Risk Factor Searching Heuristics for SNP Case-Control Studies

1. February 21, 2008 Risk Factor Searching Heuristics for SNP Case-Control Studies Dumitru Brinza Department of Computer Science & Engineering University of California at San Diego

2. Outline • SNPs, Genotypes, Common Complex Diseases • Disease Association Search in Case-Control Studies • Computational challenges • Significance and Reproducibility of RF • Genetic model / Atomic Risk Factor • Maximum Odds Ratio Risk Factors • Exhaustive Search • Complimentary Greedy Search Algorithm • K-Relaxed and Weighted Atomic Risk Factor • WCGS Algorithm for finding K-ARF and W-ARF • Dataset • Results • Conclusions

3. SNP, Haplotypes, Genotypes • Human Genome – all the genetic material in the chromosomes,length 3×109 base pairs • Difference between any two people occur in 0.1% of genome • SNP – single nucleotide polymorphism site where two or more different nucleotides occur in a large percentage of population. • Diploid – two different copies of each chromosome • Haplotype – description of a single copy (expensive) • example: 00110101 (0 is for major, 1 is for minor allele) • Genotype – description of the mixed two copies • example: 01122110 (0=00, 1=11, 2=01)

4. Heritable Common Complex Diseases • Monogenic disease • Mutated gene is entirely responsible for the disease • Break the pathway, no another compensatory pathway • Typically rare in population: < 0.1%. • Complex disease • Interaction of multiple genes • One mutation does not cause disease • Breakage of all compensatory pathways cause disease • In case of cancer – breakage of several cell functions cause disease, e.g., cell-growing and cell-checking systems • Hard to analyze - 2-gene interaction analysis for a genome-wide scan with 1 million SNPs has 1012 pair wise tests • Multiple independent causes • There are different causes and each of these causes can be result of interaction of several genes • Each cause explains certain percentage of cases • Common diseases are Complex: > 0.1%. • In NY city, 12% of the population has Type 2 Diabetes

5. DA Search in Case/Control Study Given: a population of n genotypes each containing values of m SNPs and disease status Disease Status SNPs -1 -1 -1 -1 1 1 1 1 0101201020102210 0220110210120021 0200120012221110 0020011002212101 1101202020100110 0120120010100011 0210220002021112 0021011000212120 Case genotypes: Control genotypes: Find:risk factors (RF) with significantly high odds ratio i.e., pattern/dihaplotype significantly more frequent among cases than among controls

6. Challenges in Disease Association • Computational – Scalability • Interaction of multiple genes/SNP’s • Too many possibilities – obviously intractable • Multiple independent causes • Each RF may explain only small portion of case-control study • Statistical – Reproducibility • Search space / number of possible RF’s • Adjust to multiple testing • Searching engine complexity • Adjust to multiple methods / search complexity

7. Addressing Challenges in DA • Computational– Scalability • Constraint model / reduce search space • Negative effect = may miss “true” RF’s • Heuristic search • Look for “easy to find” RF’s • May miss only “maliciously hidden” true RF • Statistical – Reproducibility • Validate on different case-control study • That’s obvious but expensive • Cross-validate in the same study • Usual method for prediction validation

8. Significance of Risk Factors OR= TP/FP TN/FN Original Case Control Have RF Case True Positive False Positive (TP) (FP) Control False Negative True Negative (FN) (TN) Significance of Risk Factors • Relative risk (RR) • cohort study • Odds ratio (OR) – case-control study • P-value • binomial distribution • multiple testing adjustment of the p-value: • more searching  more findings by chance

9. Reproducibility Control • Multiple-testing adjustment • Bonferroni: • adjusted p = # possibilities x unadjusted p • easy to compute but overly conservative  • SNP’s are linked – difficult to take in account • Randomization • 10000 times repeat: • Randomly permute disease status • Find the best RF using the same method • adjusted p = # times RF has higher OR than found • computationally expensive but ideally accurate 

10. Risk/Resistance factors • Previous works model Risk/resistance factor =one SNP with fixed allele value 0 1 1 0 1 2 1 0 2 case 0 1 1 1 0 2 0 0 1 case 0 0 1 0 0 0 0 2 1 case 0 1 1 1 1 2 0 0 1 case 0 0 1 0 1 2 1 0 2 case 0 1 0 0 1 1 0 0 2 control 0 1 1 0 1 2 0 0 2 control present in 5 cases : 1 control Third SNP with fixed allele value 1 is a risk factor with frequency among case individuals higher than among control individuals.

11. Genetic Model 1 4 2 5 End Product 3 Genetic Model Cellular Pathway • Breaking1 & 2 does not imply disease because of compensatory link 3 • Breaking1 & 2 & 3 imply disease = “atomic” risk factor • Breaking 1 & 2 & 3 or 4& 5 imply disease = “complex” RF • Several causes of disease (ARFs) 1 & 2 & 3 or 4 & 5 • ARF ↔ multi-SNP combination (MSC)

12. Multi-SNP Combination and Cluster • Multi-SNP combination (MSC) • a subset ofSNP-columns of S (set of SNPs) • With fixed values of these SNPs, 0, 1, or2 0 1 1 0 1 2 1 0 2 case 0 1 1 1 0 2 0 0 1 case 0 0 1 0 0 0 0 2 1 case 0 1 1 1 1 2 0 0 1 case 0 0 1 0 1 2 1 0 2 case 0 1 0 0 1 1 0 0 2 control 0 1 1 0 1 2 0 0 2 control x x 1 x x 2 x x x MSC present in 4 cases : 1 control Cluster= subset of genotypes with the same MSC

13. MORARF formulation • Maximum Odds Ratio Atomic Risk Factor • Given: genotype case-control study • Find: ARF with the maximum odds ratio • Number of RF is enormous large • Constrain searching among Atomic Risk Factors

14. Exhaustive Searching Approaches • Exhaustive search (ES) • For n genotypes with m SNPs there are O(3km) k-SNP MSCs • Exhaustive Combinatorial Search (CS) • Drop small (insignificant) clusters • Search only plausible/maximal MSC’s Case-closure of MSC: • MSC extended with common SNPs values in all cases • Minimum cluster with the same set of cases i i 0 1 1 0 1 2 1 0 2 case 0 1 1 0 1 2 1 0 2 case Case-closure 2 0 1 1 0 2 0 0 1 case 2 0 1 1 0 2 0 0 1 case 0 0 1 0 0 0 0 2 1 case 0 0 1 0 0 0 0 2 1 case 0 1 1 0 1 2 0 0 2 control 0 1 1 0 1 2 0 0 2 control 0 1 1 0 1 2 0 1 2 control 0 2 1 0 1 2 0 1 2 control x x 1 x x 2 x x x x x 1 x x 2 x 0 x Present in 2 cases : 2 controls Present in 2 cases : 1 control

15. Cases Controls Cases Controls Cases Controls Exhaustive Combinatorial Search • Exhaustive Combinatorial Search Method (CS): • Searches only among case-closed MSCs • Avoids checking of clusters with small number of cases • Alternating Combinatorial Search method (ACS): • Find significant MSCs faster than ES • Still too slow for large data • Further speedup by reducing number of SNPs • Indexing:compress S by extracting most informative SNPs • Use multiple regression method

16. Heuristics for MORARF • Clusters with less controls have higher OR => MORARF includes finding of max control-free cluster • max control-free cluster contains max independent set problem => NP-hard • max control-free cluster can be transformed to Red-Blue Set Cover Problem • Cannot be reasonably approximated in polynomial time for an arbitrary S • Red-Blue Set Cover Problem includes weighted set-cover problem • The best known approximation algorithm for the weighted set-cover problem is greedy heuristic

17. Complimentary Greedy Search(CGS) • Intuition: • Greedy algorithm for finding maximum independent set by removing highest degree vertices • Fixing an SNP-value • Removes controls -> profit • Removes cases -> expense • Maximize profit/expense! • Algorithm: • Starting with empty MSC add SNP-value removing from current cluster max # controls per case • Result is maximum control free cluster  MORARF Cases Controls

18. OR after each iteration of CGS The value of OR of ARF with 95% CI on i-th iteration of CGS on lung-cancer dataset

19. Complimentary Greedy Search(CGS) • Comparison with optimum: • For the small dataset of Tick-borne encephalitis we were able to find an optimal solution for MORARF using ILP. • CGS founds the same solution. • We can assume that CGS founds the optimal or close to optimal solution.

20. Randomized CGS Repeat 100 times and choose the best MSC Empty MSC Empty MSC CASES 1/4 1/2 CONTROLS 1

21. 5 Data Sets • Crohn's disease (Daly et al ):inflammatory bowel disease (IBD). Location: 5q31 Number of SNPs: 103 Population Size: 387 case: 144 control: 243 • Autoimmune disorders (Ueda et al) : Location: containing gene CD28, CTLA4 and ICONS Number of SNPs: 108 Population Size: 1024 case: 378 control: 646 • Tick-borne encephalitis (Barkash et al) : Location: containing gene TLR3, PKR, OAS1, OAS2, and OAS3. Number of SNPs: 41 Population Size: 75 case: 21 control: 54 • Lung cancer (Dragani et al) : Number of SNPs: 141 Population Size: 500 case: 260 control: 240 • Rheumatoid Arthritis (GAW15) : Number of SNPs: 2300 Population Size: 920 case: 460 control: 460

22. Search Results Comparison of 5 methods searching ARF on 5 real datasets

23. Validation Results 2-fold Cross-validation = % of best MSC on the training validated on testing half (p < 5%) Random-validation = the same but testing is allowed to overlap with training Significance = % of best MSC on the training half significant after MT-adjustment Double Significance = % of best MSC on the training half significant after MT-adjustment that are also significant on the testing half

24. Generalization of ARF wild type mutation P P (a) Atomic Risk Factor P P (b) 1-Relaxed Atomic Risk Factor P P (c) Weighted Relaxed Atomic Risk Factor

25. k-Relaxed Atomic Risk Factor • k-MSC • MSC with n SNPs • a subset ofSNP-columns of S (set of SNPs) • With fixed values of these SNPs, 0, 1, or2 • Threshold k • k-neighborhood of MSC = at most k mismatches 0 1 1 0 1 2 1 0 2 case 0 1 1 1 0 2 0 0 1 case 0 0 0 0 0 2 0 2 2 case 0 1 1 1 1 2 0 0 1 case 0 0 1 0 1 0 1 0 2 case 0 1 0 0 1 1 0 0 0 control 0 1 1 0 1 2 0 0 1 control 1-MSC x x 1 x x 2 x x 2 present in 5 cases : 1 control k-Cluster = subset of genotypes satisfying k-MSC

26. Example of 1-MSC MSC1 Sick individuals k-MSC MSC2 k-Cluster

27. MORRARF Formulation • Maximum Odds Ratio k-RARF • Given: genotype case-control study and constant k • Find: k-RARF with the maximum odds ratio • MORRARF includes MORARF => harder • k-CGS Algorithm: • CGS with objective computed for the k-cluster instead of cluster

28. Weighted k-Relaxed ARF • Weighted k-MSC • k-MSC with weights on each SNP 0 1 1 0 1 2 1 0 2 case w(2)=2+1-1 0 1 1 1 0 2 0 0 0 case w(3) 0 0 0 0 0 2 0 2 2 case w(0)=1-1 0 1 1 1 1 2 0 0 0 case w(3) 0 0 1 0 1 0 1 0 2 case w(1) 0 1 0 0 1 1 0 0 0 control w(0) 0 1 1 0 1 2 0 0 1 control w(3) x x 1 x x 2 x x 2 MSC weights 0 0 2 0 0 10 0 -1 and k = 2 present in 3 cases : 1 control Weighted k-cluster = subset of genotypes within a weighted distance k from weighted k-MSC

29. MORWRARF Formulation • Maximum Odds Ratio WRARF • Given: genotype case-control study • Find: Weighted k-RARF with the maximum odds ratio • MORWRARF includes MORARF => harder • WCGS Algorithm: • Two move CGS with objective computed for the k-cluster instead of cluster

30. One iteration of Greedy Methods CGS/k-CGS WCGS (∆D/∆H)max Step backward ∆D ∆H H = number of controls H = number of controls ∆H ∆H (∆H/∆D)max step forward (∆H/∆D)max step forward ∆D ∆D D = number of cases D = number of cases (a) (b) Cluster content

31. Cluster content Tick-borne encephalitis 240 H=# Health in k-cluster 260 S = # Sick in k-cluster

32. Behavior of Greedy Heuristics (a) Lung cancer (b) Rheumatoid Arthritis (c) Tick-borne encephalitis (d) Crohn's disease

33. Search Results for 3 Greedy methods

34. Validation Results Cross-validation = % best MSC on the training half validated on testing half (p < 5%) Random-validation = the same but testing is allowed to overlap with training Significance = % best MSC on the training half significant after MT-adjustment Double Significance = % of best MSC on the training half significant after MT-adjustment that are also significant on the testing half

35. Conclusions • Approximate search methods find more significant RF’s • RF found by approximate searches have higher cross-validation rate • Significant MSC’s are better cross-validated • WCGS has finds significant MSC’s when no other methods could find anything