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Microarray preprocessing and quality assessmentWolfgang HuberEuropean Bioinformatics Institute
H. Sueltmann DKFZ/MGA
Which genes are differentially transcribed?
same-same tumor-normal
log-ratio
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
05
10
15
log
feat
ure
inte
nsity
(a.
u.)
arrays, colour channels
Scatterplot, colored by PCR-plateTwo RZPD Unigene II filters (cDNA nylon membranes)
PCR plates
PCR plates
PCR plates: boxplots
print-tip effects
-0.8 -0.6 -0.4 -0.2 0.0 0.2
0.0
0.2
0.4
0.6
0.8
1.0
41 (a42-u07639vene.txt) by spotting pin
log(fg.green/fg.red)
F
1:11:21:31:42:12:22:32:43:13:23:33:44:14:24:34:4
q (log-ratio)
F(q
)
spotting pin quality decline
after delivery of 3x105 spots
after delivery of 5x105 spots
H. Sueltmann DKFZ/MGA
spatial effectsspatial effects
R Rb R-Rbcolor scale by rank
spotted cDNA arrays, Stanford-type
another array:
print-tip
color scale ~ log(G)
color scale ~ rank(G)
10 20 30 40 50 60
1020
3040
5060
1:nrhyb
1:nr
hyb
1 2 3 4 5 6 7 8 910111213141516171823242526272829303132333435363738737475767778798081828384858687888990919293949596979899100
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Batches: array to array differences dij = mediank |hik -hjk|
arrays i=1…63; roughly sorted by time
A complex measurement process lies between mRNA concentrations and intensities
o tissue contamination
o clone identification and mapping
o image segmentation
o RNA degradation
o PCR yield, contamination
o signal quantification
o amplification efficiency
o spotting efficiency
o ‘background’ correction
o reverse transcription efficiency
o DNA-support binding
o hybridization efficiency and specificity
o other array manufacturing-related issues
The problem is less that these steps are ‘not perfect’; it is that they vary from array to array, experiment to experiment.
Statistics 101:
bias accuracy
p
reci
sio
n
v
aria
nce
Basic dogma of data analysis
Can always increasesensitivity on the cost of specificity,
or vice versa,
the art is to - optimize both- then find the best trade-off.
X
X
X
X
X
X
X
X
X
ratios and fold changes
Fold changes are useful to describe continuous changes in expression
1000
1500
3000
x3
x1.5
A B C
0
200
3000
?
?
A B C
But what if the gene is “off” (below detection limit) in one condition?
ratios and fold changes
The idea of the log-ratio (base 2)0: no change
+1: up by factor of 21 = 2 +2: up by factor of 22 = 4 -1: down by factor of 2-1 = 1/2 -2: down by factor of 2-2 = ¼
What about a change from 0 to 500?- conceptually- noise, measurement precision
A unit for measuring changes in expression: assumes that a change from 1000 to 2000 units has a similar biological meaning to one from 5000 to 10000.
ratio compression
Yue et al., (Incyte
Genomics) NAR (2001)
29 e41
How to compare microarray intensities with each other?
How to address measurement uncertainty (“variance”)?
How to calibrate (“normalize”) for biases between samples?
Sources of variationamount of RNA in the biopsy efficiencies of-RNA extraction-reverse transcription -labeling-fluorescent detection
probe purity and length distributionspotting efficiency, spot sizecross-/unspecific hybridizationstray signal
Calibration Error model
Systematic o similar effect on many measurementso corrections can be estimated from data
Stochastic
o too random to be ex-plicitely accounted for o remain as “noise”
iik ika a
ai per-sample offset
ik additive noise
bi per-sample normalization factor
bk sequence-wise probe efficiency
ik multiplicative noise
exp( )iik k ikb b b
ik ik ik ky a b x
The two component model
measured intensity = offset + gain true abundance
The two-component model
raw scale log scale
“additive” noise
“multiplicative” noise
B. Durbin, D. Rocke, JCB 2001
Parameterization
(1 )
y a b
y a
x
xb e
two practically equivalent forms
(<<1)
a systematic background
same for all probes (per array x color)
per array x color x print-tip group
random background
iid in whole experiment
iid per array
b systematic gain factor
per array x color per array x color x print-tip group
random gain fluctuations
iid in whole experiment
iid per array
Important issues for model fitting
Parameterization (model complexity)variance vs bias
"Heteroskedasticity" (unequal variances) weighted regression or variance stabilizing
transformation
Outliers use a robust method
AlgorithmIf likelihood is not quadratic, need non-linear
optimization. Local minima / concavity of likelihood?
Models are never correct, but some are useful
True relationship:
1 2 22
(0, 0.15 )y x x N
Model: linear dependence Model: quadratic dependence
variance stabilizing transformations
Xu a family of random variables with EXu=u,
VarXu=v(u). Define
var f(Xu ) independent of u
1( )
v( )
x
f x duu
derivation: linear approximation
0 20000 40000 60000
8.0
8.5
9.0
9.5
10
.01
1.0
raw scale
tra
nsf
orm
ed
sca
le
variance stabilizing transformations
f(x)
x
variance stabilizing transformations
1( )
v( )
x
f x duu
1.) constant variance (‘additive’)
2( ) sv u f u
2.) constant CV (‘multiplicative’) 2( ) logv u u f u
4.) additive and multiplicative
2 2 00( ) ( ) arsinh
u uv u u u s f
s
3.) offset2
0 0( ) ( ) log( )v u u u f u u
the “glog” transformation
intensity-200 0 200 400 600 800 1000
- - - f(x) = log(x)
——— hs(x) = asinh(x/s)
2arsinh( ) log 1
arsinh log log2 0limx
x x x
x x
P. Munson, 2001
D. Rocke & B. Durbin, ISMB 2002
W. Huber et al., ISMB 2002
raw scale log glog
difference
log-ratio
generalized
log-ratio
constant partvariance:
proportional part
glog
ii k i k i ka a L a i p e r - s a m p l e o ff s e t
L i k l o c a l b a c k g r o u n d p r o v i d e d b y i m a g e a n a l y s i s
i k ~ N ( 0 , b i2 s 1
2 )
“ a d d i t i v e n o i s e ”
b i p e r - s a m p l en o r m a l i z a t i o n f a c t o r
b k s e q u e n c e - w i s el a b e l i n g e ffi c i e n c y
i k ~ N ( 0 , s 22 )
“ m u l t i p l i c a t i v e n o i s e ”
e x p ( )ii k k i kb b b
i k i k i k i ky a b x
m e a s u r e d i n t e n s i t y = o ff s e t + g a i n * t r u e a b u n d a n c e
parameter estimation (vsn package)
2Yarsinh , (0, )iki
k ki kii
aN c
b
:
o maximum likelihood estimator: straightforward – but sensitive to outliers
o model is for genes that are unchanged; differentially transcribed genes act as outliers.
o robust variant of ML estimator, à la Least Trimmed Sum of Squares regression.
o works well as long many genes are not differentially transcribed (<50% throughout the intensity range)
“usual” log-ratio
'glog' (generalized log-ratio)
1
2
2 21 1 1
2 22 2 2
log
log
xx
x x c
x x c
c1, c2 are experiment specific parameters (~level of background noise)
Variance Bias Trade-Off
Est
imat
ed l
og
-fo
ld-c
han
ge
Signal intensity
logglog
Variance-bias trade-off and shrinkage estimators
Shrinkage estimators:a general technology in statistics:pay a small price in bias for a large decrease of variance, so overall the mean-squared-error (MSE) is reduced.
Particularly useful if you have few replicates.
Generalized log-ratio is a shrinkage estimator for fold change
“Single color normalization”
n red-green arrays (R1, G1, R2, G2,… Rn, Gn)
within/between slidesfor each slide i=1…n
calculate Mi= log(Ri/Gi), Ai= ½ log(Ri*Gi)normalize Mi vs Ai
Then normalize M1…Mn
all at oncenormalize the combined matrix (R, G)
then calculate log-ratios or any other contrast you like
What about non-linear effects?
o Good data operate in the linear regime, where fluorescence intensity increases proportionally to target abundance (see e.g. Affymetrix dilution series)
Two reasons for non-linearity:
o At the high intensity end: saturation/quenching. This can and should be avoided experimentally - loss of data!
o At the low intensity end: background offsets
0 20 40 60 80 100
02
04
06
08
01
00
original scale
x
y
1 2 5 10 20 50 1002
05
01
00
log/log scale
x
y
10 1y x
Non-linear or affine linear?
Response curveLockhart et. al. Nature Biotechnology 14 (1996)
Gene expression matters
Probe set summaries for Affymetrix expression analysis
genechips
Probe set summarization - data and notation
PMijg , MMijg = Intensities for perfect match and mismatch probe j for gene g in chip i
i = 1,…, n one to hundreds of chips
j = 1,…, J usually 11 or 16 probe pairs
g = 1,…, G 6…30,000 probe sets.
Tasks: calibrate (normalize) the measurements from different
chips (samples)summarize for each probe set the probe level data, i.e.,
16 PM and MM pairs, into a single expression measure.
compare between chips (samples) for detecting differential expression.
Expression measures: MAS 4.0
Affymetrix GeneChip MAS 4.0 software used AvDiff, a trimmed mean:
o sort dj = PMj -MMj o exclude highest and lowest valueo J := those pairs within 3 standard
deviations of the average
1( )
# j jj J
AvDiff PM MMJ
Expression measures MAS 5.0
Instead of MM, use "repaired" version CTCT= MM if MM<PM
= PM / "typical log-ratio" if MM>=PM
"Signal" =
Tukey.Biweight (log(PM-CT))
(… median)
Tukey Biweight: B(x) = (1 – (x/c)^2)^2 if |x|<c, 0 otherwise
Expression measures: Li & Wong
dChip fits a model for each gene
where
– i: expression index for gene i
– j: probe sensitivity
Maximum likelihood estimate of MBEI is used as expression measure of the gene in chip i.
Need at least 10 or 20 chips.
Current version works with PMs only.
2, (0, )ij ij i j ij ijPM MM N
Expression measures RMA: Irizarry et al. (2002)
o Estimate one global background value b=mode(MM). No probe-specific background!
o Assume: PM = strue + b
Estimate s0 from PM and b as a conditional expectation E[strue|PM, b].
o Use log2(s).
o Nonparametric nonlinear calibration ('quantile normalization') across a set of chips.
RMA
Additive model for probe effects Pj and expression value Ei
log2Sij = Ei + Pj + εij
Estimate Ei using robust procedure
0 200 400 600
0.00
00.
010
X=B+S
X
Den
sity
B
SX
-2 -1 0 1 2 3
-4-2
0
x
f(x)
log2xglog2x
0 50 100 200
-50
050
150
RMA background correction
xS
0 50 100 200
-50
050
150
VSN background correction
x
Svs
n0 50 100 200
-2-1
01
2
log2
x
log 2
Srm
a
0 50 100 200
-2-1
01
2
glog2
x
glog
2Svs
n
bioc/Courses/bioc_R_intro/vsn_vs_bgcorrect.R
ArrayQualityMetrics
R package by Audrey Kauffmann (EBI)
Collaboration with Alvis Brazma, Misha Kapushesky (ArrayExpress)
EU Project "Emerald" http://www.microarray-quality.org
A probe effect normalisation for tiling arrays
Huber et al., Bioinformatics 2006
Genechip S. cerevisiae Tiling Array
4 bp tiling path over complete genome(12 M basepairs, 16 chromosomes)
Sense and Antisense strands6.5 Mio oligonucleotides 5 m feature size
manufactured by Affymetrixdesigned by Lars Steinmetz (EMBL & Stanford Genome Center)
RNA Hybridization
Before normalization
Probe specific response normali-zation
2log ii
i
yq
s
2
( )glog i i
ii
y b sq
s
2log iy
2log is
remove ‘dead’ probes
2glog
i ii
i
PM MMq
s
S/N
3.22
3.47
4.04
4.58
4.36
Probe-specific response normalization
si probe specific response factor. Estimate taken from DNA hybridization data
bi =b(si ) probe specific background term. Estimation: for strata of probes with similar si, estimate b through location estimator of distribution of intergenic probes, then interpolate to obtain continuous b(s)
2
( )glog i i
ii
y b sq
s
Estimation of b: joint distribution of (DNA, RNA) values of intergenic PM probes
log2 DNA intensity
log
2 R
NA
in
ten
sity unannotated
transcripts
background
b(s)
After normalization
Cross-Hybridization
One or several probes from a probeset hybridize with intended transcripts.
Reasons:
sequence similarity (gene families)
errors in gene annotation (e.g. mis-assignment of exons)
pure chance
Casneuf et al., Bioinformatics 2006
Affymetrix CDF
Custom CDF
without sequence-similar genes (E<10-10 )
all genes
How to disentangle correlation
because of functional relatedness and
because of cross-hybridization?
Meta-correlation
for probe set pair X and Y, the Spearman correlation between
alignment scores of X’s reporters to Y's target transcript
and
the signal correlation coefficient of X's reporters to Y's probe set expression value.
Significant positive metacorrelation in cases of off-target sequence similarity
Metacorrelation examplesProbe set
summariesX' reporters off-target alignment vs
signal correlationX: extensin-like family protein
Y: belongs to a zinc finger family.
0.63
0.89
0.70X, Y: functions unknown
0.62
X: protein kinase
Y: ATP citrate lyase
neighbours
0.3
0.55
Caveat RMA
Simulated periodic probeset
White noise
9 of B,2 of A
RMA (median polish) summariesAB=-0.07AC=0.73
Histogram of x
x
Fre
qu
en
cy
-2 0 2 4 6
0.0
1.0
2.0
3.0
Histogram of x
x
Fre
qu
en
cy
-2 0 2 4 6
0.0
1.0
2.0
3.0
set.seed(0xdada)a=c(rnorm(6), rnorm(5, mean=4))b=a; b[6]=b[7]myhist=function(x){ hist(x, col="orange", breaks=seq(-3,6,by=0.8),ylim=c(0,3)) abline(v=median(x), col="black",lwd=3)}par(mfrow=c(2,1))myhist(a)myhist(b)
Median is robust, but can be sensitive to small wiggles in the data
References
Bioinformatics and computational biology solutions using R and Bioconductor, R. Gentleman, V. Carey, W. Huber, R. Irizarry, S. Dudoit, Springer (2005).
Variance stabilization applied to microarray data calibration and to the quantification of differential expression. W. Huber, A. von Heydebreck, H. Sültmann, A. Poustka, M. Vingron. Bioinformatics 18 suppl. 1 (2002), S96-S104.
Exploration, Normalization, and Summaries of High Density Oligonucleotide Array Probe Level Data. R. Irizarry, B. Hobbs, F. Collins, …, T. Speed. Biostatistics 4 (2003) 249-264.
Error models for microarray intensities. W. Huber, A. von Heydebreck, and M. Vingron. Encyclopedia of Genomics, Proteomics and Bioinformatics. John Wiley & sons (2005).
Differential Expression with the Bioconductor Project. A. von Heydebreck, W. Huber, and R. Gentleman. Encyclopedia of Genomics, Proteomics and Bioinformatics. John Wiley & sons (2005).
Acknowledgements
Anja von Heydebreck (Darmstadt)Robert Gentleman (Seattle)Günther Sawitzki (Heidelberg)Martin Vingron (Berlin)Annemarie Poustka, Holger Sültmann, Andreas
Buness, Markus Ruschhaupt (Heidelberg)Rafael Irizarry (Baltimore)Judith Boer (Leiden) Anke Schroth (Heidelberg)Friederike Wilmer (Hilden)Lars Steinmetz (Heidelberg)Tineke Casneuf (Gent)
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