Spatial Smoothing and Hot Spot Detection for CGH data using the Fused Lasso

1. Spatial Smoothing and Hot Spot Detection for CGH data using the Fused Lasso Pei Wang Cancer Prevention Research Program, PHS, FHCRC Joint work with Robert Tibshirani, Stanford University, CA Pei Wang, pwang@fhcrc.org

2. Outline DNA copy number alterations and Array CGH experiments. Detect copy number alterations using Fused Lasso regression. Simulation and real data examples. Jointly model copy number alterations and disease out comes using Fused Lasso regression. Pei Wang, pwang@fhcrc.org

3. DNA Copy Number In normal human cells: DNA copy number = 2 • Genome instability => Copy number alterations. Pei Wang, pwang@fhcrc.org Alberson and Pinkel, Hum. Mol. Gen., 2003

4. DNA Copy Number In cancer researches, knowledge of copy number aberrations helps to Identify important cancer genes. Reveal different tumor subtypes with different mechanism of initiation and/or progression. Predict tumor prognosis, and improve clinical diagnosis Pei Wang, pwang@fhcrc.org

5. Array CGH • array Comparative Genomic Hybridization. Scan machine reports the for each spot on the chips, which correspond to: Pei Wang, pwang@fhcrc.org

6. Array CGH • Array CGH has been implemented using a wide variety of techniques. • BAC array : produced from bacterial artificial chromosomes; • cDNA microarray: made from cDNAs; • oligo array: made from oligonucleotides (Affy, Agilent, Illumina). • Output from array CGH experiment: Pei Wang, pwang@fhcrc.org

7. Goal • Identify genome regions with DNA copy number alterations An example segment of CGH data from a GMB primary tumor (Bredel et al.2005). Pei Wang, pwang@fhcrc.org

8. Goal • Identify genome regions with DNA copy number alterations Estimated copy number from fused lasso regression shows copy number alteration regions. Raw CGH data. Pei Wang, pwang@fhcrc.org

9. Method • Denote the log2 ratio measurement of a chromosome (or chromosome arm) as . • Assume: = log2( true copy number / 2) + ei • = + ei , • We are interested in recovering . • Property of : • (1) =0 for genome regions without alterations; • >0 or <0 for regions of gain/loss. • (2) Profile { } has strong spatial correlation along index i. Pei Wang, pwang@fhcrc.org

10. Method • We are interested in finding coefficients satisfying (1) Lasso constraint --- detect alteration regions; (2) Fused constraint --- account for the spatial correlation. Pei Wang, pwang@fhcrc.org

11. lasso & fused lasso • lasso Regression (Tibshirani 1996) • fused lasso Regression (Tibshirani et al. 2004) Pei Wang, pwang@fhcrc.org

12. Method • Apply fused lasso on aCGH data: (1) Solve the optimization. (2) Choose the tuning parameters. (3) Control the False Discovery Rate (FDR). Pei Wang, pwang@fhcrc.org

13. Method • Apply fused lasso on aCGH data: (1) Solve the optimization. (2) Choose the tuning parameters. (3) Control the False Discovery Rate (FDR). Pei Wang, pwang@fhcrc.org

14. 1. Solve the optimization2. Choose the tuning parameter • For the general fused lasso regression: • Use SQOPT by Gill et al. to solve the quadratic programming problem with sparse linear constraints (Tibshirani et al., 2004) Pei Wang, pwang@fhcrc.org

15. 1. Solve the optimization2. Choose the tuning parameter • For the special application on CGH array: • - Pathwise coordinate optimization(Jerome Friedman et. al. Tech Report) • A modification of original Coordinate-wise descent algorithm (Shooting procedure) (Fu 1998, Daubechies et al. 2004). • The running time is only 1/100 of the quadratic programming Pei Wang, pwang@fhcrc.org

16. 1. Solve the optimization 2. Choose the tuning parameter • Estimates s1 and s2 from pre-smoothed version of the data: • s1 controls the overall copy number alteration amount of the target chromosome --- using heavily smoothed Y. • s2 controls the frequency of the copy number alterations on the target chromosome --- using moderately smoothed Y. Pei Wang, pwang@fhcrc.org

17. Other Method • Lai et. al. 2005 provides a thorough review of statistical methods for aCGH analysis. • Simple smoothing with Lowess • Hidden Markov Model (Fridlyand et al. 2004) • Top Down: Circular Binary Segmentation (Olshen et al. 2004, Venkatramanet al. 2007) • Bottom-up: Cluster along chromosomes (Wang et al. 2005) • Dynamic Programming: CGHseg (Picard et al. 2005) • Denoising using wavelet (Hsu et al. 2005) • And many others. Pei Wang, pwang@fhcrc.org

18. Other Method • Lai et. al. 2005 provides a thorough review of statistical methods for aCGH analysis. • Simple smoothing with Lowess • Hidden Markov Model (Fridlyand et al. 2004) • Top Down: Circular Binary Segmentation (Olshen et al. 2004, Venkatramanet al. 2007) • Bottom-up: Cluster along chromosome (Wang et al. 2005) • Dynamic Programming: CGHseg (Picard et al. 2005) • Denoising using wavelet (Hsu et al. 2005) • And many others. Pei Wang, pwang@fhcrc.org

19. General smoothing methods are not typically useful for analyzing CGH data, because their results can be difficult to interpret. • Fused lasso regression can also be viewed as a smoothing approach; but, it is able to capture the structure of the CGH data very well. Pei Wang, pwang@fhcrc.org

20. Comparison of Fused lasso with three segmentation methods: CGHseg (Picard et. al. 2005) CLAC (Wang et.al. 2005) CBS (Olshen et.al. 2004) Pei Wang, pwang@fhcrc.org

21. Simulation Example • Further comparison of fused lasso results with the three segmentation methods on simulation data sets from Lai et al. 2005. • Total length of chromosome segment: 100 • Four Different aberration width: 5, 10, 20, 40. • Signal to Noise ratio is equal to 1. • Normal region: x~ N(0, 0.25); • Alteration region: x~N(0.25, 0.25). • For each width, simulate 100 independently chromosomes. • Evaluation process: • 1. Estimate copy number using different methods. • 2. Apply different thresholds on the estimated copy numbers, and calculate • TPR = # of correct calls / # of total aberration. • FPR = # of false calls / # of total normal probes. Pei Wang, pwang@fhcrc.org

22. The TPR-FPR curves for the fours methods under different window sizes. Pei Wang, pwang@fhcrc.org

23. Real Data Example Breast Cancer Cell line MDA157 (Pollack 2002) Pei Wang, pwang@fhcrc.org

24. Computation Time Comparison of the speed of the four Methods: Data Simulation: 1. Pre-specify chromosome length p=100, 500, 1000, 2000. 2. Random sample 50 genome segments of length p from 17 Breast Cancer CGH arrays. 3. Apply each method on the 50 segments, and record the CPU time. (seconds) Pei Wang, pwang@fhcrc.org

25. Applying Fused Lasso on CGH: • gives an appropriate way to model aCGH data. • has favorable performance compared to other method. • is computationally efficient. Pei Wang, pwang@fhcrc.org

26. Applying Fused Lasso on CGH: • provides an appropriate model for aCGH data. • has favorable performance compared to other method. • is computationally efficient. • Provides a flexible frame work for aCGH analysis in more complicated settings. Pei Wang, pwang@fhcrc.org

27. Joint Model • Study copy number alterations and disease outcomes. • Model: Interested in finding disease associated genes. Pei Wang, pwang@fhcrc.org

28. Joint Model • Study copy number alterations and disease outcomes. • Model: Interested in finding disease associated genes. • Naïve method (Two-Steps): • 1. call gains and losses for each individual array; • 2. use the estimated copy numbers to look for disease associated genes. Pei Wang, pwang@fhcrc.org

29. Joint Model • Naïve method (Two-Steps): • 1. call gains and losses for each individual array; • 2. use the estimated copy numbers to look for disease associated genes. • Drawbacks: • 1. Loss information after first round of data processing. • 2. “Smoothing adds to already existing among neighboring values, thus causing the within-class covariance to be even more jagged… increase the computational cost with zero benefit in classification performance” (Hastie et al. 1995 Ann. of Stat.) Pei Wang, pwang@fhcrc.org

30. Joint Model • Joint modeling: Pei Wang, pwang@fhcrc.org

31. Compare different approaches on a simulation data set. • Simulate genome segment with p=50 genes for n=30 samples: • - true copy numbers - noise CGH measurements • Generate psuedo phenotype for each sample using two pre-selected non-adjacent genes. • Look for disease associated genes with different methods. Varying the tuning parameter t and produce ROC curves for each method. • Repeat for 200 times and plot the mean ROC curve. Pei Wang, pwang@fhcrc.org

32. Summary • Fused Lasso Regression can be used to characterize the spatial structure of array CGH data. • - Tibshirani & Wang, Biostatistics (In press) • - google-> tibshirani -> click on cghFlasso under software • The flexible framework of the regression model can be easily extended to solve other problems involving CGH data. Pei Wang, pwang@fhcrc.org

33. Acknowledgment • Stanford University, Department of Statisitcs Robert Tibshirani, Jerry Friedman, Trevor Hastie. • Stanford University, Department of Pathology Jonathan Pollack. Pei Wang, pwang@fhcrc.org