gene mapping with bayesian variable selection and mcmc michael swartz [email protected]
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Gene Mapping with Gene Mapping with Bayesian Variable Bayesian Variable
Selection and MCMCSelection and MCMCMichael Swartz
OutlineIntro to GeneticsIntro to Gene mapping, Association studiesThe Conditional logistic regression model
for Gene mappingBayesian Model Selection
Stochastic Search Variable Selection Stochastic Search Gene Suggestion (SSGS)
Performance on Simulated Data SSGS vs the MLE.
Intro to GeneticsIntro to Genetics
Picture book of Genetics
Chromosomes: Line up genes
Gene: A specific coding region of DNA
Locus: a gene’s position
Alleles:
Genotype: Both
Molecular Marker: A polymorphic locus with a known position on the chromosome
Haplotype: One
Linkage Violates Mendel’s Second
law: Genes segregate independently
Genes that co-segregate in the recombinant gametes are linked.
Biological source of linkage: Meiosis -- the process of cell division that produces haploid gametes.
Linkage
Allows us to measure genetic distance
Linkage Disequilibrium
Association of alleles in a population
Gene Mapping: Association Gene Mapping: Association StudiesStudies
Data: The Case-Parent Triad
Collect Haplotype information on the Parents (G) as well as the case (g) so we have information about the transmitted and non transmitted haplotypes. Model the probability of transmission.
Gene Mapping By Association
Transmission Disequilibrium Test (TDT) Uses transmitted and non-transmitted alleles in case parent
triads to jointly test for linkage and linkage disequilbrium Based on McNemar’s test for case-control data Tests for association between two loci at a time
Log-linear models Also used for case-control data TDT triads can be modeled with Conditional Logistic
Regression for case control data. (Self, et al, 1991, Thomas, et al., 1995)
Extends the TDT to multiple loci
Advantages to a log-linear model
Using a Bayesian model we can incorporate genetic association between the markers.
Easy to analyze multiple lociEasy to consider Gene X Gene interactionsEasy to consider haplotypesEasy to consider environmental effectsEasy to consider Gene X Envrionment effects
Advantages to a log-linear model
Using a Bayesian model we can incorporate genetic association between the markers.
Easy to analyze multiple lociEasy to consider Gene X Gene interactionsEasy to consider haplotypesEasy to consider environmental effectsEasy to consider Gene X Envrionment effects
Coding the Triads (Thomas et al., 1995; Schaid 1996)
Ex: 3 diallelic loci.Recall gip and GT
ip from the case-parent triad.
For the Logistic Regression model we use Zi= gim+gif.
This is known as GTDT coding scheme (Schaid 1996)
Using Haplotypes in Conditional Logistic Regression is one way to examine Complex Diseases using Triads
10
01
01
01
10
01
ifim gg
Sampling Distribution for Triads
( | ) ( | , , )
( | , , )* ( | , )
( | , )
( | , , )
m f
m f m f
m f
m f
P D g P D g G G
P g D G G P D G G
P g G G
P g D G G
The Sampling distribution: a Conditional Logistic Function
(Thomas et al., 1995, Self et al., 1991)
**
),(
),(
)|(
)|(,,|
Ggifim
ifim
Ggi
iifimi ggRR
ggRR
gDP
gDPDGGgP
i
where G* is the set of all possible transmitted genotypes given the parents’ genotypes (“Pseudo-Controls”):
1
1 1
, explAL
m f la lal a
RR g g g
0 0 1 0 0 1 1 1* , , , , , , ,im if im if im if im ifG G G G G G G G G
and
Identifiability for Conditional Logistic Regression Parameters
Gene Mapping with Conditional Logistic Regression (CLR) uses categorical covariates (genotpye or haplotype)
For identifiability, we must define a reference category for each locus
Choose the most prevalent allele at each locus as it’s reference allele.
Calculating Prevalence from Triads (Thomas, 1995)
Let Cla denote the number of haplotypes in the case that carry allele a at locus l.
Likewise, let Pla denote the number of haplotypes in the parents that carry allele a at locus l.
If N denotes the total number of triads, then the prevalence of allele a at locus l can be calculated by: (Pla – Cla)/2N
Using CLR to infer genesFrequentist
Make Inference on the Maximum Likelihood Estimates for the parameters in the CLR model.
• Requires numerical optimization
• Prepackaged in STATA clogit command.
Bayesian Calculate Posterior Distribution and make inference
from the appropriate summaries• Requires Markov Chain Monte Carlo posterior simulation
• Implemented in Stochastic Search Gene Suggestion (SSGS)
Bayesian Model SelectionBayesian Model Selection
Use a Hierarchical Bayesian method
( ) ( ) ( ), | | | ,P Data P P f Data
Make inferences from the variable posterior:
Hierarchical Bayesian setup for Variable Selection
is an indicator vector of the variables, and () is the vector of coefficients for model .
| , |Data Data
Advantages to Bayesian Hierarchical Modeling
Account for prior informationAllow for Bayesian Variable Selection
TechniquesMake inference from model posterior No multiple testing because discussing pure
probabilities
Linear Regression: Introduce a latent variable to indicate covariate’s importance.
Hierarchy – allows prior information to enter the model and be updated by the data Likelihood: Y|,2 ~ Nn(X, 2I) Model Prior: ~ Binomial(p) Parameter Priors:
• | ~ Np(0,DR D )
• 2| ~ IG(/2, /2) /2 ~ 2
Stochastic Search Variable Selection(George and McCulloch, 1993)
Full Conditionals for , , and 2 recognizable Gibbs Sampling
Generalized to Various GLMs (George, McCulloch, and Tsay, 1996; Ntzoufras, Forster, and Dellaportas, 2000; and a few others).
Stochastic Search Variable Selection(Continued)
( ) ( ) ( ), | | | | ,P Data P P P f Data
Stochastic Search Gene SuggestionExtends Stochastic Search Variable Selection
(George and McCulloch, 1993)
Introduces two latent variables to indicate a gene’s importance in the model: one for loci and one for alleles.
Induces a hierarchy that allows prior information about genes to enter the model Genetic structure Genetic correlation
The hierarchical nature allows the data to update the probability of including a particular gene
Priors for Gene Suggestion Use two priors for gene suggestion
One indicator vector for locus selection: =(1,…,L),
where pl = P(Locus l is associated with the disease)
L
llpBernoulli
1
One indicator vector for allele selection given each locus: . Each element [la] pertains to a particular allele at locus l.
1
1 1
lAL
lal a
Bernoulli q
where qla= P(Allele a at locus l causes disease)
Prior for allele main effects (|,):Allelic dependence in model selectionPrior for main effects models the genetic
dependencies between loci and alleles
RDD,0MVN,|
where LLALA kkkk ,,,,,,Diag 1111 1
D
with each kla defined as
0* if0
1 * if*
lalala
lalalala λ
λck
How SSGS worksExploits MVN Covariance matrix DRD (George
and McCulloch, 1993) If = 0, then la focuses the probability of la around 0
if = 1, then lacla expands the probability of la to cover reasonable values
Automatic methods for choosing and c in paperSubjectively
choose la such that -3la < la < 3la implies la =0
choose cla such that 3lacla covers reasonable values for la
Model information contained in P(| Data)R based on Linkage Disequilbrium can be helpful for
gene mapping
L
The Prior Covariance Matrix
Define the Diagonal Blocks {lili} using the covariance for a multinomial distribution using allele frequencies assuming they are constant across generation.
Determine the off-diagonal blocks {lilj}{ij} using the allelic disequilibirium between the alleles at locus i and locus j: .
Define R = L-1
bababa jijiji ppp
Sampling from the Posterior
No full conditional for updating Use Hybrid Gibbs sampling and Metropolis-Hastings
Algorithm to construct a Markov Chain. Full conditionals for updating and Metropolis Hastings acceptance ratio for updating by locus
For a given model, sample repeatedly from Metropolis Hastings before proposing a new model Even model iterations generated by independence MLE proposal Odd model iterations generated by random walk proposal
,,,|,|
),,,|,,(
DGGgp
DGGgP
fm
fm
Gibbs Sampling ComponentP(i=1| (-i), , , g, Gm, Gf) = P(i=1| (-i), )
= a1/(a0+a1) a1 = f(| i=1, (-i), )*f((-i), i=1)
a0= f(| i=0, (-i), )*f((-i), i=0)
P(i=1| (-i), , , g, Gm, Gf) = P(i=1|(-i), ) = b1/(b0+b1) b1 = f(| i=1, (-i), )*f((-i), i=1)
b0 = f(| i=0, (-i), )*f((-i), i=0)
Metropolis Hastings Component (by locus)
1*11
*1**
;,,|,,,|
;,,|,,,|
tllll
tll
tl
lltlllll
qLp
qLp
MH Ratio:
Two different proposal Distributions: MLE independence proposal conditional on other loci
Random Walk symmetric proposal conditional on other loci
llll
tll DHDMVNq
|,1
,|1* ,|
ll
tll
tll DHDMVNq
|,1
,1
)(|1* ,|
SSGS Flow Chart
Using a Bayesian Model, we simply summarize the posterior in a meaningful way
The MCMC sample is a large sample from our posterior
Thus we can summarize gene’s importance by using the marginal posterior probability of inclusion for each gene
Use the median model threshold: P(la) > .5
Finding Genes
1( is important) =
# laall iterations
p laiterations
Simulating DataSimulating Data
Simulated DataUsed genetic data simulated for Genetic
Analysis Workshop 12 (GAW12)Used Chromosome 1 from isolated population
Microsatellite markers simulated 1 cM apart, with 4-16 alleles
Simulated without influence from selection reference: Wijsman, E.M. Almasy, L., Amos, C.I., Borecki, I.,
Falk C.T., King, T.M., Martinez, M. M., Meyers, D., Neuman, R., Olson, J.M., Rich, S., Spence, M.A., Thomas, D. C., Vieland, V.J., Witte, J. S., MacCluer, J.W. (2001) Genetic Analysis Workshop 12: Analysis of Complex Genetic Traits: Applications to Asthma and Simulated Data. Genet Epidemiol 21(supp 1):S1-S853
Using GAW 12 Data: Model Simulation
Simulate directly from model Use the conditional logistic regression function to determine
probability of transmission of the genes• The parents determine the 4 possible children
• Treat each child as a category in a multinomial distribution
• Calculate the probability of each child using a conditional logistic regression function with specified ’s
• Draw 1 sample from the corresponding multinomial distribution to determine the affected genotype for the triad.
Know the right answers for Analyze the data twice
Independent R = I Dependent R – based on HWE & LD
Simulation 1: Model Simulation
3 loci with a total of 20 alleles, close together A1=4; A2=11; A3=5
GAW 12 Chromosome 1 Loci 9, 11, and 12Genetic Covariance Present
Average D’ for 3 loci span from 0.133 to 0.256 90% of || [0.005,0.386] ; median = 0.012
True Model: g2, g14, g16
True Betas: {2=2.74, 14=3.63, 16 =4.39; -
(2,14,16)=0} 200,000 iterations
Running STATAData was collected in TriadSTATA needs pseudocontrols enumeratedAssuming no recombination, construct each
Z vector (sum of the haplotypes) of the possible children given the parents
Obtain MLE and confidence intervals: Run clogit on the data stratified by family (only the 4 children are present in each stratification)
Preparing for SSGS
Label the haplotypes in the parents as transmitted or non transmitted
Calculate the MLE’s and Fisher’s information using STATA to define the proposal distribution for even iterations
Define the initial values for (mle) (= l) (= 1)
Simulation 1: Model Simulation
Independent Prior p = q = 0.5 = 0.2, c = 10 None of the ’s failed the
Heidelberger and Welch test for stationarity
Total models visited: 302
Dependent Priorp = q = 0.5 = 0.2, c = 10None of the ’s failed
the Heidelberger and Welch test for stationarity
Total models visited: 6046
Simulation 1: Suggested Genes
00.1
0.2
0.30.4
0.5
0.6
0.70.8
0.9
1
g1 G2* g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 G14* g15 G16* g17
DPIP
Method Suggested Genes
Dependent Prior g2, g14, g16
Independent Prior g2, g14, g16
MLE (95% CI) g2, g4, g5, g10, g12, g14, g16
Simulation 1: Estimation Intervals
Using GAW 12 Data: Disease Simulation
Simulate a disease Pick alleles at a marker that cause the disease Simulate disease based on a determined
penetrance,(P(D|genes)) sporadic risk (P(D|normal), and dominance
Know which alleles should be suggested by SSGS, but not the true
Analyze the data twice Dependent R – based on HWE & LD Independent R = I
Simulation 2: Simulated Disease3 loci from GAW 12 chromosome 1: Locus 1
A1=6, Locus 2 A2=8, Locus 8 A8 = 4Genetic Correlation:
Average D’ values span from 0.084 to 0.29 90% of || [0.0003,0.259]; median = 0.005
Penetrances:• P(D|L1a3,L1a3) =0.4• P(D|L8a2,L8a2) =0.6• P(D|L8a4,L8a4) =0.4• P(D|L8a2,L8a4) =0.5• P(D|any other genes) = 0.05
True model: g3, g14, g15 200,000 iterations
Simulation 2: Suggested Genes
Method Suggested Genes
Dependent Prior g3, g14, g15
Independent Prior g13, g14, g15, Missed g3
MLE (95% CI) g1, g3, g13, g14, g15
00.10.20.30.40.50.60.70.80.9
1
g1 g2 G3* g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 G14*G15*
DPIP
Sensitivity Analysis
What we learned Today Extending the TDT to a conditional logistic regression
model has many advantages: analyze multiple loci Bayesian setting can incorporate genetic association and more!
We can find genes using Maximum likelihood estimation and inference for the parameters of the CLR model using STATA
We can improve the estimates of MLE by using SSGS with a prior that accounts for genetic association
SSGS has some sensitivity to prior: lower prior, less genes
ReferencesBarbieri, M.M., and Berger, J. O. (2004), Optimal
Predictive Model Selection, Annals of Statistics 32, to appear.
Schaid, D. (1996) General Score tests for Associations of Genetic Markers with Disease Using Cases and Their Parents. Genetic Epidemiology. pp. 423-449
Self, S.G., et al. (1991) On estimating HLA/disease association with applications to a study of Aplastic Anemia. Biometrics, pp.53-61.
Thomas, D. C., et. al. (1995) “Variation in HLA-associated risks of Childhood Insulin Dependent Diabetes in the Finnish population: II. Haplotype Effects” Genetic Epidemiology. pp. 455-466.
SSGS dissertation: https://epi.mdanderson.org/~mswartz/
Papers Extending SSVSChipman, H. (1996) “Bayesian variable selection
with related predictors”. The Canadian Journal of Statistics pp. 17-36.
George, E. I., McCulloch, R.E., and Tsay, R.S. (1996). Two approaches to bayesian model selections with applications” Bayesain Analysis in Econometrics and Statistics-Essays in honor of Arnold Zellner. (Eds. D.A. Berry, K.A. Chaloner, and J.K. Geweke). New York: Wiley pp. 339-348.
Ntzoufras, I. Forster, J.J., and Dellaportas, P. (2000) “Stochastic Search Variable Selection for Log-Linear Models” Journal of Statistical Computations and Simulations. pp.23-37