1. Department of Biostatistics
Bayesian Variable Selection in Semiparametric Regression Modeling with Applications to Genetic Mappping
Fei Zou
Department of Biostatistics

2. Outline Introduction Bayesian semi-parametric QTL Mapping Results
Experimental crosses
Existing QTL Mapping Methods
Bayesian semi-parametric QTL Mapping
Results
Remarks and Conclusions

4. Overview One gene one trait: very unlikely
The vast majority of biological traits are caused by complex polygenes
Potentially interacting with each other
Most traits have significant environmental exposure components
Potentially interacting with polygenes

5. Experimental Crosses: F2
Parents P2

6. Experimental Crosses F2 Backcross(BC) P2 P1 P1 P2 F1 P1 F1 F1 F2: BC:AA AA BB BB F1 P1 F1 F1 AA AB AB AB BB AB AB AB AA AB
F2:
BC:

7. QTL Data Format 0: homozygous AA, 2: homozygous BB,
1: heterozygote AB.
Marker positions:

8. Linkage Analysis Data structure:
Marker data (genotypes plus positions)
Phenotypic trait(s)
Other nongenetic covariates, such as age, gender, environmental conditions etc
Quantitative trait loci (QTL): a particular region of the genome containing one or more genes that are associated with the trait being assayed or measured

9. QTL Mapping of Experimental Crosses
Single QTL Mapping
Single marker analysis (Sax, 1923 Genetics)
Interval mapping: Lander & Botstein (1989, Genetics)
Multiple QTL mapping
Composite interval mapping (Zeng 1993 PNAS, 1994 Genetics; Jansen & Stam, 1994 Genetics)
Multiple interval mapping (Kao et al., 1999 Genetics)
Bayesian analysis (Satagopan et al., 1997 Genetics)

10. Single QTL Interval Mapping
For backcross, the model assumes
QTL analysis:
If QTL genotypes are observed, the analysis is trivial: simple t-test!
However, QTL position is unknown and therefore QTL genotypes are unobserved

11. Interval Mapping For QTL between markers
QTL genotypes missing: can use marker genotypes to infer the conditional probabilities of the QTL genotypes for a given QTL position
Profile likelihood (LOD score) calculated across the whole genome or candidate regions using EM algorithm
In any region where the profile exceeds a (genome-wide) significance threshold, a QTL declared at the position with the highest LOD score.

12. Profile LOD

13. Multiple QTL Mapping
Most complicated traits are caused by multiple (potentially interacting) genes, which also interact with environment stimuli
Single QTL interval mapping
Ghost QTL (Lander & Botstein 1989)
Low power

14. Multiple QTL Mapping
Composite interval mapping (Zeng 1993, 1994; Jansen & Stam1993): searching for a putative QTL in a given region while simultaneously fitting partial regression coefficients for "background markers" to adjust the effects of other QTLs outside the region
which background markers to include; window size etc
Multiple interval mapping (Kao et al 1999): fitting multiple QTLs simultaneously
Computationally intensive; how many QTLs to include?

15. Multiple QTL Mapping
Bayesian methods (Stephens and Fisch 1998 Biometrics; Sillanpaa and Arjas 1998 Genetics; Yi and Xu 2002 Genetic Research, and Yi et al Genetics): treat the number of QTLs as a parameter by using reversible jump Markov chain Monte Carlo (MCMC) of Green (1995 Biometrika)
change of dimensionality, the acceptance probability for such dimension change, which in practice, may not be handled correctly (Ven 2004 Genetics)

16. Multiple QTL Mapping
Alternative, multiple QTL mapping can be viewed as a variable selection problem
Forward and step-wise selection procedures (Broman and Speed 2002 JRSSB)
LASSO, etc
Bayesian QTL mapping
Xu (2003 Genetics), Wang et al (2005 Genetics) Huang et al (2007 Genetics): Bayesian shrinkage
Yi et al (2003 Genetics): stochastic search variable selection (SSVS) of George and McCulloch (1993 JASA)
Yi (2004 Genetics): composite model space of Godsill (2001 J. Comp. Graph. Stat)
Software: R/qtlbim by Yi’s group

17. Multiple QTL Mapping Limitations of existing QTL mapping methods
do not model covariates at all or only model covariate effect linearly
do not model interactions at all or model only lower order interactions, such as two way interactions

18. The multiple QTL mapping is a very large variable selection problem: for p potential genes, with p being in the hundreds or thousands, there are possible main effect models, possible two-way interactions and possible higher order (k > 2) interactions.

19. Semiparmetric Multiple (Potentially Interacting) QTL Mapping
Goal: map multiple potentially interacting QTLs without specifically model all potential main and higher order interaction effects
Semiparametric model:
where function is unspecified, QTL genotypes and represent all non- genetics factors/covariates.
When equals : non-explicitly modeling the two way interaction between genes 1 and 2 and the gene-environmental interaction between gene 3 and covariate 1.

20. Bayesian Semi/non-parametric Methods
Dirichlet process (Muller et al. 1996)
Splines (Smith and Kohn 1996; Denison et al and DiMatteo et al. 2001)
Wavelets (Abramovich et al JRSSB)
Kernel models (Liang et al 2007)
Gaussian process (Neal 1997; 1996)
Gaussian process priors have a large support in the space of all smooth functions through an appropriate choice of covariance kernel.
Gaussian process is flexible for curve estimation because of their flexible sample path shapes
Gaussian process related to smoothing spline somehow (Wahba 1978 JRSSB)

21. Prior Specification on
A Gaussian process such that all possible finite dimensional distributions follow multivariate normal with mean 0 and covariance function
where , s and s are hyperparameters and

22. Hyperparameter defines the vertical scale of variations, i. e
Hyperparameter defines the vertical scale of variations, i.e., controls the magnitude of the exponential part. Hyperparameters related to length scales which characterize the distance in that particular direction over which y is expected to vary significantly
controls the smoothness of : when the posterior mean of almost interpolates the data while centered around the prior mean function if
When = 0, y is expected to be an essentially constant function of that input variable xj, which is therefore deemed irrelevant (Mackay 1998).

23. Priors on
The original papers on the Gaussian process (Mackay 1998; Neal 1997) did not view this method as an approach for variable selection and imposed a Gamma prior on the parameters. However, does provide information about the relevance of any QTL with value near zero indicating an irrelevant QTL.
For variable selection purpose, we can impose the following Gamma mixture priors on

24. Prior Specifications
Inverse Gamma distributions are used for the priors of and

25. Simulations Set ups: backcross population 200 or 500 individuals
151 evenly spaced markers at 5cM intervals
Four QTLs with varying heritabilities:
Main effect model: all four QTL act additively
Main plus two way interactions
Four way interactions only

32. n=500 and pure 4 way-interaction model

33. n=500 and pure additive model

34. Real Data Analysis A mouse study # samples: 187 backcross samples
# markers: 85 with average marker distance 20 cM
Phenotypes: inguinal, gonadal, retroperitoneal and mesenteric fat pad weights

37. Remarks
For studies with large # of samples and/or large # of markers, MCMC converges very slowly
We employed the hybrid Monte Carlo method, which merges the Metropolis-Hastings algorithm with sampling techniques based on dynamics simulation.
We also estimated the maximum a posteriori (MAP) via conjugate gradient method (Hestenes et al 1952 J. Research of National Bureau of Standards)
point estimate

38. Real Study: Cardiovascular Disease
2655 tag SNPs from roughly 200 selected candidate genes for cardiovascular disease
820 individuals
Non-genetic covariates: gender, smoking status, age

40. Remarks
Semiparemetric mapping is powerful in mapping multiple (potentially interacting with higher orders) QTL
Picks up genes related to the trait regardless of their marginal main effects or joint epistasis effects
Cannot readily differentiates genetic contributions
main effect? interaction? or both?
Fine tuned parametric model with selected genes

41. Remarks and Future Research
How to extend the methodologies to human genome-wide association (GWA) studies, where hundreds of thousands of markers are available
Is it possible?
potential solutions: pathway analysis; data reduction techniques
How to extend the method to human pedigree analysis where mixed effect model is used for correlated family members?
Use inheritance vector: so far results are very promising

42. Acknowledgement Joint work with Funding support Hanwen Huang
Haibo Zhou
Fuxia Cheng
Ina Hoeschele
Funding support
NIH R01 GM074175