Use of the Half-Normal Probability Plot to Identify Significant Effects for Microarray Data

1. Use of the Half-Normal Probability Plot to Identify Significant Effects for Microarray Data C. F. Jeff Wu University of Michigan (joint work with G. Dyson)

2. Outline • Current Methods • Proposed Methodology • Analysis Plan • Example • Conclusions

3. What are microarrays? • Two major types • Oligonucleotide gene chips • Spotted glass arrays • Perfect match (PM) and mismatch (MM) probes are spotted onto a gene chip • ~20 probes make up a probe set (or gene) • MM probe for each gene has the middle base set to the complement of its PM probe • Hybridize labeled RNA corresponding to PM probes • Glass arrays involve the competitive hybridization of two RNA pools to cDNA spotted onto a glass slide • Typically thousands on genes on a slide

4. Multiplicity Problem • When we make more than one comparison in a hypothesis testing situation, p-value interpretation falls through • Control of family error rate is necessary in order to preserve nominal type I error rate • Various approaches to correct the chance of making a type I error for multiplicity, including Tukey, Bonferroni and Holms

5. Microarray Analysis Techniques • Westfall Young step down (WY) • Significance Analysis of Microarrays (SAM) • Empirical Bayes (EB) • Bayesian (MCMC) • Mixture Modeling • Dimension reduction techniques • Machine learning

6. and Westfall Young (WY) • Compute ranks of original test statistic rjsuch that • Construct b balanced permutations of the samples, computing the same test statistic as above for each b • Compute • Repeat B times and calculate the adjust p-value as • Less conservative than Bonferroni

7. Significance Analysis of Microarrays (SAM) • Use a t-like statistic • Use balanced permutation method from previous slide to estimate null distribution, assuming all effects are null • Call genes that fall outside D bars significant

8. Half-Normal Analysis

9. Microarray Specific Problem

10. Analysis Plan • Robust measures of location and scale • Summary statistic • Two half-normal plots (for upward-regulated and downward-regulated genes) • Segment determination • Find • insignificant, borderline, significant • Repeat the procedure, using as base

11. Robust Measures of Location and Scale • Perform transformation and suitable normalization • Compute median and Maximum Absolute Deviation (MAD) for each gene • Reasonable estimates • Less affected by outliers than mean and SD • Interested in robustness rather than efficiency

12. Summary Statistic • Compute quasi two-sample t-statistic using robust values from above: • c is chosen to minimize for the middle 100*(1-2e)% of the ssl. • Tusher et al. (2001) chose c to minimize the coefficient of variation • Efron et al. (2001)used the 90th percentile of the gene standard error estimates for c

13. Two Half-Normal Plots • Construct two half-normal plots: one for the p positive and r negative ssl. • Run the procedure separately on each set • Denote the ordered p positive effects by • Plot abssiagainst half-normal distribution quantiles, i.e. the points • Goal: obtain set of noise effects • Yield a baseline against which to test the rest of the effects

14. Segment Determination: • Given b, initialize null set as points abss1: abssk • Regress null set on 1:k half-normal quantiles (Q1:Qk) • Produce predicted values at the remaining quantile values (Qh:h>k) • Compute predicted statistics with • Find

15. Segment Determination: (cont) • The initial null set of k genes becomes k + m (= ) null genes • Now re-do the segment determination procedure, using the k + m genes as base null set • Continue until no new genes are added • Do for each k less than p-1 • Store the end point • Set the most frequent to

16. Sample • Let k = 200, total effects = 500 • First 200 ordered positive effects regressed on first 200 half-normal quantiles • Test ordered effects 201 to 500 using absolute value of predicted statistics • For example, effect 239 is the largest h less than the t-critical value • So would initially be 239 • Redo the above, with k = 239 effects; so we test effects 240 to 500 • Say statistic 242 is the largest h less than t-critical value based on new regression line • So the new would be 242 • Redo the above again with k = 242, test effects 243 to 500 • No statistics are less than t critical value • So is 242

17. Example

18. Find • Will test all effects after using same statistics • To adjust for multiple testing, define NC as the number of consecutive significant effects necessary to call all subsequent effects significant • Use the Bonferroni adjustment (does not require independence): • Instead of doing thousands of comparisons, only need to do NC to determine significance • Define • Now we have identified the change points in the graph for segment detection

19. Example: Downward- regulated Speed Mouse Data

20. Example: Downward Regulated Speed Mouse Data (cont)

21. Error Rate Estimation: FDR • False Discovery Rate (FDR) is the expected proportion of falsely rejected hypotheses • Permute the condition labels, maintaining balance • Example: 8 replicates in conditions A and B • Each A’ and B’ will have 4 replicates from A and 4 from B • Compute the robust statistics, keeping the same c from the actual data • Determine the average number of effects that fall above the positive or below the negative boundary of the significant sets • Divide that number by the total number of called significant effect

22. Speed Data: Analysis and Comparison • WY found 8 genes significant, with Type I error = 0.05

23. Lemon Data: Analysis and Comparison • WY found 253 genes significant, with Type I error = 0.05

24. Conclusions • Proposed a new method for determining differential expression in genes • Dealt with the multiplicity problem by using only a small subset of genes • Can extend to other large data sets • Allow scientists to play a role in sequential decision making • Incorporate a priori knowledge of experiment with selection of c