Microarray normalization, error models, quality

Wolfgang HuberEMBL-EBIBrixen 16 June 2008

Oligonucleotide microarrays

Base Pairing

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 oligonucleotidesPerfect match (PM): A probe exactly complementary to the

target sequenceMismatch (MM): same as PM but with a single homomeric

base change for the middle (13th) base (G-C, A-T) Probe-pair: a (PM, MM) pair.Probe set: a collection of probe-pairs (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

• about 100 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

bias

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-linearity

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.

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 (“variance”)?

How to calibrate (“normalize”) for biases between samples?

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

- Conservative: rank transformation looses information - may yield less power to detect differentially expressed genes

- Aggressive: when 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.

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 78 kg, working against a gravitation field of strength 9.81 m/s2 :

What is wrong with microarray data?

(2465 - 559) · 78 · 9.81 m kg m/s2

= 1 458 433 kg m2 s-2

= 1 458.433 kJ

vsn normalisation

Robust affine regression normalisation (n-dim.) with the additive-multiplicative error model

Error model:

Trey Ideker et al. (2000) JCB

David Rocke and Blythe Durbin:

(2001) JCB, (2002) Bioinformatics

Use for robust affine regression normalisation:

W. Huber, Anja von Heydebreck et al. (2002) Bioinformatics

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

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, Var Xu = v(u).

Define

Var f(Xu ) independent of u

( )v( )

x

duf x

u

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 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 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

dif

fere

nc

e re

d-g

reen

rank(average)

“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 fold change

Background correction

Background correctionBackground correction

Irizarry et al. Biostatistics 2003

0 pm

500 fm 1 pm

750 fm

Background correction: RMA, Irizarry et al. (2002)Background correction: RMA, Irizarry et al. (2002)

~ 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

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

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

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cy

-2 0 2 4 6 8 10

02

46

x

Fre

qu

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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

• Show example report

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)