Gene set analysis for longitudinal gene expression data.

Abstract

Gene set analysis (GSA) has become a successful tool to interpret gene expression profiles in terms of biological functions, molecular pathways, or genomic locations. GSA performs statistical tests for independent microarray samples at the level of gene sets rather than individual genes. Nowadays, an increasing number of microarray studies are conducted to explore the dynamic changes of gene expression in a variety of species and biological scenarios. In these longitudinal studies, gene expression is repeatedly measured over time such that a GSA needs to take into account the within-gene correlations in addition to possible between-gene correlations.
We provide a robust nonparametric approach to compare the expressions of longitudinally measured sets of genes under multiple treatments or experimental conditions. The limiting distributions of our statistics are derived when the number of genes goes to infinity while the number of replications can be small. When the number of genes in a gene set is small, we recommend permutation tests based on our nonparametric test statistics to achieve reliable type I error and better power while incorporating unknown correlations between and within-genes. Simulation results demonstrate that the proposed method has a greater power than other methods for various data distributions and heteroscedastic correlation structures. This method was used for an IL-2 stimulation study and significantly altered gene sets were identified.
The simulation study and the real data application showed that the proposed gene set analysis provides a promising tool for longitudinal microarray analysis. R scripts for simulating longitudinal data and calculating the nonparametric statistics are posted on the North Dakota INBRE website http://ndinbre.org/programs/bioinformatics.php. Raw microarray data is available in Gene Expression Omnibus (National Center for Biotechnology Information) with accession number GSE6085.

Full text

Page 1

METHODOLOGY ARTICLEOpen Access
Gene set analysis for longitudinal gene
expression data
Ke Zhang1*, Haiyan Wang2*, Arne C Bathke3, Solomon W Harrar4, Hans-Peter Piepho5and Youping Deng6
Abstract
Background: Gene set analysis (GSA) has become a successful tool to interpret gene expression profiles in terms
of biological functions, molecular pathways, or genomic locations. GSA performs statistical tests for independent
microarray samples at the level of gene sets rather than individual genes. Nowadays, an increasing number of
microarray studies are conducted to explore the dynamic changes of gene expression in a variety of species and
biological scenarios. In these longitudinal studies, gene expression is repeatedly measured over time such that a
GSA needs to take into account the within-gene correlations in addition to possible between-gene correlations.
Results: We provide a robust nonparametric approach to compare the expressions of longitudinally measured sets
of genes under multiple treatments or experimental conditions. The limiting distributions of our statistics are
derived when the number of genes goes to infinity while the number of replications can be small. When the
number of genes in a gene set is small, we recommend permutation tests based on our nonparametric test
statistics to achieve reliable type I error and better power while incorporating unknown correlations between and
within-genes. Simulation results demonstrate that the proposed method has a greater power than other methods
for various data distributions and heteroscedastic correlation structures. This method was used for an IL-2
stimulation study and significantly altered gene sets were identified.
Conclusions: The simulation study and the real data application showed that the proposed gene set analysis
provides a promising tool for longitudinal microarray analysis. R scripts for simulating longitudinal data and
calculating the nonparametric statistics are posted on the North Dakota INBRE website http://ndinbre.org/
programs/bioinformatics.php. Raw microarray data is available in Gene Expression Omnibus (National Center for
Biotechnology Information) with accession number GSE6085.
Background
Molecular biology, which is targeted at studying biologi-
cal systems at a molecular level, has provided rich infor-
mation of individual cellular components and their
contributions to biological functions over the last 50
years. Our understanding of genes and their functions
has been accelerated in the last decade by microarray
experiments, which identify genes that are induced or
repressed in a specific biomedical condition [1-3]. The
multiplicity and heterogeneity of these gene expression
profiles revealed that even a simple biological process or
a molecular function in a cell requires co-operations of
hundreds or even thousands of genes. Nonetheless,
decoding this kind of gene interaction and networking
in a biological process is hampered by the complexity of
biological systems.
Instead of looking at individual genes, researchers
started to interpret biological phenomena in terms of
groups of genes, or gene sets. For example, Segal et al.
(2004) mined a large number of cancer expression pro-
files and deduced 456 cancer-related modules (gene
sets) which are selected by combining with the knowl-
edge of transcriptional pathways and gene ontology [4].
The development of new statistical tools enables us to
test whether a gene set is activated in the microarray
dataset of interest. An important contribution is made
by Subramanian et al. (2005) who proposed the gene set
enrichment analysis (GSEA) to assess the significance of
a set of genes. Their idea is that the genes that
* Correspondence: ke.zhang@med.und.edu; hwang@ksu.edu
1School of Medicine & Health Sciences, University of North Dakota, Grand
Forks, ND 58202, USA
2Department of Statistics, Kansas State University, Manhattan, KS 66506, USA
Full list of author information is available at the end of the article
Zhang et al. BMC Bioinformatics 2011, 12:273
http://www.biomedcentral.com/1471-2105/12/273
© 2011 Zhang et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.

Page 2

cooperate in a biological function have similar patterns
in transcriptional levels such that the statistical power of
assessing a gene set is higher than that of individual
genes [5].
GSEA relies on permutation tests to identify the sig-
nificant gene sets that have distinct gene expression
between treatment groups. It works in three steps. First,
all genes are ranked according to their statistics for the
treatment effect. For example, a t-statistic can be used
to compare two classes of samples. A score is assigned
to each gene set using a weighted Kolmogorov-Smirnov-
like statistic that sums up the ranks of the genes. Sec-
ondly, the class labels of the samples are permuted for a
number of times, and gene set scores are calculated for
each new label assignment. The permutation of sample
labels preserves the inherent correlation between genes.
Because the permutation is conducted under the null
hypothesis of no treatment differences, the P value of
each observed score can be determined empirically by
the null score distribution. Thirdly, if more than one
gene set is tested, the P values should be adjusted for
multiple tests. GSEA is often applied for hundreds of
gene sets, for which the false discovery rate (FDR) is
recommended.
Ever since GSEA was introduced, it has drawn a wide
attention from the biomedical and biostatistical commu-
nities. A number of alternative and extended versions of
gene set analysis method (GSA) have been proposed in
the last few years that use a variety of score systems and
randomization procedures to resample data [6,7]. For
instance, Efron et al. proposed a GSA method, which is
based on a more powerful statistic maxmean to score
gene sets [8]. In the case of two sample classes, max-
mean is the maximum absolute value between the aver-
age of the positive t-statistics, and also the average of
the negative t-statistics. Before permutation test, the
maxmean score should be restandardized by centering
and scaling its mean and standard deviation using ran-
domized gene sets.
Despite their enormous success, all these aforemen-
tioned GSA methods have limited applications in micro-
array samples with dependence. A permutation test has
to rely on the assumption of sample independence. This
assumption presents a barrier to extend GSA to the
fast-growing area of longitudinal microarray experi-
ments, which repeatedly profiles the gene expression of
a same object over time. Longitudinal microarray
experiments allow researchers to investigate dynamic
behavior of biological processes, such as cell cycles, cell
proliferation, oncogenosis, and apoptosis. The temporal
component is an inherent part of the study. Such time
course experiments pose novel challenges for statistical
analyses because effective methods have to take into
account both a large number of genes and within-gene
correlations. Most of the analyses in literature carry out
repeated measures analysis of individual genes followed
by FDR control [9-12].
It is desirable to apply repeated measures analysis
methods, such as a linear mixed effects model (LME) or
generalized estimating equations (GEE), to gene sets.
Tsai and Qu (2008) assessed subsets of genes by apply-
ing a non-parametric time-varying coefficient model
[13]. The within-gene correlation was taken into
account by the quadratic inference function (QIF) that
is derived from GEE. Both LME and GEE achieve their
asymptotic distributions when the number of replica-
tions goes to ∞. However, the large sample size assump-
tion is usually not applicable due to the high cost of
microarray experiments. Rather, there is often a rela-
tively large number of genes in a gene set compared to
the sample size, a curse of dimensionality problem. An
effective GSA method should also be robust against
deviation from the normal distribution because gene
expression data may be largely skewed, and the normal
or log-normal distribution does not provide a close fit
to the data [14,15]. Furthermore, to allow variability
between genes, heteroscedastic correlation structures
should be assumed for different genes.
In this paper we propose a GSA method for assessing
the expression patterns of gene sets from longitudinal
microarray data. The method employs a couple of novel
nonparametric statistics that work for small sample size
as long as we maintain a relative large number of genes
in a set (large p, small n). The method is robust with
respect to non-normality and heteroscedastic correlation
structures. To ensure extensive application, unbalanced
designs are allowed in our model. For example, unba-
lanced data may occur when the data are pooled from
different versions or manufacturers of arrays.
The genes in a signal transduction pathway are often
highly correlated in that the expression of one gene is
regulated by the other gene in this pathway. To ensure
an unbiased analysis, we need to take into account the
correlation among genes. Permutation method has been
widely used in GSA to provide a robust test that pre-
serves between-genes correlations. For example, Tsai
and Chen (2009) used permutation test with the Wilks’
Λ statistic for their multivariate analysis of GSA [16].
To take into account the correlations among genes
within a gene set, we also present a permutation-based
test for our proposed statistics.
The outline of this paper is as follows. Our main results
are presented in section Results and Discussion. In subsec-
tion Model and Hypotheses, we describe the model and
assumptions. In the subsection of Simulation study, we
present the simulation results of type I error estimates and
power analysis for our proposed methods. In subsection
Results on real data, we describe an application of our
Zhang et al. BMC Bioinformatics 2011, 12:273
http://www.biomedcentral.com/1471-2105/12/273
Page 2 of 12

Page 3

method to a recent longitudinal microarray study in which
the gene expression profiles of murine T cells in the pre-
sence or absence of interleukin-2 (IL-2) were repeatedly
collected. A number of functional gene sets were tested to
investigate IL-2 signaling over time. The test statistics and
their asymptotic results for a large number of genes but
small replications are provided in subsection Test statistics
of section Methods. Subsection Permutation tests
described the permutation-based test with our proposed
nonparametric statistics. Finally, we provide mathematical
proof for the asymptotic results of our test statistics in
Appendix.
Results and Discussion
Model and hypotheses
In a longitudinal design for microarray studies, global
transcriptional levels of each object were repeatedly mea-
sured at multiple time points under various conditions,
such as different drug doses, genotypes, and chemical
environments. Our goal is to find whether the transcrip-
tion levels of a set of genes show a dynamic pattern that
differs between conditions. We enumerate all the condi-
tions using i = 1,..., I and refer them as treatments. If the
number of genes in a gene set is relatively small com-
pared to the number of sample replications, the methods
for repeated measures analysis, such as LME and GEE,
are able to test the variation among treatments under
certain distributional assumptions. Both LME and GEE
provide efficient model parameter estimates when the
assumed covariance matrices can be estimated consis-
tently. However, when the number of genes plus the
number of time points is much larger than the number
of replications, consistent estimates of the large covar-
iance matrices are no longer available, especially if multi-
ple large covariance matrices need to be estimated when
empirical evidence suggests heteroscedasticity is present.
We will focus our e ort on the latter case.
For a gene set, let Xikl= (Xi1kl,..., XiJkl)’ be the tran-
scriptional levels of the kthgene (or probe) of the lth
replicate in treatment i, where k = 1,..., K, l = 1,..., nik,
and i = 1,..., I. The expression of this gene is measured
at J time points with subscript j to enumerate the jth
repeated measurement. Denote μik= E(Xikl) and Σik=
Var(Xikl) = (si, k, jj’)J×Jto be the gene specific mean and
covariance matrix. Each individual gene has its own
transcriptional activity, therefore, each gene has its
unique correlation structure. The heteroscedastic covar-
iances for different treatments and different genes allow
us to take into account of the different mechanisms that
different genes respond to a treatment. This is more
realistic than assuming a common covariance matrix in
that many of the genes are not responsive to a specific
stimulus while the responsive genes could exhibit
different temporal dependence. An example is that a sti-
mulus specific regulator gene or transcription factor
tends to be activated at the early stage of the stimulus
and the downstream genes of the regulator will respond
at a later stage. We leave the joint distribution of Xikl
unspecified and assume the observations from different
treatments or replicates are independent.
Let ¯ μi·= K−1?K
for the ithtreatment. Let a be the I × J matrix with ith
row being ¯ μ?
trast of the treatments can be stated as
k=1μikbe the mean expression profile
iThe hypothesis of no effect for the con-
H0(Treatment) : L1α1J= 0p,
(0:1)
where L1is a p × I contrast matrix with full row rank, 1J
is the J-dimensional vector of ones, and 0pis a p-dimen-
sional vector of zeros. The contrast matrix is convenient
to assess the effect of a specific treatment factor if the
treatment consists of multiple factors. Typical contrast
matrix for a single treatment factor with I levels is an (I -
1) by I matrix L1= (1I-1| - diag(I - 1)), where the first col-
umn is 1I-1, a column vector of ones, and the remaining
columns are -diag(I - 1), the negative of the identity matrix
of dimension (I - 1). For I = 3, the above L1is
?1 −1 0
This particular contrast matrix basically specifies that
all the treatment means some treatments averaged over
the whole time period and over all genes are identical.
Differences could arise if the mRNA transcriptions of
some genes are activated or inhibited by the treatment.
Genes could have distinct expression trends over time.
The hypothesis of no effect for a contrast among the
treatment by time interactions can be expressed as
M3=
1 0 −1
?
.
H0(Interaction) : L2[Vec(α) − Vec(αPJ) − Vec(PIα)] = 0q,(0:2)
where Vec() function transforms a matrix into a vector
by concatenating all columns, PIis the projection matrix
I−11I1?
rank. An example of the contrast matrix is the Kro-
necker product MI⊗MJthat specifies that all interac-
tions are zero, where MI= (1I -1| - diag(I - 1)). For
example, with I = 3, J = 4, the Kronecker product con-
trast matrix for interaction effect is
⎛
⎜
⎜
I, and L2is a q × (IJ) contrast matrix with full row
M3⊗ M4=
⎜
⎜
⎜
⎝
⎜
1 −1 0
1 0 −1 0 −1 0 1 0 0 0 0 0
1 00 −1 −1 0 0 1 0 0 0 0
1 −1 0
1 0 −1 0
1 00 −1 0 0 0 0 −1 0 0 1
0 −1 1 0 0 0 0 0 0
0 0 0 0 0 −1 1 0 0
0 0 0 0 −1 0 1 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎟
.
Zhang et al. BMC Bioinformatics 2011, 12:273
http://www.biomedcentral.com/1471-2105/12/273
Page 3 of 12

Page 4

We present a summary of notations that are used in
the rest of the manuscript. Denote σ2
i,k,j= Var(Xijkl), and
Xijk. = n−1
ik
nik
?
ι=1
Xijkl,? Xij.. = K−1
X’ikl1J,? Xi... = K−1
K
?
K
?
k=1
Xijk.,
? Xi·k. = (nikJ)−1
We consider a couple of novel nonparametric statis-
tics for hypotheses testing. A linear mixed effects model
(LME) and generalized estimating equations (GEE) are
often used for testing hypotheses (0.1) and (0.2) by
assuming an appropriate correlation structure. The sta-
tistics for both LME and GEE achieve their asymptotic
distributions when the number of samples goes to infi-
nity. Thus, theoretically LME and GEE are not suited to
large p, small n problems such as microarray data. This
motivated us to propose new statistics that converge to
their limiting distributions when the number of genes
goes to infinity. The statistics should be robust for non-
normal distributions, heteroscedastic correlation struc-
tures, and unbalanced experiment designs. Two novel
Wald statistics are proposed for null hypotheses (0.1)
and (0.2) in the method section. Their asymptoticity is
proved in Appendix.
nik
?
l=1
k=1
? Xi·k.
Simulation study
This section will present our simulation study to evalu-
ate the proposed nonparametric test statistics (NP) in
various settings. First, we calculate the estimated type I
error rate at level 0.05 for our nonparametric statistics.
The type I error will be examined for samples generated
from normal, exponential, Poisson and Cauchy distribu-
tions after introducing within-subject correlations. Sec-
ond, we will compare the power of the NP statistics
with linear mixed-effects model (LME) and generalized
estimating equations (GEE). The type I error and the
power analysis are used to validate our NP statistics.
Thirdly, we will calculate the estimated type I error and
power of the permutation test with our statistics for cor-
related genes and compare the results with GEE on data
from normal, exponential, and Poisson distributions. All
calculations and simulations were carried out with R
programming and the results were based on 1000 itera-
tions. The LME and GEE methods were implemented
by using gls and geese functions from R packages nlme
and geepack, respectively ([17,18]).
(a) Type I error rate analysis based on asymptotic
distribution with simulated data
In this section, we evaluate the specificity of our pro-
posed test (NP) based on type I error rates for simulated
data from various distributions. The number of time
points per gene we simulated is either 2 or 5. As
balanced design is only a special form of unbalanced
design, here we only consider unbalanced design in that
four fifths of genes having 4 replications and the
remaining one fifth of genes having 6 replications. First,
we examined the proposed test statistic for no gene
expression variations across treatments. A data matrix X
of n rows and J columns were randomly generated with
each row representing observations from the same gene
over J time points. The n is the sum of the number of
replications for all genes across all treatment groups.
The rows were generated from identical distribution
such that the null hypothesis of no expression changes
across treatments is satisfied. To allow a wide variety of
data types, we use normal, exponential, Poisson, and
Cauchy distributions to generate random samples. For
normal, exponential, and Poisson distributions, the
mean of random data was set to 2. The normal distribu-
tion was given a standard deviation of 1. The Cauchy
distribution had a location parameter of 0, and a scale
parameter of 1. Unstructured within-gene correlations
were then generated from a uniform distribution on (0,
0.5).
Identical unit variance is used for data under the null
hypotheses. We used the Cholesky decomposition (via R
function chol) to produce the lower half triangular
matrix h for the covariance matrix Σ. Thus the data
matrix Y = Xh has the desired covariance structure and
it is used for subsequent data analysis. The matrix Y
had equal means across rows. However, at different time
points (across columns), the values from the same gene
could vary.
Table 1 gives the estimated type I error rates for data
with unstructured correlation using the asymptotic dis-
tribution of the test statistic for treatment. For normal,
exponential, and Poisson distributions, the error rates
for at least 5 genes and 2 or 5 time points were in high
agreement with the expected level a = 0.05. The error
rate for Cauchy distribution failed to converge to 0.05 as
the number of genes increases. This happens because
the test requires finite fourth central moments while
Cauchy distribution does not have finite moments.
The next test was concerned with the interaction of
treatment and time effect. Under the null hypothesis of
no interaction, we generated random data as follows.
Given the value yijfor probe i at the jthtime point, the
random observation at the (j + 1)thtime point can be
obtained by
yi,j+1= ρyij+ εij,
(0:3)
where εijis a random variable with mean 2(1 - r).
Thus the mean of Xi, j+1is 2, which is the same as that
of Xij. For the Poisson distribution, we first generated
Zhang et al. BMC Bioinformatics 2011, 12:273
http://www.biomedcentral.com/1471-2105/12/273
Page 4 of 12

Page 5

the mean values with the iterative algorithm (0.3), and
then used the means to generate random integer num-
bers. An unstructured correlation was introduced to the
repeated measures for each gene similarly as was gener-
ated for the test of no treatment effects. The type I
error rates at a level 0.05 were shown in Table 2. Nor-
mal, exponential, and Poisson distributions had error
rates close to 0:05 when the number of genes was above
50. Cauchy distribution did not converge to 0.05.
(b) Power analysis based on asymptotic distribution with
simulated data
To evaluate the proposed NP statistics, we calculated
the estimated power curves for three methods, NP, LME
and GEE. Data were simulated for 4 treatment groups
and 3 replicates. As shown in Tables 1 and 2, the num-
ber of genes being 50 or above achieves expected error
rates. Therefore, we used 50 genes for all of the power
analysis in this subsection. Each gene was repeatedly
measured at 4 time points. Random data were generated
in the similar way as for type I error simulation study.
Log-normal distribution was assumed so that the data
were first generated by a normal distribution and were
then taken exponential transformation. An unstructured
correlation was introduced between time points for each
gene as described in the simulation study subsection.
For LME and GEE, gene expression levels were mod-
eled as the response variables with treatment group and
time as fixed effects. The variable subject, which pro-
vides measurements for all genes at all time points, are
modeled as a random effect. Unstructured correlation
structure cannot be estimated in LME and GEE model
fitting due to the number of replications being small. In
this part of the simulation, compound symmetry corre-
lation structure was assumed for LME and working
independence correlation structure was used for GEE.
First, we conducted a power analysis for the treatment
effect. The means of the normal distributions are differ-
ent between the treatment groups under alternative
hypothesis, and the standard deviation of the normal
distribution for each gene is randomly generated by a
uniform distribution in (0, 3). The mean differences Δ
between groups range from 0 to 2.5 to generate the
power curves. Thus in each experiment, the logarithm
of the mean of treatment group 2 is Δ higher than that
of group 1, and that of group 3 is Δ higher than group
2, and so on. The three power curves for NP, LME, and
GEE were shown in Figure 1. NP outperformed GEE
and NP for all Δ. When Δ = 0.7, NP has 91% power,
whereas LME has 60% power and GEE has 70% power.
Next, we conducted power simulation analysis for the
test of no treatment and time interaction. The results
were similar to that for the treatment effect. So we do
not present the results here.
(c) Type I error and Power analyses for the permutation
test
We further conducted simulation study for the permuta-
tion test with our NP statistics by generating random
data that had both within-gene correlation over time
and between-gene correlation within a gene set.
Random data were generated for two treatment
groups with three time points. In order to show the
effects of sample size on the power, the number of repli-
cates for a group varied from 5 to 50. Random data
were generated in the same way as for power analysis of
NP statistics described earlier except that an AR(1) cor-
relation structure with correlation coefficient 0.5 was
introduced to gene-gene relationship. Gene sets with 20,
50 and 100 genes were generated following normal,
exponential and Poisson distributions. Since linear
mixed effects model is not valid for exponential or Pois-
son distributions, we compare the permutation-based
NP statistics with GEE. For this part of the simulation,
Table 1 Estimated Type I errors for the test of no
treatment effect based on asymptotic distribution
#time.points#genesnormalexponentialPoisson Cauchy
250.060
0.047
0.053
0.048
0.044
0.043
0.040
0.053
0.052
0.046
0.063
0.052
0.052
0.050
0.063
0.048
0.054
0.058
0.053
0.057
0.042
0.021
0.024
0.026
0.019
0.021
0.020
0.020
10
20
30
40
50
100
55 0.056
0.053
0.047
0.060
0.050
0.044
0.062
0.052
0.055
0.045
0.058
0.049
0.041
0.047
0.059
0.057
0.066
0.050
0.047
0.041
0.050
0.032
0.025
0.020
0.014
0.018
0.016
0.023
10
20
30
40
50
100
The data from the same gene have unstructured correlation.
Table 2 Estimated type I error of the test of no
treatment by time interaction at 0
#genes normalexponentialPoisson Cauchy
50.087
0.074
0.061
0.070
0.071
0.064
0.037
0.043
0.048
0.057
0.103
0.082
0.063
0.071
0.060
0.052
0.051
0.050
0.040
0.046
0.099
0.064
0.050
0.063
0.065
0.056
0.048
0.052
0.051
0.048
0.046
0.035
0.024
0.019
0.019
0.011
0.012
0.018
0.022
0.013
10
20
30
40
50
100
200
500
1000
The data from the same gene followed unstructured correlation. For each
simulation, there are two time points.
Zhang et al. BMC Bioinformatics 2011, 12:273
http://www.biomedcentral.com/1471-2105/12/273
Page 5 of 12