Bayesian Inference of Epistatic Interactions in Casecontrol Studies

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Bayesian Inference of Epistatic Interactions in
Case-control Studies

• Yu Zhang Jun S Liu
• Nature Genetics 39, 1167 - 1173 (2007)
• Presented by Yixuan Chen
• 2/29/08

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Outlines

• Epistasis Background
• Methods
• The Bayesian marker partition model
• MCMC sampling
• B statistic
• Results
• Epistasis models and simulations
• Comparisons
• Genome-wide association study of AMD
• Discussions

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Epistasis

• Epistasis is a phenomenon whereby the effects of
  a given gene on a biological trait are masked or
  enhanced by one or more other genes.
• For complex traits such as diabetes, asthma,
  hypertension, etc., the presence of epistasis is
  a particular cause for concern.
• An increasing number of reports have indicated
  the presence of multilocus interactions in many
  human complex traits.

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Genome-wide Epistasis

• The number of possible interaction combinations
  is astronomical.
• It is a daunting task to catch one or a very
  few disease-related interactions.

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BEAM Algorithm

• Bayesian Epistasis Association Mapping
• A Bayesian partitioning model
• B statistic and conditional B statistic
• BEAM is significantly more powerful.
• A genome-scale epistasis mapping is both feasible
  and desirable.

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Notations

• Suppose Nd cases and Nu controls were genotyped
  at L SNP markers.
• D(d1,,dNd) case genotypes
• U(u1,,uNu) control genotypes
• di(di1,,diL)
• ui(ui1,,uiL)

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Marker Partitioning

• The L markers are partitioned into 3 groups
• group 0 contains markers unlinked to the disease
• group 1 contains markers contributing
  independently to the disease risk
• group 2 contains markers that jointly influence
  the disease risk (interactions).

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Group Membership

• Let I(I1,,IL) indicate the membership of the
  markers with Ij0, 1 and 2, respectively.
• The goal is to infer the set j Ij gt 0.
• Let l0, l1, l2 denote the number of markers in
  each group (l0 l1 l2L)
• Let D0, D1 and D2 denote case genotypes of
  markers in group 0, 1 and 2, respectively.

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The Bayesian marker partition model

• The likelihood model assumes independence between
  markers in the control population.
• The genotype frequencies of each biallelic marker
  in group 1 in the disease population
• The likelihood of D1 is
• nj1, nj2, nj3 are genotype counts of marker
  j

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Dirichlet Distribution

• A family of continuous multivariate probability
  distributions parameterized by the vector a of
  positive reals
• The probabilities of K rival events are xi.

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Dirichlet Prior for Group 1

• Assume a Dirichlet(a) prior for ?j1,?j2,?j3,
  where a(a1,a2,a3)
• Integrate out T1 and obtain the marginal
  probability

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Dirichlet Prior for Group 2

• 3l2 possible genotype combinations with frequency
• nk the number of genotype combination k
• Assume a Dirichlet(ß) prior distribution of
• Integrate out T2

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Dirichlet Prior for Group 0

• Same distributions as controls
• The genotype frequencies of the L markers in the
  control population
• njk and mjk are the numbers of individuals with
  genotype k at marker j in D and U

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Dirichlet Prior for Group 0 (c1)

• Assume Dirichlet priors for ?j with parameters
• Integrate out T

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Posterior Distribution

• The posterior distribution of I

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Markov chain Monte Carlo

• Markov chain Monte Carlo (MCMC) methods are a
  class of algorithms for sampling from probability
  distributions
• Based on constructing a Markov chain that has the
  desired distribution as its stationary
  distribution.
• The state of the chain after a large number of
  steps is then used as a sample from the desired
  distribution.
• The quality of the sample improves as a function
  of the number of steps.

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Metropolis-Hastings algorithm
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M-H in BEAM

• Two types of proposals are used
• Randomly change a markers group membership
• Randomly exchange two markers between groups 0, 1
  and 2.
• The proposed move is accepted according to the
  M-H ratio, which is just a ratio of Gamma
  functions.
• To improve the sampling efficiency
• Set a lower bound on the number of markers in
  group 2
• Gradually reduce this bound to 0 during burn-in
• This forces the algorithm to explore the space of
  high-order interactions.
• Also used an annealing strategy in burn-in
  iterations
• a temperature set high initially and gradually
  reduced to 1

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The simulated data on Model 4 contains 1,000
markers from 1,000 cases and 1,000 controls, with
MAF 0.1 and marginal effect size 0.4. Run BEAM
for 150,000 burn-ins plus 200,000 samplings in
three chains, with prior p1p21/3.
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Circles denote the overall posterior
probabilities of associations, with marginal and
joint associations combined. Plus signs denote
posterior probabilities of marginal associations.
Three circles on the top correspond to the three
simulated disease markers having interaction
effects.
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B Statistic

• A hypothesis-testing procedure to check each
  marker or set of markers for significant
  associations
• For each set M of k markers to be tested, the
  null hypothesis is that markers in M are not
  associated with the disease.
• Here, k1,2,3, represents single-marker, two-way
  and three-way interactions, etc.

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B Statistic (c1)

• Define the B statistic for the marker set M
• P0(DM, UM) and PA(DM, UM) are really the Bayes
  factors
• the marginal probabilities of the data with
  parameters integrated out from our Bayesian model
• under the null and the alternative models,
  respectively

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Bayes Factor

• Given a model selection problem between two
  models M1 and M2, on the basis of a data vector
  x. The Bayes factor K is given by
• p(x Mi) the marginal likelihood for
  Mi.
• Similar to a likelihood-ratio test
• instead of maximizing the likelihood
• Bayesians average it over the parameters

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B Statistic (c2)

• Choose both P0 and PA as an equal mixture of two
  distributions
• One that assumes independence among markers in M,
  Pind(X), of the form of P(D1I)
• The other a saturated joint distribution of
  genotype combinations across all markers in M,
  Pjoin(X), as P(D2I)
• Under the null hypothesis, the B statistic is
  asymptotically distributed as a shifted ?2 with
  3k1 degrees of freedom

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Conditional B Statistic

• A set of k (2,3,) markers may include t(ltk)
  markers that are significant through either
  marginal or partial interaction associations.
• The asymptotic null distribution of BMT is a
  shifted ?2 with 3k3t degrees of freedom.

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Simulated Epistasis Models
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Simulated Epistasis Models (c1)
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Simulated Epistasis Models (c2)

• Model 6 is a 6-way interaction model
• Denote the genotypes of each SNP by 0, 1, and 2
• Code each genotype combination over 6 disease
  loci by integers between 0728
• Assign disease effect ? 50 to genotype
  combinations 4, 5, 7, 111, 114, 253, 254, 360,
  387, 603, and 630.
• ? 50 so that these genotype combinations can
  explain a non-trivial portion (gt10) of cases.

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Simulated Epistasis Models (c3)

• 50 data sets for each disease model were
  simulated under each setting
• Marker minor allele frequencies (MAF) are
  uniformly in 0.05, 0.5.
• Each untyped disease locus is linked to one
  genotyped marker
• The remaining markers are unlinked

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Comparison Algorithms

• The stepwise logistic regression approach
• All markers are individually tested and ranked
  for marginal associations.
• The top 10 of markers are selected, among which
  all k-way (k2 or 3) interactions are tested and
  ranked
• A ?2 test with two degrees of freedom to test for
  single-marker associations.
• A stepwise B-stat
• the same search strategy as stepwise logistic
  regression
• use the B statistic for testing significance

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Power Calculation

• Define the power of each method as the proportion
  of 50 data sets in which all truly associated
  markers are identified and show statistically
  significant associations (adjusted P values below
  0.1) with the disease.

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A Hierarchical Procedure to Declare Significance

• Marginal associations
• report all markers with significant marginal
  associations after a Bonferroni correction for
  the number of markers, L.
• 2-way interactions
• after the Bonferroni correction for L(L-1)/2
  tests
• report all significant novel 2-way interactions
• Neither markers has been reported earlier
• if one marker has been reported earlier
• compute its conditional B-statistic (or the
  conditional log likelihood ratio (LLR) for
  logistic regression)
• report the interaction if significant after a
  Bonferroni correction.

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A Hierarchical Procedure to Declare Significance
(c1)

• 3-way interactions
• report all novel 3-way interactions that are
  significant after a Bonferroni correction for
  L(L-1)(L- 2)/6 tests
• if t1 or 2 markers were already found
  significant
• calculate the conditional B-statistic (or the
  conditional LLR)
• report the interaction if it is still significant
  after a Bonferroni correction.
• All p-values were estimated by a chi-square
  distribution with d3k3t degrees of freedom, for
  k1,2,3, t0,1,2, tltk, and adjusted by Bonferroni
  corrections.

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Results

• The power for detecting marginal associations was
  not compromised by using the more complex models.

BEAM (B), the stepwise B-stat (S), the stepwise
logistic regression (L) the 2-d.f. ?2 test (C)
Each data set contains 1,000 markers. Black bars
represent the power for 1,000 cases and 1,000
controls. Gray bars represent the power for 2,000
cases and 2,000 controls.
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BEAM (B), the stepwise B-stat (S), the stepwise
logistic regression (L) the 2-d.f. ?2 test (C)
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BEAM (B), the stepwise B-stat (S), the stepwise
logistic regression (L) the 2-d.f. ?2 test (C)
Each data set contains 1,000 markers. Black bars
represent the power for 1,000 cases and 1,000
controls. Gray bars represent the power for 2,000
cases and 2,000 controls.
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Results (c1)

• BEAM performs better especially when either
  disease allele frequencies or marginal effects
  were small.
• The power of all methods decreases with the decay
  of the LD (measured in r2) between disease loci
  and associated markers.

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Type I Errors

• All three epistasis mapping methods made similar
  amounts of type I errors.
• At the 0.1 significance level, they all made 10
  type I errors (after Bonferroni correction) when
  searching only for marginally significant
  markers.
• All methods made much fewer than 10 type I
  errors when searching for interactions.

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Impact of mismatch in allele frequencies and LD

• The power of association mapping can be greatly
  hampered by the discrepancy of allele frequencies
  between unobserved disease loci and associated
  genotyped markers.
• Investigated the impact based on model 2
• MAFs at two interacting disease loci were both
  0.1
• The marginal effect size per disease locus was
  0.5
• One linked marker had the matched MAF, whereas
  the other had an MAF ranging from 0.05 to 0.5.
• The LD between disease loci and associated
  markers was controlled to range from D0.7 to
  D1.

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Genome-wide association study of AMD

• The AMD (age-related macular degeneration) data
  set contains 116,204 SNPs genotyped for 96
  affected individuals and 50 controls.
• Remove nonpolymorphic SNPs and those that
  significantly deviated from Hardy-Weinberg
  Equilibrium (HWE)
• Remove additional SNPs containing more than five
  missing genotypes.
• After the filtration, 96,932 SNPs remained.

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Prior Calibration

• With only 146 individuals and 100,000 SNPs, the
  posterior probability of associations for each
  marker is strongly influenced by the choice of
  priors, although the order of these probabilities
  is nearly invariant.

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Simulation Based on AMD Data
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Comparison with other epistasis mapping approaches

• MDR identifies k-way interactions through an
  exhaustive search and evaluates the association
  between each interaction and the disease by
  cross-validations.
• Logic regression infers a tree-based relationship
  between the disease status and a set of
  markers.It evaluates the detected associations by
  permutation tests.
• BGTA uses a bootstrap-type resampling screening
  procedure to select markers, and those markers
  with return frequencies greater than the third
  quartile plus 1.8 times the interquartile range
  are deemed disease-associated markers.

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Discussions

• The BEAM algorithm has two essential components
• a Bayesian epistasis inference tool implemented
  via MCMC
• a novel test statistic for evaluating statistical
  significance
• A natural advantage of the Bayesian approach
• incorporate prior knowledge about each marker
• quantify all information and uncertainties in the
  form of posteriors
• Evaluating the statistical significance of a
  candidate finding via P values
• more robust to model choice and prior assumptions
• can give the scientist peace of mind

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Discussions (c1)

• The power of epistasis mappings depends
  critically on
• sample size
• effects of disease mutations
• any discrepancy in allele frequencies between
  disease loci and associated markers
• There are several issues that may affect the
  accuracy
• population substructures
• genotyping errors
• disease heterogeneities

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THANK YOU