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Microarray normalization, error models, quality
Wolfgang HuberEMBLBrixen 15 June 2009
Brief historyLate 1980s: Poustka, Lennon, Lehrach: cDNAs spotted on nylon membranes
1990s: Affymetrix adapts microchip production technology for in situ oligonucleotide synthesis („commercial and heavily patent-fenced“)
1990s: Brown lab in Stanford develops two-colour spotted array technology („open and free“)
1998: Yeast cell cycle expression profiling on spotted arrays (Spellmann) and Affymetrix (Cho)
1999: Tumor type discrimination based on mRNA profiles (Golub)
2000-ca. 2004: Affymetrix dominates the commercial microarray market
Since ~2003: Nimblegen, Illumina, Agilent (and many others)
Throughout 2000‘s: CGH, CNVs, SNPs, ChIP, tiling arrays
Since ~2007: Next-generation sequencing (454, Solexa, ABI Solid,...)
Oligonucleotide microarrays
Base Pairing
Ability to use hybridisation for constructing specific + sensitive probes at will is unique to DNA (cf. proteins,
RNA, metabolites)
Oligonucleotide microarrays
5µm5µm
Millions of copies of a Millions of copies of a specificspecificoligonucleotide probe oligonucleotide probe molecule per patchmolecule per patch
Image of array after hybridisation and stainingImage of array after hybridisation and staining
up to 6.5 Mioup to 6.5 Miodifferent probe patchesdifferent probe patches
Target - single strandedTarget - single stranded cDNAcDNA
Oligonucleotide probeOligonucleotide probe
**
**
*
1.28cm1.28cm
GeneChipGeneChip
Hybridized Probe CellHybridized Probe Cell
Probe sets
Terminology for transcription arrays
Each target molecule (transcript) is represented by several oligonucleotides of (intended) length 25 bases
Probe: one of these 25-mer oligonucleotidesProbe set: a collection of probes (e.g. 11) targeting the
same transcript
MGED/MIAME: „probe“ is ambiguous!Reporter: the sequenceFeature: a physical patch on the array with molecules
intended to have the same reporter sequence (one reporter can be represented by multiple features)
Image analysis
• several dozen pixels per feature• segmentation• summarisation into one number representing the intensity level for this feature
CEL file
array data
samples:mRNA fromtissue biopsies,cell lines
arrays:probes = gene-specific DNA strands
2.93
1.67
0.72
0.6
5.8
1.12
tissue B
3.314.2MCAM
0.671.32LAMA4
0.120.01CASP4
1.02.2ALDH4
1.81.1VIM
2.120.02ErbB2
tissue Ctissue A
fluorescent detection of the amount of
sample-probe binding
Why do you need ‘normalisation’?
From: lymphoma dataset
vsn package
Alizadeh et al., Nature 2000
Systematic drift effects
MA-plotM
A
2
2
2
2
log
log ( )
1 1log2 2log
1 1
A RG
RM
G
RA
GM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
05
10
15
log 2
inte
nsity
arrays / dyes
5 10 15
0.0
00
.05
0.1
00
.15
0.2
00
.25
8 arrays from the lymphoma data (Alizadeh 2000)
log2intensity
De
nsi
ty
log2 Cope et al. Bioinformatics 2003
Non-linearityspike-in data
ratio compression
Yue et al., (Incyte
Genomics) NAR (2001)
29 e41
nominal 3:1
nominal 1:1
nominal 1:3
A complex measurement process lies between mRNA concentrations and intensities
o RNA degradation
o quality of actual probe sequences (vs intended)
o image segmentation
o amplification efficiency
o scratches and spatial gradients on the array
o signal quantification
o reverse transcription efficiency
o cross-talk across features
o signal "preprocessing"
o hybridization efficiency and specificity
o cross-hybridisation
o labeling efficiency
o optical noise
The problem is less that these steps are ‘not perfect’; it is that they vary from array to array, experiment to experiment.
Preprocessing Terminology
Calibration, normalisation: adjust for systematic drifts associated with dye, array (and sometimes position within array)
Background correction: adjust for the non-linearity at the lower end of the dynamic range
Transformation: bring data to a scale appropriate for the analysis (e.g. logarithm; variance stabilisation)
Log-ratio: adjust for unknown scale (units) of the data
Existing approaches differ in the order in which these steps are done, some are exactly stepwise („greedy“), others aim to gain strength by doing things simultaneously.
Why do you need statistics?
tumor-normal
Which genes are differentially transcribed?
same-same
log-ratio
Statistics 101:
bias accuracy
p
recis
ion
vari
an
ce
Basic dogma of data analysis
Can always increase sensitivity
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
How to compare microarray intensities with each other?
How to address measurement uncertainty?
How to calibrate (“normalize”) for systematic differences between samples?
How to deal with non-linearity (esp. at the lower end, „background“)
Questions
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”
Quantile normalisation
Quantile normalisation
Ben Bolstad 2001
1e
2e
d
d
d
data("Dilution")nq = normalize.quantiles(exprs(Dilution))nr = apply(exprs(Dilution), 2, rank)for(i in 1:4) plot(nr[,i], nq[,i], pch=".", log="y", xlab="rank",
ylab="quantile normalized", main=sampleNames(Dilution)[i])
6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
before
log2(exprs(Dilution))
De
nsi
ty
6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
after quantile normalisation
log2(nq)
De
nsi
ty
Quantile normalisation is: per array rank-transformation followed by replacing ranks with values
from a common reference distribution
Histogram of log2(nq[, 1])
log2(nq[, 1])
Fre
qu
en
cy
6 8 10 12 14
05
00
01
00
00
15
00
02
00
00
25
00
03
00
00
Quantile normalisation
+ Simple, fast, easy to implement
+ Always works, needs no user interaction / tuning
+ Non-parametric: can correct for quite nasty non-linearities (saturation, background) in the data
- Always "works", even if data are bad / inappropriate
- May be conservative: rank transformation looses information - may yield less power to detect differentially expressed genes
- Aggressive: if there is an excess of up- (or down) regulated genes, it removes not just technical, but also biological variation
loess normalisation
"loess" normalisationloess (locally weighted scatterplot smoothing): an
algorithm for robust local polynomial regression by W. S. Cleveland and colleagues (AT&T, 1980s) and handily available in R
Local polynomial regression
0
22 1
1 1
1
Global polynomial regression
( ) ...
applied to data ( , ),..., ( , ), with equal weights
resulting in global fit ( ,..., )
Local polynomial regression around
with w
pp
n n
p
y x a x a x a x a
x y x y
a a
1
eights ( - )
resulting in local fit ( ( ),..., ( ))
b
p
h x
a v a v
bandwidth b
Robust regression
2 4 6 8 10
51
01
52
0
x
y
lmrlm
2
1
1
1, ,
OLS: ( ) min
M-est.: ( ) min
LTS: { ( ) | } min
n
i ii
n
i ii
i i i n
y f x
M y f x
y f x
F
C. Loader
Local Regression and Likelihood
Springer Verlag
loess normalisation
before after
• local polynomial regression of M against A• 'normalised' M-values are the residuals
local polynomial regression normalisation in >2 dimensions
n-dimensional local regression model for microarray normalisation( ) ( )
: log-intensity of gene in condition , replicate
: baseline value gene ( -value)
: effect of treatment on gene
( ) : intensity-dependent normalisation fu
kij k ij k ik k kij
kij
k
ik
ij k
Y
Y k i j
k A
i k
nction for array
( ) : intensity-dependent error scale function
: i.i.d. error term
k
kij
ij
An algorithm for fitting this robustly is described (roughly) in the paper. They only provided software as a compiled binary for Windows. The method has not found much use.
Estimating relative expression
(fold-changes)
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.…. data reduction
Many data are measured in definite units:
• time in seconds• lengths in meters• energy in Joule, etc.
Climb Mount Plose (2465 m) from Brixen (559 m) with weight of 76 kg, working against a gravitation field of strength 9.81 m/s2 :
What is wrong with microarray data?
(2465 - 559) · 76 · 9.81 m kg m/s2
= 1 421 037 kg m2 s-2
= 1 421.037 kJ
Two component error model and variance
stabilisation
ik i ika a
ai per-sample offset
ik additive noise
bi per-sample gain factor
bk sequence-wise probe efficiency
ik multiplicative noise
exp( )ik i 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
The additive-multiplicative error model
Trey Ideker et al.: JCB (2000)
David Rocke and Blythe Durbin: JCB (2001), Bioinformatics (2002)
Use for robust affine regression normalisation: W. Huber, Anja von Heydebreck et al. Bioinformatics (2002).
For background correction in RMA: R. Irizarry et al., Biostatistics (2003).
Parameterization
(1 )
y a b x
y a b x e
two practically equivalent forms
(<<1)
a: average background
on one array, for one color, the same for all features
also dependent on the reporter sequence
background fluctuations
same distribution in whole experiment
different distributions
b: average gain factor on one array, for one color, the same for all features
intensity dependent
gain fluctuations same distribution in whole experiment
different distributions
variance stabilizing transformations
Xu a family of random variables with
E(Xu) = u and Var(Xu) = v(u). Define
Var f(Xu ) does not depend on u
Derivation: linear approximation,relies on smoothness of v(u).
( )v( )
x
duf x
u
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 transformation
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 0
0( ) ( ) arsinhu u
v u u u s fs
3.) offset 20 0( ) ( ) log( )v u u u f u u
the “glog” transformation
P. Munson, 2001
D. Rocke & B. Durbin, ISMB 2002
W. Huber et al., ISMB 2002
2 2
2 2
e
glog ( , ) log2
glog ( ,1) log 2 arsinh( )e
x x cx c
x x
raw scale log glog
difference
log-ratio
generalized
log-ratio
constant partvariance:
proportional part
glog
dif
fere
nc
e re
d-g
reen
rank(average)
Parameter estimation
2Yarsinh , (0, )iki
k ki kii
aN c
b
:
o maximum likelihood estimator: straightforward – but sensitive to deviations from normality
o model holds 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 as <50% of genes are differentially transcribed (and may still work otherwise)
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
Least trimmed sum of squares regression
0 2 4 6 8
02
46
8
x
y
n
2 2
( ) ( )i=1
( )i iy f x
minimize
- least sum of squares - least trimmed sum of squares
P. Rousseeuw, 1980s
“usual” log-ratio
'glog' (generalized log-ratio)
1
2
2 21 1 1
2 22 2 2
log
log
x
x
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 log fold change
Variance-bias trade-off and shrinkage estimators
Same-same comparison
log-ratio
glog-ratio
Lines: 29 data points with observed ratio of 2
Fig. 5.11 from Hahne et al.‘s useR-book
Linear and Non-linear
linear affine linear “genuinely” non-linear
Always affine?
vsn provides a combination of glog-transformation and affine between-array* normalisation
What if you want to normalise for genuine non-linear effects, and still use the nice transformation?
Set parameter calib in vsn2 function to none (default: affine) and do your own normalisation beforehand (do not (log-)transform). The vignette shows an example for use with quantile normalisation.
* print-tip groups or other stratifications are also possible
Background correction
Background correctionBackground correction
Irizarry et al. Biostatistics 2003
0 pm
500 fm 1 pm
750 fm
RMA Background correction
~ log-normal with mean and sd read off values
~ exponential
closed form expression for [ | ],
ˆ use this as ( 0).
(NB, [ 0] 1 is not realistic)
PM B S
B MM
S
E S PM
s
P S
Irizarry et al. (2002)
Background correction: Background correction:
raw intensities x
biased background correction
s=E[S|data]
unbiased background correction
s=x-b
log2(s) glog2(s|data)
?
Comparison between RMA and VSN background correction
vsn package vignette
Dilution data
vsn package vignette
Summaries for Affymetrix genechip probe sets
Data and notationPMikg , MMikg = Intensities for perfect match and
mismatch probe k for gene g on chip i
i = 1,…, n one to hundreds of chips
k = 1,…, J usually 11 probe pairs
g = 1,…, G tens of thousands of probe sets.
Tasks: calibrate (normalize) the measurements from different chips (samples)summarize for each probe set the probe level data, i.e., 11 PM and MM
pairs, into a single expression measure.compare between chips (samples) for detecting differential
expression.
Expression measures: MAS 4.0
Expression measures: MAS 4.0
Affymetrix GeneChip MAS 4.0 software used AvDiff, a trimmed mean:
o sort dk = PMk -MMk o exclude highest and lowest valueo K := those pairs within 3 standard deviations of
the average
1( )
# k kk K
AvDiff PM MMK
Expression measures MAS 5.0
Expression measures MAS 5.0
Instead of MM, use "repaired" version CT
CT = MM if MM<PM
= PM / "typical log-ratio" if MM>=PM
Signal = Weighted mean of the values log(PM-CT)
weights follow Tukey Biweight function
(location = data median,
scale a fixed multiple of MAD)
0 20 40 60 80 100
0.0
0.4
0.8
Tukey Biweight
x
w
Expression measures: Li & Wong
Expression measures: Li & Wong
dChip fits a model for each gene
where
i : expression measure for the gene in sample i
k : probe effect
i is estimated by maximum likelihood
2, (0, )ki ki k i ki kiPM MM N
dChip
RMA
bi is estimated using the robust method median polish (successively remove row and column medians, accumulate terms, until convergence).
Expression measures RMA: Irizarry et al. (2002)Expression measures
RMA: Irizarry et al. (2002)
2log ki k i kiY a b
2, (0, )ki k i ki kiY N
However, median (and hence median polish) is not always so robust...
See also: Casneuf T. et al. (2007), In situ analysis of cross-hybridisation on microarrays and the inference of expression correlation. BMC Bioinformatics 2007;8(1): 461
x
Fre
qu
en
cy
-2 0 2 4 6 8 10
02
46
x
Fre
qu
en
cy
-2 0 2 4 6 8 100
24
6
- median- trimmed mean (0.15)
Probe effect adjustment by using gDNA reference
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
Quality assessment
Quality Assessment and Control
arrayQualityMetrics package by Audrey Kauffmann
This afternoon!
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).
Normalization and analysis of DNA microarray data by self-consistency and local regression. T.B. Kepler, L. Crosby, K. Morgan. Genome Biology. 3(7):research0037 (2002)
Statistical methods for identifying differentially expressed genes in replicated cDNA microarray experiments. S. Dudoit, Y.H. Yang, M. J. Callow, T. P. Speed. Technical report # 578, August 2000 (UC Berkeley Dep. Statistics)
A Benchmark for Affymetrix GeneChip Expression Measures. L.M. Cope, R.A. Irizarry, H. A. Jaffee, Z. Wu, T.P. Speed. Bioinformatics (2003).
....many, many more...
Acknowledgements
Anja von Heydebreck (Darmstadt)Robert Gentleman (Seattle)Günther Sawitzki (Heidelberg)Martin Vingron (Berlin)Rafael Irizarry (Baltimore)Terry Speed (Berkeley)Judith Boer (Leiden) Anke Schroth (Wiesloch)Friederike Wilmer (Hilden)Jörn Tödling (Cambridge)Lars Steinmetz (Heidelberg)Audrey Kauffmann (Cambridge)