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) 022Course Announcements
Why are we so different?
- HW #1 is out
- Project proposal
- Due next Wed
- 1 paragraph describing what you’d like to work on for the class project.
Any observable characteristic or trait
- Different “phenotype”
- Disease susceptibility
- Drug responses
- Different “genotype”
- Individual-specific DNA
- 3 billion-long string
DNA – 3 billion long!
- Appearance, Personality, Disease susceptibility, Drug responses, …
- Which sequence variation affects a trait?
- Better understanding disease mechanisms
- Personalized medicine
Parkinson’s disease? 0.3%
Heart disease? 20.1%
Colon cancer? 6.5%
A different person
1 2 3 4 5 … 3,000
- 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.
- Statistical methods for mapping QTL
- What is QTL?
- Experimental animals
- Analysis of variance (marker regression)
- Interval mapping (EM)
1 2 3 4 5 … 3,000
:Interval mapping [Lander and Botstein, 1989]
IM: the mixture model
- 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.
Nearest flanking markers
M1 QTL M2
0 7 20
IM: estimation and LOD scores
- 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?
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).A simulated example
LOD score curvesInterval mapping
- 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
- Greater computational effort (doing EM for each position)
- Requires specialized software
- More difficult to include covariates
- Only considers one QTL at a time
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)
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).
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.
Only ~3% of chance that the genomic position gets LOD score≥1.LOD thresholds
More on 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)
- 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)
Permutation distribution for a traitModeling multiple QTLs
Trait variation that is not explained by a detected putative QTL.
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
Multiple marker model
- 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
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)Multiple marker model
- Constant variance
- Assuming normality
- µ(x1,…,xp) = µ + ∑j ∆jxj + ∑j,k wj,kxjxk
- 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
x2Mapping QTL as model selection
y = w1 x1+…+wN xN+ε
minimizew (w1x1 + … wNxN - y)2?
- Select the class of models
- Additive models
- Additive with pairwise interactions
- Regression trees
Y = w1 x1+…+wN xN+ε
minimizew (w1x1 + … wNxN - y)2+model complexity
Lasso* (L1) Regression
- Search model space
- Forward selection (FS)
- Backward deletion (BE)
- FS followed by BE
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
* Tibshirani, 1996Model selection
- Compare models
- 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
Review: genetic variation
- Basic concepts
- Haplotype, haplotype frequency
- Recombination rate
- Linkage disequilibrium
- Haplotype reconstruction
- Parsimony-based approach
- EM-based approach
- 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
Let’s consider the history of two neighboring alleles…History of two neighboring alleles
Alleles that exist today arose through ancient mutation events…History of two neighboring alleles
- One allele arose first, and then the other…
Haplotype: combination of alleles present in a chromosomeRecombination can create more haplotypes
No recombination (or 2n recombination events)
- Consider N binary SNPs in a genomic region
- There are 2N possible haplotypes
- But in fact, far fewer are seen in human population
A combination of alleles present in a chromosome
Each haplotype has a frequency, which is the proportion of chromosomes of that type in the populationMore 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?
- 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.; 12:195-219, 1997.
- Lynch, M. and Walsh, B. Genetics and analysis of quantitative traits. Sinauer Associates, Sunderland, MA, pp. 431-89, 1998.
- Broman, K.W., Speed, T.P. A review of methods for identifying QTLs in experimental crosses, 1999.