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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

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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


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


Source: Affymetrix website


Data for one probe set, one array

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


Box plot showing array and probe effects


Modeling probe effects

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


Model for one gene in multiple arrays


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


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).


Residuals of the fitting


Model fitting amounts to fixing ’s and regress to estimate 


Fig 1.5 Array outlier: large standard errors of 4


Fig. 1.6 Probe outlier: large standard errors of 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

  • 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.


Source: Affymetrix website


Finding Confidence Interval of Fold Change


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


Functional significant clusters

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


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

  • R. Sasik and J. Corbeil (UCSF)

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:

  • 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 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.


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.


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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