Tom Kepler Santa Fe Institute Normalization and Analysis of DNA Microarray Data by

1. Tom Kepler Santa Fe Institute Normalization and Analysis of DNA Microarray Data by Self-Consistency and Local Regression kepler@santafe.edu

3. Rat mesothelioma cells control Rat mesothelioma cells treated with KBrO2

4. Normalization • Method to be improved: • Assume that some genes will not change under the treatment under investigation. • Identify these core genes in advance of the experiment. • Normalize all genes against these genes assuming they do not change

5. Normalization • New Method: • Assume that some genes will not change under the treatment under investigation. • Choose these core genes arbitrarily. • Normalize (provisionally) all genes against these genes assuming they do not change. • Determine which genes do not change under this normalization. • Make this set the new core. If this core differs from the previous core, go to 3. Else, done.

6. Error Model I = spot intensity [mRNA] = concentration of specific mRNA c = normalization constant

7. Error Model I = spot intensity [mRNA] = concentration of specific mRNA c = normalization constant  = lognormal multiplicative error

8. Error Model I = spot intensity [mRNA] = concentration of specific mRNA c = normalization constant  = lognormal multiplicative error index 1, i: treatment group index 2, j: replicate within treatment index 3, k: spot (gene)

9. Y = log spot intensity  = mean log concentration of specific mRNA  = treatment effect (conc. specific mRNA)  = normalization constant  = normal additive error index 1, i: treatment group index 2, j: replicate within treatment index 3, k: spot (gene)

10. Model: Identifiability constraints: Estimate by ordinary least squares:

11. Model: Identifiability constraints: But note: cannot identify between a and d

12. Self-consistency: The weight wk(d) is small if the kth gene is judged to be changed; close to one if it is judged to be unchanged. Procedure is iterative.

16. Failure of Model

18. Generalized Model The normalization aij(xk) and the heteroscedasticity function gij(xk) are slowly varying functions of the intensity, x. Estimate by Local Regression

19. Local Regression data

20. Predict value at x=50: weight, linear regression

21. Predict whole function similarly

23. Compare to known true function

27. Simulation-based Validation 1. Reproduce observed bias.

28. Simulation-based Validation 2. Reproduce observed heteroscedasticity.

29. Test based on z statistic:

30. Choice of significance level: expected number of false positives: E(false positives) = a N But minimum detectable difference increases as a gets smaller

31. a E(fp) min diff min ratio 0.05 250 0.916 2.5 0.01 50 1.09 3 0.001 5 1.29 3.6 0.0001 0.5 1.61 5

32. Validation of method against simulated data 3. Hypothesis testing: Simulated from stated model bias “-fold change” Proportion changed spots “rate false pos.” = mean observed / expected

33. Simulated data: mis-specified model — multiplicative + additive noise

34. Validation of method against simulated data 4. Hypothesis testing: Simulated from “wrong” model: additive + multiplicative noise. bias “-fold change” Proportion changed spots

35. Acknowledgments Lynn Crosby North Carolina State University Kevin Morgan Strategic Toxicological Sciences GlaxoWellcome

36. Santa Fe Institute www.santafe.edu  postdoctoral fellowships available (apply before the end of the year) kepler@santafe.edu