Statistical Methods for Analyzing
Ordered Gene Expression
Microarray Data
Shyamal D. Peddada
Biostatistics Branch
National Inst. Environmental
Health Sciences (NIH)
Research Triangle Park, NC
An outline
Ordered gene expression data
Common experimental designs
A review of some statistical methods
An example
Demonstration of ORIOGEN
–
a software for ordered
gene expression data
Some examples of ordered
gene expression data
Comparison of gene expression by:
–
various stages of cancer
Normal

Hyperplasia
–
Adenoma
–
Carcinoma
–
tumor size
New tumor
–
Middle Size
–
Large tumor (with necrosis)
–
dose of a chemical (dose

response study)
–
duration of exposure to a chemical (time

course
experiments)
–
dose & duration
Some commonly used experimental
designs
Experimental unit: Tissues/cells/animals
Single chemical/treatment
–
Dose response study
–
Time course study
single dose but responses obtained at multiple time
points after treatment
experimental units are treated at multiple time points
using the same dose.
–
Dose response x Time course study
Multiple doses at multiple time points
Multi chemicals/treatments
Possible objectives
–
Investigate changes in gene expression at certain
biologically relevant category.
E.g. Hyperplasia to Adenoma to Carcinoma
E.g. “early time point” to “late time point” since the
exposure to a chemical
–
Identify/cluster genes with similar expression profiles
over time/dose.
Correlation coefficient based methods
Correlation coefficient based methods match genes with
similar
observed
patterns of expression across dose/time
points.
Gene 1
Gene 2
Correlation coefficient based methods
A number of variations to this general principle
exist in the literature. Here we outline some
prominent ones.
A.
Chu et al. (Science, 1998):
Pre

select a set of biologically relevant patterns of
gene expressions over time.
Identify a sample of about 3 to 8 genes for each
pattern.
Compute the correlation coefficient of each candidate
gene in the microarray data with the above pre

selected genes.
Cluster each candidate gene into the cluster with
highest correlation coefficient
Correlation coefficient based
methods …
B.
Kerr and Churchill (PNAS, 2001):
They correctly recognized the uncertainty associated with
Chu et al. ‘s clustering algorithm. Hence they proposed a
bootstrap methodology to evaluate Chu et al.’s clusters.
C.
Heyer et al. (Genome Research, 1999):
Rather than using the standard correlation coefficient
between genes, they employ jackknife version which
robustifies against outliers.
Unlike Chu et al.’s strategy, they classify genes on the basis
of pairwise correlation coefficients.
Correlation coefficient based
methods …
Strengths
Familiarity among biologists
Easy to compute and interpret (although it is often
misinterpreted too!)
Weakness
Non

linearity in the data can lead to misinterpretation
Outliers and influential observations can affect the numerical
value of the correlation coefficient.
Heterogeneity between genes can also affect the numerical value
of the correlation coefficient.
It is also important to note that correlation coefficient is
typically estimated on the basis of a very small number of
points.
Regression based procedures
Basic assumption among these methods:
The “conditions” are numerical,
e.g. dose or time
Polynomial regression
Liu et al. (BMC Bioinformatics, 2005)
For each gene Liu et al. fitted a quadratic regression model:
They cluster each gene into a particular cluster depending
upon the sign and statistical significance of the regression
parameters.
If for a gene none of the regression coefficients are
significant then such a gene is declared un

important.
t
g
g
g
g
t
g
t
t
Y
,
2
2
,
1
,
0
,
,
Polynomial regression
Liu et al. (BMC Bioinformatics, 2005)
Strengths:
Biologists are reasonably familiar with quadratic regression
analysis.
Regression coefficients are easy to interpret.
For small number of doses or time points and for evenly spaced
doses, a quadratic model may be a reasonable approximation.
An easy to use EXCEL based software is available.
Polynomial regression
Liu et al. (BMC Bioinformatics, 2005)
Two major limitations because it is fully parametric:
1. Departure from quadratic model is common:
In such cases the
quadratic model
may not be correct.
2. Normality assumption need not be valid.
Time
“Semi

parametric” regression methods
Several authors have tried semi

parametric regression
approach to gene expression data.
E.g.
deHoon et al. (Bioinformatics, 2002)
Bar

Joseph et al. (PNAS, 2003, Bioinformatics, 2004)
Luan and Li et al. (Bioinformatics, 2003)
Storey et al. (PNAS, 2005)
Storey et al. (2005)
Basic idea:
For each gene, they fit mixed effects model with a B

spline
basis. This methodology is largely based on Brumback and
Rice (JASA, 1998).
Statistical significance of each gene is evaluated using an F
like test statistic with P

value (q

value) determined by
bootstrap.
Storey et al. (2005)
Strengths:
It is semi

parametric
A user friendly software called EDGE is available
Limitations:
It does not perform well for “threshold” patterns of gene
expression
The “conditions” should be numerical
Unequal dose or time spacing can have an impact on the
performance of the procedure
O
rder
R
estricted
I
nference for
O
rdered
G
ene
E
xpressio
N
(ORIOGEN)
Peddada et al. (
Bioinformatics
, 2003, 2005)
Simmons and Peddada (
Bioinformation
, 2007)
Temporal Profile /Dose Response
Pattern of the
(unknown)
mean expression of a gene
over time (dose) is known as the
temporal profile
(
dose
response
) of a gene.
–
ORIOGEN: uses mathematical (in)equalities to describe a
profile.
)
(
Some Examples
Null profile:
6
5
4
3
2
1
Examples Continued …
Up

down profile with maximum at 3 hours
6
5
4
3
2
1
Examples Continued …
Non

increasing profile
Cyclical profile
6
5
4
3
2
1
6
5
4
3
2
1
ORIOGEN
Step 1 (Profile specification)
:
Pre

specify the
shapes
of profiles of interest.
Some Examples Of Pre

specified
Profiles
ORIOGEN …
Step 2 (profile fitting)
: Fit each pre

specified profile
to each gene using the estimation procedure
described in:
Hwang and Peddada (1994,
Ann. of Stat.
)
A Brief Description Of The Estimation
Procedure …
Definitions
Linked parameters:
Two parameters are said to be linked if
the inequality between them is known
a priori
.
Nodal parameter:
A parameter is said to be nodal if it is
linked to all parameters in the graph.
For any given profile, the estimation always starts at the
nodal parameter.
Pool the Adjacent Violator Algorithm
(PAVA)
Hypothesis:
Observed data
Isotonized data (PAVA)
Estimation: The General Idea
1
2
4
5
3
1
2
5
4
3
3 is the only nodal parameter
Estimation Continued …
From this sub

graph we estimate 1 and 2.
1
2
3
Step 3:
Determine the norm of a gene corresponding
to each temporal profile.
This is defined as the maximum (studentized) difference
between estimates corresponding to linked parameters.
Peddada et al. (2001,
Biometrics
).
A Measure of “Goodness

of

fit”
Norm
l
l
An Example
Observed data:
1, 1.5, 2, 2.5, 1.5, 2.25
Two pre

specified temporal profiles:
(a)
(b)
Example Continued …
Fit under profile (a)
1, 1.5, 2.25, 2.25, 1.875, 1.875
Fit under profile (b)
1, 1.5, 2, 2.5, 1.875, 1.875
Example Continued …
norm for profile (a) is:
2.25

1 = 1.25
norm for profile (b) is:
2.5

1 = 1.5
l
l
“Best Fitting” Profile
Step 4:
Identify the profile with the largest norm.
In the example, profile (b) has larger norm than profile (a) .
Hence profile (b) is a better fit than (a).
Statistical Significance
Step 5:
Statistical significance:
P

value
for statistical significance is obtained using the
bootstrap methodology:
Illustration …
MCF

7 breast cancer cell treated with
17

estradiol (Lobenhofer et al.,
2002,
Mol. Endocrin
.).
Gene expressions were measured after:
1hr, 4hrs, 12hrs, 24hrs, 36hrs and 48hrs
of treatment.
# of genes on each chip = 1900.
# of samples at each time point = 8
Available softwares
Linear Regression Method (Liu et al., 2005)
EDGE (Storey et al., 2005)
EPIG (Chao et al., 2008)
ORIOGEN (Peddada et al., 2006)
Concluding remarks
Methodology
Freely available
software
Applicable
to ordinal
“conditions”
Repeated
measures
and
correlated
data
Model
assumptions
Linear Regression
Yes
No
No
Linear
regression
EPIG
Yes
No
?
No
EDGE
Yes
No
Yes
No
ORIOGEN
Yes
Yes
Yes
No
Some open problems
ORIOGEN is potentially subject to Type III error. How do
we control FDR & Type III error.
How to deal with
–
Dependent samples?
–
Covariates?
Order restricted inference in the context of mixed effects
linear models.
Acknowledgments
–
Leping Li
–
David Umbach
–
Clare Weinberg
–
Ed Lobenhofer
–
Cynthia Afshari
Software developers at
Constella Group
–
(late) John Zajd
–
Shawn Harris
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