A Shrinkage Regression Approach to Tackle the HLA Region

1. A Shrinkage Regression Approach to Tackle the HLA Region Charlotte Vignal Variable Selection Workshop Vienna, July 26th 2008

2. Outline • Overview of the HLA system and the challenge of analysing data from the HLA region • Multivariate association test using a Bayesian-inspired shrinkage regression approach • Application to the rheumatoid arthritis case-control study • Conclusion

3. The Human Leukocyte Antigen System • A genomic region found in almost all vertebrates, the major histocompatibility complex (MHC) - gene composition and arrangement vary between species (below) • In humans, the MHC is the HLA system • A set of genes encoding proteins essential to immune response • Major role in histocompatibility and protection against pathogens MOUSE RAT CHIMPANZEE HUMAN Kelley et al. Immunogenetics (2005)

4. The Challenge • Susceptibility to many complex disorders maps to the HLA region • High degree of correlation within the region hampers the identification of causal variants • Widely used approaches test the effect of one genetic variable at a time • Require methods that allow the detection of (possibly multiple) causal variants among highly correlated data

5. Multi-SNP Methods can be more Powerful than Single-SNP Analyses • Multivariate logistic regression • Problematic when nVars >> nObs • Stepwise procedures can be unstable in presence of many highly-correlated terms • Shrinkage method using Bayesian logistic regression • A variable selection approach • Based on the Least Absolute Shrinkage and Selection Operator approach (LASSO) (Tibshirani 1996) • Fast implementation using the Bayesian Binary Regression (BBR) software for text-categorisation analysis (Genkin et al. 2004, http:/www.stat.rutgers.edu/~madigan/BBR)

6. Bayesian Logistic Regression for variable selection • Each coefficient βj has a Laplace prior distribution with mode 0 and prior variance ν=2/λ2, where λ is the penalty factor • Mode 0 encodes a prior belief of no effect • The prior variance determines the strength of this belief and hence the sparseness of the fitted model • The maximum a posteriori (posterior mode) estimates are often zero or else shrunk towards zero • Terms with non-zero are included in the final model, and treated as significant • The value of gives a (shrunk) measure of effect size

7. The Density of the Laplace Distribution p(x) x ! Effect size estimates are biased towards zero; Over-shrinking true effects can lead to non-causal correlated variables to be retained

8. ApplicationTheRheumatoid Arthritis Dataset • RA is an autoimmune disease and a complex disorder • Estimated genetic contribution of ~30-50% • The HLA region is strongly implicated in RA susceptibility • Genetic associations reported with a biomarker called the shared epitope (SE) defined by a class of alleles at HLA-DRB1 • The mechanism by which RA is determined is still unknown • Is the SE association the only HLA effect predisposing to RA? • The subjects: 842 RA cases and 957 controls(but 774 cases and 945 controls with no missing data analysed) • The independent variables: • 2,302 genetic markers, a continuous variable coded as 0, 1 and 2 based on the number of allele copies • The shared epitope, a continuous variable coded as 0, 1, 2 based on the number of shared epitope positive (SE+) alleles

9. The Effect of Shared Epitope on RA • The presence of SE is strongly associated with RA • Increasing risk for RA associated with the number of SE+ allele copies • The objective: to investigate the presence of additional causal variants in the HLA region, possibly correlated with SE

10. Specification of the Penalty λ • Cases and controls permuted 100 times for each λwithin each SE group (i.e. SE effect retained) • SE (additive term) included in each model • λ selected if false positive per model < 1 • λ = 62 was selected for further analyses

11. The Effect of Shrinking a True Effect R2 between each genetic variables and SE across the HLA region • In blue are the genetic variables selected by BLR in addition to SE • Three variables selected are correlated with SE • Shrinking a known effect may cause correlated SNPs to be selected

12. The Effect of Shrinkage on True Effects • To investigate the effect of shrinkage,SE included twice (SE & SEfake) in the model: • When SE and SEfake are shrunk, both variables retained • Shrinking a known effect may cause correlated SNPs to be selected • When SE is not shrunk, only SE is retained • Correlated SNPs could be eliminated • The shrinkage factor was not applied to SE in subsequent analyses (λ = 0)

13. BLR and Correlated Data • Can the BLR approach distinguish positive effects from spurious associations in presence of correlation? • 4 variables correlated with SE were used to evaluate error rates and power • Records of each variables re-distributed in cases and controls to achieve different size of OR while maintaining correlation with SE • Error rate and power assessed by permuting cases and controls • Error rate: frequency of the variables selected beyond SE & the simulated correlated variables over 100 permutations • Power: frequency of the simulated variables over 100 permutations

14. Power • Selection of simulated variables correlated with SE • variables moderately correlated with SE selected if OR> 2 • variables highly correlated with SE selected if OR> 5

15. Error Rate • Selection of simulated variables correlated with SE • Under the null,expect 1 false positive per analysis (λ = 62) • Analysis generates 1 to 2 false positives per analysis

16. ATT- BLR Results Comparison • Data were analysed by Armitage Trend Test (ATT) and BLR • With λ=62, BLR identified 10 SNPs • Single-point analysis using ATTidentified 109 associated SNPs at • α= 4.34e-04 = 1/2302 • Variables selected by BLR are not correlated with SE

17. Additional AnalysisThe NEG Distribution • Data re-analysed using the normal-exponential-gamma (NEG) prior with parameters set to expect 1 false positive per model (Hoggart et al. PLoS (2008)) ! NEG has heavier tails to allow sparser solutions

18. Additional AnalysisThe NEG Distribution • NEG identified 4 variables; of which three (snp271, snp384, snp545) were also retained by DE • Variables identified with NEG prior are less correlated among themselves and with SE than those selected using DE • Three of the selected variables are in genes/region reported to contribute to RA susceptibility: BAT1 and HLADQA1/DQB1

19. Conclusions • BLR appears to perform better than single-point association analysis (ATT) when data are correlated • Computationally efficient • Identifies fewer positive results (10 vs.109) • Correlation might be more effectively handled • Simulation analyses confirm reasonable power and error rate • Three variables identified by both DE and NEG priors lie in genes previously implicated in RA • Results suggest the presence of independent RA-associated effects in the HLA region

20. Acknowledgements • David Balding, Imperial College, UK • Clive Hoggart, Imperial College, UK • Aruna Bansal, GSK, UK • The Genetics Division at GSK