Model-based analysis of oligonucleotide arrays, dChip software Statistics and Genomics – Lecture 4...

Model-based analysis of oligonucleotide arrays, dChip

software

Statistics and Genomics – Lecture 4

Department of Biostatistics

Harvard School of Public Health

January 23-25, 2002

Cheng Li

(Joint work with Wing Wong)

Source: Affymetrix website

Custom software: raw image

Custom software: getting representative value of a probe cell

Normalization is needed to minimize non-biological variation between arrays

Normalization methods

• Current software uses linear normalization

• Nonlinear curve fitting based on scatter plot is still inadequate because 1) effects of differentially expressed genes

may be “normalized” 2) regression phenomenon and asymmetry

Regression phenomenon and asymmetry

Invariant set normalization method

• A set of points (xi, yi) is said to be order-preserving if yi < yj whenever xi < xj

• The maximal order-preserving subset can be obtained by dynamic programming

• If a gene is really differentially expressed, it’s cells tend not to be included into an large order-preserving subset

• Our method is based on an approximately order preserving subset, called “Invariant set”

Fig. 2.9 Normalization of a pair of replicated arrays

Figure 2.10. Two different samples. The smoothing spline in (A) is affected by several points at the lower-right corner, which might belong to differentially expressed genes. Whereas the “invariant set” does not include these points when determining normalization curve, leading to a different normalization relationship at the high end.

A pair of split-sample replicate arrays

adeno-268t1(dup b4)-A93-3 normalized

0 2000 4000 6000

adeno-268t1(dup b4)-A93-3

0 2000 4000 6000 8000 10000 12000

adeno-268t2d-A189-4

0 2000 4000 6000

adeno-268t1(dup b4)-A93-3

0 2000 4000 6000 8000 10000 12000

PM/MM differences eliminate background and cross-hybridization signals

Data for one probe set, one array

Lockhart et al. (1996) Nature Genetics, Vol 14: 1675-1680

Validation experiments suggest Average Differences are linear to mRNA concentrations at certain dynamic range

Data for one gene in many arrays

Box plot showing array and probe effects

Modeling probe effects1) Probes sequence has different hybridization efficiency

2) cross hybridization, SNP, alternative splicing

3) Probe position effect, 3’ bias

Probe effects can dominate biological variation of interest

Previous method : use multiple probes, average to reduce “noise”

Our methods: statistical models for probe effects, “meta-analysis”, learning algorithms, estimation of expression level conditional on knowledge of probe effect

Principal component analysis (42 points in 20-space)

suggests the data matrix has approx. rank 1

Comp. 1Comp. 2Comp. 3Comp. 4Comp. 5Comp. 6Comp. 7Comp. 8Comp. 9Comp. 10

0.9420.952 0.961 0.968 0.973 0.978 0.982 0.985 0.988

0,0,0,0:constraint

jpair probe zingcharacteri factors scaling: ,

i, chipin l)index(leve expression :

j,pair probefor levelion hybridizat-rossbaseline/c:

j, probe i, chipfor valuesPM and MM observed : and 10

ijjijijij

ijjijij

Model for one gene in multiple arrays

5 10 15 20

Figure 1.1. Black curves are the PM and MM data of gene A in the first 6 arrays. Light curves are the fitted values to model (1). Probe pairs are labeled 1 to 20 on the horizontal axis.

Using PM/MM Differences

• PM/MM differences eliminate most background and cross-hybridization signals

• Affyemtrix’s GeneChip software is using average differences as basis for determining fold changes, and their validation showed average differences are linear to mRNA concentrations at certain dynamic range

errors normal random :

ilityidentifiabfor J constraint subject to

, jpair probefor index y sensitivit probe :

level expresion true toalproportion be toassumed

level, differnece PM/MM tocomparable

, iarray for index level expression :

:gene onefor model tivemultiplica An

ijjiijijij MMPMY

Model for PM/MM differences (1.2)

Figure 1.2. Black curves are the PM-MM difference data of gene A in the first 6 arrays. Light curves are the fitted values to model (2).

5 10 15 20

Probe pair

5 10 15 20

Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

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Probe pair

5 10 15 200

Probe pair

5 10 15 20

Multiplicative model fits the data well, explained variance: 95%

Residuals of the fitting

Model fitting amounts to fixing ’s and regress to estimate

),0(~,1, 2 NJjy jjjj

Var( and )~

E( with, ~ 2

)1/())((ˆ with)/ˆ( )~

( Error Std 222 JobserevedfittedJj

5 10 15 20

theta1000 2000 3000 4000 5000

250 array 4

Fig 1.5 Array outlier: large standard errors of 4

Fig. 1.6 Probe outlier: large standard errors of 17

phi0.0 0.5 1.0 1.5 2.0 2.5

probe 17

5 10 15 20-4

5 10 15 20

Also see gene 6898

Fig. 1.4 Array outlier image shows that the model automatically handles image contamination

Compare Model-based expression with Average Difference

• The array set 5 has 29 pair of arrays replicated at split-mRNA level

• The differences between the replicated arrays provides a opportunity to assess different expression calculation method

Figure 2.5. Log (base 10) expression indexes of a pair of replicate arrays (array 1 and 2 of array set 5) for MBEI method (A) and AD method (B). The center line is y=x, and the flanking lines indicate the

difference of a factor of two.

(A) (B)

Figure 2.6. Boxplots of average absolute log (base 10) ratios between replicate arrays stratified by presence proportion for (A) MBEI method, (B) AD method.

Finding Confidence Interval of Fold Change

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let,known with),(~ˆ),,(~ˆ Suppose

Expression 1 Std Error 1 Expression 2 Std Error 2 Fold Change Lower CB Upper CB

Gene 1 859.635 41.7808 347.57 36.0887 2.47327 2.06844 3.02672 Gene 2 405.72 31.2305 164.014 44.2505 2.47369 1.66938 4.49127

Gene 3 283.931 28.5281 114.705 18.4661 2.47531 1.83926 3.48466

Gene 4 45.9821 64.2419 18.5727 84.5308 2.47579 0 Infinity

Gene 5 225.178 57.489 90.9045 36.1766 2.47709 1.18104 7.48749

Gene 6 247.002 50.6518 99.6642 19.5384 2.47834 1.51079 4.0211

Gene 7 49.9739 21.5345 20.1514 23.5651 2.47992 0.487603 Infinity Gene 8 276.491 18.6883 111.373 36.1004 2.48256 1.59069 5.34635

Gene 9 436.071 32.9779 175.384 21.0669 2.48638 1.98665 3.18811

Gene 10 75.6914 17.7215 30.4395 17.9707 2.48662 1.07209 86.1656

Gene 11 80.673 25.3085 32.4314 16.9626 2.4875 0.960787 18.1833

Gene 12 181.528 42.4837 72.8751 28.1787 2.49094 1.24668 7.11945 Gene 13 1122.28 99.2835 449.889 63.2821 2.49456 1.92075 3.35055

Gene 14 168.234 40.629 67.4387 30.2982 2.49462 1.17639 9.81547

Table 2.1 Using expression levels and associated standard errors to determine confidence intervals of fold changes

Resampling hierarchical clustering using standard errors of model-based expression

Incorporate biological knowledge and database when analyzing microarray data

Right picture: Gene Ontology: tool for the unification of biology, Nature Genetics, 25, p25

Found 13 structural protein genes out of a 49-cluster (all: 198/2622, PValue: 1.00e+000)

Functional significant clusters

Problems with LWR model:

• LWR model:

• The expression index can still be negative.

• Genes with negative index can still be classified as present.

ij ij i j jPM MM

Statistical analysis of high-density oligonucleotide arrays: a multiplicative noise

R. Sasik and J. Corbeil (UCSF)

Slides prepared by Xuemin Fang

Statistical model:

• Based on the same assumption as the LW model, that PM intensity is directly proportional to the concentration ci of the transcript, . Write the relation in the form

• Our model is• where

• Least squared estimation of the parameters.• Constraint:

~ij j iPM c

2 2 2log ~ log logij j iPM c

ij j i ij

~ (0, )ij N

0ij jij

Algorithm -- When analyzing a batch of ns samples:

• Normalize all samples to the first one on the list by requiring the sum of all PM intensities be the same as that of the first sample.

• Select the background probes using Naef’s method (MM is used in this step).

• Subtract the median of the background probe intensity from every PM probe in the array.

• Probes that become negative are eliminated. • Fit the model and probes contributes most to the sum of

squares are eliminated.• Normalize again and repeat 1-5, until the distribution of

residuals is Gaussian.

Bias, variance and fit for three measures of expression: AvDiff, Li & Wong's,

AvLog (PM -bg)

Rafael Irizarry, Terry Speed (Johns Hopkins)

Slides prepared by Xuemin Fang

A background plus signal model:

• Here represents background signal caused by optical noise and non-specific binding.

• The mean background level is represented with and the random component with .

• The transcript signal contains a probe affinity effect , the log expression measures , and an error term.

• Both error terms and are independent standard normal.

ijn ijn ijnPM bg s 2log ( )in ijnbg z

2log ( )ijn n jn ijns jn

ijnz ijn

Expression index:

• A naïve estimate of is given by

with the mode of the log2(MM) distribution.

• An estimate of this distribution is obtained using a density kernel estimate.

log( )J

in n ijn ij

2log ( )i

Acknowledgement

Data source:

Stan Nelson (UCLA)Sven de Vos (UCLA) Dan Tang (DFCI)Andy Bhattacharjee (DFCI)Richardson Andresa (DFCI) Allen Fienberg (Rockefeller)

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