1. Statistical Methods for Quantitative Trait Loci (QTL) Mapping II
Lectures 5 – Oct 12, 2011
CSE 527 Computational Biology, Fall 2011
Instructor: Su-In Lee
TA: Christopher Miles
Monday & Wednesday 12:00-1:20
Johnson Hall (JHN) 022

2. Course Announcements HW #1 is out Project proposal Due next Wed
1 paragraph describing what you’d like to work on for the class project.

3. Any observable characteristic or trait
Why are we so different?
Any observable characteristic or trait
Human genetic diversity
Different “phenotype”
Appearance
Disease susceptibility
Drug responses :
Different “genotype”
Individual-specific DNA
3 billion-long string
TGATCGAAGCTAAATGCATCAGCTGATGATCCTAGC…
TGATCGTAGCTAAATGCATCAGCTGATGATCGTAGC…
……ACTGTTAGGCTGAGCTAGCCCAAAATTTATAGCGTCGACTGCAGGGTCCACCAAAGCTCGACTGCAGTCGACGACCTAAAATTTAACCGACTACGAGATGGGCACGTCACTTTTACGCAGCTTGATGATGCTAGCTGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATTCACTTTTACGCAGCTTGATGACGACTACGAGATGGGCACGTTCACCATCTACTACTACTCATCTACTCATCAACCAAAAACACTACTCATCATCATCATCTACATCTATCATCATCACATCTACTGGGGGTGGGATAGATAGTGTGCTCGATCGATCGATCGTCAGCTGATCGACGGCAG……
TGATCGCAGCTAAATGCAGCAGCTGATGATCGTAGC…

4. Motivation Which sequence variation affects a trait? … cell cell
Appearance, Personality, Disease susceptibility, Drug responses, …
Which sequence variation affects a trait?
Better understanding disease mechanisms
Personalized medicine
Sequence variations XX AG
XXX
GTC
Different instruction
Instruction
ACTTCGGAACATATCAAATCCAACGC
DNA – 3 billion long! …
cell
Obese? 15%
Bold? 30%
Diabetes? 6.2%
Parkinson’s disease? 0.3%
Heart disease? 20.1%
Colon cancer? 6.5% :
cell
A different person
A person

5. QTL mapping
Data
Phenotypes: yi = trait value for mouse i
Genotypes: xik = 1/0 (i.e. AB/AA) of mouse i at marker k
Genetic map: Locations of genetic markers
Goals: Identify the genomic regions (QTLs) contributing to variation in the phenotype.
Genotype data
Phenotype data
… ,000
3000 markers
mouse
individuals :
…011
…001
…010 :
…101
…100 1 :

6. Outline Statistical methods for mapping QTL QTL? What is QTL?
Experimental animals
Analysis of variance (marker regression)
Interval mapping (EM)
QTL?
… ,000
mouse
individuals : 1 :

7. Interval mapping [Lander and Botstein, 1989]
Consider any one position in the genome as the location for a putative QTL.
For a particular mouse, let z = 1/0 if (unobserved) genotype at QTL is AB/AA.
Calculate P(z = 1 | marker data).
Need only consider nearby genotyped markers.
May allow for the presence of genotypic errors.
Given genotype at the QTL, phenotype is distributed as N(µ+∆z, σ2).
Given marker data, phenotype follows a mixture of normal distributions.

8. IM: the mixture model Nearest flanking markers M1/M2
99% AB
M QTL M2
65% AB
35% AA
Let’s say that the mice with QTL genotype AA have average phenotype µA while the mice with QTL genotype AB have average phenotype µB.
The QTL has effect ∆ = µB - µA.
What are unknowns?
µA and µB
Genotype of QTL
35% AB
65% AA
99% AA

9. IM: estimation and LOD scores
Use a version of the EM algorithm to obtain estimates of µA, µB, σ and expectation on z (an iterative algorithm).
Calculate the LOD score
Repeat for all other genomic positions (in practice, at 0.5 cM steps along genome).

10. A simulated example
Genetic markers
LOD score curves

11. Interval mapping Advantages Disadvantages
Make proper account of missing data
Can allow for the presence of genotypic errors
Pretty pictures
High power in low-density scans
Improved estimate of QTL location
Disadvantages
Greater computational effort (doing EM for each position)
Requires specialized software
More difficult to include covariates
Only considers one QTL at a time

12. Statistical significance
Large LOD score → evidence for QTL
Question: How large is large?
Answer 1: Consider distribution of LOD score if there were no QTL.
Answer 2: Consider distribution of maximum LOD score.
Null hypothesis – assuming that there are no QTLs segregating in the population.
Null distribution of the LOD scores at a particular genomic position (solid curve) and of the maximum LOD score from a genome scan (dashed curve).
Null distribution of the LOD scores at a particular genomic position (solid curve)
Only ~3% of chance that the genomic position gets LOD score≥1.

13. LOD thresholds
To account for the genome-wide search, compare the observed LOD scores to the null distribution of the maximum LOD score, genome-wide, that would be obtained if there were no QTL anywhere.
LOD threshold = 95th percentile of the distribution of genome-wide max LOD, when there are no QTL anywhere.
Methods for obtaining thresholds
Analytical calculations (assuming dense map of markers) (Lander & Botstein, 1989)
Computer simulations
Permutation/ randomized test (Churchill & Doerge, 1994)

14. More on LOD thresholds Appropriate threshold depends on:
Size of genome
Number of typed markers
Pattern of missing data
Stringency of significance threshold
Type of cross (e.g. F2 intercross vs backcross)
Etc

15. An example
Permutation distribution for a trait

16. Modeling multiple QTLs
Trait variation that is not explained by a detected putative QTL.
Advantages
Reduce the residual variation and obtain greater power to detect additional QTLs.
Identification of (epistatic) interactions between QTLs requires the joint modeling of multiple QTLs.
Interactions between two loci
The effect of QTL1 is the same, irrespective of the genotype of QTL 2, and vice versa
The effect of QTL1 depends on the genotype of QTL 2, and vice versa

17. Multiple marker model Let y = phenotype, x = genotype data.
Imagine a small number of QTL with genotypes x1,…,xp
2p or 3p distinct genotypes for backcross and intercross, respectively
We assume that
E(y|x) = µ(x1,…,xp), var(y|x) = σ2(x1,…,xp)

18. Multiple marker model Constant variance Assuming normality Additivity
σ2(x1,…,xp) =σ2
Assuming normality
y|x ~ N(µg, σ2)
Additivity
µ(x1,…,xp) = µ + ∑j ∆jxj
Epistasis
µ(x1,…,xp) = µ + ∑j ∆jxj + ∑j,k wj,kxjxk

19. Computational problem
N backcross individuals, M markers in all with at most a handful expected to be near QTL
xij = genotype (0/1) of mouse i at marker j
yi = phenotype (trait value) of mouse i
Assuming addivitity,
yi = µ + ∑j ∆jxij + e which ∆j ≠ 0?
Variable selection in linear regression models

20. Mapping QTL as model selection
Select the class of models
Additive models
Additive with pairwise interactions
Regression trees … x1 x2 xN w2 w1 wN
Phenotype (y)
y = w1 x1+…+wN xN+ε
minimizew (w1x1 + … wNxN - y)2 ?

21. Linear Regression Search model space
minimizew (w1x1 + … wNxN - y)2+model complexity
Search model space
Forward selection (FS)
Backward deletion (BE)
FS followed by BE … x1 x2 xN w1 w2 wN w2 w1 wN
parameters
Phenotype (y)
Y = w1 x1+…+wN xN+ε 21

22. Lasso* (L1) Regression minimizew (w1x1 + … wNxN - y)2+  C |wi|
L1 term
minimizew (w1x1 + … wNxN - y)2+  C |wi|
Induces sparsity in the solution w (many wi‘s set to zero)
Provably selects “right” features when many features are irrelevant
Convex optimization problem
No combinatorial search
Unique global optimum
Efficient optimization … x1 x2 x1 x2 xN L2 L1 w1 w2 w2 w1 wN
parameters
Phenotype (y)
* Tibshirani, 1996 22

23. Model selection Compare models Assess performance
Likelihood function + model complexity (eg # QTLs)
Cross validation test
Sequential permutation tests
Assess performance
Maximize the number of QTL found
Control the false positive rate

24. Outline Basic concepts Haplotype reconstruction
Haplotype, haplotype frequency
Recombination rate
Linkage disequilibrium
Haplotype reconstruction
Parsimony-based approach
EM-based approach

25. Review: genetic variation
Single nucleotide polymorphism (SNP)
Each variant is called an allele; each allele has a frequency
Hardy Weinberg equilibrium (HWE)
Relationship between allele and genotype frequencies
How about the relationship between alleles of neighboring SNPs?
We need to know about linkage (dis)equilibrium

26. Let’s consider the history of two neighboring alleles…

27. History of two neighboring alleles
Alleles that exist today arose through ancient mutation events…
Before mutation A
After mutation A C
Mutation

28. History of two neighboring alleles
One allele arose first, and then the other…
Before mutation A G C G
After mutation A G C G C C
Mutation
Haplotype: combination of alleles present in a chromosome

29. Recombination can create more haplotypesG C C
No recombination (or 2n recombination events)
Recombination A G C C A C C G

30. A G C G C C A G C G C C A C Recombinant haplotype
Without recombination A G C G C C
With recombination A G C G C C A C
Recombinant haplotype

31. Haplotype A combination of alleles present in a chromosome
Each haplotype has a frequency, which is the proportion of chromosomes of that type in the population
Consider N binary SNPs in a genomic region
There are 2N possible haplotypes
But in fact, far fewer are seen in human population

32. More on haplotype What determines haplotype frequencies?
Recombination rate (r) between neighboring alleles
Depends on the population
r is different for different regions in genome
Linkage disequilibrium (LD)
Non-random association of alleles at two or more loci, not necessarily on the same chromosome.
Why do we care about haplotypes or LD?

33. References Prof Goncalo Abecasis (Univ of Michigan)’s lecture note
Broman, K.W., Review of statistical methods for QTL mapping in experimental crosses
Doerge, R.W., et al. Statistical issues in the search for genes affecting quantitative traits in experimental populations. Stat. Sci.; 12: , 1997.
Lynch, M. and Walsh, B. Genetics and analysis of quantitative traits. Sinauer Associates, Sunderland, MA, pp , 1998.
Broman, K.W., Speed, T.P. A review of methods for identifying QTLs in experimental crosses, 1999.