

A function to reorder the clones within a slide so that duplicates are
listed sequentially.

\subsubsection{Usage}

order.clones(x)

\subsubsection{Arguments}

\begin{longtable}{rp{8cm}}

x: & a matrix of 13056 rows corresponding to all the clones on a slide. The number of columns corresponds to the number of slides in an
experiment.The data in x are the log2 ratios. \\
y: & the number of clones on each subgrid of a microarray.\\

\end{longtable}

\subsubsection{Details}

We exclude any clones with a replicate that has a missing value in the
experiment and reorder the matrix so that duplicate clones in a slide
are sequential:

\subsubsection{Script}

<<>>=
order.clones <- function(x, y){
  rep.a<-rep(c(TRUE,FALSE),each=y)
  rep.b<-rep(c(FALSE,TRUE),each=y)
  replicates <-  cbind(x[rep.a,],x[rep.b,])
  replicates <- na.omit(replicates)
  reordered.clones <- NULL
  for(i in 1:ncol(x)){
    y<-c(rbind(replicates[,i],replicates[,i+ncol(x)]))
    reordered.clones<-cbind(reordered.clones,y)
    reordered.clones
  }
}
@


\subsection{Identification of differential expression}

We then compute the probabilities for differential expression:


<<>>=
adaptive.bayesian <- stat.bayesian( M= order.clones(dataB.A.NoBGS.s$ratios, y= 544),
nb=6, nw=2)
fixed.bayesian <- stat.bayesian( M= order.clones(dataB.F.NoBGS.s$ratios, y= 544), nb=6,
nw=2)
histogram.bayesian <- stat.bayesian( M= order.clones(dataB.H.NoBGS.s$ratios, y= 544),
nb=6, nw=2)
genepix.bayesian <- stat.bayesian( M= order.clones(Res.genepix, y= 544), nb=6, nw=2)
@

The number of genes with a log odds ratio of more than zero for each
method was as follows:
 
<<>>=
length(which(adaptive.bayesian$lods>0))
length(which(fixed.bayesian$lods>0))
length(which(histogram.bayesian$lods>0))
length(which(genepix.bayesian$lods>0))
@

We plot the volcano plot for each method of segmentation with the log2
ratios (M) on the x axis and the lods, probabilities of differential
expression, on the y axis.


\begin{center}
<<fig=TRUE,echo=FALSE>>=
par(mfrow=c(1,4), las=1)
par(moi=c(0.5,0,0,0))
plot(adaptive.bayesian$Xprep$Mbar,adaptive.bayesian$lods, xlim=c(-1,1.5),
ylim=c(min(adaptive.bayesian$lods),18.5), ylab="Lod", xlab="M", main="A")
text("n=345",x=-0.95, y=18.5, adj=c(0,0))
abline(h=0)
plot(fixed.bayesian$Xprep$Mbar,fixed.bayesian$lods, xlim=c(-1,1.5),
ylim=c(min(fixed.bayesian$lods),18.5), ylab="", xlab="M", main="F")
text("n=967",x=-0.95, y=18.5, adj=c(0,0))
abline(h=0)
plot(histogram.bayesian$Xprep$Mbar,histogram.bayesian$lods, xlim=c(-1,1.5),
ylim=c(min(histogram.bayesian$lods),18.5), ylab="", xlab="M", main="H")
text("n=832",x=-0.95, y=18.5, adj=c(0,0))
abline(h=0)
plot(genepix.bayesian$Xprep$Mbar,genepix.bayesian$lods, xlim=c(-1,1.5),
ylim=c(min(genepix.bayesian$lods),18.5), ylab="", xlab="M", main="G")
text("n=944",x=-0.95, y=18.5, adj=c(0,0))
abline(h=0)
@
\end{center}

\begin{thebibliography}{20}

\bibitem{Ihaka1996}
Ihaka, R. and Gentleman, R. (1996) {R}: {A} language for data analysis and
  graphics.
\newblock \emph{J. Comp. Graph. Stat.}, \textbf{5}, 299--314.

\bibitem{Dudoit2002}
Dudoit, S., Yang, Y.~H., Callow, M.~J. and Speed, T.~P. (2002) Statistical
  methods for identifying differentially expressed genes in replicated {cDNA}
  microarray experiments.
\newblock \emph{Stat. Sin.}, \textbf{12}, 111--139.

\bibitem{Brown2001}
Brown, C.~S., Goodwin, P.~C. and Sorger, P.~K. (2001) Image metrics in the
  statistical analysis of {DNA} microarray data.
\newblock \emph{Proc. Natl. Acad Sci U.S.A.}, \textbf{98}, 8944--9.

\bibitem{Yue2001}
Yue, H., Eastman, P.~S., Wang, B.~B., Minor, J., Doctolero, M.~H., Nuttall,
  R.~L., Stack, R., Becker, J.~W., Montgomery, J.~R., Vainer, M. and Johnston,
  R. (2001) An evaluation of the performance of {cDNA} microarrays for
  detecting changes in global {mRNA} expression.
\newblock \emph{Nucleic Acids Res.}, \textbf{29}, E41--1.

\bibitem{Bland1986}
Bland, J.~M. and Altman, D.~G. (1986) Statistical methods for assessing
  agreement between two methods of clinical measurement.
\newblock \emph{Lancet}, \textbf{1}, 307--10.

\bibitem{Jenssen2002}
Jenssen, T.~K., Langaas, M., Kuo, W.~P., Smith-Sorensen, B., Myklebost, O. and
  Hovig, E. (2002) Analysis of repeatability in spotted {cDNA} microarrays.
\newblock \emph{Nucleic Acids Res.}, \textbf{30}, 3235--44.

\bibitem{Lee2000}
Lee, M.~L., Kuo, F.~C., Whitmore, G.~A. and Sklar, J. (2000) Importance of
  replication in microarray gene expression studies: statistical methods and
  evidence from repetitive {cDNA} hybridizations.
\newblock \emph{Proc. Natl. Acad. Sci. U.S.A.}, \textbf{97}, 9834--9.

\bibitem{Lonnstedt2002}
L{\"o}nnstedt, I. and Speed, T. (2002) Replicated microarray data.
\newblock \emph{Stat. Sin.}, \textbf{12}, 31--46.

\end{thebibliography}

\end{document}