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Model-based analysis of oligonucleotide arrays, dChip software. Cheng Li (Joint work with Wing Wong). Statistics and Genomics – Lecture 4 Department of Biostatistics Harvard School of Public Health January 23-25, 2002. Source: Affymetrix website. Custom software: raw image.

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Model based analysis of oligonucleotide arrays dchip software

Model-based analysis of oligonucleotide arrays, dChip software

Cheng Li

(Joint work with Wing Wong)

Statistics and Genomics – Lecture 4

Department of Biostatistics

Harvard School of Public Health

January 23-25, 2002






Normalization methods
Normalization methods between arrays

  • 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



Invariant set normalization method
Invariant set normalization method between arrays

  • 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”



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 (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.


Source: Affymetrix website (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.


Data for one probe set one array
Data for one probe set, one array (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.

PM/MM differences eliminate background and cross-hybridization signals


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

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


Data for one gene in many arrays
Data for one gene in many arrays linear to mRNA concentrations at certain dynamic range


Box plot showing array and probe effects
Box plot showing array and probe effects linear to mRNA concentrations at certain dynamic range


Modeling probe effects
Modeling probe effects linear to mRNA concentrations at certain dynamic range

1) 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
Principal component analysis linear to mRNA concentrations at certain dynamic range(42 points in 20-space) suggests the data matrix has approx. rank 1


Model for one gene in multiple arrays linear to mRNA concentrations at certain dynamic range


Figure 1.1. Black curves are the PM linear to mRNA concentrations at certain dynamic range 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
Using PM/MM Differences linear to mRNA concentrations at certain dynamic range

  • 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


Model for PM/MM differences (1.2) linear to mRNA concentrations at certain dynamic range


Figure 1. linear to mRNA concentrations at certain dynamic range2. 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).


Residuals of the fitting
Residuals of the fitting linear to mRNA concentrations at certain dynamic range


Model fitting amounts to fixing s and regress to estimate
Model fitting amounts to fixing linear to mRNA concentrations at certain dynamic range’s and regress to estimate 


Fig 1.5 Array outlier: large standard errors of linear to mRNA concentrations at certain dynamic range4


Fig. 1.6 Probe outlier: large standard errors of linear to mRNA concentrations at certain dynamic range17

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 automatically handles image contamination

  • 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) 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.

(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.


Source: Affymetrix website 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.


Finding confidence interval of fold change
Finding Confidence Interval of Fold Change 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.


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



Incorporate biological knowledge and database when analyzing microarray data

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


Functional significant clusters microarray data

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


Problems with lwr model

Problems with LWR model:

  • LWR model:

  • The expression index can still be negative.

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

Slides prepared by Xuemin Fang


Statistical model
Statistical model: a multiplicative noise model

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

  • Our model is

  • where

  • Least squared estimation of the parameters.

  • Constraint:


Algorithm when analyzing a batch of n s samples
Algorithm -- When analyzing a batch of a multiplicative noise modelns 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
A background plus signal model: AvDiff, Li & Wong's,

  • 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.


Expression index
Expression index: AvDiff, Li & Wong's,

  • 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.


Acknowledgement AvDiff, Li & Wong's,

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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