MatchOnly Integral Distribution (MOID) Algorithm for highdensity oligonucleotide array analysis
 Yingyao Zhou^{1} and
 Ruben Abagyan^{1, 2}Email author
DOI: 10.1186/1471210533
© Zhou and Abagyan; licensee BioMed Central Ltd. 2002
Received: 01 November 2001
Accepted: 22 January 2002
Published: 22 January 2002
Abstract
Background
Highdensity oligonucleotide arrays have become a valuable tool for highthroughput gene expression profiling. Increasing the array information density and improving the analysis algorithms are two important computational research topics.
Results
A new algorithm, MatchOnly Integral Distribution (MOID), was developed to analyze highdensity oligonucleotide arrays. Using known data from both spiking experiments and nochange experiments performed with Affymetrix GeneChip^{®} arrays, MOID and the Affymetrix algorithm implemented in Microarray Suite 4.0 (MAS4) were compared. While MOID gave similar performance to MAS4 in the spiking experiments, better performance was observed in the nochange experiments.
MOID also provides a set of alternative statistical analysis tools to MAS4. There are two main features that distinguish MOID from MAS4. First, MOID uses continuous P values for the likelihood of gene presence, while MAS4 resorts to discrete absolute calls. Secondly, MOID uses heuristic confidence intervals for both gene expression levels and fold change values, while MAS4 categorizes the significance of gene expression level changes into discrete fold change calls.
Conclusions
The results show that by using MOID, Affymetrix GeneChip^{®} arrays may need as little as ten probes per gene without compromising analysis accuracy.
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/UniGene contained 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 [5–7]. 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 highdensity oligonucleotide GeneChip http://www.affymetrix.com for 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 MatchOnly 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 nochange 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 PMonly information is based on the following observations: MAS4 essentially discards the oneone 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 mismatchfree gene chip (GNFHS1) in July 1999, the matchonly expression analysis idea was proposed in other independent studies as well. Li (submitted, 2001) adjusted their previous modelbased analysis of oligonucleotide arrays [8] to PMonly 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 nochange experiments, where results are unambiguous. Modelbased 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 modelbased 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 noisetolerance 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.
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 downregulated fold changes (captured by R_{err2}) than upregulated ones (captured by R_{err1}). MAS4 is more asymmetric than MOID in this aspect.
NoChange experiments
Reduced Probe Set Simulation
In the studies above, we demonstrated the feasibility of a matchonly 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.
Noise Tolerance of both Algorithms
ProbeSpecific Effect
MOID is based on the assumption that the background for probes is mainly nonprobespecific. The results indicate that this assumption is sound overall. However, it is clear that by better understanding probespecific 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.com also 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
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.
 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}.
Relative errors of MAS4 and MOID measured by 366 spiking fold change data points.
Algorithm  R _{err1}  R _{err2}  R _{err} 

MAS4  0.23  0.37  0.30 
MOID  0.25  0.32  0.28 
Conclusions
Comparison between MAS4 and MOID.
MAS4  MOID  

Mismatch used  Yes  No 
Sensitivity to probe set size  SuperOlympic filtering is turned off when remaining probe number is less than 8.  Confidence intervals gradually increase as probe set size drops. 
Relative error (Spiking tests)  30%  28% 
Minimum of probes required (Spiking tests)  10  10 
Nochange test results (True fold change = 1.0)  80% of fold change values in [0.80, 1.34]  80% of fold change values in [0.92,1.22] 
Normalization  Constant based on average intensity. Normalize once.  Allow both constant and curve normalization. Iterative scheme. 
Problematic genes (Bright mismatches)  May underestimate expression level  May overestimate expression level 
Expression Level  Average difference (may be negative). No confidence interval evaluations  Expression value (always positive) Lower bound and upper bound (with probe set size taken into account. 
Present Call  A, P, M (discrete, unstable). Rules were derived from 50 μm and 24 μm feature arrays, may need update for 20 μm)  Continuous LogP 
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
The user should refer to the Affymetrix web site http://www.affymetrix.com and the documentations that come with the MAS4 software for technical details regarding Affymetrix array and MAS4 algorithm.
Affymetrix Oligonucleotide Chip
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 mismatchsubtracted match intensities, the conclusion stays more or less the same.
Despite the fact that the intensity distribution of a set is nonGaussian, 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 nonprobespecific, 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 nonprobespecific cross hybridization data. In fact, on GNFHS1, 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. Prefiltering 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 nonprobespecific 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 7 is 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 KolmogorovSmirnov 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 KS 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 KS 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_{ Sk } = _{ 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_{Sk = 0.1}  I_{B = pct}, 0)
and
E_{ ku }^{0} (0.9) = max (I_{Sk = 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.
 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 precompute 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 nonnegligible intrinsic spread as well; this may be further complicated by alternative splicing and tissuespecific 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 factorunrelated signals, therefore could further increase the nonnormal 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 thej 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 "superOlympic 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 superOlympic filtering process mentioned above. Based on this observation, the oneone 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 probespecific mismatch signals, we hypothesize that contributions of the mismatch probes are mainly probenonspecific, 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 :

matchonly integral distribution
 MAS4 :

Affymetrix algorithms implemented in its Microarray Suit 4.0 software
 PM :

perfect match
 DMM :

mismatch
Declarations
Acknowledgements
We thank David Lockhart for introducing us to the field of gene expression and many helpful discussions about the MAS4 analysis algorithms; Lisa Wodicka and Helin Dong for providing spiking data; David Balaban, and Paul Kaplan for stimulating discussions about the MOID algorithm; Elizabeth Winzeler for providing yeast genomics experiments for curve normalization study. Chris Benner for careful reading of the manuscript.
Authors’ Affiliations
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