Gene Mapping with Gene Mapping with Bayesian Variable Bayesian Variable

Selection and MCMCSelection and MCMCMichael Swartz

mswartz@stat.tamu.edu

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)

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),(

),(

)|(

)|(,,|

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

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),,,|,,(

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**

;,,|,,,|

;,,|,,,|

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tll

tl

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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

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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