understanding principle component approach of detecting population structure jianzhong ma pi:...
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Understanding Principle Component Approach of
DetectingPopulation Structure
Jianzhong MaPI: Chris Amos
Introduction
• Analysis of association between markers and disease causing loci because of strong linkage (i.e. linkage disequilibrium) is more efficient than linkage analysis
• When samples arise from different ethnic groups, or an admixed population, spurious association occurs, resulting in false positives
Introduction• Genomic control approach (GC)• Transmission/disequilibrium test (TDT)• Structured association (MCMC)• Principle component approaches
– Traditional: marker-oriented– Eigenstrat: sample-oriented
• Eigenstart Theory: implemented in EIGENSTART and HelixTree
Eigenstrat References
• 1. Price, Alkes L., Patterson, Nick J. Plenge, Robert M. Weinblatt, Michael E. Shadick, Nancy A. Reich, David. (2006). ユ Principal Components Analysis Corrects for Stratification in Genome-Wide Associations Studies ユ . Nature Genetics 38, 904-909.2.
• Patterson N, Price AL, Reich D (2006) Population Structure and Eigenanalysis PLoS Genet 2(12): e190. doi:10.1371/journal.pgen.0020190.
Eigenstart Theory: Model• Data for association
test:
MK1 MK2 ……. MKN
Ind1 g11 g12 ……. g1N
…………………………………..
IndM gM1 gM2 ……. gMN
Eigenstrat Theory: Model
• Define random vector with M components for the M individuals
• Values of genotypes of the M Individuals at any marker are a special realization of this random vector
Eigenstrat Theory: Model
• The randomness is from both drawing genotypes and choosing allele frequency
• Under this model, genetically independent individuals will not be independent to each other
• Covariance between individuals from different subpopulations are smaller than that from the same subpopulations
Eigenstrat Theory: Model
• Only population properties of the PCA are considered (no sample properties considered), in order to gain some theoretical guidelines for interpreting PC-PC plots
Case 1: one-subpopulation
Covariance matrix
Case 1: one-subpopulation
Large eigenvalue
Eigenvector:
Small eigenvalue
Case 1: one-subpopulation
Large eigenvalue reflects co-variation of individuals
Small eigenvalues reflect variations between individuals
Neither is for population stratification!
Case 1: one-subpopulation
Zero-mean transform
Case 2: two-subpopulations
Random vector
Covariance matrix
Case 2: two-subpopulation• There are two large eigenvalues, with
corresponding eigenvectors having constant values for individuals in the same subpopulations. --- They are mixture of variances caused by stratification and intra-population co-variations
• Small eigenvalues are the same as in homogenous population
Case 2: Two-subpopulation
Zero-mean transform
Case 2: two-subpopulation
The two large eigenvalues and corresponding eigenvectors
Case 2: two-subpopulation case
2* Reflecting variation caused by
stratificationIf there are only two subpopulations, do NOT plot a PC vs PC figure; only the eigenvector of the largest eigenvalue shows the population structure.
Case 3: Three subpopulations
There are now three sub-populations
Case 3: Three subpopulations
Zero-mean transform
Case 3
General Case: K subpopulations
Summary• Only large eigenvalues reflect variations caused by
stratification• There are K-1 large eigenvalues if there are K
subpopulations• If there are merely two subpopulations, only the
eigenvector of the first largest eigenvalue tells the population structure; no two-dimensional PC-PC plot should be inspected
• In the case of multiple subpopulations, all K-1 vectors of the large eigenvalues should be carefully inspected in order to classify individuals into K subpopulations and infer the inter-population relationships
……
First off, if you choose as many components as there are markers, if thatユ s possible, you will wind up subtracting out ALL effects, thus getting nothing from your tests!
The best answer consists of first simply obtaining the components themselves and their corresponding eigenvectors. (Do this either while running uncorrected tests or from the separate PCA window.)
Then look at the pattern of the eigenvalues. If the first few are very large compared with the remaining eigenvalues, then use that many components in a second analysis in which you DO apply the PCA technique.
…….
Helix Manual:
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