Background
Genomics sequencing projects have rapidly generated
tremendous amount of information. At the time of writing,
the NCBI UniGene database [ 1 ]
http://www.ncbi.nlm.nih.gov/UniGenecontained 96,109 Homo
sapiens clusters and 85,047 Mus musculus clusters.
Predictions from the Human Genome Project [ 2 ] and Celera
Genomics [ 3 ] suggest there are about 26,000-40,000 human
genes. Other recent studies suggest that these numbers may
be an underestimation and that the human genome appears
more complicated [ 4 ] . Understanding the functions of
such a large number of genes has been an unprecedented
challenge for functional genomics research. As the array of
hope in recent years, gene expression array technology has
quickly grown into a powerful tool to chart a gene atlas in
various biological sources and under various conditions in
a massively parallel manner [ 5 6 7 ] . Facing the
challenge of annotating such a huge amount of genomic data,
increasing array information density and improving analysis
algorithms have become two critical research areas to
ensure that gene expression profiling proceeds in an
efficient and cost effective manner.
Take an Affymetrix high-density oligonucleotide GeneChip
http://www.affymetrix.comfor example. Firstly, its human
U95 series chip consists of 5 chip types with 12,000 coding
clusters each, which makes it expensive to profile all the
human genes in samples of interest. Can a gene chip take
more genes? Comparing its U95 chip and Human 6800 chip,
Affymetrix has already increased chip information density
by 20% by reducing the number of probe pairs per gene from
20 to 16. Since demand for higher information density has
still not been met, it is of interest to study the probe
number effect in detail. Secondly, most optional research
efforts focus on the downstream statistical and clustering
analysis. However, on the upstream side, Affymetrix chip
users are still dependent on the Microarray Suite ®software
that comes with the measurement system to interpret raw
data. The Affymetrix algorithm implemented in its
Microarray Suite 4.0 package (referred hereafter as MAS4)
uses empirical rules derived from its internal research
data to assign absolute calls for the significance of gene
presence and assign fold change calls for the significance
of expression variations. Such discrete categorizations are
not the most appropriate language to describe quantities of
continuous nature. Although it is well known that fold
change numbers have defined behaviors of uncertainty, there
are very few studies in this area. How does one assign
statistical significance to expression analysis results?
This work presents our preliminary research results for the
two questions raised above.
The Affymetrix gene chip layout used in this study
contains the same number of perfect match (PM) probes and
mismatch (MM) probes. MAS4 uses differences between these
two types of probes for gene expression signals. The
primary goal of Match-Only Integral Distribution (MOID)
algorithm is to discard mismatch information, which allows
immediate doubling of the chip information density. In this
study, the performance of both algorithms were benchmarked
using 366 known fold change values derived from 34 spiking
experiments. Their false positive tendencies were assessed
by no-change expression experiments. Computer simulations
were used to study their noise tolerances, and to determine
the minimum number of probes required for chip
analysis.
The idea of using PM-only information is based on the
following observations: MAS4 essentially discards the
one-one correspondence between a PM and its MM partner (for
details, see materials and methods on MAS4 algorithm for
absolute analysis) and still gives satisfactory
interpretation, which suggests the contribution of MM
probes might be approximated in a nonspecific manner
overall. After we designed the first mismatch-free gene
chip (GNF-HS1) in July 1999, the match-only expression
analysis idea was proposed in other independent studies as
well. Li (submitted, 2001) adjusted their previous
model-based analysis of oligonucleotide arrays [ 8 ] to
PM-only calculations and found their results correlate well
with that of using PM and MM information. In addition, the
idea is endorsed by recent studies of Naef
et al [ 9 ] and Irizarry
et al (2002, in prepare)
http://biosun01.biostat.jhsph.edu/~ririzarr/papers/.
One difficulty in comparing algorithms for gene
expression analysis is the lack of "known" results. Here we
overcome the problem by resorting to a spiking set and set
of no-change experiments, where results are unambiguous.
Model-based methods [ 8 ] generally require a reasonable
numbers of training experiments of the same chip type,
among which probes under study give significantly large
signals in at least some experiments. Considering the fact
that the 34 spiking experiments used were obtained by three
different chip types, and in addition some experiments are
replicates under different concentrations and hybridization
conditions, it is impractical to sufficiently train
model-based algorithms in this study. We limit our study to
MOID and MAS4, where training is not required.
In the materials and methods section, we summarize
Affymetrix chip technologies and describe the MOID
algorithms in detail. MAS4 algorithms for both absolute
expression analysis and comparison analysis are included
afterwards. The benchmarks used for algorithm comparison
were explained and comparison results were shown in the
results session. Issues such as noise-tolerance of both
algorithms and further reduction of the probe set size are
also included. We discuss generalization of the
normalization algorithm, which may be of interest to other
researchers. Finally, the main differences between MAS4 and
MOID are summarized in a tabular form as conclusions.
Results and Discussion
Spiking experiments
Spiking experiments were done by adding to tissue
samples a certain number of control genes with known
concentrations. Since signal intensity is not exactly
proportional to gene concentration across different probe
sets, only the fold change values of each gene between
comparisons are considered reliable and used to benchmark
both algorithms. We are fortunate to have access to 34
spiking experiments done previously in GNF for a
hybridization protocol study, where 366 independent known
fold change numbers were derived. These experiments
covered three Affymetrix chip types: Hu35kSubA, Hu6800,
and Mu11kSubA. True experimental fold change values,
f
exp , range from 1.0 to 10.0. The raw
data (34 CEL files) and experimental fold change values
are available from
http://carrier.gnf.org/publications/MOID.
We applied both algorithms to these array data and the
fold change numbers calculated were compared to known
experimental values. Because of the way we defined the
base and experiment,
f
exp are always no less than 1.0. For
MAS4, we used all the default parameters used by the
Microarray Suite 4.0 program in the calculation. For
MOID, we found using 70 percentile for
both pct and
pct
f
(defined in materials and methods) gave optimal
performance. Figure 1shows the comparison results (data
available from http://carrier.gnf.org/publications/MOID).
Two algorithms give virtually similar performance.
In the comparison, we defined relative error,
R
err , as the following to benchmark
each algorithm,
R
err1 = <|(
f
calc -
f
exp> )/
f
exp |>
all pairs ,
R
err2 = <|(1/
f
calc - 1)/
f
exp ) ·
f
exp |>
all pairs ,
and
R
err = <|(
f
exp /
f
calc -
f
calc /
f
exp )|>
all pairs .
Where
f
exp and
f
calc are fold change values from
experiment and calculation, respectively.
R
err1 and
R
err2 are defined symmetrically for
relative errors of
f
exp and 1/
f
exp , reflecting the fact that base
and experiment can be defined arbitrarily, and the final
R
err is defined as the average of the
two. Each relative error number is an average over all
the 366 data points.
It seems the two algorithms have very close
performance Table 1. They all have greater error margins
for calculating down-regulated fold changes (captured by
R
err2 ) than up-regulated ones
(captured by
R
err1 ). MAS4 is more asymmetric than
MOID in this aspect.
No-Change experiments
Another set of experiments used for benchmarking
algorithms is the comparison between two experiments
where mRNA prepared from the same tissue sample was
hybridized twice under slightly different conditions.
Those fold change numbers that deviate largely from 1.0
are considered to be false positives. When we applied the
algorithms to ten experiments done with either human
brain or human lung replicate samples (same mRNA,
slightly different hybridization conditions), the results
were all similar. Figure 2shows one of the typical
comparison results. When only "Present" genes (defined by
MAS4) were taken into account, 80% of fold change numbers
were spread across 0.54 for MAS4 and only 0.30 for MOID.
Clearly MOID assigns much less false positive fold
changes than MAS4 does in this comparison. If all the
genes were counted, the spreads of MAS4 and MOID results
would be 0.89 and 0.29 for the same data sets. This test
suggests MOID is more robust than MAS4.
Reduced Probe Set Simulation
In the studies above, we demonstrated the feasibility
of a match-only gene chip design based on the MOID
algorithm. In order to further increase chip information
density, it is of common interest to understand how many
probes are sufficient for expression analysis and push
for the lower limit of the size of a probe set. This is
done via computer simulations.
In the simulation, for each probe set, a subset of
n
r
probes were randomly chosen to be used in the
calculation. Both the spiking and no-change calculations
presented above were repeated with the selected subset,
as we gradually reduced
n
r
. Figure 3and figure 4show the results for the
spiking calculation and no-change calculation,
respectively. As the graph suggested, accuracy of MOID is
essentially unaffected while reducing the number of
probes down to ten. This result enables us to almost
triple the amount of clusters one can put on a gene chip
using MOID design. Combined with other new design ideas,
MOID lays the foundation for the first universal human
chip that contains 75,000 UniGene clusters (release 116).
The results have recently been validated and led to many
interesting discoveries, which provides indirect support
for MOID (to be submitted).
Noise Tolerance of both Algorithms
Computer simulations also enable us to study the
behaviors of both algorithms under noise disturbance. In
simulations, a subset of
n
n
probes were randomly chosen, their PM values were
multiplied by ten to mimic the effect of serious noise
effect. We repeated the calculations for both spiking
experiments and no-change experiments, while gradually
increasing
n
n
. Figure 5and figure 6show the results for spiking
calculation and no-change calculation, respectively. As
the graph suggested, both MAS4 and MOID can resist noise
perturbation up to 2 cells out of 16, the test favors
MOID slightly.
Probe-Specific Effect
MOID is based on the assumption that the background
for probes is mainly non-probe-specific. The results
indicate that this assumption is sound overall. However,
it is clear that by better understanding probe-specific
behaviors, the analysis accuracy can be further improved.
For instance, probe response factors might be derived by
accumulating and studying many experiments. If there are
a sufficient number of experiments, where the target gene
is significantly present, probe response factors may be
retrieved by some statistical modeling. A recent study [
8 ] provides an important example in this direction.
Researchers at Corimbia http://www.corimbia.comalso
developed some proprietary methods to identify "good"
probes and assign different weights to them to improve
data analysis. Efforts can be made along similar lines
with the MOID algorithm in the future. If probe response
factors can be calculated, the "bad" but "stable" probes
can be scaled, and the distribution may be expected to be
closer to normal, therefore expression levels and their
uncertainties can be calculated in a more statistically
sound manner.
Future studies for more accurate background
subtraction models may also improve the fold change
distribution; therefore yield better statistics for fold
change evaluation as well. The relative error in spiking
experiments (28%) suggests that there is room for future
improvement.
LogP and Absolute Calls
In the derivation of LogP (see materials and methods,
Figure 7), it is assumed that the statistics of any
absent probe can be described by the general background
curve
B. The
P value should not be literally
interpreted as accurate probabilities, since it is only
as good as the underlining assumption. Users should also
bear in mind that a
P value is for the null hypothesis
that the discrepancy between observables and generic
mismatches is generated by noise, which is different from
the likelihood of gene presence conditioned on the
observed discrepancy. However, LogP values more or less
serve as a statistical indicator for the sorting of genes
by their significance. The discrete absolute calls in
MAS4 are determined in a completely different empirically
manner. Although higher LogP values have a higher
tendency of being assigned "Absent" in MAS4, the exact
correspondence between the two varies from experiment to
experiment. Roughly, LogP values above -3.0 have a
greater chance to be called "Absent" than "Present" by
MAS4. Users should be aware not to over interpret this
correspondence. One interesting observation we had in
this study was that discrete calls by MAS4 are not as
stable as LogP values. Sometimes a gene can be reassigned
to "Present" from "Absent" by MAS4 due to a small
perturbation in the underlying quantities without going
through a "Marginal" transitional stage.
Curve Normalization
It is clear by observing various experiments that the
response of intensity to signal varies in different
intensity regions. Occasionally, the normalization
constant
n
f
does not cover well for all intensity values. A more
generalized normalization procedure should use a
normalization curve instead of a constant factor. The
MOID normalization algorithm can be easily modified to
the following (see [ 10 ] for a similar approach):
Step 1: include all genes in the normalization gene
list.
Step 2: using the
E
k
values for all the genes in the list, generate
integral intensity distributions for both experiments. A
normalization curve
NF(I) is constructed in such a way
that the two integral distributions are identical after
normalization.
Step 3: normalize the data sets using
NF obtained in the previous step;
refine the gene list by further excluding those genes
whose intensity values changed by more than a certain
fold, >
f
max or < 1/
f
max , between the two data sets.
Step 4: repeat Step 2 and update
NF to
NF', until one of the following
conditions is met:
1) the maximum number of iterations,
Itr
max , is reached;
2) max(|
NF (
I )/
NF '(
I ) - 1|) ≤
max , where
max is a small predefined
threshold;
3) size of the normalization gene list drops below a
predefined threshold,
Sz
min .
This algorithm offers a chance to correct
non-linearity in the chip system to a certain extent. To
demonstrate the algorithm, we applied the procedure to
two measurements, where genomics DNA sample of yeast
s288c strain were hybridized onto Affymetrix YG_S98
arrays. The second experiment was a repeat of the first
one after about 50 days, where some of hybridization and
scanning parameters had been changed over the course. It
is shown in Figure 8that the curve normalization
procedure out-performed constant normalization as
expected.
Conclusions
MOID algorithm allows at least double or even triple the
information density of current (U95 human chip) Affymetrix
high-density oligo nucleotide arrays without compromising
analysis accuracy. Table 2summarizes feature comparisons
between MOID and MAS4.
It should be noted that at the time of this study,
Affymetrix U95 chip uses 16 probe pairs per set. As
Affymetirx is planning on further reducing probe set size
in their next design, the density improvement by using MOID
may be less than the estimation given above.
Materials and Methods
Affymetrix Oligonucleotide Chip
At the time of this study, Affymetrix synthesizes
25-mer oligonucleotides on a 640 × 640 array with 20 μm
feature size using photolithographic fabrication
techniques (some data used in this study were collected
from some previous generation arrays of 540 × 540 cells
and 24 μm feature size). Each cell (also called a probe)
in the array contains the same oligonucleotide sequences.
As shown in Figure 9, typically a set of 16-20 probes are
designed for each targeting cluster (called a probe set,
or sometimes a "gene".). An expression value will be
derived per probe set as the result of analysis. Probes
taken as fragments from a target sequence are called
perfect matches (PM). Multiple matches per set serve as
independent signal detectors and provide a possibility to
capture statistical uncertainties. For each match, there
is also a corresponding mismatch (MM) probe, whose
sequence differs from its match by a single base in the
middle. Mismatches are meant to detect cross
hybridization components of their corresponding matches,
and are used by MAS4 for probe-specific background
subtraction. There are also several Affymetrix control
sets on each chip used for quality references, which were
used in the spiking experiments to validate and refine
hybridization protocols.
Intensity Distribution
In an ideal case, all the probes in a set should give
similar signals and serve as replicate measurements. One
common rule in probe design is to select probes with
similar melting temperatures to minimize the variances
among probes. The chemistry involved in a hybridization
experiment is often too complicated to be predicted
computationally at the current stage, therefore in
reality probe intensity distribution for a set is usually
fairly wide and has a long tail, which causes all kinds
of difficulty for data analysis.
One usual attraction in expression analysis is to
assume a normal distribution for probe intensities. This
hypothesis can be examined by the following calculation.
For each probe set, we applied a linear transformation to
probe intensities, so that the mean and the standard
deviation of the intensities in the set were normalized
to zero and one, respectively. Then all the resulting
match intensities were used to generate an overall
distribution. Our calculation shows the normal
distribution assumption is not an appropriate one. When
similar analysis was applied to mismatch intensities and
mismatch-subtracted match intensities, the conclusion
stays more or less the same.
Despite the fact that the intensity distribution of a
set is non-Gaussian, it is still crucial to find out the
distribution function for match or mismatch intensities
to generate meaningful statistics tests, if such
distributions actually exist. The authors tried several
analytical functions on match and mismatch probe
intensities. Unfortunately, the distributions seemed to
be quite "bad", the tail portions even fade out slower
than the extreme value distribution. It was found that
distributions of mismatch signals also significantly
depended on the sample and experimental conditions. One
may reasonably suspect that cross hybridization,
intensity saturation, overall sample concentration, chip
production irregularities, and miscellaneous noise may
all cause a change in the signal distribution. The study
seems to suggest the distribution of cross hybridization
should be taken from each experiment as a result of
measurement instead of using an
a priori determined analytical
expression. This guideline is used in MOID algorithm.
MOID algorithm: hypothesis and principles
As mentioned in the intensity distribution analysis,
MOID assumes cross hybridization behavior of probes is
mainly non-probe-specific, and therefore all the match
cells can share the same cross hybridization background.
MOID primarily aims at saving 50% of the chip space
currently dedicated to mismatch cells in the Affymetrix
design under this study. It was also discussed before
that cross hybridization distributions should be taken
from experimental data instead of from some analytical
function. Based upon the general belief that there is
always a significant amount of genes unexpressed in any
mRNA experiment, we will use the match cell signals from
the 5% darkest probe sets in the current public
Affymetrix arrays to prove the point that it is possible
for MOID not to use any mismatch signal. For the next
generation gene chip designed based on MOID algorithm,
some designated probes may be used as generic mismatches
to collect non-probe-specific cross hybridization data.
In fact, on GNF-HS1, 16,460 probes from 476 viral genes
are used for such a purpose.
As discussed, the distributions of quantities in
expression analysis are usually asymmetric and contain a
long tail, regardless whether it is match intensity,
mismatch intensity, or average intensity difference. For
such abnormal distributions, concepts like mathematical
average and standard deviation are not the most
appropriate statistical terms to use. Pre-filtering the
data, like what MAS4 does, certainly helps reshape the
distribution, but by no means this can bring the
distribution back to normal without throwing out a
significant portion of measurements. The MOID algorithm
is specifically designed to avoid these problems by using
percentile instead of mean and using confidence intervals
instead of standard deviation in all possible cases.
MOID algorithm for significance of gene
presence
A non-probe-specific integral background distribution
(representing noise, cross hybridization, and possible
other factors),
B (
I ), is derived from the
intensities of the 5% darkest probe sets (or from generic
mismatches in the future).
I stands for intensity;
B satisfies boundary conditions:
B (0) = 0 and
B (∞) = 1. For each probe set
k, the match signals are sorted and
the integral distribution,
S
k
(
I ), is generated as well. Figure
7is a schematic diagram.
If the gene represented by a probe set is absent, the
intensity data from matches are due to pure background
contribution, therefore
S
k
is likely to be close to
B. We choose the Kolmogorov-Smirnov
test to determine likelihood that the observed signal
distribution,
S
k
, could be explained by
B. If we define
d
k max as the maximum vertical
distance between
B and
S
k
, according to K-S statistics, the probability of
observing discrepancies greater than
d
k max is determined by a P
value [ 11 ] ,
where
n
k
is the effective number of data points in the K-S
statistics, which is the number of PM for gene
k in our case.
P carries the meaning of
probability, therefore is a number between 0 and 1.
Practically, Log
10
P , a negative value, is used as
our final representation. In this way, MOID uses a
continuous LogP criterion to replace MAS4 absolute calls.
Those signal sets that can be easily explained by noise
are assigned a LogP value closer to zero.
MOID algorithm for expression level
calculation
MOID uses the horizontal distance between
S
k
and
B to represent gene expression
level (Figure 7). Since the darkest probes may be more
likely caused by their poor binding properties to the
target gene, and the brightest probes may be more likely
caused by serious cross hybridization issues, different
parts of the integral distribution tend to have different
qualities. Therefore, the MOID algorithm uses the
horizontal distance measured at a certain percentile,
pct, instead of the whole curve.
Based on the analysis of the spiking experiments
discussed later, it is empirically determined that 70% is
the optimal percentile for signal retrieval. We also
tried to use the average of horizontal distances from
several
pct, but without significant
improvement. Therefore the expression level
E
k
, related to gene concentration for gene
k, is defined as
E
k
= max(
I |
S
k
=
pct -
I |
B =
pct , 0).
Instead of using the standard deviation, we take
confidence intervals directly from the distribution curve
S
k
, with the background subtracted. E.g., 80%
confidence intervals can be represented by a lower bound
(
E
kl
) at 10 percentile and an upper bound (
E
ku
) at 90 percentile of the distribution, i.e.,
E
kl
0(0.1) = max (
I |
S
k
= 0.1 -
I |
B =
pct , 0)
and
E
ku
0(0.9) = max (
I |
S
k
= 0.9 -
I |
B =
pct , 0).
As one might expect, increment of the probe set size
helps narrowing down the confidence intervals in a manner
determined by the statistics of probe distribution.
However, this piece of information is unknown and we took
an
ad hoc approach by modifying the
boundaries formulae as the following:
and
MOID algorithm for normalization
The most common way of understanding gene functions by
expression profiling is to study the change of their
expression levels in various disease states and cellular
environments. The observed expression level is a product
of complicated protocols, and is subjected to various
factors from sample preparation to final image scanning.
Therefore, it is a common practice to normalize
expression data drawn from multiple profiles before
making any reliable interpretation.
The normalization procedure in MAS4 aims at
normalizing the average intensities of probe sets
(excluding the top and bottom 2%). To avoid possible
signal contamination, MOID takes similar approach by
normalizing the probe sets between 10 and 90 percentiles.
However, it is noticed that an ideal normalization factor
should be derived from only those genes that do not
change their expression levels between the two
experiments. MAS software provides the possibility to use
a list of housekeeping genes for such purposes; however,
it certainly requires careful downstream research to
validate any such a list. Some common normalization
criteria were summarized in a recent study by Zien et al.
[ 12 ] . MOID uses a heuristic bootstrap method to
identify an approximate list of unchanged genes between
two data sets:
Step 1: include all genes as the normalization gene
list.
Step 2: sort the
E
k
values (already background subtracted) for all the
genes in the list; the interesting portion of expression
values (between 10% and 90% in our case) is retrieved and
average intensity is calculated for each experiment,
respectively. The initial normalization factor
nf is calculated by the ratio of
the two average intensities.
Step 3: normalize the data sets using
nf obtained in the previous step;
refine the gene list by further excluding those genes,
whose intensity values changed by more than a certain
fold, >
f
max or < 1/
f
max , between the two data sets.
Step 4: repeat Step 2 and update
nf to
nf', until one of the following
conditions is met:
1) the maximum number of iterations,
Itr
max , is reached;
2) |
nf/nf' - 1| ≤
max , where
max is a small predefined
threshold;
3) size of the normalization gene list drops below a
predefined threshold,
Sz
min .
In practice, a single iteration is usually sufficient
for most comparisons. User can adjust threshold
parameters towards their preferred stringent levels. In
this study we use
f
max = 2.0,
max = 0.05,
Itr
max = 5,
Sz
min = 1000.
MOID algorithm for comparison analysis
Different probes within the same probe set generally
respond very differently to mRNA sample fragments. We
examined various probe properties such as melting
temperature, nucleotide base composition, possible
sequence motifs, and potential secondary structures in
order to understand the cause of such diverse response
properties. That study did not find any conclusive factor
that explains observed probe signal distribution
satisfactory. Although it seems rather difficult to
pre-compute and predict probe responses, it is possible
to derive some features afterwards for a probe based on a
significant amount of expression data where the target
gene of interest is present [ 8 ] . Instead of using a
modeling approach to explain various signal
contributions, MOID uses a new approach to avoid the
affect of diversities of probe response factors in
calculating expression fold changes.
Let us assume any two probes in a probe set
k of size
n
k
always respond to their target gene concentration
with factor
r
1 and
r
2 , respectively. That is, if the gene
is present at concentration
c
1 and
c
2 in two hybridizations, the two
probes in average give signal intensities
r
1
c
1 and
r
2
c
1 in the first experiment,
r
1
c
2 and
r
2
c
2 in the second experiment. As in
MAS4, fold change is calculated by the ratio between
r
1
c
2 +
r
2
c
2 and
r
1
c
1 +
r
2
c
1 , where the result is essentially
dominated by the probes with the largest response
factors, and the statistics of the ratio becomes
difficult to estimate. However, we observe that by taking
the ratio of the two signals for each probe individually,
one essentially is looking at
n
k
independent measurements of
c
2 /
c
1 for that probe set. This opens a
possibility for obtaining a distribution of fold change
values.
If one assumes most probes have a rather stable
intrinsic response factor,
r, a fold change number can be
calculated from each probe
i in the set independently, which
hopefully removes the affect of the unknown response
factors
r
i
. However, it seems the response factor of each
probe has a non-negligible intrinsic spread as well; this
may be further complicated by alternative splicing and
tissue-specific cross hybridization issues. In addition,
the
n
k
ratio numbers calculated for a probe set are not
normally distributed (it is well known that even the
ratio of two normally distributed values is no longer
normally distributed). The current background subtraction
algorithm may not take fully into account response
factor-unrelated signals, therefore could further
increase the non-normal component. To overcome this
difficulty, MOID uses a percentile,
pct
f
, of the integral distribution formed by the
n
k
fold change numbers,
F
k
(
f ), which satisfied the boundary
conditions:
F (0) = 0 and
F (∞) = 1. Empirically, 70% is used
for
pct
f
as determined by spiking experiments discussed
later. The confidence intervals are also directly taken
from relevant percentiles in the distribution
corresponding to the lower bound and upper bound,
respectively. The effect of the number
n
k
is taken into account in the same heuristic manner
as we did previously for absolute analysis. The final
formulae are:
MAS4 Algorithm for Absolute Analysis
After a sample is hybridized to probes on a chip, the
chip is scanned and fluorescent signals are collected and
stored in an Affymetrix DAT file. For cell
i, after filtering out boundary
pixels,
N
i
numbers of pixels are used to calculate an average
intensity value
I
i
and standard deviation σ
i
. Hereafter we index a probe set by
k, which contains
n
k
probe pairs. The calculated intensity values for the
j th pair in the
k th probe set are denoted as
PM
kj
for PM and
MM
kj
for MM. Background intensity,
bg, is derived from the average
intensity of the 2% darkest cells. Noise level,
Q, is determined as
where average is done over all background cells.
MAS4 assumes that the cross hybridization component of
a match is captured by its mismatch companion, therefore
the difference,
PM
k
-
MM
k
, is used to derive expression levels. MAS4 also
uses a "super-Olympic scoring" procedure to iteratively
filter out outliers, defined as those probe pairs with
difference values more than three times the standard
deviation away from the mean. After filtering, the
average difference value for a probe set,
D
k
, is used to represent the expression level of that
particular gene.
The significance of gene presence is determined by an
empirical absolute call decision matrix coupled with
proprietary MAS4 algorithms, parts of which are described
in the Microarray Suite documentation. As the result, a
probe set is marked as either "Present", "Absent", or
"Marginal".
MAS4 calculates average difference by
D
k
= <
PM
kj
-
MM
kj
>
j
,
which is essentially
D
k
= <
PM
kj
>
j
- <
MM
kj
>
j
,
where the average is taken over all the probe pairs
surviving the super-Olympic filtering process mentioned
above. Based on this observation, the one-one
correspondences between
PM
kj
and
MM
kj
, typically with an average correlation coefficient
around 0.8, are essentially lost during such averaging.
Since MAS4 is successful in various applications despite
the fact it averages out probe-specific mismatch signals,
we hypothesize that contributions of the mismatch probes
are mainly probe-nonspecific, if the pairwise
correspondence between
PM
kj
and
MM
kj
are not enforced. This observation led to the idea
of only using match probes in the MOID algorithm
discussed above.
MAS4 Algorithm for Normalization
MAS4 introduces a normalization factor (a scaling
factor in MAS4 serves the same purpose). The goal of
normalization is to reduce false positives in the
expression fold change calculation and ensure
housekeeping genes are marked as unchanged. Since it is
usually unknown
a priori what genes sustain their
expression level during the two experiments, different
heuristic normalization schemes may be adopted and it is
unclear at this point which particular approach is
optimal. MAS4 normally derives the normalization factor
by scaling average
D
k
for all genes with expression levels within the
central 96% to a certain target intensity constant. This
algorithm is based on the assumption that the total copy
of mRNA per cell is a conserved number among
experiments.
MAS4 Algorithm for Comparison Analysis
In comparison of two chips, we follow Affymetrix
convention of calling the reference chip measurement as
the base (
b ), the other one as the
experiment (
e ). After the above normalization
procedure, the fold change value for gene
k, f
k
, is essentially calculated by dividing
D
k
(
e ) with
D
k
(
b ). MAS4 uses a filtering process
to ensure that a common subset of "good" probe pairs
between
b and
e are used to recalculate
D
k
. Very weak expression signals (sometimes negative)
are rounded to the noise level mentioned before. The
final formula for
f is:
f
k
= [
D
k
(
e ) + max (
Q -
D
k
(
b ), 0)] / max (
D
k
(
b ),
Q ), if
D
k
(
e ) ≥
D
k
(
b ),
or
f
k
= max (
D
k
(
e ),
Q ) / [
D
k
(
b ) + max (
Q -
D
k
(
e ), 0)], otherwise.
An empirical difference call decision matrix and a
proprietary MAS4 algorithm, partly described in the
Microarray Suite 4.0 documentation, determine the
significance of the fold change. As a result, the probe
set is marked as "Increase", "No Change", "Decrease",
"Marginally Increase", or "Marginally Decrease". It is
obvious that the fold change number is only meaningful if
the probe set is clearly present at a level above the
noise in both data sets.
List of Abbreviations
MOID - match-only integral distribution
MAS4 - Affymetrix algorithms implemented in its
Microarray Suit 4.0 software
PM - perfect match
DMM - mismatch
