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Using multi-level Bayesian hierarchical model to detect related multiple SNPs within multiple genes to disease risk

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Using multi-level Bayesian hierarchical model to detect related multiple SNPs within multiple genes to disease risk

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USING MULTI-LEVEL BAYESIAN HIERARCHICAL MODEL
TO DETECT RELATED MULTIPLE SNPS WITHIN MULTIPLE GENES
TO DISEASE RISK

by

Lewei Duan

A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(BIOSTATISTICS)

May 2013

Copyrights 2013 Lewei Duan
i

DEDICATION

To my family and friends

ii

ACKNOWLEDGEMENTS
I am grateful to Dr. Duncan Thomas, Dr. Paul Marjoram, Dr. David Conti, and Dr.
Stanley Azen for their invaluable guidance and support.

iii

TABLE OF CONTENTS
DEDICATION ………………………………………………………………………………….………………. i
ACKNOWLEDGEMENTS …………………………………………………………… …….……………… ii
LIST OF TABLES……………………………………………………………………………………………… v
LIST OF FIGURES ………………………………………………………………………………………….. vi
ABBREVIATION ………………………………………………………………………………………..……vii
ABSTRACT …………………………………………………………………………………………..……… viii
CHAPTER 1: INTRODUCTION .................................................................................................1
Previous Approaches to Variant Selection .........................................................................2
Motivation for the Multi-level Bayesian Hierarchical Model ........................................4
Organization ................................................................................................................................6
CHPATER 2: MOTIVATING EXCAMPLE: THE WECARE STUDY OF DNA DAMAGE
RESPONSE PATHWAY IN SECOND BREAST CANCER FOLLOWING
RADIOTHERAPY .........................................................................................................................7
DNA Double-Strand Breaks Response Pathway and Breast Cancer ...........................7
WECARE study design ..............................................................................................................9
Summary .................................................................................................................................... 13
CHAPTER 3: THE GENERAL BAYESIAN FRAMEWORK ................................................ 15
Bayesian Framework ............................................................................................................. 15
Bayes Factors ........................................................................................................................... 19
Bayesian Hierarchical Models Structure .......................................................................... 21
iv

CHAPTER 4: MULTI-LEVEL BAYESIAN HIERARCHICAL MODEL: A NOVEL
APPROACH FOR PATHWAY ANALYSIS OF GENETIC ASSOCIATION STUDIES ..... 25
Hierarchical Modeling: Applications to Genetic Data .................................................. 25
Multi-level Bayesian Hierarchical Model: Statistical Methods .................................. 29
Model I ........................................................................................................................................ 31
Model II ...................................................................................................................................... 34
Fitting the Model ..................................................................................................................... 35
Posterior Summarization ...................................................................................................... 37
CHAPTER 5: SIMULATION STUDIES .................................................................................. 39
CHAPTER 6: APPLICATION TO THE WECARE DATA ................................................... 43
CHPATER 7: DISCUSSION ..................................................................................................... 49
REFERENCES ............................................................................................................................ 72

v

LIST OF TABLES
Table 1: Bayes Factos’s Interpretation for association (Kass and Raftery, 1995) .
...................................................................................................................................................... 58
Table 2: Frequency Distribution of SNPs based on their MAFs ................................ 58
Table 3: Bayes Factors in SNP-level and Gene-level from Model I and Model II for
Selected SNPs. .......................................................................................................................... 59
Table 4: Association Between Selected Variants in DNA-Damage Response
Genes and CBC risk ................................................................................................................. 60

vi

LIST OF FIGURES
Figure 1: Directed acyclic graph describing the structure of the model.. .............. 61
Figure 2: Graphical representation of the A matrix derived from the Gene
Ontology.. ................................................................................................................................... 62
Figure 3: Comparison of estimated parameters based on different priors. .......... 63
Figure 4: The correlation analysis n between Estimated and True parameters. 64
Figure 5.1: Simulation analysis for Model I. Sensitivity/Specificity based on
proportions of the number of SNPs (a). Sensitivity based on Bayes Factors of
SNPs (b). Bayes factors of genes (c) ................................................................................. 65
Figure 5.2: Simulation analysis for Model II. Sensitivity/Specificity based on
proportions of the number of SNPs (a). Sensitivity based on Bayes Factors of
SNPs (b). Bayes factors of genes (c). ................................................................................ 66
Figure 6.1: Simulation analysis for Model I by MAF. Sensitivity/Specificity based
on proportions of the number of SNPs categorized by MAF (Left panels).
Sensitivity based on Bayes Factors of SNPs categorized by MAF (Right panels).
...................................................................................................................................................... 67
Figure 6.2: Simulation analysis for Model II by MAF. Sensitivity/Specificity based
on proportions of the number of SNPs categorized by MAF (Left panels).
Sensitivity based on Bayes Factors of SNPs categorized by MAF (Right panels).
...................................................................................................................................................... 68
Figure 7: Trace plots of Conditional LogLikelihood for Model I (a) and for Model
II (b).. .......................................................................................................................................... 69
Figure 8: Gene-Disease association analysis.. ............................................................... 70
Figure 9: SNP inclusion analysis.. ...................................................................................... 71

vii

ABBREVIATIONS

Abbreviation Description
BF Bayes Factor
BRI Bayesian Risk Index
CBC Contralateral Breast Cancer
CDCV Common Disease Common Variants
CI Confident Interval
DSB Double-strand Breaks
GO Gene Ontology
GWAS Genome-Wide Association Scan
Gy
HRR
Gray
Homologous Recombination Repair
IR Ionizing Radiation
LD linkage disequilibrium
MCMC Markov Chain Monte Carlo
MLE Maximum Likelihood Estimation
MRN
NHEJ
MRE11A-RAD50-NBN complex
Non-homologous End Joining
OR
RERF
Odds Ratio
Radiation Effects Research Foundation
RR Relative Risk
RT Ratiation Treatment
SNP Single Nucleotide Polymorphism
SSGS Stochastic Search Gene Suggestion
UBC Unilateral Breast Cancer
WECARE Women's Environmental Cancer And Radiation Epidemiology Study

viii

ABSTRACT
We proposed a novel statistical method to investigate the involvement of
multiple genes thought to be part of a common pathway for a particular disease. If a
gene is identified to be associated with the disease, we are also interested in
discovering which single nucleotide polymorphisms (SNPs) within this gene are
responsible for this association. Here we present a Bayesian hierarchical modeling
strategy that allows for multiple SNPs within each gene, with external prior
information at either the SNP or gene level. The model involves variable selection at
the SNP level through latent indicator variables and Bayesian shrinkage at the gene
level towards a prior mean vector and covariance matrix that depend on external
information. The entire model is fitted using Markov Chain Monte Carlo methods.
Simulation studies show good ability to recover the underlying model. The method
is applied to data on 504 SNPs in 38 candidate genes involved in DNA damage
response in the Women's Environmental Cancer And Radiation Epidemiology study
(WECARE) of second breast cancers in relation to radiotherapy exposures.
Key words: pathway models; candidate genes; hierarchical Bayes models; breast
cancer; DNA damage response; WECARE study.

1

CHAPTER 1: INTRODUCTION
There is a growing literature on methods for pathway modeling, motivated in
large part by an interest in mining genome-wide association scan (GWAS) data for
commonalities across related genes that individually may not achieve genome-wide
significance, but in the aggregate may point to novel pathways (see [Wang et al.
2007] for a review of gene set enrichment analysis and alternatives). Our goal here
is more modest, guided by an a priori selection of strong candidate genes (Thomas
2010a). Like other methods of pathway analysis, however, we aim to exploit
external knowledge about the biological function of each gene and the relationships
between them (Thomas 2010ab).
A major focus of investigation in recent years has been the search for genetic
risk factors for diseases of interest. However, genetic association studies using
conventional statistical tools have been hindered by complications of multiple
comparisons or selective reporting. Hierarchical modeling may improve the
analyses of genetic data by increasing the precision of the risk estimates and
reducing the likelihood of false positives via shrinkage toward to the prior mean
(Hung et al 2007).
The identification of susceptibility and/or resistance alleles in candidate-
gene studies provides direct evidence that these genes and their biological pathways
are relevant to specific diseases in humans. High-throughput genotyping
technologies such as whole-genome SNP scans have made it possible to find genetic
2

variants that contribute to the risk of common diseases in the past few years
(Hunter 2005). An agnostic GWAS is currently underway within the Women’s
Environmental Cancer And Radiation Epidemiology (WECARE) study, but we
focused on methods for pathway analysis in candidate gene association studies. The
aim of our study is to provide a comprehensive modeling strategy for examining the
effects of all genes and their SNPs in a pathway, and to apply our approaches to
variants selection and gene association study on WECARE data.
Previous Approaches to Variant Selection
Selection of predictors to be included in a multiple regression model has long
been a crucial problem in statistics, from both frequentist and Bayesian
perspectives. Variable selection via Gibbs sampling was proposed for selecting
promising subsets (George and McCulloch 1993) by embedding the regression setup
in a hierarchical normal mixture model where latent variables are used to identify
subset choices. The variables with higher posterior probabilities are identified as
the promising subsets of predictors. The Gibbs sampler is used to indirectly sample
from this multinomial posterior distribution on the set of possible subsets. The
more frequently the subsets appear in the Gibbs sample, the higher the probability
is that they have predictive value.
George and McCulloch (1997) also compared various approaches for prior
specification and posterior computation for Bayesian hierachical variable selection.
They pointed out that a variety of Markov Chain Monte Carlo (MCMC) algorithms
3

can be constructed based on the Gibbs sampler and Metroplis-Hastings algorithms,
or even a combination of both, for the pupose of posterior exploration. They pointed
out that it is more appropriate to average predictions over the posterior distribution
rather than using predictions from any single model, when the goal is prediction.
Clyde et al. (1996) and Raftery et al. (1993) have discussed the potential of
prediction averaging in the face of uncertainty about variable selection. This concept
has been implemented in our model.
Most of the previous methodological literature on the design of genome-wide
association studies and most applications have used statistical significance as the
sole criterion. However, various investigators have recognized that giving higher
priority to a targeted subset of SNPs with greater biological plausibility of true
association can lead to higher predictive power (Bostein and Risch 2003; Sohns et
al. 2009; Tintle et al. 2009). Information from previously published investigations
indicating the effects of SNPs can be considered for studies that attempt to replicate
association findings (Rebbeck et al. 2004). Methods that prioritize the choice of
genetic variants to be genotyped in molecular epidemiological studies can provide
some useful guidance to optimally select variants for candidate gene association
studies. For instance, 21 SNPs selected by GWAS for association with primary breast
cancer were genotyped in WECARE study population along with other SNPs in DNA
damage response pathway, and several of these were found to increase risk for
contralateral breast cancer (Teraoka et al. 2011).
4

Preliminary snalysis of the genetic information based on previous GWAS
findings can provide evidence that will help prioritize the strongest constellations of
results. Using pathway analysis of GWAS results to prioritize genes and pathways
within a biological context is one of the common analytic methods to combine
preliminary GWAS statistics to identify genes, alleles and pathways for deeper
investigations (Cantor et al. 2010). Moreover, Lewinger et al. (2007) illustrated a
hierarchical regression approach that would allow appropriate use of multiple
sources of prior knowledge, so call “prior covariates”, without prejudging therir
informativeness. If at least some of the prior covariates have predictive value, the
ranking by posterior expectations performs better at selecting the true positive
association than a simple ranking of p-values. For instance, the large amount of
markers in GWAS scan enables the data to suggest which prior covariates are
correlatted with the strength of the marker-disease association.
Motivation for the Multi-level Bayesian Hierarchical Model
Our approach is motivated by many of the hierarchical methods used in
previous studies. For instance, our model is motivated in part by SSGS (Swartz et al
2006) to implement a stochastic search with a prior covariance structure to search
for genes associated with disease, for which using a prior covariance structure
improves variable selection under the normal linear model. Our model is also
motivated in part by recent work on methods for testing associations with multiple
rare variants using Bayesian methods in cancer data (Wilson 2010b, Quintana et al.
5

2011), and in next generation sequencing data (Quintana and Conti 2013), where
testing any single variant is nearly impossible because of their rarity and the
enormous multiple comparisons penalty. This motivates our choice of a burden
index for gene-level associations comprising simple 1/0/+1 weights with model
averaging across their uncertainty distribution. An extensive review of hierarchical
Bayesian methodology for variable selection in regression models can be found in
Rockova et al. (2012).
Our starting point is a model for multiple rare variants proposed by Quintana
et al. (2011), which collapses all the variants with a gene into a single “burden” type
index, similar to quite a number of other recent proposals (see Basu and Pan [2011]
for a recent review and comparison by simulation), but extended to allow for both
deleterious and protective effects and to explicitly allow for uncertainty about which
variants to include in the model (and which direction for those that are included) by
Bayesian model averaging. Hoffman and Witte (2010) introduced a step-up variable
selection approach that allows for deleterious and protective effects, but does not
consider model uncertainty except in the form of a permutation procedure for the
overall significance test, so is unable to assess the importance and direction of
particular variants or alternative models. Chen et al. (2010) describe a somewhat
similar model that combines variable selection at the SNP level with shrinkage at the
gene level. In this thesis, we extend this approach to multiple genes, incorporating
prior covariates and prior gene-gene similarity information in a Bayesian
hierarchical modeling framework.
6

Organization
The thesis is organized as follows. Chapter 1 (this chapter) outlines the
statistical problem and introduces methods previously used for variable selection.
We provide an overview of the motivations and inspirations for our new models.
Chapter 2 presents a motivating example: the WECARE study of DNA damage
response pathway in second breast cancer following radiotherapy. We describe the
WECARE study design, and summarize our study aim and strategy in this chapter.
Chapter 3 provides background of Bayesian framework, and introduces general
concepts and components of hierarchical Bayes model and some of their
applications. Chapter 4 briefly reviews the applications of hierarchical model in
genetic studies, and proposes our novel models: multi-level Bayesian hierarchical
models for pathway analysis in gene-disease association studies. We describe the
likelihood, define the hierarchical priors, and introduce the implementation of the
Markov chain to explore the posterior distribution. Next, in chapter 5, we evaluate
the performance of our multi-level Bayesian hierarchical models on simulated data.
In chapter 6, we examine the results from our multi-level Bayesian hierarchical
models applied to the real WECARE data. Finally, we conclude in chapter 7 with a
discussion of the results and implications of our modeling strategies, as well as
potential future work.
7

CHPATER 2: MOTIVATING EXCAMPLE: THE WECARE STUDY OF DNA
DAMAGE RESPONSE PATHWAY IN SECOND BREAST CANCER
FOLLOWING RADIOTHERAPY
DNA Double-Strand Breaks Response Pathway and Breast Cancer
Various kinds of DNA damage, such as double-strand breaks (DSB), can result
in development of cancer. Ionizing radiation (IR), for example, is known to cause
DSBs. Such damage invokes various DNA damage response mechanisms including
homologous recombination repair (HRR, a type of genetic recombination in which
nucleotide sequences are exchanged between two similar or identical molecules of
DNA), non-homologous end joining DNA repair (NHEJ, in contrast to HRR, repairs
DSBs in DNA by directly ligating the break ends without the need for a homologous
template), apoptosis (biochemical events lead to characteristic cell changes and
death) and cell cycle arrest (a regulatory process that halts progression through the
cell cycle during one of the normal phases (G1,S,G2,M)).
The invocation of DNA damage response pathways results in activation of
several DNA damage response molecules, some of which play pivotal roles in
sensing DNA damage and initiating the subsequent DNA repair process. These can
repair the damage or cause cell cycle arrest or apoptosis to protect the tissue from
disease, but can also permanently misrepair the damage in a manner that can lead
to cancer. In human cancer cells, some of the proteins in the DNA damage response
pathways are mutated or non-functional. The survival and proliferation of the
8

cancer cells are reliant on an impaired DNA damage response pathway, which
provides a therapeutic concept for IR treatment. Germline variation in these genes
can also lead to an inherent susceptibility to cancer or sensitivity to exposures like
IR, modifying an individual’s risk following expos ure. It is this latter constitutional
genetic susceptibility to second cancers following radiotherapy that is investigated
here, rather than the response of the first cancer to treatment.
Based on a comprehensive program of medical follow-up of survivors of the
atomic bombings of Hiroshima and Nagasaki, Japan, the Radiation Effects Research
Foundation (RERF) has produced quantitative estimates of cancer risk from
exposure to IR. In particular, RERF studies have found an excess risk of breast
cancer in female atomic bomb survivors (Land 1995), decreasing with increasing
age at exposure (Tokunaga, Land et al. 1987).
The influence of IR to breast cancer risk has been further explored in many
other studies, notably various cohorts of medical patients exposed to diagnostic or
therapeutic radiation. For instance, in the WECARE Study, Stovall et al. (2008)
showed an association with radiation dose to the contralateral breast, modified by
age at exposure and time since exposure.
The regulation of DSBs repair pathway induced by IR involves many genes in
a complex structure. Mutations in ATM, MRE11A, RAD50, and NBN lead to genetic
disorders and deficiency that are associated with increased cellular sensitivity to IR
(Helleday et al. 2008). The MRE11A-RAD50-NBN (MRN) complex recognizes the
DSB and stabilizes the broken strands of DNA at the break (Lee and Paull 2005,
9

Paull and Lee 2005). MRN recruits ATM to the damage site and activates ATM to
phosphorylate a number of downstream targets (Lee et al. 2010). These
downstream targets including CHEK2, NBN, MDC1, and TP53BP1, amplify the
damage signal by stabilizing proteins such as MRN at the DSB sites and recruiting
MDC1, TP53BP1, and others. This process activates multiple signaling cascades and
invokes several DNA damage response mechanisms including cell-cycle checkpoint
arrest, DNA repair, and apoptosis (Lavin 2008).
Thus, implications of variation in genes involved in the DNA DSB response
pathway become candidate risk factors for radiation-induced breast cancer. The
WECARE Study (Bernstein et al. 2004) is a typical example of a study aimed at a
comprehensive examination of genes involved in a particular pathway, in our case,
ionizing radiation induced DSB.
WECARE study design
Genetic epidemiology has been historically dominated by the use of family-
based designs from which inherited susceptibility can be inferred. With the advent
of methods for assessing DNA-sequence variability directly, association studies
using unrelated individuals became more popular. With evidence accrued from
large association (case-control) studies, it is possible to leverage much more
accurate predictions about the impact of genes on disease risk by aggregating the
results from individual rare variants on the basis of characteristics shared by groups
of variants (Capanu and Begg 2011). The WECARE study was designed with the
10

purpose of examining the joint roles of radiation exposure and genetic susceptibility
in the etiology of breast cancer. It is a multicenter, population-based, nested case-
control study, where cases are women with asynchronous contralateral breast
cancer (CBC) and controls are women with unilateral breast cancer (UBC)
(Bernstein et al 2004). Participants were identified, recruited, and interviewed
through four population-based cancer registries in the United States that are part of
the National Cancer Institute’s Surveillance, Epidemiology, and End Results
program: the Los Angeles County Cancer Surveillance Program; Cancer Surveillance
System of the Fred Hutchinson Cancer Research Center (Seattle); State Health
Registry of Iowa; and Cancer Surveillance Program of Orange County/San Diego-
Imperial Organization of Cancer Control (Orange County/San Diego). The fifth
registry from which participants were recruited was the Danish Cancer Registry
(Bernstein et al., 2004).
708 women with CBC were selected as cases from a cohort of 52,536 women
with histological confirmed breast cancer reported to one of the five population-
based cancer registries. Cases were required to meet the following criteria: (1)
diagnosed between January 1
st
, 1985 and December 31
st
, 2000 with UBC followed
by a second primary, in situ or invasive, breast cancer in the contralateral breast,
diagnosed at least 1 year later; (2) resided in the same study reporting area for both
diagnoses; (3) had no previous or intervening cancer diagnosis; (4) were under age
55 years at the time of diagnosis of the first primary breast cancer; (5) were alive at
the time of contact; and (6) provided informed consent, completed an interview, and
11

provided a blood sample. A 1-year interval between first and second breast cancer
diagnosis was used to rule out synchronous disease (Begg et al. 2008, Bernstein et al
2004, Borg et al 2010).
1399 women in the cohort of survivors of a first breast cancer serve as
controls. They were selected based on the same criteria as cases except that they
were not diagnosed with CBC by December 31
st
, 1999, and that they had not had
prophylactic mastectomy of the contralateral breast during the at-risk interval. (The
time between cases’ two di agnoses defined the “at risk interval”.) Two controls
were individually matched to each case on year of birth (in 5-year strata), year of
diagnosis (in 4-year strata), registry region, and race/ethnicity. In addition, cases
and controls were counter-matched on registry-reported radiation exposure to
improve statistically efficiency (Langholz and Goldstein 1996, Langholz et al. 2003),
so that each case-control triplet contained two radiation exposed women and one
unexposed (Bernstein et al. 2004). Counter-matching brings the marginal full cohort
exposure information into the sample, and hence is a more efficient sampling design
that increases the variability in exposure values over that of random sampling. In
particular, the counter-matching technique implemented in the WECARE study
results in the following situation in each study set – 1) for each exposed case, one
exposed and one unexposed control were selected from the relevant stratum, 2) for
each unexposed case, two exposed controls were selected. Here, the exposure status
refers to the indicator of radiation treatment as recorded in the cancer registry
where the subjects were identified. The counter-matching technique ensured that
12

each triplet contributed to the analysis, avoiding the situation where all members of
a matched set had the same radiation status.
The only difference between an analysis of a counter-matched sample and
that of a simple random sample is the inclusion of the weights in the model. The
weights depend on the number of exposed and unexposed subjects in the risk set.
An offset term is the log of the counter-matching weights, and is added to the log-
linear model with a coefficient fixed at one. An offset term is included in all analyses
to account for the counter-matched sampling design.
The original WECARE study was designed to focus on mutations in the ATM
gene, which plays a central role in recognition of DSBs (Bernstein et al. 2003,
Bernstein et al. 2005, Bernstein et al. 2006). Concannon and colleagues (2008) have
shown that in the same study population, there is an increased CBC risk associated
with rare variants in ATM that are predicted to be deleterious by evolutionary
conservation and a reduced CBC risk associated with some common ATM variants.
Furthermore, a statistically significant increase in CBC risk (Relative Risk [RR] = 2.0,
95%CI 1.1-3.9) has been found among those women who carry rare deleterious
ATM variants and were exposed to radiation treatment to the contralateral breast,
compared non-carriers who didn’t receive the radiation treatment (Bernstein et al.,
2010).
Many genes (i.e., BRCA1, BRCA2, CHEK2, ATM) found to be associated with
increased susceptibility to breast cancer function within a common biochemical
pathway involved in signaling and coordinating DSBs response (Thompson and
13

Easton 2004). The WECARE study was then extended to include BRCA1, BRCA2, and
CHEK2, which are all involved in DNA damage responses, and later still to a broader
set of 38 candidate genes involved in this and other DSB damage responses. The
main effects of ionizing radiation have been previously reported (Stovall et al. 2008,
Langholz et al. 2009), as have marginal associations with ATM (Bernstein et al. 2006,
Concannon et al. 2008), BRCA1/2 (Begg et al. 2008, Bernstein et al. 2010, Borg et al.
2010, Figueiredo et al. 2010, Capanu et al. 2011, Quintana et al. 2011), CHEK2
(Mellemkjaer et al. 2008), and interactions of radiation with ATM (Bernstein et al.
2010) and of BRCA1/2 with treatments and reproductive factors (Poynter et al.
2010, Reding et al. 2010), amongst other risk factors.
Summary
Ionizing radiation causes DSBs in DNA, which can invoke several DNA
damage response mechanisms, such as HRR, NHEJ, apoptosis and cell cycle arrest.
We selected 38 candidate genes that are involved in DSB damage response pathway.
We are interested in exploring whether there are associations between some of the
candidate genes and the outcome. If there is an association between a gene and the
disease, we want to find out which SNPs within this gene are driving this
association. In what follows, multilevel posterior probabilities, conditional
probabilities and Bayes Factors are used to address these questions. The SNP
inclusion probabilities are grouped based on which genes they belong to. The gene
inclusion probabilities are the posterior probability that at least one SNP within this
14

gene is associated with the disease. This gene inclusion posterior probability can be
defined as either the sum of the posterior probabilities of all models that include at
least one of the SNPs within the given gene, or simply as one minus the posterior
probability that none of the SNPs within the corresponding gene is included.

15

CHAPTER 3: THE GENERAL BAYESIAN FRAMEWORK
Bayesian Framework
The WECARE data involves multiple genes and environmental factors, which
interact in complex ways. Most association studies for discovery of novel main
effects and interactions in the previous analyses of WECARE data were based on
standard frequentist approaches. Lately, Bayesian methods have been used in
related WECARE investigations. Capanu et al. (2011) presented a hierarchical
statistical modeling approach involving psedo-likelihood estimation of the relative
risk parameters with Bayesian estimation of the variance compotents, to assess
risks of rare BRCA1 and BRCA2 variants to CBC with WECARE data. Quintana et al.
(2011) introduced the Bayesian risk index (BRI) methodology incorporating model
uncertainty in investigating the aggregate effects of multiple rare variants in BRCA1
and BRCA2 on risk of developing CBC with WECARE data.
The Bayesian framework can be applied to a wide range of statistical models.
Many advantages have been found in using Bayesian methods to model the
relationships in an integrated “systems biology” manner, particularly in a genome -
wide setting (Wilson et al. 2010ab). For instance, it is intuitive to make an inference
within a Bayesian framework, and the Bayesian models are extremely flexible.
Hence, the Bayesian approaches are particularly suited for solving complex genetic
questions.
We chose a Bayesian framework for application to the WECARE study
because it provides a very natural setting for incorporating complex structures and
16

multiple parameters. In addition, the Bayesian framework allows us to incorporate
external information in the analysis in various ways, such as by specifying prior
probability distributions for the parameters of interest and integrating known
correlation information among covariates into its variance-covariance matrix for a
specified distribution.
Bayes rule is fundamental to a Bayesian approach:

where θ denotes a vector of the parameters and D denotes observed data;
represents the joint probability; and represent conditional
probabilities. In the frequentist approach, the parameters of the model are assumed
fixed and unknown; instead, the Bayesian framework treats the parameters as
random variables. Hence, we specify both the prior distribution of the parameter
and the likelihood of the observed data in order to define the joint probability
model. Intuitively, Bayes rule allows us to update our current beliefs about the
unknown parameter θ given observed data D via the conditional
probability .
Previous experiences with genetic studies revealed that it is relatively
infrequent to detect SNP associations. The approach we implemented to explore
multiple hypothesis integrates potential submodels into a single hierarchical model.
Bayes Factors (BFs) are used to compare composite hypotheses in a Bayesian
variable selection framework. Hence, the values of BFs derived from the model
17

depend on the prior distribution over models. Therefore, selection of appropriate
priors on the model space is important.
There are many ways of formalizing a prior in Bayesian approaches. A
simpler prior formalization is preferred. There are many forms of prior
incorporation, such as lasso (Tibshirani, 1996), BIC (Chen and Chen, 2008), and
Beta-Binomial prior (Wilson et al, 2010b). The “Spike and Slab” prior was
introduced as a Bayesian approach to subset selection for the regression coefficients
in linear regression model (Mitchell and Beauchamp, 1988). This prior assumes
that the regression coefficients are mutually independent with a two-component
mixture distribution made up of a uniform flat distribution (the slab) and a
degenerate distribution at zero (the spike).
Ishwaran and Rao (2005) introduced a Bayesian rescaled spike and slab
hierarchical model, which was specifically designed for the multi-group gene
detection problem for DNA microarray analysis. This rescaled spike and slab model
reduces model uncertainty and is effective for prediction. They showed that the
spike and slab model identified important biological signals and minimized false
detection. Ishwaran et al. (2010) also applied Bayesian spike and slab models to
compute the estimators in correlated high dimensional problems based on weighted
generalized ridge regression, and demonstrated the effectiveness of this model.
Spike and slab models and their applications have become more and more popular
for challenging problems in different fields over the last decade. One of the most
important advantages for these models is selective shrinkage (Ishwaran and Rao,
18

2011), which allows the posterior mean to shrink toward zero only for non-
informative variables.
The prior selection for our model is in part inspired by the concept of a
“spike and slab” prior distribution. We incorporated directions in the “spike and
slab” concept to our model. Instead of a two-component mixture distribution, the
prior on SNP inclusion indicators in our model follows a 3-point distribution
( 1,0,+1), in which most regression coefficients are zero and a few coefficients have
some effects with effective or protective directions.
Markov chain Monte Carlo methods have been widely used to approximate
the posterior distribution of the models (for review and application of MCMC
methods, see [Gilks et al. 1998]). Basically MCMC methods entail iterative sampling
of each unknown in the model (e.g., latent variables or parameters) in turn,
conditional on the observed data and the current values of all the other unknowns;
this process continues for many iterations, ultimately yielding samples from the
joint posterior distribution of all parameters given the data after a “burn in” period
for convergence. Specifically, we use the Metropolis-Hastings algorithm (MH) for
obtaining a sequence of random samples from the multi-dimensional probability
distribution. Basically, a random walk MH algorithm explores models in the
neighborhood of the current model by proposing a new model based on a random
change to the current model, which is then accepted or rejected with a defined
probability (Robert and Casella, 2004).
19

Bayes Factors
Jeffreys (1935, 1961) developed the Bayesian approach for quantifying the
evidence in favor of a scientific hypothesis. The centerpiece of this approach was to
compose an index comparing the predictions made by two competing scientific
theories. This approach introduced statistical models to compare the probability of
the data according to each of the two hypotheses, and used Bayes’ theorem to
compute the posterior probability that one of the hypotheses is favored. On the
basis of Jeffreys’ Bayesian approach, Kass and Raftery (1993, 1995) defined the
Bayes Factor (BF) as the ratio of the posterior odds of alternative hypothesis to its
prior odds. For our analysis, we use Bayes factors at the SNP-level and the gene-
level to measure the change of the evidence provided by data for one hypothesis to
the other.
Assume two hypothesis (

and

) are to be compared, where pr(

) = 1 –
pr(

) . Let D represent data, and the posterior probability pr(

|D)= 1  pr(

|D).
Based on Bayes’s theorem, we can derive

(i=1,2),
therefore,

where BF is defined as

20

Specifically, we are interested in exploring which gene is associated with the
outcome of interest (contralateral breast cancer), and we address the following
hypotheses:

: There is no association between the gene of interest and the outcome

: At least one SNP within the gene is associated with the outcome of
interest.
We are also interested in finding out which SNP is driving the association:

: There is no association between the SNP of interest and the outcome

: The SNP of interest is associated with the outcome.
We can therefore calculate the Bayes factor for each of the above hypotheses
respectively:

Bayes Factors offer a way of evaluating evidence in favor of a null hypothesis,
and provide a way of incorporating external information into the evaluation of
evidence about a hypothesis. The advantages of Bayes factors over conventional
significance tests include that the Bayes factors are very general and do not require
alternative models to be nested.
We use Bayes Factors at both the SNP-level and the gene-level to compare
the posterior odds provided by data to their prior odds of a pair of hypothesis. The
detailed formula can be found in the section on posterior summarization in Chapter
21

3. Our interpretation of the significance of the analysis results (Table 1) is based on
the Bayes Factors summary of the evidence proposed by Jeffreys (1961), as
modified by Kass and Raftery (1995). We are essentially looking for genes and SNPs
which yield Bayes Factors greater than 3, or equivalently, 2ln(BF) greater than 2, as
candidate variables for positive associations with disease.
Bayesian Hierarchical Models Structure
Conventional genetic analyses often ignore the external information about
the set of plausible models or their parameters. Hierarchical modeling offers a
solution to problems of multiple inference by incorporating higher-level “p rior”
models into a conventional analysis (Greenland 1993). By “borrowing strength”
from the similarities in one’s data, the hierarchical modeling approach can give
estimates that are more plausible and stable than conventional approaches (Witte
1997).
A typical hierarchical Bayes modeling setup might involve two levels. The
first level might be a traditional regression model for the outcome of interest in
relation to various main effects and interactions. The second level can be a model for
the first level regression coefficients in relation to prior covariates that describe
characteristics of the variables, in our case, the pathways they act in. The prior
covariates can be derived from various pathway ontology databases such as Gene
Ontology, Ingenuity Pathways Analysis, Protein Analysis Through Evolutionary
Relationships, or literature mining. The second level model could include in
22

information on gene networks in the covariance of the first level coefficients. One
can add additional levels to allow for SNPs within genes, different variables
accounting for each environmental factor, or to distinguish each type of main effect
or interaction (Thomas 2010b).
Take the analysis of GWAS data as an example. GWASs were designed to
decipher the genetic basis of complex phenotypes. They first investigate hundreds
of thousands of SNPs across the genome, and then further evaluate the most-
promising SNPs with additional subjects, for replication or a joint analysis. The
conventional approach ignores the extensive information known about the SNPs
and entails simply selecting SNPs with the smallest association p-values using a
standard maximum-likelihood test (Satagopan et al. 2002). Instead of assuming
each SNP is equally causal, one can quantitatively incorporate existing information
about the SNPs into the analysis in a weighted false discovery approach, using
prespecified weights (Pe'er et al. 2006, Roeder et al. 2006, Sun et al. 2006).
However, as the number of prior covariates grows, the chance of sparse strata also
grows, it become increasingly difficult to evaluate data with these approaches.
Application of a hierarchical modeling framework avoids these limitations. It
combines various types of prior information simultaneously, and distinguishes true
causal variants from noise more clearly by estimating the weights using the
predictive value of the covariates in a multiple regression fashion, using all the
GWAS data (Chen and Witte 2007).
23

Assume one collected data on multiple related exposures of interest X and an
outcome of interest Y. One wants to estimate the coefficients β for the effects of X on
Y. Conventionally, one would estimate β from a generalized linear model for the
expectation of Y conditional on X,

(1)
where

is a monotonic differentiable strictly increasing link function between
random and systematic components, and Y has mean E(Y|X) and variance

.
Conventional analytic approaches to estimating β using (1) include: 1) fitting a full
model containing all of the exposures; 2) reducing a full model with a preliminary
testing algorithm; and 3) constructing numerous one-at-a-time models.
Unfortunately, none of the approaches properly address issues of multiple
comparisons (Morris 1983, Thomas et al. 1985).
Hierarchical modeling provides a coherent framework for multiple inference
problems, using shrinkage estimation to improve estimation accuracy (Greenland
1993; Witte et al. 1994). Prior knowledge about gene functions, protein interactions,
and disease pathways have been used in various hierarchical modeling approaches
for a single study (Capanu and Begg 2011, Capanu et al. 2011, Chen and Thomas
2010, Conti et al. 2003, Hoffmann et al. 2010, Hung et al. 2007, Lewinger et al. 2007,
Quintana et al. 2011, Shahbaba et al. 2012, Thomas et al. 2009a, Wilson et al.
2010b), and has been incorporated into joint modeling of related studies with
different designs (Li et al. 2013).
24

Assuming that one also has information about the expectation and variances
or covariances of the parameter of interest (β), hierarchical modeling uses higher
level “priors” to model β as random variables whose joint distribution is a function
of hyperparameters. Such additional information can be used in a second-stage
generalized linear model for the expectation of β conditional on this information,

(2)
where

is a strictly increasing link function, β has mean and variance

, and Z
is a second-stage design matrix expressing the similarities between the β.
One can combine results from the different level models by taking weighted
averages to obtain hierarchical estimates (posterior estimates). Weights in this
context reflect how well each stage was able to estimate that level’s parameters (see
[Chen and Witte 2007] for the implementation of hierarchical models on GWAS data
structures).

25

CHAPTER 4: MULTI-LEVEL BAYESIAN HIERARCHICAL MODEL: A
NOVEL APPROACH FOR PATHWAY ANALYSIS OF GENETIC
ASSOCIATION STUDIES
Hierarchical Modeling: Applications to Genetic Data
The hierarchical modeling approach has been applied in a variety of
situations (Witte 1994, Greenland 1992), particularly in association studies of
candidate genes or regions (Thomas et al 1992, Witte 1997, Kim et al 2001, Conti
and Witte 2003, Hung et al. 2004, Liu et al 2005). Thomas et al. (1992) presented a
hierarchical empirical-Bayes approach for testing associations with large numbers
of candidate genes. The simulation results indicated that hierarchical empirical-
Bayes was superior to maximum likelihood with or without haplotype effects. Witte
(1997) concluded that hierarchical modeling generally gives better estimates than
conventional maximum likelihood as long as the higher level models can provide a
reasonable approximation to reality.
Thus, hierarchical modeling is remarkably well suited to genetic analysis,
where a large amount of information was collected with relevant descriptors for
each variant (Witte 1997). Theoretical and simulation work has demonstrated the
potential improvement possible when incorporating higher level models, especially
for evaluation of large amounts of data on a limited number of subjects (Morris
1983, Greenland 1993, Witte 1994, Witte and Greenland 1996). This is precisely the
26

situation faced by GWAS. Chen and Witte (2007) illustrated how a hierarchical
method can be used to determine an optimal ranking of SNPs for follow-up in GWAS.
One can enrich the overall GWAS signal by including existing information and
borrowing strength from similarities among SNPs in a hierarchical model.
Mapping the genes for a complex disease requires finding multiple genetic
loci that may contribute to the onset of the disease. Hierarchical modeling has been
successfully adapted to such genetic epidemiology research. In order to model the
probability of transmitting a risk allele to a diseased child for a case-parent triad
data, Swartz et al. (2006) proposed Stochastic Search Gene Suggestion (SSGS) -- a
hierarchical Bayesian model using conditional logistic regression likelihood and
specifically incorporating genetic covariance. Hung et al (2007) applied an
empirical-Bayes approach allowing the prior knowledge regarding the evolutionary
biology and physicochemical properties of the variant to be incorporated into the
hierarchical model, in order to improve the estimation of risk for specific variants
involved in DNA repair and cell cycle control pathways. Wilson (2010b) presents an
efficient Bayesian hierarchical model search strategy for multilevel inference on SNP
associations. This method searches over the space of genetic markers and
alternative genetic parameterizations, and shows higher power to detect association
than standard procedures in simulation studies. These hierarchical Bayesian
models use techniques incorporating prior information into the risk estimates,
stochastically search variables to explore the posterior distribution on the model
27

space, and use Bayesian model averaging methods to make inference about the
importance of the variables.
In addition, Capanu et al. (2008) have employed hierarchical modeling using
pseudo-likelihood and Gibbs sampling methods to estimate the relative risk of
individual rare variants using data from a case-control study and showed that one
can draw strength from the aggregating power of hierarchical models to distinguish
the variants that contribute to cancer risk. They further studied the statistical
properties of the method and investigated the validity of these hierarchical
modeling techniques (Capanu and Begg 2011). They concluded that hierarchical
modeling is a promising strategy to interpret the evidence from future association
studies that involve sequencing of known or suspected cancer genes (Capanu et al.
2011).
In GWASs, there have been successes in identifying SNPs associated with
complex and common diseases; in breast cancer, many genome-wide association
studies have also been conducted (Easton et al. 2007, Hunter et al. 2007, Gold et al.
2008, Thomas et al. 2009c). The premise of these studies, which involve the use of
common SNPs, is that the major influences on the risk of common diseases (such as
cancer) are likely to involve common variants. This premise has been challenged,
since only very small proportion of the overall phenotypic variance was explained
by the discovered disease-susceptibility loci for most common diseases (Gorlov et al.
2008, Maher 2008, Schork et al. 2009). Multiple rare variants may influence a trait
either on their own or synergistically with common variants (Bansal et al 2010).
28

Quintana et al. (2011) sought to avoid the Common Disease Common Variant
(CDCV) hypothesis by constructing a Bayesian risk index based on multiple rare
variants within a region. Further evidence shows that selected rare variants in
known causal gene confer very high risk, while common variants appear to convey
little or no risk; in particular, the genetic risk of breast cancer may be primarily
caused by rare variants (Capanu et al. 2011). Moreover, the improvements of
hierarchical modeling estimates over conventional logistic regression increase as
the variants become rarer (Capanu and Begg 2011).
Both common variants and rare variants contribute to individual
susceptibility to common complex disease, because the allelic architecture of
susceptibility variants has a wide range of allele frequencies and effect sizes (Wen et
al. 2004, Azzopardi et al. 2008, McCarthy 2009, Schork et al. 2009). The usual
association tests for common variants are underpowered for detecting low
frequency variants. Several collapsing methods have been proposed to increase the
power to model multiple markers simultaneously by combining a set of rare
variants and testing their collective frequency in cases vs. controls (Li and Leal
2008, Madsen and Browning 2009, Morgenthaler and Thilly 2007). However, a
major drawback of these methods is that they neglect the directions of the effect, i.e.,
protective or risk causing, of individual variants. Extensive reviews of statistical
methods for rare variants can be found in Asimit and Zeggini (2010), Morris and
Zeggini (2010), and Bansal et al. (2010). Powerful alternative tools are required to
detect variants with various frequencies associated with disease of interest.
29

Multi-level Bayesian Hierarchical Model: Statistical Methods
We propose a novel approach for pathway analysis of genetic association
studies: Multi-level Bayesian hierarchical modeling. The first (subject)-level model
specification can be described as a logistic regression model that relates the mean of
binary outcome (disease status) to a subset of predictor variables:

where Y is a binary outcome variable for an individual,

denotes some general
structure or parameterization of a set of covariates,

is the effect of

on the
outcome of interest Y, and

indicates whether covariate j is included in the model.
The regression coefficients  j and variable inclusion indicators I j in this model are
then further modeled in the higher levels of the hierarchy. By averaging across the
distribution of inclusion indicators, one can draw inferences about the probability
and magnitude of the effects of individual genes or SNPs.
Treating each polymorphism as independent may be appropriate in a GWAS
study, since little or no external information about most SNPs is known. However, in
a pathway-driven study, the combination of external knowledge such as the
association of gene with disease, and gene set information such as gene-gene
correlations, may be used as prior biological knowledge about gene function to
facilitate more powerful analysis (for review of using pathway information in
30

candidate gene studies, see Thomas et al. [2009], Volk et al. [2008], Wang et al.
[2010], Wilson et al. [2010a]). Hence, we specify the relations among the observed
data with three stages to create a joint probability model.
Our model is based on a hierarchical Bayes framework comprising a subject-
level model for the association between genes and disease and a gene-level model
for the regression coefficients of the gene-disease association parameters. The
subject-level model is specified in terms of a SNP-burden for each gene (the SNP
level), comprised of the number of positively associated SNPs minus the number of
negatively associated SNPs.
If covariates are highly correlated, the posterior probability of an association
will be diluted or distributed across several correlated covariates due to competing
evidence for an association. Hence, we used Pearson correlation coefficients among
SNPs within a gene to eliminate the diffusion of the significance among SNPs with
strong linkage disequilibrium (LD). If a set of SNPs within the same gene have a
correlation coefficient beyond some arbitrary value, one of the SNPs is selected to be
included in the model with equal probability. Since there is no standard threshold,
we chose a Pearson correlation coefficient of 0.8 as a cut point.
After the initial screening, the choice of whether a SNP is included or not and,
if included, its direction is governed by prior probabilities that in principle vary
across genes or across SNPs within genes. In the simulation, we assume the
probability of SNP inclusion and its direction is inversely related to the size of the
gene. Each gene has a log relative risk coefficient that also has a prior distribution
31

that depends upon a vector of external gene-specific “prior covariates” and a
covariance matrix that depends upon external gene-gene connection information.
The overall model is represented as a directed acyclic graph in Figure 1. For the
simulations and the analysis of the real WECARE data, we extracted the information
from Gene Ontology (GO) for the 38 WECARE candidate genes to construct the prior
covariate and correlation information, as described in more detail in the simulation
section.
Model I
Level 1: The subject-level model for case-control data is a conditional logistic model
of the form

where

denotes the case-control status for the ith individual; X denotes a vector of
fixed covariates (confounders) and is its coefficient vector; G denotes a function
of the raw SNP genotypes for gene g containing NS g SNPs,

with weights

∈ {-1,
0, 1}. In WECARE study, an offset term is the log value of the counter-matching
weights, as explained in Chapter 2.
Note that the exponential function of  in the logistic model ensures that the
effects of each gene will be positive, thereby avoiding the label-switching problem
that would arise if the signs of  g and all the W gs were reversed for a given gene.
This also avoids having to deal with truncated normal distributions if  g were
constrained to be positive.
32

Level 2: The logistic model has regression coefficients  given by the gene-level of
the hierarchical model:
 g = Z g´  + b g + e g (3)
where
 = (  0,…,  NZ) ~ N(0,V

I)
b = (b 1,…b NG) ~ N(0, 
2
A)
e = (e 1,…,e NG) ~ N(0, 
2
I)
We incorporate information regarding the relations between the factors into
the design matrix Z, here structured as a gene by pathway matrix of binary values,
each indicating whether a gene is in a particular pathway. Basically, Z contains
second-stage covariates for each of the genetic factors. π is a column vector of
coefficients corresponding to these higher-level effects, and is assigned a normal
probability distribution. We incorporate the prior gene-gene connection
information in A matrix for b with a multivariate normal distribution centered at
zero. The term e is included as a residual error, also given a zero mean independent
normal distribution, with 
2
specifying the residual variance of the second-stage
covariates.
Level 3: The SNP-level deterministic model is designated to obtain G, where each
gene is uniquely determined by SNP inclusion parameters in the model and by sets
of previous states of these SNPs. G serves as a design matrix of genetic factors (we
may consider additional environmental factors and interaction terms in the first
33

level of the model, if needed) for the individuals within the study. In other words,
the function serves as a risk index for each gene:

(4)
where weights W = 1, 0, or +1 have probabilities

(5)
and

denotes the average number of SNPs within a gene; we assigned c to be the
minimum number of SNPs within any gene;

denotes the number of SNPs within
gene g. This form of prior probabilities on the SNP indicator variables keeps the
expected number of SNPs included in the model to be roughly similar across genes,
while allowing genes with more SNPs to have similar probabilities of being included
as genes with fewer SNPs. For now, we treat  as fixed parameters, but these too
could be given hyperpriors.
The posterior estimates for the association parameters resulted from the
three-stage hierarchical Bayesian analysis are an inverse-variance weighted average
between the conventional estimates from the logistic regression only and the
estimated conditional second-stage means, Zπ . Between the maximum likelihood
first-stage estimates and the second stage prior estimates, the weights will favor the
one with smaller variance. This intuitive weight adjustment is one of the important
34

differences between Bayesian hierarchical approach and the single-stage logistic
regression analysis.
Finally, the variance components have standard inverse gamma hyperprior
distributions:

2
~ IG(df e,E)

2
~ IG(df b,B)
Model II
Model II shares many similarities with Model I. The differences are stated
below.
Level 1: The subject-level model for case-control data is also a conditional logistic
model, but the coefficient of each gene effect is

, instead of the exponential of

in
Model I:

,

Level 2: The second level of the hierarchical model still aims to obtain the logistic
regression coefficient

, but is in the form:

where denotes the probability density of normal distribution, and denotes the
cumulative density of normal distribution. This is a proper density for

, since
35

thus the probability distribution of

integrates to one.
Level 3: The third level of Model II remains the same as Model I (Equation 4) with
the same probabilities for the weights (Equation 5).
Fitting the Model
Model I
Model fitting is performed in series of MCMC steps. The methods to update each
parameter in each iteration are described below.
 The coefficients ( ) of subject-level confounders are updated using single
Newton-Raphson iteration towards the maximum likelihood estimate (MLE)
of , following a random multivariate normal update to sample the new .
The procedure is based on the approximation that the likelihood for is
quadratic with flat priors.
 Selection of SNPs to include in the model involves evaluating the three
posterior probabilities for d = {-1, 0, +1} and selecting

with the
corresponding probability
[W s = d | Y,S,W; ]  [Y | {G g({W gs=d,W
gs,S g),G
g}; {  gsd, 
g}]  [W s=d |  d, NS g]
where  gsd is a single Newton step iteration towards the MLE of  g if W s were
set to d.
36

 Update the vector of regression coefficients  using a multivariate
Metropolis-Hastings move with proposal ´ ~ N( , 

I) and acceptance
probability
[  | Z, ,b,Y,S,W, 
2
]  [Y | G(W,S); ]  [  | Z´  + b, 
2
I]
Note: an alternative possibility, not yet implemented, would be to use
[  | Z, ,A,Y,S,W, 
2
]  [Y | G(W,S); ]  [  | Z´ , 
2
I+ 
2
A]
 Update the vector of random effects b with a similar Metropolis-Hastings
move with acceptance probability
[b | ,Z,, 
2
, 
2
,A]  [  | Z´ +b, 
2
I]  [b | 0, 
2
A]
Again, the  vector and this step could probably be completely omitted
simply by using [  | Z´ , 
2
I+ 
2
A] in lieu of [  | Z´  + b, 
2
I] above and in the
various other updates below.
 Update the prior regression coefficients  by a simple linear regression and
taking a multivariate normal around its MLE:
[ | ,Z,b, 
2
]  [  | Z´  + b, 
2
I] [ | 0,V

I]
 Update the variances 

and 

using a Metropolis-Hastings move with
proposals ln( ´) ~ N(ln(), 

) and similarly for , with acceptance
probabilities
[,  | ,Z, ,A]  [  | Z´ , 
2
I+ 
2
A] [ 
2
] [ 
2
]
 As noted above, for now we’re treating the s as fixed, but these too could be
given prior distributions and estimated as well.
37

Model II
Model fitting for Model II is similar to Method I except for some details in
updating

s and s. For sampling

s, we continue to use Metropolis-Hastings as
in Model I, but the prior contributions for genes with

= 0 are based on the
cumulative density function (CDF) rather than on the probability density function
(PDF) of the distribution. The likelihood contribution is still the same as in Model I.
s are updated using Newton-Raphson as in Model I, except that (1) instead of
resetting s to zeros before updating them to (score x inverse information) in
Method I, we accumulate score and information contributions as additions to the old
s in Model II; (2) we accumulate the first and second derivatives of the cumulative
normal for genes with

.
Posterior Summarization
We tabulate the following quantities:
 For each SNP: the posterior probability of W gs = 1,0,+1 and Bayes Factor

where the first factor are posterior probabilities given the data D and the
second are the priors.
 For each gene: the posterior probability that at least one SNP is included in
the model and its Bayes factor
38

We also tabulate the posterior means and SDs of each

and a significance
test as approximately

, along with the mean
number of SNPs included in the model.
 For the other parameters, , , 
2
, 
2
: we simply tabulate the posterior means
and SDs.
 Finally, we tabulate the posterior distributions of numbers of SNPs and
numbers of genes with at least one SNP included in the model. We still have
to work out the corresponding prior distributions, which are essentially
mixtures of binomials, so we have only approximate BFs for these quantities.

39

CHAPTER 5: SIMULATION STUDIES
To evaluate the performance of our model, we conducted simulation studies
based on the structure of the WECARE Study data. Specifically, we used the real
SNP, covariate, and counter-matching offset data for each risk set and reassigned the
case status in each risk set based on an assumed relative risk model. We assigned
the with their estimated values from the real data and randomly assigned weights
W gs to simulate SNPs. Log relative risk coefficients  g for each gene were simulated
under the level 2 of the models (Equation 3). The simulated values of the variances
in this level were assigned as  =  = 0.5.
The analysis was performed on a total of 504 SNPs in 38 genes (ranging from
1 to 51 SNPs per gene) involved in DNA damage response pathways (i.e., DNA
repair, cell cycle checkpoint control, and apoptosis). Using the Gene Ontology, we
extracted 860 terms relating to biological process or molecular function annotated
to any of these 38 genes and selected four of these GO terms (DNA damage
checkpoint, MRE11 complex, double-strand break repair via nonhomologous end
joining, and negative regulation of cell cycle) as prior covariates in the Z matrix.
These GO terms were assigned coefficients  = 0.25, 0.5, 0.75, and 1 respectively.
The intercept  0 was set to 2 for Model I, and was set to 0 for Model II. All 860 GO
terms were used to construct a correlation matrix A for the similarity in the ways
each pair of genes was described in the GO (Figure 2).
40

The resulting gene indices G g(W,S) and the corresponding  g, along with the
real X i and estimated  coefficients and offset terms, were then used to compute
each subject’s relative risk. These were then used to randomly assign which
member of each risk set would be designated as the case. The following estimates
are based on multiple data replicates for each of several parameter choices; each
realization of the data uses 1000 MCMC scans for tabulation after a burn-in of 500
scans. For Model I, we simulated 10 parameter sets and 10 data replicates for each
of parameter set. Simulation analysis for Model II was based on single randomly
chosen parameter set and simulated data set.
The SNP inclusion indicators were assigned +/- 1 or 0 with probabilities
given by Equation 5 with

and c = 1.
We plotted estimated parameters versus simulated parameters from Model I
(Figure 3a) and Model II (Figure 3b) based on SNP inclusion prior φ=0.05. The
plots suggested that the values of the parameters both our model estimate yield
adequate correlation with the true parameters.
We selected different values for φs to ensure that fewer than 10% of the
SNPs were included in the model. We compared the estimated exp(β)s from Model I
(Figure 4a), and estimated βs in Model II (Figure 4b), respectively, based on
different SNP inclusion φ priors (0.025, 0.05, 0.1), where the data and parameter
was simulated on a fixed value ( 

=  + = 0.05 and c=1). Both graphs suggest that
the estimated parameters are insensitive to priors.
41

We summarized the BFs for SNP inclusion using Kass and Raffery’s
classification (>3, >20, >150; or equivalent to 2ln(BF) >2, >6, >10, ) (Figure 5.1ab for
Model I and 5.2ab for Model II). Each column represents the average posterior
probabilities for truly negative SNPs, most common truly null SNPs, and truly
positive SNPs. Each color represents the estimated probability of SNPs inclusion and
their directions, given the true SNP inclusions and their directions. Truly associated
SNPs had much larger posterior probabilities of being associated on average (and in
the right direction) than null SNPs (Figure 5.1a and 5.2a). Specificity is high for both
models; however, the sensitivity of Model II is almost double the sensitivity of Model
I (Figure 5.1b and 5.2b).
In addition to SNP associations, we also summarized simulation results for
gene-level associations (Figure 5.1c and 5.2c). Both models identified genes with
positive to strong association. Red stars indicate the gene having at least one SNP
included in the model. As shown, not every gene with at least one SNP included was
identified as a strong candidate by the models.
We summarized the frequency distribution of SNPs based on their minor
allele frequency (MAFs) (Table 2). After filtering highly correlated SNPs, 384 SNPs
were employed for the simulation analysis. About 66% of SNPs included in each
model were common variants (MAF>0.05); about a quarter of the SNPs are
uncommon variants (0.01<=MAFs<0.05); and less than 9% of SNP included in each
model the analyses are rare alleles (MAF<0.05). By examining the posterior
probabilities and Bayes factors by categories of MAF (Figure 6.1 and 6.2), we
42

discovered that both of our models worked satisfactorily in identifying common
variants but not rare variants -- the sensitivity increased along with the categories,
while the specificity remained high (Figure 6 left panels). The histograms based on
SNP Bayes Factors categorized by MAFs also show an increase in sensitivity (Figure
6 right panels). Model II appeared to yield a higher sensitivity than Model I, with an
approximate doubling for common variants if the true SNP indicator was 1, and
with an approximate tripling for common variants if the true SNP indicator was 1.
The data management procedures were conducted with SAS 9.2 (SAS
Institute Inc., Cary NC), R (http://www.R-project.org/), and Microsoft Excel (2007);
the analysis procedures were conducted with C/C++ (Microsoft Visual Studio 2008).

43

CHAPTER 6: APPLICATION TO THE WECARE DATA
Using the same settings as for the simulation studies, we applied our models
to the real WECARE data with the aim of identifying candidate genes from a set of
genes thought to be involved in DSB response pathways -- which of them were
associated with second primary breast cancer, and among those associated, which
SNPs within the genes are driving these associations.
Among 2017 breast cancer survivors, 708 were cases with second primary
breast cancer, and 1399 served as controls, matched on age, race and latency of the
disease. Since we focused on the main effects of genes but not GxE interactions, we
did not put restrictions based on radiotherapy or radiation dose. Furthermore, no
restrictions based on ATM or BRCA genotypes were applied. Missing genotypes
were treated as non-carriers. The missing data problem can be resolved by using
Monte Carlo maximum likelihood if the complete data likelihood is known
(Thompson and Guo 1991). The MCMC method provides a natural solution by
imputing values for the missing data at each iteration; sampling from their full
conditional distribution given the available data (Gilks et al 1998). Among the 504
SNPs in 38 genes from candidate gene panels, 384 SNPs were employed in the
analysis after eliminating highly correlated SNPs within a gene. Twelve variables
(age, menarche, menopause, family history, pregnancy, histology, treatment, the
FGFR2 GWAS-identified SNP (not thought to be involved in DSB damage response),
and known deleterious variants in ATM, BRCA1, BRCA2, and CHECK2) were treated
44

as fixed covariates; an offset term to adjust for the counter-matched design was also
included.
We constructed prior covariates and gene-gene connections from the Gene
Ontology. 860 GO terms in the categories of Molecular Function or Biological
Process that were annotated to any of the 38 genes were used to construct the
correlation matrix A. More specifically, the diagonal elements of the A matrix are all
one, and the off-diagonal elements reflect the correlation between a pair of genes
given all 860 GO term values for each gene. We selected 4 of these GO terms (DNA
damage checkpoint, MRE11 complex, DSB repair via NHEJ, negative regulation of
cell cycle) associated with DNA DSBs response to be included as prior covariates Z.
The results of real data analysis were based on 10,000 iterations after 4,000
iterations burn-in. The trace plots of joint probability for both models appeared to
have converged (Figure 7), however, the trace plots for some individual β and b in
both models appeared unstable (not shown); the trace plots of some other
parameters estimated such as and π were highly stable (not shown).
Model I identified one gene (MDC1) with strong association (2ln(BF) > 6),
one gene (RAD51) with positive association (2ln(BF) > 2), and two genes (ATM and
NBN) with weak associations (2ln(BF) > 0) with second primary breast cancer
(Figure 8b). This finding matched the results for the probabilities of gene inclusion
(Figure 8a), where the highest probability of gene inclusion identified the same 4
candidate genes as identified by gene-level Bayes Factors. Model II identified only
one gene (MDC1) with positive association based on gene-level Bayes Factors
45

(Figure 8d), as well as quite a few genes with weak association. MDC1 also had the
highest posterior probability of being included in the model (Figure 8c).
Based on Bayes Factors, we identified SNPs driving the gene-disease
associations with strong evidence (2ln(BF) > 6) in genes NBN and RAD51, and with
positive evidence (2ln(BF) > 2) in genes ATM, CHECK2, MDC1, MRE11A, using Model
I (Figure 9a). For Model II, we identified SNPs driving the gene-disease associations
with positive evidence in genes ATM, FANCA, LIG4, MDC1, NBN, and RAD51 (Figure
9b). Both models identified 9 SNPs with positive evidence for driving gene-disease
association. Among those identified SNPs from both models, 4 SNPs (rs4713354,
rs2269705, rs9297757, rs1801320) are in common.
Table 3 presents the Bayes Factors for selected SNPs identified by Model I
and Model II, as well as those identified by a previous WECARE publication (Brooks
et al. 2012). The log relative risk estimates (lnRR) are based on the adjusted
conditional multivariate logistic regression from each model.
Seven of the nine SNPs identified by Model I have been found associated with
breast cancer risk in previous investigations. More specifically, among these seven
identified SNPs, five were selected by Brooks et al. (2012) on WECARE data, and two
were selected by other breast cancer studies. Brooks et al. (2012) identified six
SNPs that have a significant association (p-value < 0.05 uncorrected for multiple
components) with CBC risk, among 152 SNPs in 6 genes (CHEK2, MRE111A, MDC1,
NBN, RAD50, TP53BP1). Five (rs6005861 in CHEK2, rs4713354 in MDC1,
rs13447682 in MRE11A, rs9297757 and rs3736640 in NBN) of the six SNPs Brooks
46

and colleagues identified were also identified by our model (Model I). Model I also
identified SNP rs1800057, a variant in ATM, which was previously shown to be
associated with a statistically significant reduction in CBC risk (Concannon et al.
2008). Furthermore, rs1801320 (135G>C) identified by Model I is a SNP in the 5'-
untranslated region (UTR) of the RAD51 gene. Investigators have found mixed
results for its role in breast cancer risk from previous studies (Antoniou et al. 2007,
Yu et al. 2011). In addition to these previously distinguished SNPs, Model I also
identified rs4987951 in ATM and rs2269705 in MDC1, about which no previous
evidence was found for association with breast cancer risk.
Model II identified rs4713354 in MDC1, rs9297757 in NBN, and rs1801320 in
RAD51, which were also identified using Model I; and these have been shown to be
associated with breast cancer risk in previous studies, as discussed above. In
addition, Model II identified rs664677 in ATM, which was not identified by Model I,
but a recent meta-analysis (Shen et al. 2012) has shown its association with
increased breast cancer risk. Model II also identified rs7187436 in FANCA,
rs9468811 in MDC1, rs1555902 and rs11620361 in LIG4, but no previous literature
was found with evidence for their associations with breast cancer risk. Nevertheless,
these could be novel findings from our models.
Table 4 lists the numbers of pairs of the homozygous reference allele,
heterozygous allele, and homozygous risk allele for cases (CBC) and controls (UBC),
respectively, for all the SNPs identified by our models and by Brooks et al. (2012).
We also report the estimated s from simple logistic regression for each
47

selected SNP, adjusted for age, menarche, menopause, family history, pregnancy,
histology, treatment, the FGFR2 GWAS-identified SNP, deleterious variants in ATM,
BRCA1,BRCA2,CHECK2 and offset term. Based on the adjusted simple logistic
regression, among fifteen selected SNPs, nine SNPs were statistically significant
associated with CBC risk (PC41, rs1800057, v_IVS14m55, rs6005861, rs11620361,
rs4713354, rs2269705, rs13447682, rs3736640, rs3736640, rs1801320; p values
<0.05), three were found marginally significant (rs1555902, rs11620361,
rs9297757; 0.05 <=p values <0.1). Eight of the nine variants selected by Model I
were statistically significant (rs1800057, v_IVS14m55, rs6005861, rs4713354,
rs2269705, rs13447682, rs3736640, and rs1801320; p values < 0.05), and one was
marginal significant (rs9297757, p = 0.097). Four of the nine variants selected by
model II were significant (PC41, rs4713354, rs2269705, rs1801320; p values <
0.05), three of the rest were marginally significant (rs1555902, rs11620361,
rs9297757; 0.05 <=p values < 0.1). Four of the six variants selected by Brooks et al.
(2012) are significant (rs6005861, rs4713354, rs13447682, rs3736640; p values <
0.05), one is marginal significant (rs9297757, p value = 0.097).
Among these selected variants with statistical significance and marginal
significance from adjusted simple logistic regression, based on their relative risks,
we identified that PC41, rs11620361, rs4713354 and rs269705 are casual variants,
and that rs1800057, v_IVS14m55, rs6005861, rs1555902, rs13447682, rs9297757,
rs3736640 and rs1801320 are protective variants. It is interesting to note that both
PC41 and v_IVS14m55 are SNPs in the same gene (ATM), and that both rs1555902
48

and rs11620361 are SNPS in the same gene (LIG4), but these SNPs within the same
gene produced opposite effects in terms of CBC risk.

49

CHPATER 7: DISCUSSION
We proposed two models based on a Bayesian hierarchical modeling
framework for multi-level variable selection, motivated by WECARE study to
identify effective (causal or protective) genes associated to CBC risk. CBC is a
complex disease involving multiple genetic and environmental factors that
complicate mapping the genetic components of these diseases. Our multi-level
Bayesian hierarchical selection methods are applicable to these types of
investigations. These methods are extremely flexible, allowing for model
uncertainty to be taken into consideration. Additionally, the hierarchical nature of
the model provides means to incorporate a priori knowledge about genes (such as
known covariance structure) into the model, which can improve variable selection
among multiple predictors. These approaches can be used in a variety of fields,
particularly in genetic epidemiology, including identifying variants associated with
diseases of interest and their directions of effects.
Using our models, we are able to identify which genes within a pathway are
associated with the outcome of interest and, given that certain genes are associated,
which SNPs are responsible for these associations. As demonstrated in the WECARE
study application section, both of the two Bayesian hierarchical models were able to
detect some positive to strong evidence of gene-disease associations, and also
pinpoint specific SNPs that are most likely driving these associations. By performing
a model search to determine which variants to include and using Bayesian model
50

averaging techniques to calculate posterior quantities of the interest, our Bayesian
hierarchical models demonstrate reasonable sensitivity with high specificity.
Simulation study shows that model II demonstrated much higher sensitivity than
Model I. Both models show high specificity.
Additional simulation studies could help to elucidate the performance of our
methods. A set of independent genetic association study-based simulations could be
further developed to examine the power of the multi-level Bayesian hierarchical
methods. Including some alternative variable selection approaches for power
comparison with our methods would be a suitable avenue for future research. The
alternatives include penalized regression methods such as Lasso (Tibshirani, 1996),
Elastic Net (Zou and Hastie 2005) and Group Bridge (Huang et al. 2009), and basic
Bayesian model uncertainty methods such as choosing the “non -informative” prior
distribution (uniform prior, binomial prior, or beta-binomial prior) over individual
models. The marginal true positive rates (TPR) and false positive rate (FPR) as the
proportion of causal and non-causal predictors respectively can be calculated for
each of the methods; ROC curves can be plotted for power analysis.
Since it seems unreasonable to consider the known deleterious variants in
ATM, BRCA1/2, and CHEK2 as exchangeable with the tagging SNPs, we have treated
these variants as fixed covariates, along with age, menarche, menopause, FH,
pregnancy, histology, treatment, FGFR, forcing these covariates into all models.
Unfortunately, this precludes borrowing strength across all the variants within
these genes—i.e., given that we know that some variants in these genes are
51

deleterious, it would seem more likely that there would be other causal variants in
the same genes, and that if these four genes have similar prior covariate values Z g,
that should inform the estimation of the corresponding  gs and draw the estimates
of s for other genes that are highly correlated with them in the A matrix towards
the  g values for these genes.
Our model framework uses a simple MH algorithm to sample models of
interest from the enormous space of possible models. Both of our models take
approximately 3 hours to complete on a single processor when performing 10,000
iterations (after 4000 iterations burn-in) of the current MH/MCMC algorithm on the
WECARE data. The trace plots showed that the overall likelihood and most of the
higher-level model parameters reached convergence well before burn-in, although
some of the more specific gene-level parameters remained somewhat unstable.
Our results are encouraging from a broad perspective. Each model we
proposed identified nine SNPs that are possibly driving the gene-disease
associations; four of them are in common for both models. Model I identified five
SNPs that were discovered by a previous analysis on WECARE data, and two SNPs in
previous general breast cancer risk studies. Model II identified two SNPs discovered
by previous WECARE analyses, and two SNPs discovered in previous studies for
potential association with general breast cancer risk. It is possible that both of our
models identified some novel SNPs responsible for gene-breast cancer association
that were not detected in previous investigations. Although Model I identified more
SNPs with previously discovered association with breast cancer risk than Model II
52

did in the real data analysis, the simulation study suggested that Model II had more
statistical power than Model I. Further studies of both of these models would be
useful for identifying novel candidate genes and SNPs associated with disease risk.
Our models use latent indicator variables at the SNP level to do variable
selection, and Bayesian shrinkage at the gene level towards a prior mean vector and
covariance matrix that depend on external information. By incorporating these
biological covariates about the gene structures in the WECARE study, we showed
evidence of associations of CBC with the genes ATM, CHEK2, MDC1, MRE11A, NBN,
RAD51 (Model I), or with the genes ATM, FANCA, LIG4, MDC1, NBN, RAD51 (Model
II).
The premise of the WECARE study was that the power to detect main effects
of relatively rare genetic mutations and their interactions with environmental
factors would be considerably enhanced by restricting consideration to women with
a first primary breast cancer and then studying the determinants of developing a
second primary breast cancer. In the current models, once we adjusted for the
possible confounding variables, we focused on identifying main effects within the
variants of interest. However, if we ignore the interactions of genes and
environment, but merely estimate the contributions of genes and environment
separately, we might not be correctly estimating the population attributable risk of
the disease, because a proportion of the disease might be explained by the joint
effects of genes and environment (Hunter 2005). Extending Lewinger et al.’ s (2007)
hierarchical Bayes prioritization approach, Sohns et al. (under review) developed a
53

novel empirical hierarchical Bayes approach to detect GxE interactions in GWAs. It
can serve as one of the recommended methods for follow-up SNP selection,
particularly when G-E associations are suspected.
Deficiencies in cellular response to DNA damage can predispose to cancer.
Ionizing radiation is known to cause cluster damage and DSBs that pose problems
for cellular repair process, and hence is a breast cancer carcinogen. Many genes
encode products that are essential for the normal cellular response to DSBs, but
predispose to breast cancer when mutated. The carriers of certain haplotypes may
be susceptible to the DNA-damaging effects of radiation therapy associated with
radiation-induced breast cancer. These genetic factors and radiation exposure,
individually or via interaction, may contribute to the development of radiation-
induced CBC. A previous WECARE analysis found an interaction between ATM and
radiation treatment contributing to the risk of developing CBC -- women who carry
rare ATM missense variants predicted to be deleterious and were exposed to
radiation had a statistically significant higher risk of CBC compared with unexposed
women who carried the wild-type genotype, or compared with unexposed women
who carried the same deleterious ATM missense variant (Bernstein et al. 2010).
The WECARE study design was based on counter-matching on radiotherapy
status, increasing the informativeness of the collected dosimetry data by increasing
the variability of radiation dose within the case-control sets, and allowing for
unbiased estimation of the main effects and interactions of interest. The focus on
DNA DSB damage response pathway genes was entirely predicated on an interest in
54

DSBs induced by ionizing radiation. Thus, we had stronger priors for gene-radiation
interactions than for main effects of these genes, and the ability to detect gene-
radiation therapy interaction was consequently enhanced (Bernstein et al. 2004).
Radiation therapy is highly plausible at the biological level in gene-environment
interaction for the WECARE study. It is worth noting that we tried to implement the
environment (radiation therapy or radiation dose) and gene-environment
interaction term as extension to our models, but by far, no interesting findings were
detected. Even in the post-GWAS era, analysis of GxE interaction remains one of the
greatest challenges. Our models are highly extensive — it is of future interest to
investigate possible gene-environment interactions with further refinement to our
models.
In our current models, we aim to detect both common and rare variants that
contribute to the gene-disease association; hence, we gave the same weight to
common variants, uncommon variants, and rare variants within a gene. Based on
the simulation results, our models appear to identify effects of common variants
with higher accuracy than for rare variants (Figure 6). However, this result may be
due to the unbalanced number of the common, uncommon, and rare variants across
the SNP data (Table 2). For common variants (and perhaps in candidate gene
studies for uncommon variants), it may be possible to allow each SNP to have its
own regression coefficient from some continuous distribution, but constraints
would be needed if both SNP- and gene-level parameters were to be estimated to
ensure identifiability.
55

The Bayesian hierarchical framework provides a flexible and powerful basis
for selecting multiple genes within a pathway and selecting multiple SNPs within
each gene, incorporating external prior information at either the SNP or gene level.
Capanu et al. (2011) and Quintanta et al. (2013) proposed similar models
incorporating external information at the SNP level. In this thesis, we have only used
external information at the gene level, since the GO does not provide any annotation
of specific variants within genes. However, there are many ways to classify SNPs a
priori, such as simple indicators for whether they are coding or noncoding variants,
or the predictions of programs like SIFT (Ng and Henikoff 2003) and PolyPhen (Xi et
al. 2004) based on predicted effects on protein conformation or evolutionary
conservation. Such information could easily be incorporated into a multinomial
logistic or probit model for the inclusion probabilities  s (Quintana et al., 2013). The
current version of our program treats the deleterious SNP inclusion probability  +
and protective SNP inclusion probability 

as fixed constants dependent on the size
of the gene the SNP belongs to, but in principle, other prior distributions on the
model specific coefficients can be incorporated into the framework. For instance,  +
and 

could be assigned prior Beta or Beta-Bionomial distributions, subject to the
constraint that  + + 

< 1. In reality, we might want to keep the SNP inclusion
probability less than 10%, but this can be adjusted according to the parameters of
the prior distribution.
56

The information that can be incorporated into an analysis via prior
covariates is extremely flexible. In particular, for our motivating application of
genetic association studies, a vast amount of external biological information exists
for the variants under consideration, as discussed in Conti et al. (2009). Many of
methods reviewed in Cooper and Shendure (2011) used a combination of
evolutionary, biochemical and structural information to guide the estimation. We
expect an increase in power if more information were implemented into our model
framework.
In summary, we applied multi-level hierarchical modeling in a Bayesian
framework incorporating a priori knowledge about genes involved in radiation
induced DNA DSB pathway into the CBC risk estimates. We applied these models to
WECARE data, and identified potentially effective genes and SNPs associated to CBC
risk, based on Bayes Factor cutoffs. The results of the analysis of real WECARE data
are encouraging; however, independent replication is required to confirm the
associations found in this study. A replication GWAS study (WECARE II) using 2100
CBC cases and 2100 UBC controls is currently in progress. Further investigation of
the association of the effects of gene-radiation interaction and risk estimates is
possible.
To systematically study the effects of rare variants is one of the goals of our
model development in future application. Although the data we used contain mainly
common variants, and the identification test accuracy appears to be satisfactory
57

only for common variants, our models are in principle equally applicable to data
containing rare variants, e.g., from next generation sequencing data.

TABLES

Table 1: Bayes Factos’s Interpretation for association (Kass and Raftery, 1995)
Bayes Factor (BF) (HA:H0) 2ln(BF) Evidence against H0
1 to 3 0-2 Not worth more than a bare mention
3 to 20 2 to 6 Positive
20 to 150 6 to 10 Strong
>150 > 10 Very strong

Table 2. Frequency Distribution of SNPs based on their MAFs

SNPs
MAFs Model I
a
Model II
b

<0.01 34 (8.9%) 34 (8.8%)
>=0.01 and <0.05 96 (25.0%) 95 (24.7%)
>=0.05 254 (66.1%) 256 (66.5%)
Total 384 385
a
Average number of SNPs included in the model I from 10 data replicates for each of the 10 replicates of parameter sets
b
Number of SNPs included in model II from single replicate of data and parameter set

58

Table 3: Bayes Factors in SNP-level and Gene-level from Model I and Model II for Selected SNPs.
Gene rs#

Model I

Model II
BFsnp 2lnBFsnp BFgene 2lnBFgene lnRR
d
BFsnp 2lnBFsnp BFgene 2lnBFgene lnRR
d

ATM
PC41
b
0.00 n/a
1.41 0.68 0.68
2.73 2.01
1.58 0.91 0.11
rs1800057
a

4.58 3.04

1.67 1.03
v_IVS14m55
a

9.04 4.40

1.99 1.38
CHEK2
rs6005861
a,c

7.00 3.89
0.36 -2.02 0.60
1.41 0.69
1.27 0.48 0.10
FANCA
rs7187436
b

eliminated
0.00 n/a n/a
2.87 2.11
1.69 1.05 0.08
LIG4
rs1555902
b

1.23 0.41
0.25 -2.78 0.34
2.96 2.17
1.63 0.98 0.10
rs11620361
b

1.38 0.64

3.17 2.31
MDC1
rs9468811
b

0.69 -0.74
20.71 6.06 0.54
2.76 2.03
6.00 3.58 0.48
rs4713354
a,b,c

9.72 4.55

13.48 5.20
rs2269705
a,b

15.91 5.53

8.41 4.26
MRE11A
rs13447682
a,c

5.70 3.48
0.52 -1.30 0.57
1.42 0.70
0.99 -0.03 0.07
NBN
rs14448
c

0.20 -3.22
2.62 1.93 0.45
1.62 0.96
1.45 0.75 0.11
rs9297757
a,b,c

27.33 6.62

5.49 3.41
rs3736640
a,c

4.14 2.84

1.32 0.56
RAD51
rs1801320
a,b
21.38 6.12
3.51 2.51 0.53
3.32 2.40
1.89 1.27 0.14
a
SNPs identified by model I based on Bayes Factors. Only those SNPs with 2ln(BF)exceeding 2 were selected.
b
SNPs identified by model II based on Bayes Factors. Only those SNPs with 2ln(BF)exceeding 2 were selected.
c
SNPs identified by Brooks et al 2012 based on per-allele RR (log-additive model), adjusted for age at first diagnosis and counter-
matching weight. Only those SNPs with p value for trend < 0.05 were selected.
d
is estimated by subject-level adjusted conditional logistic regression: in Model I is equivalent to

; in Model
II is equivalent to

.

59

Table 4. Association Between Selected Variants in DNA-Damage Response Genes and CBC risk
Gene rs#
Homozygous;
reference allele
Heterozygous
Homozygous;
risk allele
lnRR
d

p value
e

Case
(CBC)
Control
(UBC)
Case
(CBC)
Control
(UBC)
Case
(CBC)
Control
(UBC)
(95% CI)
ATM
PC41
b
207 473 361 671 140 255
0.15 (0.01, 0.28) 0.038
rs1800057
a
680 1322 28 76 0 1
-0.47 (-0.95, -0.01) 0.046
v_IVS14m55
a
674 1278 34 121 0 0
-0.66 (-1.32, -0.25) 0.002
CHEK2
rs6005861
a,c
680 1311 27 86 1 2
-0.40 (-0.85, 0.06) 0.086
FANCA
rs7187436
b
263 468 323 677 122 254
-0.07 (-0.21, 0.07) 0.313
LIG4
rs1555902
b
573 1098 130 280 5 21
-0.21 (-0.43, 0.01) 0.065
rs11620361
b
469 986 216 375 23 38
0.18 (0.00, 0.36) 0.050
MDC1
rs9468811
b
674 1324 34 73 0 2
0.15 (-0.30, 0.60) 0.502
rs4713354
a,b,c
535 1116 157 267 16 16
0.47 (0.26, 0.68) <0.001
rs2269705
a,b
589 1220 113 175 6 4
0.50 (0.25, 0.76) <0.001
MRE11A
rs13447682
a,c
690 1343 18 54 0 2
-0.56 (-1.12, -0.01) 0.046
NBN
rs14448
c
640 1215 60 171 8 13
-0.11 (-0.40, 0.18) 0.447
rs9297757
a,b,c
651 1233 148 52 5 18
-0.26 (-0.58, 0.05) 0.097
rs3736640
a,c
676 1288 32 107 0 4
-0.64 (-1.27, -0.21) 0.003
RAD51
rs1801320
a,b
646 1209 58 186 4 4
-0.31 (-0.62, 0.00) 0.048
a
SNPs identified by model I based on Bayes Factors. Only those SNPs with 2ln(BF)exceeding 2 were selected.
b
SNPs identified by model II based on Bayes Factors. Only those SNPs with 2ln(BF)exceeding 2 were selected.
c
SNPs identified by Brooks et al 2012 based on per-allele RR. Only those SNPs with p value for trend < 0.05 were selected.
d
: regression coefficients of each SNP from simple logistic regression, adjusted for age, menarche, menopause, family history,
pregnancy, histology, treatment, the FGFR2 GWAS-identified SNP, deleterious variants in ATM, BRCA1,BRCA2,CHECK2 and offset
term.
e
p-values associated with Wald-z test for estimates from simple logistic regression adjusted for fixed covariants listed in d.

60

FIGURES

Figure 1: Directed acyclic graph describing the structure of the model. Boxes describe observed data, circles represent
latent variables or model parameters, and the triangle represents a logical node.

61

Figure 2: Graphical representation of the A matrix derived from the Gene Ontology. The lower levels of the graph
indicate sets of genes with high correlations across the 860 GO terms.

62

Figure 3: Comparison of estimated exp( )s based on different priors ( ) from Model I (a); and
comparison of estimated s based on different priors ( ) from Model II (b). The black dots are
simulated true betas corresponding to each gene. The true was set to 0.05.

63

Figure 4: The correlation between Estimated Exp(Beta) and True Exp(Beta) from Model I (a) and the correlation analysis
between Estimated Betas and True Betas from Model II (b). The simulation is based on single replica of data set and
parameter set for both Model (500 burn-ins, 1000 iterations).

64

Figure 5.1: Simulation analysis for Model I. Sensitivity/Specificity based on proportions of the number of SNPs (a), Sensitivity
based on Bayes Factors of SNPs (b), Bayes factors of genes (c). The red stars indicate the gene has at least one SNP included in the
model based on simulated data. The upper panels is based on simulation analysis on 10 data replicates per parameter set for 10
replicates of parameter sets, the lower panel is based on a single replicate of data.

65

Figure 5.2: Simulation analysis for Model II. Sensitivity/Specificity based on proportions of the number of SNPs (a), Sensitivity
based on Bayes Factors of SNPs (b), Bayes factors of genes (c). The red stars indicate the gene has at least one SNP included in the
model based on simulated data. The upper panels is based on simulation analysis on 10 data replicates per parameter set for 10
replicates of parameter sets, the lower panel is based on a single replicate of data.

66

60

Figure 6.1: Simulation analysis for Model I by MAF. Sensitivity/Specificity based on proportions of the number of SNPs
categorized by MAF (Left panels). Sensitivity based on Bayes Factors of SNPs categorized by MAF (Right panels).

67

Figure 6.2: Simulation analysis for Model II by MAF. Sensitivity/Specificity based on proportions of the number of SNPs
categorized by MAF (Left panels). Sensitivity based on Bayes Factors of SNPs categorized by MAF (Right panels).

68

Figure 7: Trace plots of Conditional LogLikelihood for Model I (a) and for Model II (b). Each plot is based on 10,000
iterations with 4,000 burn-ins for real data.

69

Figure 8: Gene-disease association analysis for Model I (a,b) and for Model II (c,d). Probability of at least 1 SNP in the gene
was included in the model (a,c). Candidate genes identification based on Bayes Factors.
70

Figure 9: SNP inclusion analysis for Model I (a) and for Model II (b).
71

72

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Abstract (if available)

Abstract We proposed a novel statistical method to investigate the involvement of multiple genes thought to be part of a common pathway for a particular disease. If a gene is identified to be associated with the disease, we are also interested in discovering which single nucleotide polymorphisms (SNPs) within this gene are responsible for this association. Here we present a Bayesian hierarchical modeling strategy that allows for multiple SNPs within each gene, with external prior information at either the SNP or gene level. The model involves variable selection at the SNP level through latent indicator variables and Bayesian shrinkage at the gene level towards a prior mean vector and covariance matrix that depend on external information. The entire model is fitted using Markov Chain Monte Carlo methods. Simulation studies show good ability to recover the underlying model. The method is applied to data on 504 SNPs in 38 candidate genes involved in DNA damage response in the Women's Environmental Cancer And Radiation Epidemiology study (WECARE) of second breast cancers in relation to radiotherapy exposures.

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Asset Metadata

Creator Duan, Lewei (author)

Core Title Using multi-level Bayesian hierarchical model to detect related multiple SNPs within multiple genes to disease risk

School Keck School of Medicine

Degree Master of Science

Degree Program Biostatistics

Publication Date 05/02/2013

Defense Date 05/02/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag breast cancer,candidate genes,DNA damage response,hierarchical Bayes models,OAI-PMH Harvest,pathway models,WECARE study

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Thomas, Duncan C. (committee chair), Conti, David V. (committee member), Marjoram, Paul (committee member)

Creator Email leweidua@usc.edu,leweiduan@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-250211

Unique identifier UC11288168

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Legacy Identifier etd-DuanLewei-1645.pdf

Dmrecord 250211

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Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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