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Theoretical and experimental comparisons of gene expression indexes for oligonucleotide arrays

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Motivation: Oligonucleotide expression arrays exhibit systematic and reproducible variation produced by the multiple distinct probes used to represent a gene. Recently, a gene expression index has been proposed that explicitly models probe effects, and provides improved fits of hybridization intensity for arrays containing perfect match (PM) and mismatch (MM) probe pairs. Results: Here we use a combination of analytical arguments and empirical data to show directly that the estimates provided by model-based expression indexes are superior to those provided by commercial software. The improvement is greatest for genes in which probe effects vary substantially, and modeling the PM and MM intensities separately is superior to using the PM-MM differences. To empirically compare expression indexes, we designed a mixing experiment involving three groups of human fibroblast cells (serum starved, serum stimulated, and a 50:50 mixture of starved/stimulated), with six replicate HuGeneFL arrays in each group. Careful spiking of control genes provides evidence that 88-98% of the genes on the array are detectably transcribed, and that the model-based estimates can accurately detect the presence versus absence of a gene. The use of extensive replication from single RNA sources enables exploration of the technical variability of the array.
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BIOINFORMATICS
Vol. 18 no. 11 2002
Pages 1470–1476
Theoretical and experimental comparisons of
gene expression indexes for oligonucleotide
arrays
William J. Lemon, Jeffrey J.T. Palatini, Ralf Krahe and
Fred A. Wright
Division of Human Cancer Genetics, The Ohio State University, Columbus, Ohio, USA
Received on September 18, 2001; revised on April 22, 2002; accepted on April 30, 2002
ABSTRACT
Motivation: Oligonucleotide expression arrays exhibit sys-
tematic and reproducible variation produced by the mul-
tiple distinct probes used to represent a gene. Recently,
a gene expression index has been proposed that explic-
itly models probe effects, and provides improved fits of
hybridization intensity for arrays containing perfect match
(PM) and mismatch (MM) probe pairs.
Results: Here we use a combination of analytical argu-
ments and empirical data to show directly that the esti-
mates provided by model-based expression indexes are
superior to those provided by commercial software. The
improvement is greatest for genes in which probe effects
vary substantially, and modeling the PM and MM inten-
sities separately is superior to using the PM–MM differ-
ences. To empirically compare expression indexes, we de-
signed a mixing experiment involving three groups of hu-
man fibroblast cells (serum starved, serum stimulated, and
a 50:50 mixture of starved/stimulated), with six replicate
HuGeneFL arrays in each group. Careful spiking of control
genes provides evidence that 88–98% of the genes on the
array are detectably transcribed, and that the model-based
estimates can accurately detect the presence versus ab-
sence of a gene. The use of extensive replication from
single RNA sources enables exploration of the technical
variability of the array.
Availability: Scripts for computing the Li–Wong reduced
and full models are available in C, Splus and Perl in the
supplementary information.
Contact: fwright@bios.unc.edu
Supplementary information: http://thinker.med.ohio-state.
edu
INTRODUCTION
Oligonucleotide DNA arrays are a powerful means to
monitor expression of thousands of genes simultaneously
To whom correspondence should be addressed at Present address:
Department of Biostatistics UNC-CH, 3107B McGavran-Greenberg Hall,
CB #7420, Chapel Hill, NC 27599-7420, USA
Lipshutz et al. (1999). However, important challenges
remain in estimating expression level from raw hybridiza-
tion intensities on the array. Li and Wong (Li and Wong,
2001a,b) recently introduced a statistical model which
better fits observed patterns of hybridization than the mod-
els implicitly employed by standard commercial software
(Affymetrix, 1999). The Li–Wong model was derived
for Affymetrix GeneChip arrays, but the model-based
approach is likely to be useful in other photolithography
or ink-jet arrays (Hughes et al., 2001) in which genes are
represented by multiple oligonucleotide probes. Individual
probe effects are large and systematic, and by explicitly
fitting such effects the Li–Wong model presumably pro-
vides an improved index of gene expression (Li and Wong,
2001b). However, the extent of improvement has not been
shown directly or explored in the context of extensive
replication.
The term ‘expression index’ here describes a statistic
intended to reflect expression level for a particular gene,
whether or not it is based on an explicit model. We have
evaluated the theoretical relative efficiency of competing
gene expression indexes, and consider a framework for
empirical index comparison. To provide this comparison,
we conducted a carefully designed mixing experiment
involving the response of human fibroblasts to serum.
The experiment provides insight into technical variability
and the number of expressed genes. Supplemental plots,
primary data, perl and S-plus scripts and C programs for
decoding GeneChip files are available at our web site
http://thinker.med.ohio-state.edu. A longer version of the
manuscript is also online and includes additional figures
and detailed analyses of differentially expressed genes.
The current generation of photolithographic arrays (Mc-
Gall and Fidanza, 2001) have 250 000–500 000 probes ar-
ranged in pairs—a perfect match (PM) probe that is com-
plementary to a 25-base pair segment of mRNA and a
mismatch (MM) probe that is complementary to the same
mRNA segment except for the 13
th
nucleotide. A collec-
tion of 16–20 probe pairs, called a probe set, is used to
1470
c
Oxford University Press 2002
Comparing gene expression indexes
represent a gene.
Following Li and Wong (2001a), I denotes the number
of samples in an experiment, and J denotes the number of
probe pairs in a probe set (e.g. J = 20 for the Affymetrix
HuGeneFL array described in this study). The PM and
MM intensities for the ith sample and j th probe pair of
a given gene are modeled as
PM
ij
= ν
j
+ α
j
θ
i
+ φ
j
θ
i
+ e
MM
ij
= ν
j
+ α
j
θ
i
+ e, (1)
where θ
i
is the expression index and ν
j
is a non-specic
cross-hybridization term for the jth probe pair. The term
α
j
is the rate of increase of MM intensity with expression,
and φ
j
is the additional rate of increase in the PM intensity.
The errors e are assumed independent with variance ξ
2
.
We consider the θ
i
s as the parameters of interest, and
estimates may be obtained via least-squares, the maximum
likelihood solution if the errors are assumed to be normally
distributed.
We refer to Model (1) as the LiWong full model
(LWF) because the PM and MM values are treated
separately. However, most of the analyses in Li and
Wong (2001a) are based on the LiWong reducedmodel
(LWR) using only the differences y
ij
= PM
ij
MM
ij
=
φ
j
θ
i
+ ε. The reduced model follows directly from
Model (1), with var(ε) = σ
2
= 2ξ
2
. Fitting either
LWF or LWR requires an identiability constraint (Li and
Wong, 2001a), and we use
j
φ
2
j
= J .
The reduced model has the advantage of fewer param-
eters than the full model, but potentially ignores infor-
mation contained in the bivariate PM, MM data. This is
especially true for probe pairs in which PM and MM are
similarly sensitive to expression changes (i.e. α
j
is large
compared to φ
j
). We have veried the large and system-
atic probe effects in several datasets and found numerous
genes in which the reduced model is likely to perform
poorly compared to the full model. Nonetheless, the re-
duced model offers considerable improvement over the
most popular current indexes.
Average difference versus Li–Wong reduced
The MAS4 software (Affymetrix Microarray Analysis
Suite version 4.0, Affymetrix, Santa Clara, Calif) com-
putes the average difference(AD), the simple average of
the PMMM differences across a probe set. Assuming
Model (1) holds, we can contrast AD with LWR (Li
and Wong, 2001a), implemented in our software and in
the program DCHIP (www.dchip.org). For the limiting
situation with large sample sizes, the φs will be estimated
with great precision. If we assume the φs and σ
2
to
be known, then for a single sample (the subscript i is
suppressed) the maximum likelihood estimate is unbiased
and can be shown to be
ˆ
θ
reduced
=
j
y
j
φ
j
/J (Li and
Wong, 2001a). In contrast, AD (denoted η) is computed
as ˆη =
j
y
j
/J. AD is not constructed as an explicit
estimate of θ , and thus has considerable bias unless all
φ
j
= 1. Thus, AD can be made comparable to
ˆ
θ
reduced
by applying a correction factor to create a new index
ˆ
ˆη =
J/
j
φ
j
ˆη. It is easy to show that E(
ˆ
θ) = E (
ˆ
ˆη) = θ.
The variances of the estimates can be computed directly
as var(
ˆ
θ
reduced
) = σ
2
/J and
var(
ˆ
ˆη) =
J
2
φ
j
2
· var( ˆη) =
σ
2
J
φ
j
2
.
It can be shown that the variance of the φs across the
J probe pairs is var) = E
2
) E
2
) = 1
φ
j
2
/J
2
,so
φ
j
2
= J
2
(
1 var)
)
. Finally we
obtain the relative efciency
RE(reduced, AD) =
var(
ˆ
ˆη)
var(
ˆ
θ
reduced
)
=
J
2
φ
j
2
=
J
2
J
2
(
1 var)
)
=
1
1 var)
,
showing that LWR is superior to AD. Although the φs
are considered xed, var) is the variance of these
sensitivities across the J probe pairs. The efciency result
is quite sensible, because the LiWong reduced model
accounts for the variability in probe pair effects, while the
average difference does not.
Li–Wong full model versus reduced model
For the LiWong full model, solving the likelihood equa-
tion and computing the variance gives (see Supplementary
Appendix)
var(
ˆ
θ
full
) =
σ
2
2
j
α
2
j
+
j
j
+ φ
j
)
2
and the relative efciency compared to LWR:
RE(full,reduced) =
var(
ˆ
θ
reduced
)
var(
ˆ
θ
full
)
= 2
j
α
2
j
+
j
j
+ φ
j
)
2
J
.
It can be shown that RE(full, reduced)
2, with equality
holding when all of the αs are zero. This corresponds to
the situation in which the mismatch probe pairs are not
at all sensitive to gene expression, and so they simply add
noise in computing the difference PMMM (thus doubling
the variability). In the case where each α
j
= φ
j
, the
relative efciency can be computed to be 10.0 for any
1471
W.J.Lemon et al.
J . In practice, the result may be even more extreme, as
the αs are often greater than the φs (see estimates in
supplemental data). We believe the possible gains from
using the full model are not generally appreciated.
The Log-average
Another expression index is the Log-Average (LA), com-
puted by MAS4 as 10
j
log(PM
j
/MM
j
)
/J. This in-
dex may perform poorly, as the LiWong model indicates
that hybridization intensity increases proportionally to ex-
pression for both PM and MM. Thus, for a gene in which
the nonspecic hybridization terms ν are small and the er-
ror variance is small (an otherwise favorable situation), we
have (PM
j
/MM
j
) = α
j
+θφ
j
)/(θα
j
), which does not
depend on θ . This result and the empirical results below
suggest that LA is of limited value.
In addition to providing expression indexes, MAS4
attempts to make an absolute determination of the pres-
ence/absence of each gene (Affymetrix, 1999). In our
experience, typically fewer than 50% of the genes in an
array are called present by MAS4. We present evidence
below that in fact the vast majority of genes on the array
are expressed at detectable levels. MAS4 identies nu-
merous probe pairs in advance as potentially unreliable,
and trims additional probes having outlying intensities
before calculating AD and LA. Outlier detection is also a
major feature in Li and Wong (2001a). Under the carefully
controlled conditions of the present study, consideration
of outliers has a minor effect and such trimming was not
performed in our calculations.
MATERIALS AND METHODS
As shown in Figure 1, human broblast cells (GM 08330;
Coriell Cell Repositories) were grown according to the
distributors recommendations in media supplemented
with 20% FBS for 5 passages (27 asks). Cultures were
placed in serum-reduced media (0.1% FBS) for 48 h.
After 48 h, 9 asks were returned to 20% serum condition
(Stimulated) and cells from the other asks (Starved) were
placed in RNA-Stat60 (Qiagen, Valencia, CA) according
to manufacturers instructions. More RNA is produced
per cell in the stimulated condition, so fewer asks of
stimulated cells were needed to produce the same amount
of RNA as those of starved cells. Twenty-four hours later,
stimulated cells were harvested and placed in RNA-Stat60.
Total cellular RNA was extracted using phenol:chloroform
and was puried using RNeasy (Qiagen) according to
manufacturers specication. Extraction produced one
Stimulated sample and one Starved sample. A third RNA
sample (50:50) was produced from an equimolar mixture
of these two.
On each of three days, two aliquots of RNA were taken
from each group and processed separately as previously
described (Virtaneva et al., 2001; detailed protocols at
Human Fibroblasts
(GM 08330)
20% FBS
48h
24h
Harvest total RNA
Lys, PheDap, Thr
50:50
Add Bacterial
Control Genes
StimulatedStarved
5 passages
Dap, Thr,
Lys, Phe
Produce 50:50 group
Produce duplicates
each day for 3d
Synthesize cDNA,
cRNA; fragment
Add Hybridization
Control Genes
BioB, BioC, BioD, Cre
Hybridize
HuGeneFL
0.1% FBS
Serum starvation
Cell culture
Serum stimulation
0.1%
20%
Harvest total RNA
Gene Expression Indexes
Data Reduction
RNA extraction
20% FBS
Fig. 1. Design of the replication experiment. RNA was derived
from serum-starved and serum-stimulated broblasts in a single
extraction, and a 50:50 mixture created. Six replicate arrays were
hybridized in each group.
http://www.cancergenetics.med.ohio-state.edu/microarray/
uArrayProtocols.html). Modications are as follows. Stim-
ulated RNA samples received bacterial control genes Lys
and Phe RNAs at 0.08 ng/8 µ g total RNA, the Starved
samples received the same amount of Phe and Thr and
the 50:50 samples all four at 0.04 ng/8 µ g. Hybridiza-
tion controls BioB, BioC, BioD, Cre were added to nal
concentrations of 1.5, 5, 25 and 100 pM, respectively.
Each HuGeneFL array was loaded with 11 µg/200 µL
labelled cRNA. This produced six replicates for each sam-
ple (18 arrays) in a day-balanced procedure. To minimize
external variability, all arrays were from the same lot. To
test for a day effect, analysis of variance against median
summaries for entire arrays were performed, as well as
individual analyses for each gene. P-value plots revealed
no substantial day effect. Perl scripts were used to decode
the probe information contained in Affymetrix GeneChip
CEL les. Model-tting and statistical analyses were
performed using Splus v. 5.0, 6.0 and 2000 (Insightful,
Seattle).
Empirical comparison of expression indexes
The relative efciency results help clarify the advantages
of model-based approaches, although the results are
dependent on the applicability of the LiWong model.
1472
Comparing gene expression indexes
In practice, such dramatic efciency improvements are
not achieved, in part because the difculty in scaling
arrays across the range of intensity values adds additional
variation to all expression indexes. We now describe
an approach for comparing expression indexes based on
empirical data.
ˆ
θ
full
and
ˆ
θ
reduced
are derived from the same model, so
it may be appropriate to compare their variances directly,
e.g. across replicate arrays under a xed experimental con-
dition. Direct variance comparisons for other expression
indexes may not be meaningful. Instead, we propose to
judge disparate indexes on the basis of their correlation
with the underlying true expression. Although the true ex-
pression is generally not known, we propose a mixing ex-
periment allowing the explicit estimation of this correla-
tion. The use of correlation coefcients has intuitive ap-
peal, but can also be placed in the framework of relative
efciencies as outlined below.
Suppose the true underlying gene expression for a given
gene is τ. Consider two indexes of gene expression,
ˆ
θ and
ˆη, where
ˆ
θ = β
0
+ β
1
τ + e
θ
, e
θ
N (0
2
θ
)
ˆη = δ
0
+ δ
1
τ + e
η
, e
η
N (0
2
η
) (2)
In other words, we assume that each expression index
has a linear relationship with the true gene expression
and an uncorrelated error term. In practice the coefcients
will not be known, but note that rewriting the estimate as
ˆ
ˆ
θ = (
ˆ
θ β
0
)/β
1
gives an unbiased estimate of τ , and
var(
ˆ
ˆ
θ) = σ
2
θ
2
1
. Similarly, the variance of the unbiased
estimate
ˆ
ˆη = ( ˆη δ
0
)/δ
1
is σ
2
η
2
1
, for an overall relative
efciency of
ˆ
θ compared to ˆη:
RE(
ˆ
θ, ˆη) =
var(
ˆ
ˆη)
var(
ˆ
ˆ
θ)
=
σ
2
η
2
1
σ
2
θ
2
1
.
We also note that the overall variance of
ˆ
θ is var(
ˆ
θ) =
β
2
1
var ) + σ
2
θ
, so that the ratio of explained to residual
variance in the model is ER
θ
= β
2
1
var )/σ
2
θ
. This
ratio is r
2
/(1 r
2
), where r is the Pearson correlation
coefcient in the regression of
ˆ
θ on τ . Similarly, ER
η
=
δ
2
1
var )/σ
2
η
, and we have ER
θ
/ER
η
= RE(
ˆ
θ, ˆη), with
cancellation of the var ) term. Thus ER
θ
and ER
η
reect the respective efciencies of the indexes, quantities
that can be directly estimated from the regression of the
expression indexes on τ . The quantities can be equiv-
alently (and importantly) estimated using any predictor
variable that is a linear transformation of τ . We show
below how ER
θ
can be estimated from empirical data
using the following experimental design.
RESULTS
A mixing experiment with replication
Consider a mixing experiment involving two different
RNA sources (samples), A and C. After a single extraction
of mRNA from each sample, a third sample (B) is formed
as a 50:50 mixture of A and C, so that the true expression
for each gene in sample B is the average of that in
samples A and C. Replicate arrays are then prepared
for each of the three samples. As described above, we
can perform a regression of any expression index on a
suitable predictor variable (e.g. x = 1, 2, 3 for groups
A, B and C, respectively), with the assurance that x is a
linear transformation of the unknown true expression τ .
Within this framework, the magnitude of the correlation
coefcient for each gene is equal to that of Model (2), i.e.
r
2
ˆ
θ,τ
= r
2
ˆ
θ,x
.
Bioscaling
To make arrays comparable, most studies have used
forms of linear or nonlinear global scaling (Li and Wong,
2001b). However, such global scaling is inappropriate
if a large portion of the genes are substantially up- or
down-regulated in a set of arrays, as such scaling will
largely eliminate this important feature of the data. We
observed that median hybridization intensities were 30%
lower in Starved than Stimulated (supplemental data,
p < 0.002, KruskalWallis test). As mRNA is thought to
represent only 28% of total RNA, this is clear evidence
of reduced transcription in the Starved group. To scale
arrays while preserving evidence for large scale changes
in transcription, we used the Bio hybridization control
genes added to all samples at constant concentrations
(Materials and Methods). Probe intensities were divided
by the average intensity of these probe sets, a procedure
we term bioscaling. This moderately reduced the median
within-group coefcients of variation for all genes 6%
for LWF and 11.8% for LWR. Except where otherwise
indicated, our analyses were performed using the LWF
and LWR indexes after bioscaling, and AD and LA after
the (essentially linear) scaling using MAS4 (Affymetrix,
1999).
Efficiency and clustering results
Pairwise correlations among the arrays are shown as
supplemental gures. Median within-group coefcients
of variation (std dev/|mean|, LWF 16.2%, LWR 14.4%,
AD 26.3%, LA 27.9%) are consistent with the apparent
lower heterogeneity of model-based estimates. Figure 2
shows that model-based estimates also improve the
precision in estimating expression. With or without
scaling, the efciency r
2
/(1 r
2
) follows LWF >
LWR > AD > LA. The reduction in efciency after
bioscaling reects overcorrection in the Stimulated
1473
W.J.Lemon et al.
Unscaled Scaled
LWF
LWR
AD
LA
LWF
LWR
AD
LA
Median(r
2
/(1–r
2
))
0.0 0.5 1.0 1.5
Fig. 2. Median efciency of 7129 gene expression indexes, accord-
ing to criterion derived in text. Unscaled indexes were computed
from CEL les by the authors without probe trimming. Scaling of
AD and LA was performed by MAS4 using default settings. Scaling
of LiWong models was performed using bioscaling as described in
text. Both types of scaling reduce between-group variation (and thus
efciency) because of the global difference in mRNA expression in
Stimulated versus Starved.
group. The globally increased expression in Stimulated
produces increased cross-hybridization, producing higher
expression estimates for the Bio control genes. We note
that inferences based on the unscaled data are valid, and
view the efciency comparisons based on the scaled data
to be conservative.
Hierarchical clustering of samples was performed for
each model (supplemental gures) using the 2104 genes
called Present by MAS4 in at least 15 of the samples.
The full model classies the samples into three distinct
groups corresponding to the three mRNA groups, while
the reduced model and MAS4 estimates each produce a
classication error.
Duplicated genes
A total of 149 genes are represented twice (or more) on
the HuGeneFL array, although not necessarily with the
same set of probes. The median correlation of each pair of
duplicated genes across the 18 samples was substantially
higher for the model-based estimates (LWF median r =
0.74, LWR median r = 0.43, AD median r = 0.12, LA
median r = 0.09).
Present versus absent calls
The simple z-statistic
ˆ
θ/SE(
ˆ
θ) can be used to test for
whether a gene is present (i.e. expressed). The z-
statistic is related to the Wald statistic (Cox and Hinkley,
1974), except that in our implementation SE(
ˆ
θ) is the
conditional standard error from the likelihood equation,
i.e. assuming the nuisance parameters known (Li and
Wong, 2001a). Such an approach tends to underestimate
False Positive Rate
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
(1 Specificity)
(Sensitivity)
1 False Negative Rate
Abs Call
LA
AD
LWR
LWF
Fig. 3. ROC curves, using varying thresholds of expression indexes
in declaring spiked bacterial control genes present/absent. Lines rep-
resent LWF, LWR, MAS4 AD and MAS4 LA. Triangle represents
the overall assessment of the MAS4 absolute call.
the standard error, so that fairly large z-statistic values
may be necessary to call a gene present. This is especially
true for LWF, which contains many more parameters than
LWR. In our broblast experiment, the spiked bacterial
control genes (Lys, Phe, Dap, Thr) serve as samples of
genes known to be expressed and unexpressed (a total of
12 probe sets and 18 samples, further discussion below).
Using the criterion
ˆ
θ/SE(
ˆ
θ) > 5.0, we obtain an error
rate for LWF of 3.2% (144/144 probe sets properly called
present, 65/72 properly called absent). All but one of the
misclassied observations were due to a single probe set.
Using the same criterion, the error rate for LWR was 9.3%.
In contrast, the error rate of the MAS4 present/absent calls
was a much larger 79.7% (8/144 properly called present,
36/72 properly called absent).
In a more comprehensive comparison, we present
receiveroperator characteristic (ROC) curves in Figure 3
for detecting the control genes as present/absent. The
z-statistics based on LWF and LWR are able to detect
the genes with high sensitivity and specicity. Moreover,
using the θ estimates directly gives very similar results,
with the ROC curves almost coinciding with the respective
curves for the z-statistics. For comparison we also plot the
curves for AD and LA. The AD and LA values calculated
by MAS4 are very poorthe LA curve is worse than
chance variation (the 45
line) for much of the curve.
The curves based on our own calculations of AD and LA
perform somewhat better than those provided by MAS4.
We have determined that this effect is not due to the
scaling approaches used, and we speculate that MAS4
masking and probe trimming may in certain cases produce
inferior estimates. The present/absent absolute calls from
MAS4 appear as a single point.
1474
Comparing gene expression indexes
Few unexpressed genes
Our implementation of the LiWong models does not
constrain the parameter estimates to be positive. For LWF
and LWR, only about 0.2% of the probe sets in the entire
dataset have a negative expression index. This is lower
than in other studies, but in our experience about 15%
are negative. Intriguingly, this percentage is only slightly
higher for the Starved group than for Stimulated, and is
lowest for the 50 : 50 group (consistent with a smaller
set of genes unexpressed in both Starved and Stimulated).
The overall percentage of negative values for AD and LA
was near 20%. Moreover, the number of genes we called
present is very highapproximately 7000 for each array.
The results suggest that the model-based estimates may
be extremely sensitive to genes expressed at low levels,
and that very few genes on the array are unexpressed
or undetectable by model-based estimates. We consider
the spiked out genes as samples of unexpressed genes
under these conditions. Because each of these control
genes is represented by three non-overlapping probe sets,
these provide 6 independent unexpressed genes in each of
the Starved and Stimulated conditions. From symmetry
considerations, one would expect a gene to have
ˆ
θ<0
with 50% probability in a sample in which the gene is
truly unexpressed. The results are roughly in line with
expectationsLWF produced negative indexes for 42% of
the instances in which the control genes were spiked out,
while LWR produced 35% negative.
As an additional, conservative approach to estimating
the number of expressed genes, we treat the unexpressed
genes as a contaminating population of unknown size.
Let U denote the unknown number of unexpressed genes.
If all genes are ranked from lowest to highest apparent
expression using an expression index, then
U
2 × (median rank of U genes among all genes),
with equality if the populations are disjoint. From a
random sample of unexpressed genes, we can calculate
an upper condence bound for the median rank, which in
turn generates a conservative upper bound for U . Other
possible statistics, such as the maximum rank, are overly
sensitive to sporadic outliers. For six observations from
a continuous distribution, the fth order statistic forms
an approximate 90% distribution-free upper condence
bound for the population median (Hollander and Wolfe,
1999).
The sample of unexpressed genes can be examined
among all genes in Figure 4, with the variance versus the
mean for LWF among Stimulated samples shown on the
rank scale. The variance clearly is lower for genes with
lower expression, and note that this relationship holds even
for the genes with the lowest expression. The relationship
between mean and variance for AD is curvedgenes
Thr
Dap
Phe
Lys
0 2000 4000 6000
rank(mean)
rank(var)
MAS4 AD
LWF
0 2000 4000 6000
0 2000 4000 6000
Fig. 4. Rank of expression index variance across the 6 Stimulated
arrays versus rank of index mean. Left: Results from LWF with data
points for spiked in/out control genes highlighted. The low ranks
of Dap and Thr indicate that few genes are unexpressed. Right:
MAS4 AD, with the same genes highlighted. Dap and Thr were
truly absent, Lys and Phe truly present.
with rank lower than 1650 have negative AD values
when averaged over the replicates. These plots suggest
that accounting for probe effects can improve detection
limits and precision for lowly expressed genes.
The estimates based on our sample of unexpressed genes
are particularly revealing. Five of the six Dap and Thr
genes have very low rank (50 or less) among the 7129
probe sets, while Lys and Phe show high ranks. The
condence bound procedure leads to a 90% upper bound
for U as 100. Out of 7129 probe sets (and 6800 genes),
this provocatively implies that over 98% of the genes on
the array are expressed in Stimulated (consistent with our
z-statistic criterion for present/absent calls). The relatively
high ranks of spiked-out genes and low ranks of spiked-in
genes using AD suggests a high background noise level,
and AD may fail to detect many expressed genes. Similar
results hold for the Starved group, with the rank-based
procedure indicating that over 88% of the genes on the
array are expressed.
Technical variability
We describe variation among our replicates as technical
variation, with essentially standard protocols followed
after extraction into three pools of RNA. Within the
Stimulated replicates, the median coefcient of variation
(standard deviation/|mean|; CV) for individual probe
intensities was 12%, comparable to that of carefully
constructed cDNA arrays (Yue et al., 2001). Unscaled
Stimulated expression indexes yielded median CVs as
follows: LWF 14%, LWR 14.9%, AD 26.3%, LA 27.9%.
The Starved replicates showed a similar pattern, but with
higher CVs due to the reduced mean in the denominator.
We have used the approximately linear relationship
1475
W.J.Lemon et al.
between log(variance) versus log(mean) of the replicates
to describe sample sizes necessary to two-fold changes
in expression values, with conservative bounds designed
to account for biological heterogeneity (further details in
supplementary data).
DISCUSSION AND CONCLUDING REMARKS
We have developed a theoretical and experimental frame-
work for evaluating indexes of gene expression in high-
density oligonucleotide arrays. We have demonstrated the
improvements provided by model-based estimates over
simple averaging methods and proposed an improved and
simple approach to present/absent gene calls. The model-
based estimates and results of our spiking experiment
suggest that the vast majority of genes on the array
are expressedthus the present/absent calls may not
be a meaningful distinction for many genes. It will be
important to investigate this phenomenon in a wider
variety of tissues to better understand the relationship
between quantity and activity of a gene. To our knowledge,
our experiment involves more extensive replication than
other studies reported thus far, and we hope that our data
will serve as a useful resource for further investigations
by statisticians and geneticists.
Several challenges remain in the evaluation of gene ex-
pression. We have noted that probe error variation appears
somewhat dependent on hybridization intensity, suggest-
ing that an expression-dependent error structure may offer
further improvements. The LiWong approach involves
tting the probe strengths as xed effects, and is techni-
cally possible using only a few arrays (e.g. 2IJ data points
for LWF, with only 3J + I + 1 parameters). However, the
articial constraints on the φs do not enable estimates of
absolute expression, or even relative comparisons of dif-
ferent genes on the same array. A mixed-model approach
incorporating random probe effects would enable the com-
parison of different genes. Our bioscaling procedure is a
step towards scaling using absolute concentrations of con-
trol genes, and should be developed further using genes
across a wide variety of spiked concentrations. Additional
study of the sequence-dependence of probe sensitivity is
also warranted. To further all of these efforts, it will be
important that genetic researchers distribute primary probe
intensity data (CEL les) with their published studies, so
that further improvements may be explored.
ACKNOWLEDGEMENTS
We thank Timothy Wise, Gustavo Leone, Daolong Wong,
Karl Kornacker, Wing H. Wong, and Cheng Li. Supported
in part by NIH GM58934 and the Solove Research
Institute.
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Gene Chip Analysis Suite User Guide
Affymetrix (1999) Gene Chip Analysis Suite User Guide. Affymetrix, Santa Clara, CA.