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Models and methods for summarizing

GeneChip probe set data

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Some Gene Expression Analysis Tasks

• Detection of gene expression – presence calls.

• Differential expression detection – comparative calls.

• Measurement of gene expression.

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Objective:

To compute probe set summaries which are good indicators of gene expression from background corrected, normalized, prefect match probe intensities for a set of arrays:

PM*ijk i=1,…,I, J=1,…,J, k=1,…K

Where i denotes probes in probe sets, j denotes arrays and k denotes probe sets.

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Affymetrix spike-in data set used for illustration - 14 genes spiked in at differentconcentrations into a common pool ofpancreas cRNA

A B C D E F G H I J K L M,N,O,P Q,R,S,T37777_at 0 0.25 0.5 1 2 4 8 16 32 64 128 256 512 1024684_at 0.25 0.5 1 2 4 8 16 32 64 128 256 512 1024 01597_at 0.5 1 2 4 8 16 32 64 128 256 512 1024 0 0.2538734_at 1 2 4 8 16 32 64 128 256 512 1024 0 0.25 0.539058_at 2 4 8 16 32 64 128 256 512 1024 0 0.25 0.5 136311_at 4 8 16 32 64 128 256 512 1024 0 0.25 0.5 1 236889_at 8 16 32 64 128 256 512 1024 0 0.25 0.5 1 2 41024_at 16 32 64 128 256 512 1024 0 0.25 0.5 1 2 4 836202_at 32 64 128 256 512 1024 0 0.25 0.5 1 2 4 8 1636085_at 64 128 256 512 1024 0 0.25 0.5 1 2 4 8 16 3240322_at 128 256 512 1024 0 0.25 0.5 1 2 4 8 16 32 64407_at 0 0.25 0.5 1 2 4 8 16 32 64 128 256 512 10241091_at 512 1024 0 0.25 0.5 1 2 4 8 16 32 64 128 2561708_at 1024 0 0.3 0.5 1 2 4 8 16 32 64 128 256 512

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Affy comment on non-responding probes:

• Affymetrix: “Certain probe pairs for 407_at and 36889_at do not work well. It is recommended that these two probe sets be excluded for final statistical tally.”

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To log or not to log?

In addition to providing good expression values, we would like the model to be easy to understand and analyse – Would like to fit a standard linear model:

• Homogeneity of variance

• Additivity

• Normality

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Homogeneity of variance

Look at association between the variance and the mean of the intensities – plot IQR of PM* across 59 replicates against the median of PM* across 59 replicates for probe sets spanning the range of intensities.

Repeat for log2(PM*).

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Intensity scale – ALL

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Log Intensity scale – ALL

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Additivity

Look at log-log plots of PM* vs concentrations for 14 spike-in fragments

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PM.bgc.norm vs Conc log-log plot grp 1

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PM.bgc.norm vs Conc log-log plot grp 2

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Suggested additive model

Log-log plots of PM* vs concentrations suggest the following model:

log2(PM*ij) = pi + cj + ij (1)

With pi a probe affinity effect, cj the log2 scale expression level for chip j, and ij an iid error term.

For identifiability we fit with constraint

i pi=0.

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Normality

We can examine residuals from a least squares fit to model specified in (1) to verify the adequacy of the model in terms of additivity of effects and stability of variance.

The shape of the distribution of the residuals can also be compared with a Gaussian distribution to see how far off we are from this ideal.

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Res vs chip effects - grp 1

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Res vs chip effects - grp 2

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Figure – qqnorm residuals form additive fit

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Res qqnorm - grp 1

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Res qqnorm - grp 2

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Analyzing the untransformed PM* values

On the untransformed scale, one can fit a multiplicative model (Li-Wong):

PM*ij = i·j + ij (2)

The model is fitted by least squares by iteratively fitting the s and the s, regarding the other set as known. Fitting steps are interleaved with diagnostic checks used to exclude points from subsequent fits.

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PM vs logConc - grp 1

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Res vs chip effects - grp 1

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qq - grp 1

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Why robust?

• Bad probes – probe outliers

• Bad chips – chip outliers

• Image artifacts – individual outliers

We would like a fitting procedure which yields good estimates in the presence of various types of outliers – individual points, probes, and chips.

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Robust estimation

• Huber, Hampel, Rousseeuw

• Gross errors, round off errors, wrong model.

• Distinction between approach based on identification and exclusion of outliers and the modeling approach.

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M estimators

• A general class of robust estimators are obtained as solutions to:

min i(Yi-Xi)

Where is a symmetric function. Or solving the following system:

i(Yi-Xi)· Xi=0

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Robust fit by IRLS for each probe set

Starting with robust fit, at each iteration:

S = mad(rij)·c – robust estimate of scale of

uij = rij/S – rescaled residuals

wij =(|uij|)/|uij| – weights used in next LS fit.

Theoretical considerations can lead to specification of . In practice, one selects function with desirable characteristics.

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Example functions

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Options for fitting models to probe sets

Recall model

log2(PM*ij) = pi + cj + ij (1)

Can fit by:• Least squares• Least absolute deviation ((x)=|x|)• IRLS using various functions

• Can also get a single chip robust probe set summary.

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Robust fit example A

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Actual vs fitted

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Starting weights

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Ending weights

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Robust analysis of multi way tables

• Tukey & co – median polish.

• Tukey – one degree of freedom test – additivity or not – no partial judgement.

• Gentleman and Wilks – effect of one or two outliers on residuals.

• C. Daniel** – estimate which cells are affected by interactions, and estimate the interactions in a set of cells.

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Modified weights

Standard IRLS procedures determines weights from each cell of the two way table individually.

We can also look at residuals across cells in a row (column), to determine a weighting adjustment for the entire row (column):

rwi =(|u|i•)/|u|i• , cwj =(|u|•j)/|u|•j

And get a composite weight for each cell”

wwij = rwi · wij

wwwij = cwj · rwi · wij

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Heuristic derivation of weights

Consider the model with interactions:

log2(PM*ij) = pi + cj + ij + ij

Can think of Tij = |rij|/S as a test statistic for

H0: ij = 0 vs H1: ij 0

and wij = (Tij)/ Tij as a transformation of this test statistic into a weight.

Similarly, one could use Ti = |ri•|/S =madi/mad to test

H0: ij = 0, j=1,…J vs H1: ij 0 for some j

and map this statistic into a weight.

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See how it does

• Look initial (individual) weights & fit vs adjusted weights and fit and then to convergence.

• Also look at probe weights in all spike-in probe sets.

• Column weights

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Starting weights

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Ending weights

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Robust fit – composite weights

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Probe Weights

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Low-weight Probes - 1

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Low-weight Probes - 2

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Low-weight Probes - 3

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Chip Weights

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Note on multi chip context

Note that the residual variance in the model without probe effects, the single chip analysis set-up, is ~ 6x the residual variance in the model with probe effects. Ie.

log2(PM*ij) = pi + cj + ij

Vs.

log2(PM*ij) = cj + ij

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Compare fits on sample probe sets

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Affy Probe Data Analysis – X hybridizing probe sets

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X-Hybe probe 3

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X-Hybe probe 3 – PM vs Phi, Theta

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X-Hybe probe 4

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X-Hybe probe 4 - PM vs Phi, Theta

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X-Hybe probe 5

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X-Hybe probe 5 – PM vs Phi, Theta

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Affy Probe Data Analysis – Spike-n probe sets

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Fit to spike 12

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Fit to spike 7

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Fit to spike 1

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Fit to spike 2

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Fit to spike 3

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Fit to spike 4

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Fit to spike 5

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What have we gained?

• Look at residuals from fit across large number of probe sets to show benefits of IRLS over median polish.

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boxplot Residuals from 1000 probe sets

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boxplot estimated chip effects for 1000 probe sets

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IQR chip effects for 1000 probe sets

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What have we gained?

• In order to have a high enough breakdown point to ignore 6 out of 16 probes, and improve on the median polish estimate, we pay a high price in variability.

Q. Can we find a better weighting scheme? Or show MP is optimal?

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References

1. Irizarry, R. et.al (2003) Summaries of Affymetrix GeneChip probe

level data, Nucleic Acids Research, 2003, Vol. 31, No. 4 e15

2. Irizarry, R. et. al. (2003) Exploration, normalization, and summaries of high density oligonucleotide array probe level data. Biostatistics, in press.

3. C. Li and W.H. Wong, Model-based analysis of oligonucleotide arrays: Expression index computation and outlier detection, Proceedings of the National Academy of Science U S A, 2001, Vol 98, pp 31-36.

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References - Robustness

1. P. J. Rousseeuw and A. M.Leroy, Robust Regression and Outlier Detection, John Wiley & Sons, 1987

2. P. J. Huber, Robust Statistics, John Wiley & Sons, 1981.

3. F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, W. A. Stahel, Robust Statistics: The approach based on influence functions, John Wiley & Sons, 1986.

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References – Robustness in multiway tables

4. J. D. Emerson, D. C. Hoaglin, Analysis of two-way tables by medians, in Understanding robust and exploratory data analysis, publisher=John Wiley \& Sons, Inc., edited by D. C. Hoaglin and F. Mosteller and J. W. Tukey, 1983.

5. N. Cook, Three-way analyses, in Exploring data tables, trends, and shapes, ed. D. C. Hoaglin and F. Mosteller and J. W. Tukey, 1985.

6. J. W. Tukey, One degree of freedom for non-additivity. Biometrics, 1949, 5, 232-242.

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References – Robustness in multiway tables

7. C. Daniel, Patterns in residuals in the two-way layout, Technometrics, 1978,20(4), 385-395.

8. J. F. Gentleman and M. B. Wilk, Detecting outliers in a two-way table: I. Statistical behavior of residuals, Tehnometrics, 1975, 17(1), 1-14.

9. J. F. Gentleman and M. B. Wilk, Detecting outliers: II. Supplementing the dierect analysis of residuals, Biometrics, 1975, 31, 387-410.