1. Statistical Methods in Microarray Data Analysis Mark Reimers, Genomics and Bioinformatics, Karolinska Institute

2. Four Recent Contributions Exploratory graphics Multiple comparisons corrections –Randomization-based significance tests Normalization –loess normalization for cDNA microarray Models for probe-level Affymetrix data –Robust estimation

3. Multiple comparisons Each gene has a 5% chance of exceeding the threshold at a p-value of.05 –Type I error 10,000 genes on a chip 500 genes should exceed.05 threshold

4. Corrections to p-Value Bonferroni correction –p i * = Np i, if Np i < 1, otherwise 1 –Too conservative! Sidak –p i * = 1 – (1 – p i ) N –Still conservative if genes are co-regulated (correlated)

5. Step-Down p-Values p-values for many genes: p 1, …, p N Order the smallest k as p (1), …, p (k) How likely are we to get k p-values this small by chance? An improvement in power over single-step procedures

6. Quantile Plot Plot sample t- scores against t- scores under random hypothesis Statistically significant genes stand out Corresponding quantiles of t-distribution Sample t-scores Changed genes

7. Volcano Plot Displays both biological importance and statistical significance log 2 (fold change) log 2 (p-value) or t-score

8. Normalization: Comparing Chips Measures differ consistently between chips due to: –Different amounts of RNA –Hybridization conditions –Scanner settings –Murphy’s Law Normalization: compensate for systematic technical differences in measurement process Re-scaling to mean or median leaves strong evidence of systematic technical variation

9. Normalization: Signal Distributions Distributions of log intensity of all probes among a set of 21 replicate chips Each color represents probe density on one chip Re-scaling would shift distribution shape to right or left on this plot

10. Density function Distribution function F 1 (x) Raw data Reference distribution F 2 (x) Formula: x norm = F 2 -1 (F 1 (x)) Quantile Normalization Assumes: gene distribution changes little

11. Visible Effect of Quantile Norm. Ratio-Intensity plots are straightened as by- product

12. Current Work Hybridization reaction varies across some chips Very common on cDNA 10%-20% of well- done Affy chips Synthetic image of ratio of individual probes to their median across chips: Yellow areas show ratios more than twice those of red areas

13. Models: Many Probes for One Gene GeneSequence Multiple oligo probes Perfect Match Mismatch5´3´ How to combine signals from multiple probes into a single gene abundance estimate?

14. Probe Variation Individual probes don’t agree on fold changes Probes vary by two orders of magnitude on each chip –CG content is most important factor in signal strength Signal from 16 probes along one gene on one chip

15. Models for Multiple Probes Issues: –Accuracy – does the model give accurate estimates of relative gene expression, when this is known? –Noise – what is the variance of replicates? –Theoretical basis – do we understand why we are doing what we do? Statistical experience with methodology Theory of hybridization process underlying observations

16. Three Competing Models Affymetrix MicroArray Suite –versions 4, and 5 dChip –Li and Wong, HSPH Bioconductor: affy package (RMA) –Bolstad, Irizarry, Speed, et al

17. Model 1: MicroArray Suite – Version 4 GeneChip ® older software uses Avg.diff with A a set of suitable pairs chosen by software –30%-40-% of probe differences can be negative

18. Model 2: MicroArray Suite – Version 5 MicroArray Suite version 5 uses MM* is an adjusted MM that is never bigger than PM Tukey biweight is a robust average procedure with weights: f(x)=c 2 /6[1-(1-x 2 /s 2 ) 3 ]; |x|<c For this (typical) example, it is not clear what the average would mean PM-MM values for probe pairs

19. Linear Models Extension of linear regression Essential features: –variance constant –errors independent –Small number of factors combine in algebraic form to give levels frequently additive

20. Model for Probe Signal Each probe signal is proportional to –i) the amount of target sample –ii) the hybridization efficiency of the specific probe sequence to the target –Each probe has a specific affinity to its gene target NB: Sensitivity need not imply Specificity 11 22 Probes 1 2 3 chip 1 chip 2

21. Robust Statistics Outlier: a measure that is far beyond the typical random variation –common in biological measures –10-15% in Affy probe sets Robust methods try to fit the majority of data points –Issue is to identify which points to down-weight or ignore Median is very robust – but inefficient –Trimmed means are almost as robust and much more efficient

22. Robust Linear Models Criterion of fit –Least median squares –Sum of weighted squares –Least squares and throw out outliers Method for finding fit –High-dimensional search –Iteratively re-weighted least squares –Median Polish

23. Why Robust Models for GeneChips? 10% - 15% of individual signals in a probe set deviate greatly from pattern Often outliers lie close together Causes: –Scratches –Proximity to heating elements –Uneven fluid flow

24. Why Robust Models for GeneChips? 10% - 15% of individual signals in a probe set deviate greatly from pattern Often outliers lie close together Causes: –Scratches –Proximity to heating elements –Uneven fluid flow

25. Li & Wong (dChip) Model: PM ij =  i  j +  ij - Original model (dChip 1.0) used PM ij - MM ij =  i  j +  ij by analogy with Affy MAS 4 Outlier removal: –Identify extreme residuals –Remove –Re-fit –Iterate Distribution of errors  ij assumed independent of signal strength

26. Robust Multi-chip Analysis Each probe responds roughly linearly –over a moderate range –some probes are outliers Linear Model: –signal =  i  j +   i amount of transcript in sample i;  j amplification of probe j Robust Fit: –identify outliers by heuristic – remove –standard robust method – iteratively re-weighted least squares

27. For each probe set, re-write PM ij =  i  j as: log( PM ij )= log(  i ) + log(  j ) Fit this additive model by iteratively re- weighted least-squares or median polish In practice, fit: Bolstad, Irizarry, Speed – (RMA) Where nlog() stands for logarithm after normalization NB. Now homoschedastic on log scale

28. It Makes a Difference Two fairly consistent genes in each of 71 samples MAS 5 values dChip values

29. Models Compared on Gene Variance Std Dev of gene measures from 20 replicate arrays Green: MAS5.0; Black: Li-Wong; Blue, Red: RMA Courtesy of Terry Speed LowAbundance:High

30. Improvement in Models Affymetrix Suite gets better every year –MAS 7 is expected to be a multi-chip model MAS 5.0 estimation does a reasonable job on probe sets that are bright –Metabolic and structural genes –These are most often reported in papers dChip and RMA do better on genes that are less abundant –Signalling proteins –transcription factors

31. Expression Comparison 1 – MAS 4 Courtesy of Terry Speed Ratio-Intensity Plot comparing two chips from spike-in experiment White dots represent unchanged genes Red numbers flag spike-in genes

32. Expression Comparison 2 – MAS 5 Courtesy of Terry Speed t-scores Theoretical t-distribution changed genes

33. Expression Comparison 3 – Li-Wong Courtesy of Terry Speed

34. Expression Comparison 4 - RMA Courtesy of Terry Speed

35. Current Work: Improving the Model How to use the MM information profitably –Combine estimates from PM and MM probes? Assessments of probe quality Accurate estimates of probe background Normalization method based on 2-d loess to correct spatial inhomogeneity

36. Relation Between PM and MM Across One Experiment Set Colored symbols are one probe

37. Probe Specific Background Horizontal lines represent probes; colored symbols correspond to arrays After subtracting individual backgrounds, ratios between corresponding arrays are more consistent between probes Fitted Data Probe BG subtracted

38. Where Are We? Affymetrix almost finished? –Probe variation ~40% => gene variation ~ 10% –RMA gives ~20% Work to be done: –Systematic biases for cDNA arrays –Platform reconciliation –Using QC and variation measures for individual probes in combined expression measures Frontiers: –Image analysis

39. Near Term Work to be Done New hybridization technologies for measuring gene expression Protein chips –More complex cross-hybridization Other high-throughput technologies –eg RNAi chips –Cell arrays Using sequence information to understand cross- hybridization

40. Integrated Analysis Integrating statistical measures of data uncertainty in machine-learning techniques for network analysis Statistical inference for pathways and gene ontology categories Robust data analysis to mine for genome- scale patterns in expression

41. Acknowledgements KI –Karin Dahlman –Yudi Pawitan –Arief Gusnanto –Lennie Fredriksson Berkeley –Terry Speed –Ben Bolstad Johns Hopkins –Rafael Irizarry

44. Affymetrix Arrays Single stranded, fluorescently labeled DNA target 20µm Each probe cell or feature contains millions of copies of a specific oligonucleotide probe Image of Hybridized Probe Array Image of Hybridized Probe Array Over 400,000 different probes complementary to genetic information of interest Oligonucleotide probe * * * * *1.28cm GeneChip Probe Array Hybridized Probe Cell

45. Evidence for Spatial Variation Synthetic Image of Affy chip

46. Loess Normalization for Areas Fit two-parameter loess smoother With 5-10 df