Background
Alizadeh
et al . [ 1 ] did a large scale,
long-term study of diffuse large B-cell lymphoma (DLBCL),
using microarray data chips. By doing cluster analysis on
this data, they were able to diagnose 96 donors with an
accuracy of 93% for this specific lymphoma; they were not
able to predict which individual patients would survive to
the end of the long-term study. The International
Prognostic Index for this disease was incorrect for 30% of
these patients.
Cluster analysis, together with other statistical
methods for identifying and correlating minimal gene lists
with outcome, have become established as the primary tools
for the analysis of microarray data in cancer studies. We
wished to test a different approach, ANN.
These two approaches to the analysis of microarray data
differ substantially in their mode of operation. In the
first examination of the data, clustering, as applied in
numerous recent cancer studies, is an unsupervised mapping
of the input data examples based on the overall pairwise
similarity of those examples to each other (here,
similarity with respect to the expression levels of
thousands of genes); the method is unsupervised in that no
information of the desired outcome is provided. Subsequent
analysis of the clusters in these studies generally
attempts to reduce the gene set to the subset of genes that
are most informative for the problem at hand. This step is
a supervised step since there is an explicit effort to find
correlations in the pattern of gene expression that match
the classification one is attempting to make among the
input examples (see Discussion for specific examples). The
input for this supervised step is the product of an
unsupervised step. As this subselection is not routinely
subjected to independent test using input examples
originally withheld from the subselection process, it is
generally not possible to judge how specifically the
subselection choices relate to this specific set of
examples as opposed to the general population of potential
examples. To the extent that the gene set employed is much
larger than the gene set that really determines the
classification, it is possible that much of the clustering
result will be based on irrelevant similarities.
On the other hand, backpropagation neural networks are a
supervised learning method that has an excellent reputation
for classification problems. During the training phase, the
ANN are supplied with both the input data and the answer
and are specifically tasked to make the classification of
interest, given a training set of examples from all
classes. That is, the ANN are constantly checking to see if
they have gotten the 'correct' answer, the answer being the
actual classification not just the overall similarity of
inputs.
Networks accomplish this by continually adjusting their
internal weighted connections to reduce the observed error
in matching input to output. When the network has achieved
a solution that correctly identifies all training examples,
the weights are fixed; it is then tested on input examples
that were not part of the training set to see if the
solution is a general one. It is only in this independent
test that the quality of the network is judged.
Investigators are not limited to a single network. It is
feasible to train a series of networks using, say, 90% of
the examples for training and holding back 10% for testing.
A different 10 % can be tested in a second network and so
on. In this way, with the training of ten networks, each
input can be found in a test set one time and can,
therefore, be independently evaluated. The data presented
below, with the exception of a few cases, are the output of
ten slightly different trained networks, operating in test
mode, which collectively evaluate the entire donor pool.
This 'round-robin' procedure was employed, in duplicate, in
every trial described throughout this work. The fact that
one ends up with 10 networks is not an impediment to
analysis since any future examples could be submitted to
all 10 networks for evaluation, with a majority poll
deciding the classification. That is, six networks in
agreement on a particular input datum would determine the
classification of that input. These networks are, of
course, likely to be very similar in that their training
sets differ only slightly.
A second major advantage of backpropagation networks
follows from the first. Not only are neural networks
trained to the specific question, rather than a loose
derivative of that question, and tested for generality, but
they can also be asked for a quantitative assessment of how
they got the correct answer. Numerical partial
differentiation of the network with respect to a given test
input example [ 2 3 ] allows one to see the network's
evaluation of the relative impact of each gene in arriving
at the correct answer for this particular input. Cluster
analysis, including the statistical correlations, has no
corresponding highly focused sight for targeting specific
similarities as opposed to non-specific similarities. To
the extent that this is true, neural networks should be
able to identify relatively small gene subsets which will
significantly outperform the initial gene sets in
classification and which will also significantly outperform
the gene subsets suggested by cluster analysis.
Results
Determining patient prognosis from microarray
data
Cluster analysis [ 1 4 ] had shown that the 4026 gene
expression panels for 40 DLBCL patients contained some
information relevant to the question of prognosis but
these authors did not make an attempt to provide survival
predictions for individual patients.
We wished to see if the neural network strategy, of
train, test, differentiate, retrain on the reduced gene
set, and retest, could produce any useful result with
respect to prognosis on an individual basis. The approach
would be: [ 1 ] use the entire gene set without
preprocessing to train a network, testing to confirm that
it had at least a good fit to the problem and, [ 2 ] use
the network's definition of the problem, by
differentiating the network, to focus on those genes most
essential to the classification. These genes would then
form the basis for training new networks with hopefully
improved performance. Over 130 networks were trained for
this study. Figure 1shows a work flow schematic for this
study. Table 1provides a summary overview of the data,
including data not shown.
Initially a network was trained to accept microarray
data on the complete panel of 4026 genes from 40
patients. This network had 12078 input neurons with a
semi-quantitative assessment of each gene, 100
middle-layer neurons, and a single output neuron. The
networks were originally designed with 3 input bits per
datum: one for sign,'-' = 1, and 2 for quantitative
degree of signal with 00 being 0 to 0.5, 01 being >0.5
to 1.0, 10 being >1.0 to 2.0, and 11 being >2.0.
Thus '011' would indicate a particular gene whose
expression, relative to control, was increased at a
magnitude >2. The training set included 30 donors,
with 10 additional donors being held back as test data.
The network was trained by processing 12 iterations of
the complete training set. The test set, drawn from a
mixture of survivors and non-survivors, was then run. The
entire process was then repeated with a different choice
of test data each time. In this round-robin fashion, all
donors serve as test data for one of the networks, and
each training set is necessarily slightly different. A
round robin series of 4 networks was generated. Data
underlying Figure 5 of the earlier report
http://llmpp.nih.gov/lymphoma/data.shtmlwere used for
training. The networks were asked to predict, based on
the 4026 gene set, which of 40 DLBCL patients would
survive to the end of the study (longest point = 10.8
yrs). Networks initially varied with from 1 to 3 errors
on 10 test patients each, for a total of 31 of 40
patients correctly predicted (data not shown). 1However,
a trained neural network can be numerically
differentiated [ 2 3 ] to show the relative dependence of
the output (classification) on each active input neuron
within an input vector. Briefly stated, the
differentiation process involves slightly perturbing the
activation (down from 1.0 to 0.85) of each active input
neuron, one at a time, to note the specific change in the
output value. In that there is one gene for each active
node, the largest change in the output points to the most
influential gene. We then trained qualitative networks,
with 2 bits per gene, on the 4026 gene set in order to
differentiate them ('1 0' for expression greater than, or
equal to, the control, '0 1' for less than the control).
The networks had 67 middle layer neurons. This coding has
the effect that there is an active neuron for each gene
in the set regardless of expression level and the total
number of active input neurons is constant from input to
input. By taking the top 25% of genes in each of 12
differentiations and requiring agreement of at least 4 of
12 patients in choosing each gene, we obtained a set of
34 genes. (These cutoff criteria are necessarily
arbitrary and are only justified by subsequent proof that
they produced gene subsets having the desired
information.) A round-robin series of 10 networks, with 4
test donors each, produced a single error (DLCL0018) in
survival predictions when trained on these 34 genes (data
not shown) 1. The second round-robin training with the
same gene set produced no errors, correctly evaluating
all 40 patients in a series of 10 test sets (Table
2).
For a second study, we took 20 patients and held them
in reserve to model information from a "follow-up" study.
Twenty networks were trained, on the 34 gene set, using
the remaining 20 patients; each had 19 patients in the
training set and 1 in the test set. Collectively, these
networks made no errors in the prognosis of 20 patients.
The data for the 20 reserve patients were then tested on
all 20 trained networks to emulate follow-up data. Out of
400 individual scores, there were 5 errors distributed
over 2 patients. A poll of the 20 networks, therefore,
produced no errors by a majority, correctly classifying
all 20 members of the follow-up group.(data not
shown)
The 34 genes are given in Table 3. In 5 of 12 cases,
the gene chosen as most influential in determining the
correct prognosis was 18593, a tyrosine kinase receptor
gene. While this gene set may not be the absolute best
possible, it clearly does contain sufficient information
for error-free predictions on these patients. The
identification of this gene set will hopefully lead
eventually to a better understanding of the interaction
of these genes in this disease as a result of future
studies.
Diagnosing lymphoma from microarray data
The diagnosis of DLBCL lymphoma by biopsy is not
trivial. Even with gene expression data, clustering
techniques produced a misreading of 7 out of 96 donors [
1 ] , a result unimproved in their hands by further
analysis of reduced gene panels. We wished to see if back
propagation neural networks could do better using the
same data set. Figure 2shows a work flow schematic for
this study. This testing over the whole donor set with
4026 genes produced 6 errors in diagnosis (data not
shown).
Thus, in the first round, ANN merely match cluster
analysis. In preparation for differentiation, a network
was trained with the same donor sets as the first network
above, but coded qualitatively. This network correctly
classified the 10 members of the test set (data not
shown). The 5 positive donors from the test set were each
used, in turn, to differentiate the network. In these
cases, the first criterion for selection was broad: the
gene had to contribute at least 10% as much as the gene
making the maximum contribution to the correct
classification; the second criterion was that 3 or more
of the donors had to agree on the selection. This
produced a subset of 292 genes. The number of genes
referenced by a given donor under identical criteria
ranged from 45 to 1448. Only 38% of the genes overlapped
the 670 gene subset identified by cluster analysis. It
was of interest to see if these genes were sufficient for
correct classification of the donors. Ten different
networks were trained with the 292 gene subset. Three
(OCI Ly1 and DLBCL0009 and tonsil) errors were produced
over 96 donors in 2 separate series (data not shown).
At this point, the neural networks were doing a
much-improved diagnosis; it remained to be seen if the
gene set could be further refined. The set of 292 genes
was then treated in two different ways: [ 1 ] it was
arbitrarily split into even and odd halves, with each
half being used to train ten new networks. [ 2 ] it was
used whole to train ten qualitative networks for further
differentiation.
Twenty different networks were then trained using a
146 gene (odd or even numbered) subset of the 292 gene
set in 2 series of 10. The odd set again produced 3
errors (data not shown). In the even set, a single error
was made over 96 donors in ten different test sets,
identifying the 'tonsil' inlier in the earlier cluster
analysis [ 1 ] as positive (Table 4). Ten additional
networks were trained on the even set with the same
result (data not shown).
The differentiation of the networks from the 292 gene
set pointed to 8 genes. Given the high accuracy of the
even 146 gene set, we also trained networks on this set
for differentiation. These pointed to 11 additional
genes. In these cases, only genes in the top 20% in
influence chosen in common by at least 25% of the
differentiated examples were considered. Networks trained
on these 19 genes produced 2 errors over 96 donors in 10
test sets (Table 5). The 19 genes, using the designation
from the initial report, are given in Table 6.
We also wished to test this gene set in the context of
a follow-up study. For this purpose, we set aside 50
donors as "follow-up" data, using the remaining 46 donors
in the usual training/testing round robin. Eleven
networks were trained, 9 with 42 training vectors and 4
test vectors and 2 with 41 training vectors and 5 test
vectors. Collectively, these produced 3 errors over 46
donors or 93% correct. The follow-up donors were then
tested on the 11 networks. A poll of these networks
showed a majority vote for 1 error or 98% correct.
Discussion
The rather remarkable conclusion of this analysis is
that there is sufficient information in a single gene
expression time point of less than 5 dozen genes to provide
perfect prognosis (out to ten years) and near-perfect
diagnosis for this set of donors. Furthermore, neural
networks, through a strategy of train and differentiate,
bring that information to the fore by progressively
focusing on the genes within the larger set which are most
responsible for the correct classifications, providing at
once a reduction in the noise level and specific donor
profiles. This focus on the specific classification problem
led to a set of 34 genes for prognosis and a second set of
19 genes for diagnosis. These sets are mutually exclusive.
The gene subsets suggested by cluster analysis [ 1 ] are
not supersets of these sets; the 670 gene set of the
initial report captured only 7 of the 19 gene set used for
diagnosis and the 148 gene staging set captured only 2 of
the 34 gene set used for prognosis. The 234 gene subset
proposed by Hastie,
et al . [ 4 ] for prognosis contains
6 of the 34 gene set. There was no overlap with the 13 gene
set identified by Shipp,
et al [ 5 ] to correlate with their
cured/fatal classes for this disease. At first, it might
seem surprising that the gene subsets identified here do
not appear to be subsets of those identified earlier by
Alizadeh
et al . But this surprise is based on
a naive intuition. The fact is that we do not know the
level of information redundancy that exists in these large
arrays. Apropos of this point, Alon
et al . [ 6 ] discarded the 1500
genes indicated by cluster analysis as most discriminatory
in their study of colon cancer and, upon reclustering,
found their diagnosis unimpaired. Likewise, it may be that
while the top 10% of relevant genes might be sufficient for
perfect classification, so might the next 10%. These sets
by definition are mutually exclusive. By extension, it is
not difficult to believe that some other large gene set
might be able to get 75% of the classifications correct
with little or no overlap with those genes in the top
10%.
We have been careful to avoid any claim that the gene
sets extracted in this procedure are the "best" gene sets.
Only in one, highly qualified sense can they be said to be
best; that is in classifying
this data set there are no other
gene sets which offer a statistically significant
improvement in classification accuracy. That is not to say
that there may not be other sets which could do as well.
Nor is there any implication that these genes are seminal
in the etiology of this disease. They may not be necessary
but they are sufficient to do this classification. They may
not be sufficient to the classification of a much larger
patient set. Forty patients are unlikely to be fully
representative of the general patient population with this
disease. It should be noted, however, that the same caveats
apply to the analysis of these data by any other
method.
There have been a number of additional studies of cancer
using microarray data for either prognostic or diagnostic
purposes. The following listing includes a brief discussion
of 7 of these studies:
(1) Shipp
et al . [ 5 ] did a study of 58 DLBCL
patients and 19 follicular lymphoma patients. They first
sought to classify DLBCL and FL patients. They clustered
6817 genes. Using their own weighted combination of
informative gene markers, they picked out 30 genes whose
expression levels would be used to do a 2-way
classification. They correctly classified 71/77 patients
for a diagnostic accuracy of 92%. They then attempted to
develop high risk and low risk groups with respect to 5
year prognosis. They used several different methods for
associating particular gene clusters with survival outcome:
Kaplan Meier analysis, Support Vector Machine, and
K-nearest neighbor analysis. They selected 13 genes as most
informative and achieved the best result with SVM modeling.
They did not explicitly state how many patients initially
sorted into the high risk/low risk groups but other data
suggest 17 and 41 respectively. The only way in which these
survival probability plots can be compared to the patient
by patient predictions presented above is to associate low
risk with survival and high risk with non-survival (Please
note:this equation was not made by any of the authors, with
the exception of [ 3 ] below, discussing risk groups). If
one makes this association, their best result is 14/58
errors for a 5 yr. survival accuracy of 76%.
(2) Rosenwald
et al . [ 7 ] did what they termed a
follow-up study on the original Alizadeh
et al . study of DLBCL patients.
However, it was not really a follow-up study because a
different chip was used for the microarray data. The
Alizadeh study identified 2 groups based on an analysis of
weighting the gene cluster groups: germinal center B
cell-like tumors which correlated with low risk and
activated B cell-like tumors which correlated with high
risk. If these groups were made survivors and
non-survivors, the prognosis accuracy would have been 75%.
In the follow-up, the authors found it necessary to
introduce a third group, consisting of patients who did not
fit either of the previous 2 categories. Although lacking
the associated gene profile, this third group had a
survival pattern much like the activated B cell-like group.
The authors used Cox proportional hazards modeling to
assign groups on the basis of the expression of 100 genes.
The 5 yr. survival for the low risk group was 60%, 35% for
the activated B cell-like group, and 39% for the 3rd group.
An improved result was obtained using 16 genes drawn from 4
signature gene groupings plus a score for BMP6 expression.
Kaplan Meier estimates of survival were determined for 4
quartiles for which the 5 yr. survival rate was
73%,71%,34%,15%. If these 4 are collapsed into 2 categories
of survivor and non-survivor, it would produce 62/240
errors for a prognosis accuracy of 74%.
(3) van't Veer
et al . [ 8 ] did a study of 78
patients with breast cancer. Starting with 5000 signature
genes, they narrowed down the gene pool to 231 genes by
examining the correlation coefficient of each gene with the
prognostic outcome. They then rank ordered these genes and
added them 5 at a time to a one-man-out test of their 77
patients for predicted outcome. This was repeated until an
optimum outcome classification was reached. This occurred
at 70 genes. A patient by patient classification based on
the weighting of these 70 genes was able to produce a
survival classification with 13/78 errors for an accuracy
of 83%.
(4) Beer
et al . [ 9 ] used clustering and Cox
hazard analysis to generate a list of 50 genes to be used
in Kaplan Meier 5 yr. projections of survival. They had 86
patients with lung cancer in the study. With 22 patients
originally assigned to the low risk group and 19 to the
high risk group, the corresponding 5 yr. survival rates
were 83% and 40%. If treated as survival categories this
would produce 12/41 errors for a prognosis classification
accuracy of 71%. Although these authors had complete 5 yr.
survival data on 41 of the patients in the study, they at
no point attempted to analyze this group specifically for
direct comparison with predictions.
(5) Khan
et al . [ 10 ] used linear neural
networks to analyze microarray data from patients with
small round blue-cell tumors. They wished to classify the 4
subcategories of this tumor. Principle Component Analysis
was used to reduce 2308 genes to 10 components. Neural
networks were trained using 2/3 of a 63 patient pool to
train and 1/3 to test in a fully cross-validated fashion.
The groups were shuffled 1250 times to produce 3750
networks. These networks correctly classified all 63
patients in a 4-way classification. The networks were
analyzed for the most influential inputs to produce a list
of 96 genes. New networks were calibrated with just these
96 genes; these again correctly classified the 63 patients
and also correctly classified the 25 patients who had been
withheld from the whole process.
(6) Dehanasekaran
et al . [ 11 ] did a study of 60
prostate biopsy samples, 24 non-tumorous,14 tumor in situ,
20 metastatic tumor. Cluster analysis of microarray data
from nearly 10,000 genes misplaced 2 samples out of 26 for
a diagnostic accuracy of 92%. The authors did not state why
they limited the clustering result to 26 samples when they
had 60. Although they performed additional analyses, they
did not involve using the array data for either diagnosis
or prognosis.
(7) Golub
et al . [ 12 ] wished to be able to
distinguish acute myeloid leukemia (AML) from acute
lymphoblastic leukemia (ALL). Starting with the expression
of 6817 genes from 38 patients, they did a 2-class
clustering. They then did a neighbor analysis to identify
1100 genes occurring above chance levels which related to
the AML/ALL distinction. They choose an informative subset
of 50 genes to weight for class assignment of the patients.
They were able to correctly classify 29/34 patients for a
diagnostic accuracy of 85%. They next attempted to use
self-organizing-maps (SOM) for 2 classes in place of the
initial clustering. This produced only 4/38 errors for 89%
diagnostic accuracy. Drawing a 20 gene predictor from these
SOM classes, they again produced 4/38 errors, maintaining a
89% accuracy. These authors also attempted to use array
data to predict clinical outcome on 15 AML patients but
without success.
The identification of specific genes associated with a
particular biological characteristic such as malignant
phenotype would be useful in many settings, [ 1 ] Precise
classification and staging of tumors is critical for the
selection of the appropriate therapy. At present,
classification is accomplished by morphologic,
immunohistochemical, and limited biological analyses.
Neural net analysis in the form of specific donor profiles
could provide a fine structure analysis of tumors
characterizing them by a precise weighting of the genes,
which they express differentially. At present, only subsets
of patients with a given type of tumor respond to therapy.
Networks trained to distinguish responders from
non-responders would allow a comparison of tumor-expressed
genes in responders and non-responders to find those genes
most predictive of response. Recently we have used neural
networks on the data of Perou
et al . [ 12 ] for classifying breast
tumors as hormonally responsive or non-responsive. Networks
that gave a perfect classification with 496 genes pointed
to a subset of 12 genes. Retraining on these 12 genes
produced no error in classifying 62 tissue samples from
their study (unpublished data). We have also analyzed the
data of Dhanasekaran,
et al [ 11 ] . Here the original set
of 9984 genes was reduced to 34 genes. Retraining on these
34 genes gave no errors in a 3-way (normal, early tumor,
metastatic disease) classification of 53 patients
(unpublished data). Given the significant impairment in the
quality of life for many patients undergoing chemotherapy
and/or radiation therapy, such prospective information
would be extremely beneficial. [ 3 ] T cell and
antibody-mediated immunotherapy may be efficacious
approaches for limiting tumor growth in cancer patients. At
present there is a paucity of known tumor rejection
antigens that can be targeted. Neural net analysis may
identify a panel of tumor-encoded genes shared by many
patients with the same type of cancer and thereby provide a
repertoire of potentially novel tumor rejection antigens. [
4 ] For many patients with autoimmune disease the target
antigen(s) is unknown. Enhanced identification of cell-type
specific markers of the target organ through neural net
profiling could identify potential target antigens as
candidate molecules for testing and tolerance
induction.
Conclusions
We believe neural networks will be an ideal tool to
assimilate the vast amount of information contained in
microarrays. The artificial networks presented here were
not selected from a large number of attempts. The networks
described here are the first or second attempts with the
data and format stated; the longest training session lasted
less than 5 minutes. Indeed, the trained neural network
may, in the form of its weight matrix, have the best
possible "understanding" of the very broad statement being
made in the microarray, a view that is accessible with the
differentiation of the network. In this study, that
viewpoint suggested a small subset of genes, which proved
sufficient to give a near-perfect classification in each of
two problems. This approach should be suitable for any
microarray study and, indeed, other global studies such as
2-D gels and mass-spec data which contain sufficient
information for training.
Methods
The data from microarray experiments are stored in
spreadsheet form, representing the positive or negative
level of expression, relative to some control state, of
1000's of genes for two or more experimental conditions. A
short software program is sufficient to translate these
data directly into a binary representation suitable as
input vectors for a neural network. The neural network
software used throughout this study was NeuralWorks
Professional II Plus v.5.3.Neural networks were trained on
the corresponding data sets, with a fraction of the data,
typically 10%, withheld for testing purposes. All open
fields in the data array were set to zero. The trained
networks were then asked to classify new test data as to
donor type. Since the gene expression levels are read
directly from the spreadsheet, their order and names are
provided by the spreadsheet. Given the large amount of
input data, these networks generally converge to a low
error level very quickly during training, often in a few
minutes or less. Subsequently additional networks were
trained with a simplified input that contained only
qualitative information in the form of a plus or minus sign
to characterize the expression of each gene in the panel.
This reduced the input size to 2 bits per gene, 01 for
below the control and 10 for above, or equal to, the
control. The output neuron was trained to output 1.0 for a
positive donor and 0.0 for a negative donor in the
diagnostic networks; for the prognostic networks 1.0
indicated a non-survivor and 0.0 a survivor. The 4026 gene
panel network was provided, respectively, 100 or 67
middle-layer neurons for the 3 bit or 2 bit per gene
inputs. With a very large number of input neurons it is
possible to overload the middle-layer neurons, effectively
always operating them at one extreme limit or the other;
this can have the undesirable effect of reducing their
sigmoid transfer function to a step function, with the loss
of the network's non-linearity. This is clearly indicated
if multiple output values are found to be exactly
identical. Networks were trained to an error level below
0.05 after which they were tested with previously unseen
data. A possible disadvantage of neural networks,
especially with a large input space and a relatively small
sample number, is overtraining. In overtraining, a network
can learn the specifics of each training example as opposed
to finding a global solution for the entire training set.
This behavior is characterized by a degradation in test
scores as training sessions are extended. Although we saw
no evidence of this in this study, we did look to see how
much additional training would be necessary to degrade the
test results in the case of the initial diagnosis networks
with 4026 genes. It was not until we doubled the training
iterations dictated by the 0.05 output error cutoff that we
saw some increased test error. At double the normal
training interval, 8 networks were unchanged, but 2
networks showed an increased error of 1. This is
suggestive, but not proof, of the onset of overtraining.
The networks trained on the reduced 34 or 19 gene sets had
6 or 4 middle-layer neurons.
To differentiate a trained network with respect to
specific inputs, a network was trained on the 4026 gene
panel with 2 bits per gene. The 5 positive donors from the
test set were each differentiated, using software that we
designed for that purpose [ 2 ] . The selected genes were
then compared among the 5 sets, with genes occurring in 3
or more instances being included in the final subset. This
requirement generated a subset of 292 genes from the
original 4026 genes. Networks were trained on this 292 gene
subset and on two 146 gene subsets, representing every
other gene from the 292 set. All were coded with 3 bits per
gene and employed networks with 25 or 12 middle-layer
neurons, respectively. Other networks were trained on the
292 gene set and the 146 'even' set, coded with 2 bits per
gene for subsequent differentiation.
The differentiation of the large panel networks trained
for prognosis arbitrarily employed more selective criteria
(see text) for subset determination with the result that a
single differentiation reduced the gene set from 4026 genes
to 34 genes. Subsequent networks demonstrated that this was
a highly effective selection.
All networks in this study were three-layer back
propagation networks trained with a learning coefficient of
0.3 and a momentum coefficient of 0.4 using the generalized
delta learning rule and the standard sigmoidal transfer
function. The cutoff, in all cases, between positive and
negative scoring was taken to be 0.05 RMS error at the
output neuron No network required more than 4 minutes
training time on a PC at 650 Mh; in the majority of cases,
the network was fully trained in less than a minute.
Training and testing a 10 network round-robin series could
generally be done in less than 20 minutes. Training was
deliberately kept to a minimum to avoid over-training. The
networks represented here were in each case the first or
second attempt result for the given problem. There was no
"data trolling."
Note
1All data not shown can be found at the site
http://research.umbc.edu/~moneill/GBMS
