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Quality control and normalization

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Wolfgang Huber

European Bioinformatics Institute

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)

log-ratio

same-same

tumor-normal

biasaccuracy

precision variance

Can always increase sensitivity on the cost of specificity,

or vice versa,

the art is to find the best trade-off.

X

X

X

X

X

X

X

X

X

(It can also be possible to increase both by better choice of method / model)

3000

3000

x3

?

1500

200

1000

0

?

x1.5

A

A

B

B

C

C

But what if the gene is “off” (below detection limit) in one condition?

Fold changes are useful to describe continuous changes in expression

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

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.

What about a change from 0 to 500?

- conceptually

- noise, measurement precision

The problem is less that these steps are ‘not perfect’; it is that they vary from array to array, experiment to experiment.

How to compare microarray intensities with each other?

How to address measurement uncertainty (“variance”)?

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

Systematic

Stochastic

o similar effect on many

measurements

o corrections can be estimated from data

o too random to be ex-plicitely accounted for

o remain as “noise”

Calibration

Error model

amount of RNA in the biopsy

efficiencies of

-RNA extraction

-reverse transcription

-labeling

-fluorescent detection

probe purity and length distribution

spotting efficiency, spot size

cross-/unspecific hybridization

stray signal

- describe the possible outcomes of a set of measurements
- Outcomes depend on:
- true value of the measured quantity
- (abundances of specific molecules in biological sample)
- measurement apparatus
- (cascade of biochemical reactions, optical detection system with laser scanner or CCD camera)

- Purpose:
- Data compression: summary statistic instead of full empirical distribution
- Quality control
- Statistical inference: appropriate parametric methods have better power than non-parametric (this has practical, financial, andethical aspects)

bi per-sample

normalization factor

bk sequence-wise

probe efficiency

hik ~ N(0,s22)

“multiplicative noise”

ai per-sample offset

eik ~ N(0, bi2s12)

“additive noise”

The two component model

measured intensity = offset + gain true abundance

“multiplicative” noise

“additive” noise

raw scale

log scale

B. Durbin, D. Rocke, JCB 2001

two practically equivalent forms (h<<1)

Xu a family of random variables with EXu=u, VarXu=v(u). Define

var f(Xu ) independent of u

derivation: linear approximation

f(x)

x

1.) constant variance (‘additive’)

2.) constant CV (‘multiplicative’)

3.) offset

4.) additive and multiplicative

- - - f(x) = log(x)

———hs(x) = asinh(x/s)

P. Munson, 2001

D. Rocke & B. Durbin, ISMB 2002

W. Huber et al., ISMB 2002

generalized

log-ratio

difference

log-ratio

variance:

constant part

proportional part

glog

raw scale

log

glog

i: arrays

k: probes

s: probe strata (e.g. print-tip, region)

“usual” log-ratio

'glog' (generalized log-ratio)

c1, c2are experiment specific parameters (~level of background noise)

Estimated log-fold-change

log

glog

Signal intensity

Shrinkage estimators:

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:

= a shrinkage estimator for fold change

There are many possible choices, we chose “variance-stabilization”:

+ interpretable even in cases where genes are off in some conditions

+ can subsequently use standard statistical methods (hypothesis testing, ANOVA, clustering, classification…) without the worries about low-level variability that are often warranted on the log-scale

difference red-green

rank(average)

- n red-green arrays (R1, G1, R2, G2,… Rn, Gn)
- within/between slides
- for (i=1:n)
- calculate Mi= log(Ri/Gi), Ai= ½ log(Ri*Gi)
- normalize Mi vs Ai
- normalize M1…Mn

- all at once
- normalize the matrix of (R, G)
- then calculate log-ratios or any other contrast you like

o Microarrays can be operated in a linear regime, where fluorescence intensity increases proportionally to target abundance (see e.g. Affymetrix dilution series)

Two reasons for non-linearity:

oAt the high intensity end:saturation/quenching. This can and should be avoided experimentally - loss of data!

oAt the low intensity end:background offsets, instead of y=k·x we have y=k·x+x0, and in the log-log plot this can look curvilinear. But this is an affine-linear effect and can be correct by affine normalization. Non-parametric methods (e.g. loess) risk overfitting and loss of power.

linear

affine linear

genuinely non-linear

Normalization :=

1. correction for systematic experimental biases

2. provision of expression values that can subsequently be used for testing, clustering, classification, modelling…

3. provision of a measure of measurement uncertainty

Quality trade-off: the better the measurements, the less need for normalization. Need for “too much” normalization relates to a quality problem.

Variance-Bias trade-off: how do you weigh measurements that have low signal-noise ratio?

- just use anyway

- ignore

- shrink

Aesthetic criteria

Logarithm is more beautiful than arsinh

Practical critera

It takes forever to run method XX. Referees will only accept my paper if it uses the original MAS5.

Silly criteria

The best method is that that makes all my scatterplots look like straight, slim cigars

Physical criteria

Normalization calculations should be based on physical/chemical model

Economical/political criteria

Life would be so much easier if everybody were just using the same method, who cares which one

Comparison against a ground truth

But you have millions of numbers – need to choose the metric that measures deviation from truth.

FN/FP: do you find all the differentially expressed genes, and do you not find non-d.e. genes?

qualitative/quantitative: how well do you estimate abundance, fold-change?

Spike-In and Dilution series

… great, but how representative are they of other data?

Implicitely, from resampling / cross-validating with the actual experiment of interest

… but isn’t that too much like Münchhausen’s bootstrap?

o Data:

Spike-in series: from Affymetrix 59 x HGU95A,

16 genes, 14 concentrations, complex background

Dilution series: from GeneLogic 60 x HGU95Av2,

liver & CNS cRNA in different proportions and amounts

o Benchmark:

15 quality measures regarding

-reproducibility

-sensitivity

-specificity

Put together by Rafael Irizarry (Johns Hopkins) http://affycomp.biostat.jhsph.edu

Scatterplot, colored by PCR-plate

Two RZPD Unigene II filters (cDNA nylon membranes)

F(q)

q (log-ratio)

spotting pin quality decline

after delivery of 5x105 spots

after delivery of 3x105 spots

H. Sueltmann DKFZ/MGA

R RbR-Rbcolor scale by rank

another array: print-tip

color scale ~ log(G)

color scale ~ rank(G)

spotted cDNA arrays, Stanford-type

Batches: array to array differences dij = madk(hik -hjk)

arrays i=1…63; roughly sorted by time

Density representation of the scatterplot

(76,000 clones, RZPD Unigene-II filters)

See: package hexbin; also, smoothscatter in package prada

Main software from Affymetrix:

MAS - MicroArray Suite.

DAT file: Image file, ~108 pixels, ~200 MB.

CEL file: probe intensities, ~106 numbers

CDF file: Chip Description File. Describes which probes go in which probe sets (genes, gene fragments, ESTs).

1LQ file: Probe sequences and intended targets in the transcriptome

DAT image files CEL files

Each probe cell: 10x10 pixels.

Gridding: estimate location of probe cell centers.

Signal:

Remove outer 36 pixels 8x8 pixels.

The probe cell signal, PM or MM, is the 75th percentile of the 8x8 pixel values.

Background: Average of the lowest 2% probe cells is taken as the background value and subtracted.

Compute also quality values.

PMijg , MMijg= Intensities for perfect match and

mismatch probe j for gene g in chip i

i = 1,…, none 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., 11 PM and MM pairs, into a single expression measure.

compare between chips (samples) for detecting differential expression.

Affymetrix GeneChip MAS 4.0 software uses AvDiff, a trimmed mean:

o sort dj = PMj -MMj

o exclude highest and lowest value

o J := those pairs within 3 standard deviations of the average

Instead of MM, use "repaired" version CT

CT=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

dChip fits a model for each gene

where

qi: expression index for gene i

fj: 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.

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.

AvDiff-like

with A a set of “suitable” pairs.

Li-Wong-like: additive model

Estimate RMA = ai for chip i using robust method median polish (successively remove row and column medians, accumulate terms, until convergence). Works with d>=2

Expression measures RMA: Irizarry et al. (2002)

log(PM/MM)

From: R. Irizarry et al., Biostatistics 2002

0

position- and sequence-specific effects wi(s):

Naef et al., Phys Rev E 68 (2003)

wi

i

Bioconductor R package affy.

Background estimation.

Probe-level normalization.

Expression measures

Two main functions: ReadAffy, expresso.

See also: gcrma, tilingArray, vsn.

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).