Bayes Factors Comparing Two Multi-Normal
Covariance Matrices and Their Application to
Microarray Data Analysis
Ainong Zhou
A Dissertation submitted to
The Faculty of
Columbian School of Arts and Sciences
of The George Washington University in partial satisfaction
of the requirements for the degree of Doctor of Philosophy
June 20, 2006
1
Outline







Microarray techniques and functional gene sets
Class comparison of gene set expression in mean
level and similarity measures
Bayes factors comparing two covariance matrices
Simulation study of proposed Bayes factors
Case study of a lung cancer microarray analysis
Conclusions
Further developments
2
DNA microarrays
•
•
•
•
•
Microarrays measure the gene expression of thousands
of genes to an entire genome in a sample by measuring
RNA abundance on one slide/chip.
Typically in an experiment, RNA is extracted from a part
(cells in culture, tumors, other.) of an organism.
The RNA is then amplified, labeled so that a scanner
can see it and then hybridized to the microarray slide.
The brightness of each spot on the microarray
represents the amount of RNA for each gene, and
hence the gene expression.
Image analysis and data normalization are needed.
3
cDNA Arrays
cDNA clones
(probes)
excitation
laser 2
scanning
laser 1
emission
PCR product amplification
purification
printing
mRNA target
overlay image and normalize
0.1nl/spot
microarray
Hybridize
target to
microarray
analysis
4
Compliments of R. Irizarry
High Density Oligonucleotide Array
GeneChip Probe Array
Hybridized Probe Cell
Single stranded,
labeled RNA target
*
*
*
*
*
Oligonucleotide probe
24µm
1.28cm
Millions of copies of a specific
oligonucleotide probe
>200,000 different
complementary probes
Image of Hybridized Probe Array
5
Compliments of R. Irizarry
Standard Statistical Tests for Differential
Expression of Individual Genes
Two Groups:
 Fold change
 Student’s t-test: gene-specific, global, or regularized.
More than two groups:
 ANOVA : gene-specific, global, or regularized F-test.
Significance and multiple testing:
 Nominal p-values
 Family-wise error rate: the probability of accumulating one or more
false-positive errors over a number of statistical tests.
 False discovery rate: the proportion of false positives among all of
the genes initially identified as being differentially expressed.
6
Analysis at the level of single gene
Why is it insufficient?
•
•
Microarrays have a well-defined internal data
structure dictated by the genetic network of the
living cell. Gene-specific analytical tools often
ignore this structure.
Identifying differentially expressed genes
becomes a challenge when the magnitude of
differential expression is small.
7
Analysis at the level of functional groups
defined a priori : why important?
•
•
By incorporating biological knowledge, we can
detect modest but coordinated expression
changes of sets of functionally related genes.
The interaction of genes in the gene set may be
more important than the changes in individual
genes.
8
Gene Sets
There are usually many sets of genes that might
be of interest in a given microarray experiment.
Examples include:
 genes in biological pathways, e.g. biochemical,
metabolic, and signaling.
 genes associated with a particular location in the
cell.
 genes having a particular function or being
involved in a particular process.
9
Class comparison of gene set activity:
currently used methods


Enrichment of members of a gene set among the top-ranked genes:
 Fisher’s exact test (Draghici et al 2003)
 Z-test (Doniger et al 2003)
 Kolmogorov-Smirnov (K-S) test (Mootha et al 2003)
Permutation test of similarity measures (Rahnenfuhrer et al 2004)
Comment: All of these methods assume all the genes are
independent, and focus on the comparison of gene expression at
mean level.
10
Biological Motivation for Comparing
Two Covariance Matrices
•
•
•
Increased/decreased variations of gene expression are
found under an abnormal condition compared to a
normal condition.
Some genes may be tightly correlated or coupled in the
normal or less severe condition but become decoupled
due to disease progression, as, for example, in cancer
patients.
Any significant difference between two covariance
matrices can be attributed to difference in either
variances or correlations and may illustrate some
important geneset-wise regulation/deregulation
associated with a specific phenotype.
11
Motivation from Observed Data
-0.05
-0.15
Observed
0.00 0.05
0.05
Variances
-0.10
Observed
Means
-0.05
0.00
0.05
-0.15
-0.05
0.05
Permuted
Permuted
Covariances
Correlations
-0.4
0.0
Observed
0.02
-0.02
Observed
0.4
-0.10
-0.02 0.00
0.02
Permuted
0.04
-0.4 -0.2
0.0
0.2
0.4
Permuted
Gene expression of the Myosin signaling pathway, Data from Bhattacharjee et al. PNAS, 2001
12
Class Comparison of
Variance/Covariance Matrices

Likelihood ratio test (Anderson 1984)
|S|
Ng
g
N
2
T  g
N 
log |S | p log
1 g

, df  12 p
p 1
It is asymptotically distributed as Chi-squared with (1/2)p(p+1)
degrees of freedom.



A likelihood-based hierarchical test of principal
components (Flury 1984,1987)
Resampling tests (Zhu et al 2002)
Bayes factors using Jeffery’s improper priors (Kim et
al 1999, 2001)
13
Why Bayes Factor?
•
•
•
The use of prior probability distributions represents a
powerful mechanism for incorporating information from
previous studies. In microarray data, this becomes more
needed as the sample sizes are usually very limited due
to the cost of microarrays and availability of samples.
Bayes factor uses information on whole likelihood
function, not just maxima.
Bayes factor can be used as easily interpretable
alternatives to p values.
14
The Bayes Factor
1.
If data D under each of the two hypotheses H₁ and H₂ is distributed
according to probability densities p(D|H₁) or p(D|H₂) and the prior
probabilities p(H₁) and p(H₂)=1-p(H₁), the posterior probabilities can
be computed as:
2.
3.
p
H1 |Dand p
H2 |D1 p
H1 |D
By the Bayes's therorem,
p
D|H1 
p
H1 
,
p
D|H1 
p
H1 p
D|H2 
p
H2 
p
D|H2 
p
H2 
p
H2 |D
.
p
D|H1 
p
H1 p
D|H2 
p
H2 
p
H1 |D
4.
5.
6.
So,
p
H 2 |D
p
H1
p
D|H 
p
H 
p
H 
2
2
2

B
21
p
H 
|D
p
D|H 
p
H 
1
1
1
where B21 is the Bayes factor.
Thus, the Bayes factor is a summary of the evidence provided by the
data in favor of one scientific theory, represented by a statistical model, as
compared to another.
15
Microarray Data
Let Xg={xgij}, i=1,...,Ng; j=1,...,p be the observed data (a Ng×p matrix) in the
g-th group, and define:
k
N  N g ,
n g N g 1
g1
Ng
1
x
g 
Ng
 x gi ,
i1
k
x
1
N
Ng
k
i1
g1
Ng
  x gi
g1 i1

S g  
x gi x

x gi x
g
g  S  S g .
k
Ng
S 0   
x gi x


x gi x

g1 i1
where xgi., i=1,...,Ng, is an independent observation (vector) of the p genes from the
gth group. In a microarray study, the observed data is either the log-transformed
absolute gene expression with 0 so far out in the tail or the log-ratios. In both cases, the
normal approximation is often met.
16
Assumptions on Data
Assume Xg={xgij}, the observed data matrix in the gth
group, has a p-dimensional normal distribution, conditional
on the mean parameter μg and the covariance matrix Σg:
p
Xg |g , g 
1
1 pNg
1 Ng
2

2

|g | 2
Ng
 1
exp
12  i

x


g 
x gi g 

gi
g
1
where xgi, i=1,...,Ng, is an independent observation
(vector) on the p genes from the g-th group.

1 N g
|g | 2
1 pN
g

2
2
1
 1
exp 12 N g 
x
x
tr
S g 
g g g 
g g 
g 
17
Prior Parameters
In p-dimensional multivariate normal data with complete observations, a conjugate
prior can be specified for the gth population by:
g |g N p 
g , 1 g 
, g IWp 
, 
.
where νg is the mean p-vector of the prior distribution of the mean, and κis a
scalar for the scale of this prior distribution.
IWp(Λ,η) denotes the inverted Wishart distribution with (η-p-1) degrees of
freedom and scale matrix Λ, a p×p positive definite matrix, whose density is:
p
g 
where
1 p
1
|| 2 |g | 2 p1exp
12 tr
g 
1
1


22 p 2 
p 
t pp1/4

1
p

t  12 
j 1

, the multivariate gamma function
 j
1
ηis a scalar for the scale parameter of the prior covariance distribution, and
Λis a p×p positive definite matrix for the shape parameter of this prior distribution.
We assume that the two populations are independent a priori, i.e.:
p
1 , 2 , 1 , 2 p
1 , 1 
p
2 , 2 
18
Hierarchical Models for Comparing
Two Covariance Matrices
Models
Conjugate Priors
Two covariances are:
H0: Equal:
1 2 
H1: Proportionally equal
1 , 2 c 2 
H2: correlationally equal
1 , 2 CC
H3: Freely different:
1 2
Np 
1 |1 , 1 
, Np 
2 |2 , 1 
, and IWp 
|, 
1 |N p 
1 , 1 
, 2 |, c N p 
2 , 1 c 2 
IWp 
, 
, c p
c
, c 
1 |N p 
1 , 1 
, 2 |, C N p 
2 , 1 CC
IWp 
, 
, C p
C
, C 
1 |1 N p 
1 , 1 1 
, 2 |2 N p 
2 , 1 2 
1 IWp 
, 
, 2 IWp 
, 
19
Bayes Factor for Testing Whether the Two
Covariance Matrices Are Equal or Not Equal
The following Bayes factor is to compare the Models H3 and H0:
B 30 
||
1
2


p

2
g1

2
p
p
N g
2
g1 
Ng
2
 g
1
N g

Ng
Ng 

2
N

2



x

x
S 
g g 
g g  


x

x
S g 
g g 
g g  
1
2

N

1
2

Ng 

. #
20
Bayes Factor for Testing Whether the Two
Covariance Matrices Are Equal or Proportional
The following Bayes factor is to compare the proportional model and
equivalent model:
2
B 10 

g1
c pN
N g


x

x
S 
g g 
g g 
 N g
N 1

N1
2
c 2 N 2

N2


x

x
1 1 
1 1 


x

x
2 2 
2 2 
1

N

2

12 
N

p
c
dc.
#
S1 
c 2 S 2 
where p(c) is any chosen positively-ranged univariate probability
distribution.
21
Bayes Factor for Testing Whether the Two Covariance
Matrices Are Proportionally or Correlationally Equal
The following Bayes factor is to compare the correlational model and
proportional model:
N 1

N1
|C| N 2
N 2

N2
12 
N



x

x
1 1 
1 1 
 1
C 1 
x

x

2 2 
2 2 C
p
C
dC
S 1 C 1 S 2 C 1 
B 21 
N 1

N1
c pN 2
c 2 N 2

N2


x

x
1 1 
1 1 


x

x
2 2 
2 2 
12 
N

p
c
dc.
S1 
c 2 S 2 
where p(c) is any chosen positively-ranged univariate distribution, and
p(C) is any chosen positively-ranged multivariate distribution.
22
Bayes Factor for Testing Whether the Two Covariance
Matrices Are Correlationally Equal or Different
The following Bayes factor is to compare the different model and
correlational model:
B 32 
||

1
2

2

 g
1 p

p
2
g1
p
2
N g

Ng
N 2

N2
2
N

2

#


x

x
g g 
g g  S g 
N 1

N1
|C|N2
Ng 



x

x
1 1 
1 1 
 1
C1 
x

x

2 2 
2 2 C
12 
Ng

12 
N

.
p
C
dC
S 1 C1 S 2 C1 
where p(C) is any chosen positively-ranged multivariate distribution.
23
Simulation Study 1
The first set of simulations is conducted to determine the power
of the Bayes factor B30 in comparison with that of likelihood ratio test
for several sets of prior parameters.
Assumption of Prior Parameters:
• The location parameters (ν1 and ν2) for the two prior population
means are assumed to be a common p-vector of 0’s. The scale
parameter κ for the prior population means is set to be 1, 15, 60, or
120 in the range of sample size to be simulated.
• The parameters (η and Λ) for the prior population covariance
matrix are chosen according to the first moment of the inverted
Wishart distribution. The shape parameter η is set to be p+2, p+4,
p+7, or p+12. The scale matrix Λ is assumed to be the product of
a positive scalar (1, 2, 4, or 8) to the identity matrix.
24
Procedure for Power Estimation
1.
2.
3.
Draw one sample (Σ) from the common inverted Wishart distribution
IW(Λ,η).
Draw one p-variate normal sample separately for each of the two
population means (μ1,μ2) with common prior mean (ν) and covariance
matrix (1/κΣ).
Generate N1 p-variate normal samples for the first group with mean μ1 and
covariance matrix Σ, and N2 p-variate normal samples for the second group
with mean μ2+ ε (ε =0 or 2, the difference between two sample means) and
covariance matrix which is equal to either:


4.
5.
6.
7.
δΣ: Proportional to the first group, δ= 1, 1.2, 1.5, or 2.0 - Scenario 1, or
ΔΣΔ: Correlationally equal to the first group, where Δ is a diagonal matrix
{σq,σq+1,...,σq+(p-1)}, σ=1.0, 1.1, 1.2, or 1.5, q=floor(-p/2) - Scenario 2.
Compute the Bayes factors (B30) and LRT statistics.
Repeat steps 1-4 for 1000 times
Set the significance limit as the 95th %tile under the null condition:
δ=1
or σ=1.
Compute the power of the Bayes factor and LRT as the proportion of B30’s
and LRT’s that are greater than the significance limit.
25
Comparison of Power between LRT and Bayes Factor (B30)
Effect of sample size (N1=N2=N*p), dimension (p), and difference of sample means (ε)
Scenario 1
p
Test
ε
0
10
15
20
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=2*p
N=3*p
N=5*p
N=10*p
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
2
5
1.2
1.5
2
0
1
1.2
1.5
2
0
1
1.2
1.5
2
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.2
1.5
2
0
1
1.2
1.5
2
1
1.2
1.5
2
1
1.2
1.5
2
0
1
1.2
1.5
2
X-axis: δ (effect size in Scenario 1), Y-axis: Power, Priors: ν1= ν2=0, κ=1, η=p+2, Λ=Ip
26
Comparison of Power between LRT and Bayes Factor (B30)
Effect of sample size (N1=N2=N*p), dimension (p), and difference of sample means (ε)
Scenario 2
p
Test
15
20
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=3*p
N=5*p
N=10*p
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
2
10
N=2*p
ε
0
5
1.1
1.2
1.5
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
1
1.1
1.2
1.5
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
X-axis: σ (effect size in Scenario 2), Y-axis: Power, Priors: ν1= ν2= 0, κ=1, η=p+2, Λ=Ip
27
Effect of Sample Size and Dimension on
the Power of LRT and Bayes Factor (B30)
•
•
In Scenario 1, B30 has greater power than LRT
when the sample sizes are 2 times of dimension
and there is no difference between the two
sample means. As the sample size increases
proportionally to the dimension or the two
sample means become different, however, LRT
becomes more powerful than B30 in most cases.
In Scenario 2, LRT shows greater power than
B30 under all conditions.
28
Comparison of Power between LRT and Bayes Factor (B30)
Effect of κ with different sample sizes (N1=N2=N*p, p=10)
Scenario 1
κ
Test
1
15
60
120
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.2
1.5
2
0
1
1.2
1.5
2
0
1
1.2
1.5
2
1
1.2
1.5
2
X-axis: δ (effect size in Scenario 1), Y-axis: Power, Priors: ν1 = ν2 = 0, η=p+2, Λ=Ip
29
Comparison of Power between LRT and Bayes Factor (B30)
Effect of κ with different sample sizes (N1=N2=N*p, p=10)
Scenario 2
κ
Test
1
15
60
120
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
1
1.1
1.2
1.5
X-axis: σ (effect size in Scenario 2), Y-axis: Power, Priors: ν1 = ν2 = 0, η=p+2, Λ=Ip
30
Comparison of Power between LRT and Bayes Factor (B30)
Effect of η with different sample sizes (N1=N2=N*p, p=10)
Scenario 1
η
Test
p+2
p+4
p+7
p+12
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.2
1.5
2
0
1
1.2
1.5
2
0
1
1.2
1.5
2
1
1.2
1.5
2
X-axis: δ (effect size in Scenario 1), Y-axis: Power, Priors: ν1 = ν2 = 0, κ = 1, Λ=Ip
31
Comparison of Power between LRT and Bayes Factor (B30)
Effect of η with different sample sizes (N1=N2=N*p, p=10)
Scenario 2
η
Test
p+2
p+4
p+7
p+12
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
1
1.1
1.2
1.5
X-axis: σ (effect size in Scenario 2), Y-axis: Power, Priors: ν1 = ν2 = 0, κ = 1, Λ=Ip
32
Comparison of Power between LRT and Bayes Factor (B30)
Effect of Λ with different sample sizes (N1=N2=N*p, p=10)
Scenario 1
Λ (Diagonal)
Test
Ip
2*Ip
4*Ip
8*Ip
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.2
1.5
2
0
1
1.2
1.5
2
0
1
1.2
1.5
2
1
1.2
1.5
2
X-axis: δ (effect size in Scenario 1), Y-axis: Power, Priors: ν1 = ν2 = 0, κ = 1,η=p+2
33
Comparison of Power between LRT and Bayes Factor (B30)
Effect of Λ with different sample sizes (N1=N2=N*p, p=10)
Scenario 2
Λ (Diagonal)
Test
Ip
2*Ip
4*Ip
8*Ip
LRT
─■─
─■─
─■─
─■─
B30
─▲─
─▲─
─▲─
─▲─
N=15
N=20
N=30
N=50
1
1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
0
1
1.1
1.2
1.5
1
1.1
1.2
1.5
X-axis: σ (effect size in Scenario 2), Y-axis: Power, Priors: ν1 = ν2 = 0, κ = 1,η=p+2
34
Effect of Prior Parameters on the Power of
LRT and Bayes Factor (B30)
•
•
•
No effect of κ is found on the power of either B30
or LRT.
The power of both LRT and B30 decreases as η
increases.
No effect of Λ is observed on the power of either
B30 or LRT.
35
Simulation Study 2
The power of the three hierarchical Bayes Factors comparing proportional vs
equal models (B10), correlationally vs proportionally equal models (B21), and
Freely different vs correlationally equal models (B32) is compared in this set of
simulations.
The power is estimated as follows:
1.
Draw one sample (Σ) from the inverted Wishart distribution IW(Λ,η).
2.
Draw one sample (c) from the log-normal distribution (0, 1).
3.
Draw one sample (C) from the multivariate log-normal distribution (0, Ip).
4.
Generate one p-variate normal sample for each of the two population means (μ1,μ2)
with common prior mean (ν) and covariance matrix (1/kΣ, c2/kΣ, or 1/κCΣC
depending on the underlying model).
5.
Generate N1 p-variate normal samples for the first group with means μ1 and
covariances Σ, and N2 p-variate normal samples for the second group with means
μ2 and variances δΣ (Scenario 1) or ΔΣΔ (Scenario 2) as before.
6.
Compute the integrants of c and C.
7.
Repeat steps 2-6 for 1000 times to estimate the integrals relative to c and C using
simple Monte Carlo simulations.
8.
Compute the Bayes factors.
9.
Repeat steps 1-8 for 1000 times.
10.
Set the significance limit as the 95th %tile when δ=1 or σ=1.
11.
Compute the power of the Bayes factors.
36
Comparison of Power between Bayes Factors B10, B21 and B32
B21
B32
─▲─
─▲─
─▲─
Scenario 1
p
2
B10
Scenario 2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
5
2
4
8
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1
1.2
1.5
2
1
1.2
1.5
2
0
1
2
4
8
X-axis: δ(Scenario 1) or σ (Scenario 2), Y-axis: Power
Sample Sizes: N1=N2=10, Priors: ν1 = ν2 = 0, κ = 1, η = p+2, Λ = Ip
37
Case Study
Bhattacharjee et al. PNAS, 2001


12,600 gene expression measurements obtained
using Affymetrix oligonucleotide arrays.
203 cancer patients and normal subjects, 5 predefined cancer subtypes, plus staging and survival
information.
38
Data Preprocessing
•
•
•
Multiple transcripts of the same genes defined
by Locuslink are averaged, which resulted in a
total of 8895 distinct genes.
Multiple specimens from the same subject are
averaged.
The data are centered separately for each gene
within each group
39
Determination of Priors
•
•
•
ν1 and ν2 are set to 0 considering that the data are centered in each group.
k is set to be 0 to minimize effect of differential means between prior and
sample means, if any.
η and Λ are set to be the maximum likelihood and unbiased estimates,
respectively from the data excluding genes that are involved in the
pathways being analyzed.
4)
The genes that are contained in a specific pathway (say, p genes) are excluded
from the genes on the chip (say, G genes).
A total of g (=(G-p)/p) exclusive subsets of genes are randomly generated from
(G-p) genes.
For each subset of genes, the sample covariance matrix (Si) was computed for
the two groups combined.
Λ is set to be the unbiased estimate expressed as:
5)
1
n
1



p 1


 , where 
1n  i

i . maximum likelihood estimator of η.
1 the
A recursive approach is used to compute
1)
2)
3)
40
Comparisons of the Myosin Signaling Pathways
between Above and Below the Median Survival
-0.05
0.05
0.00
0.05
-0.15
-0.05
0.05
Permuted
Covariances
Correlations
Observed
LR_P 0
BF_P 0.018
LR_P 0
BF_P 0.018
-0.4
0.0
0.4
Permuted
-0.02
0.02
-0.10
Observed
LR_P 0
BF_P 0.018
-0.15
Observed
0.43
-0.05
0.00 0.05
LR_P
Variances
-0.10
Observed
Means
-0.02 0.00
0.02
Permuted
0.04
-0.4 -0.2
0.0
0.2
0.4
Permuted
41
Conclusions
•
•
•
Four conjugate Bayes factors are derived to
compare two covariance matrices in four
hierarchical models.
The power of proposed Bayes factors is
evaluated with different prior parameters in
comparison to that of the likelihood ratio test.
One of the proposed Bayes factor is applied to a
microarray analysis and the gene expression of
one pathway is found to significantly differ in
variances but not in means between two survival
groups.
42
Future Studies

Sensitivity of Bayes factors to prior distributions

Bayes factors comparing principal components
of covariance matrices
43
Acknowledgments
Research Committee:
Director: Prof. Giovanni Parmigiani, Johns Hopkins University
Co-director: Prof. John Lachin, George Washington University
Readers: Prof. Reza Modarres and Naji Younes
Examination Committee:
Chairman: Prof. Efstathia Bura
GWU Members: Prof. Yinglei Lai,Reza Modarres,Naji Younes
Extramural Member: Prof. Bruce Trock, Johns Hopkins University
The Biostatistics Center, GWU
Friends and Family
44