Aimee Dudley, John Aach, Martin Steffen, George M. Church
Last modified: May 13, 2002
RNA expression data collected from microarrays is usually analyzed and presented as a set of ratios for each gene of the gene's expression level in cells grown in an experimental condition to its expression level in cells grown in a control condition. The ratios are derived from scanner-measured fluorescence intensities of control and experimental RNA or cDNA samples, each labeled with a different fluor, after co-hybridization to a single array (DeRisi, Iyer et al. 1997). (See Figure 1.) While microarray-derived expression ratio data have proved invaluable to researchers by providing sensitive and comprehensive indicators of the molecular underpinnings of cell behaviors and states, several related drawbacks to this form of data have been noted, including: (a) ratios of expression levels from an experimental and control condition are usually not convertible into absolute expression levels in the two conditions, (b) microarray ratios involving different experimental conditions are usually not comparable unless they are derived from the same control condition, (c) microarray ratios are difficult to compare with high throughput RNA expression level data derived by other methods such as Affymetrix oligonucleotide chips (Lockhart, Dong et al. 1996) and SAGE (Velculescu, Zhang et al. 1995) (see Aach, Rindone et al. 2000). Evaluation of the statistical significance of microarray ratios has also proved challenging, although analytical methods are improving (Chen, Dougherty et al. 1997; Ideker, Thorsson et al. 2000; Kerr, Martin et al. 2000; Roberts, Nelson et al. 2000; Tusher, Tibshirani et al. 2001). Use of scanner-measured intensities directly instead of ratios has been tried as a method of addressing the fundamental comparability issues but usually results in an unacceptable increase in error. This is because ratios correct for several sources of bias and noise that affect intensities, such as differences in spot size and in the efficiency of PCR reactions used to generate the spotted probes; direct use of intensities therefore circumvents these corrections and incurs increased error (Aach, Rindone et al. 2000).
In our article we demonstrate a method for overcoming these obstacles. First, instead of co-hybridizing labeled experimental and control samples on an array, we co-hybridize each sample against a standard labeled equimolar mixture of control oligos that are complementary to microarray probes. Ratios between sample intensities and the intensities of the oligo standards now measure sample RNA levels on a scale that is proportional to their absolute abundance, instead of measuring them with respect to variable and unknown abundances of RNAs in a control sample as obtained from the usual procedures. (See Figure 2.) In the article we describe the advantages and disadvantages of different kinds of equimolar standards. However, a technical difficulty in this approach is that abundances of sample RNAs and their corresponding equimolar controls, and likewise the sample and control fluorescence signals measured by a scanner, can differ by several orders of magnitude. It may therefore be difficult to read both signals within the linear range of a scanner and derive an accurate measure of their ratio. Thus, as a second step in our method, we scan each array at a series of increasing PMTGs and use software to combine measurements from the different linear intensity ranges corresponding to each scan into a common linear scale. (See Figure 3.) In the article, we describe a demonstration of this method on a dilution series spanning a range of 1:~400,000.
In this supplementary material we make available the masliner software used to generate a common linear scale from multiple scans of an array, and present usage notes and technical details relating to masliner processing.
Clicking on any of the images on the left will bring up a larger version of the image.
Figure 1 Current microarray practice involves co-hybridizing an experimental and control sample labeled with different fluors to the same array. Intenisities read from each fluor are transformed into measures of ratios of expression level. Transformation to ratios corrects for important sources of error but makes it difficult to compare expression data across experiments.
Figure 2 We propose co-hybridizing an experimenal sample with a standard equimolar mixture of oligos to the same array. Ratios of intensities now estimate absolute vs. relative expression levels, making it easier to compare expression data across experiments.
Figure 3 To permit accurate measurement of intensities of equimolar controls with RNA species whose abundance varies over several orders of magnitude, we scan an array at multiple PMTGs and consolidate intensities of each into a common linear scale using masliner software.
Figure 4 Scans of the equimolar control oligos have a different appearance than conventional microarray scans because all spots light up at approximately the same abundance.
Figure 5 A direct comparison of a scan of a conventional microarray and a scan of equimolar control oligos.
Processing through the masliner algorithm accurately reconstructs abundance measurements over the range tested (data for experiment presented in the article).
|FBP1||240,452 +/- 1,998||2,364 +/- 28||155,731 +/- 4,257||2,187 +/- 3||86.5 +/- 21.6||N.A.|
|YIL057C||937,566 +/- 3,934||16,920 +/- 28||692,420 +/- 6,156||19,136 +/- 3||45.8 +/- 13.6||1.89|
|CYB2||195,604 +/- 1,930||21,519 +/- 28||214,022 +/- 4,361||23,805 +/- 3||9.04 +/- 0.07||5.07|
|GAL3||278,301 +/- 2,064||130,901 +/- 3,780||155,210 +/- 4,256||70,519 +/- 3||2.16 +/- 0.05||4.18|
|GAL10||75,405 +/- 1,816||187,927 +/- 3,826||81,846 +/- 157||141,685 +/- 3||0.490 +/- 0.124||4.41|
|FUN34||57,190 +/- 169||390,390 +/- 4,121||42,646 +/- 157||279,828 +/- 6,681||0.1494 +/- 0.0042||3.28|
|XBP1||8,913 +/- 169||370,515 +/- 4,084||9,195 +/- 157||338,730 +/- 6,762||0.0256 +/- 0.0022||5.84|
|SIP18||2,448 +/- 169||316,364 +/- 3,991||3,442 +/- 157||426,860 +/- 6,912||0.0079 +/- 0.0002||3.24|
|HXT5||410 +/- 169||371,872 +/- 4,086||705 +/- 157||332,438 +/- 6,753||0.0016 +/- 0.0007||4.9|
a Oligos designed to hybridize to spots with low expression levels (see supplementary table III below) in the glucose cDNA hybridizations were labeled and spiked into a hybridization reaction containing glucose cDNA labeled with Cy3 and Cy5. Oligos labeled with Cy3 were added as an equimolar mixture of 0.5 pmols of each oligo in the 80 hybridization mixture. Oligos labeled with Cy5 were added as a 5-fold dilution series in the order shown from 100 pmols of FBP1 to 0.256 fmols of HXT5 in the 80 hybridization mixture. One standard deviation of the local background on each slide is shown as a representation of background fluorescence levels (Bkg. SD).
b Background-subtracted fluorescence intensities (BSIs) +/- estimated error of prediction associated with masliner linear regression(s) are shown for both fluors (Cy5 and Cy3) in two repeats of the experiment (Slides 1 and 2). BSIs and associated error estimates were normalized by total fluorescence.
c For each slide, the Cy5:Cy3 ratio was calculated. The average ratio +/- the standard deviation for the two repeats is shown. N.A. stands for not applicable.
d The relative abundance values for each oligo in the dilution series were calculated from the average ratios. Since each Cy5-labeled oligo was added at a 5-fold lower concentration than the previous oligo, the dilution was calculated by dividing the average ratio of one oligo by the average ratio of the next oligo in the series. The dynamic range of ratios detected (5.4 x 104) is the product of the dilution series; a perfect series would produce a value of 58 or 3.9 x 105.
Processing through the masliner algorithm accurately reconstructs abundance measurements over the range tested (an additional experiment using a different order of oligos in the dilution series to test concentration versus sequence specificity of Cy3 saturation).
|Average Ratioc||Actual Dilutiond||Measured Dilution|
|GAL10||105,803 +/- 1,519||165 +/- 50||95,478 +/- 1,768||241 +/- 52||520 +/- 174||N.A.||N.A.|
|GAL3||109,939 +/- 1,536||606 +/- 50||59,654 +/- 92||497 +/- 52||151 +/- 43||5||3.4|
|CYB2||140,813 +/- 1,676||4,269 +/- 50||61,439 +/- 92||3,945 +/- 52||24 +/- 12||5||6.2|
|YIL057C||107,894 +/- 1,528||26,693 +/- 50||96,537 +/- 1,772||29,246 +/- 52||3.67 +/- 0.52||5||6.6|
|FBP1||20,913 +/- 96||39,484 +/- 50||20,023 +/- 92||38,758 +/- 52||0.523 +/- 0.0092||5||7|
|FUN34||2,397 +/- 96||64,272 +/- 50||1,720 +/- 92||61,743 +/- 52||0.0326 +/- 0.0067||25||16.1|
|XBP1||743 +/- 96||65,692 +/- 50||409 +/- 92||56,412 +/- 52||0.0093 +/- 0.0029||5||3.5|
|SIP18||108 +/- 96||41,260 +/- 50||143 +/- 92||19,995 +/- 52||0.0049 +/- 0.0032||5||1.9|
|HXT5||108 +/- 96||36,711 +/- 50||220 +/- 92||27,218 +/- 52||0.0055 +/- 0.0036||5||0.88|
a Oligos designed to hybridize to spots with low expression levels (see supplementary table IV below) in the glucose cDNA hybridizations were labeled and spiked into a hybridization reaction containing glucose cDNA labeled with Cy3 and Cy5. Oligos labeled with Cy3 were added as an equimolar mixture of 0.5 pmols of each oligo in the 80 hybridization mixture. Oligos labeled with Cy5 were added as a 5-fold dilution series in the order shown from 100 pmols of GAL10 to 0.05 fmols of HXT5 in the 80 hybridization mixture. One standard deviation of the local background on each slide is shown as a representation of background fluorescence levels (Bkg. SD).
b Background-subtracted fluorescence intensities (BSIs) +/- estimated error of prediction associated with masliner linear regression(s) are shown for both fluors (Cy5 and Cy3) in two repeats of the experiment (Slides 1 and 2). BSIs and associated error estimates were normalized by total fluorescence.
c For each slide, the Cy5:Cy3 ratio was calculated. The average ratio +/- the standard deviation for the two repeats is shown. N.A. stands for "not applicable."
d The relative abundance values for each oligo in the dilution series were calculated from the average ratios. The actual dilution is the value in the dilution series that was added to the hybridization reaction. The measured dilution is the value in the dilution series that was measured by the scanner and adjusted by masliner. Since each Cy5-labeled oligo was added at a 5-fold lower concentration than the previous oligo, the dilution was calculated by dividing the average ratio of one oligo by the average ratio of the next oligo in the series. The 25-fold dilution between FBP1 and FUN34 is due to a missing data point from an oligo that did not hybridize to any of the arrays.
Background fluorescence contributed by the Cy5 and Cy3 labeled glucose cDNA sample. In this experiment, the labeled cDNA samples were hybridized to the oligo array without the addition of the spiked oligos listed in supplementary Table I. This experiment was done in parallel with the experiments presented in supplementary Table I above. Both sets of data were processed through masliner and normalized as described in the article.
|No labeled oligos|
Background fluorescence contributed by the Cy5 and Cy3 labeled glucose cDNA sample. In this experiment, the labeled cDNA samples were hybridized to the oligo array without the addition of the spiked oligos listed in supplementary Table II. This experiment was done in parallel with the experiments presented in supplementary Table II above. Both sets of data were processed through masliner and normalized as described in the article.
|No labeled oligos|
Conventional and reconstructed microarray ratio values +/- standard errors of the mean of these ratios for the 8 genes displayed in Figure 4 of the article.
|GAL1||gal||42.14 +/- 20.11||84.43 +/- 41.6||0.939 +/- 0.034||0.965 +/- 0.02|
|GAL7||gal||157.05 +/- 62.92||155.74 +/- 60.00||0.989 +/- 0.004||0.988 +/- 0.005|
|COX5A||mito||5.42 +/- 1.16||4.06 +/- 2.22||0.825 +/- 0.036||0.687 +/- 0.096|
|QCR7||mito||3.94 +/- 0.86||4.46 +/- 0.88||0.771 +/- 0.052||0.8 +/- 0.033|
|RPL3||glu||0.31 +/- 0.03||0.6 +/- 0.28||0.236 +/- 0.019||0.316 +/- 0.092|
|RPL29||glu||0.36 +/- 0.06||0.35 +/- 0.06||0.261 +/- 0.034||0.254 +/- 0.036|
|PHO88||const||0.97 +/- 0.14||1.45 +/- 0.76||0.484 +/- 0.035||0.474 +/- 0.117|
|STE5||const||1.02 +/- 0.24||3.14 +/- 2.19||0.485 +/- 0.057||0.502 +/- 0.161|
aGenes in Figure 4 were selected to represent the following types: "gal" = galactose-induced gene, "mito" = mitochondrial gene, "glu" = glucose-induced gene, "const" = constitutive gene.
bAverage of four replicates of conventional microarray ratios derived from galactose-grown experimental samples and glucose-grown reference samples. Error is standard error of the mean of the four replicates.
cReconstructed gal/glu ratio derived from four gal:oligo measurements based on galactose-grown experimental samples vs. a common oligo reference, and from four glu:oligo measurements based on glucose-grown experimental samples vs. a common oligo reference. Details on the derivation of these reconstructed gal/glu ratios and their standard errors of the mean are described in Supplemental methods below.
dAverage of four gal/(glu+gal) values +/- standard error of the mean of these values derived from the four gal/glu values whose average is given in the "gal/glu conventional" column of this table. Each gal/(gal+glu) value was computed as (gal/glu)/(1+(gal/glu)).
eReconstructed gal/(gal+glu) ratio derived from four gal:oligo measurements based on galactose-grown experimental samples vs. a common oligo reference, and from four glu:oligo measurements based on glucose-grown experimental samples vs. a common oligo reference. Details on the derivation of these reconstructed gal/(gal+glu) ratios and their standard errors of the mean are described in Supplemental methods below.
This table is supplementary to Table 1 of the article. It gives a per slide breakdown of the counts of background and saturated spots for the two replicates whose averages were given in Table 1. These replicates were used in the spiked oligo experiment described in the article.
To get these figures only spots corresponding to genes on the slides were considered (6307). Standard deviations (SDs) of the median pixel background were computed for scans performed at PMTG 65% and 75% (Cy5), and at PMTG 75% and 85% (Cy3), for the two replicate slides E2 and F2. The columns labeled "65_75" give statistics for the scan pair Cy5 65% and Cy3 75%, and those labeled "75_85" for the scan pair Cy5 75% and Cy3 85%. The masliner columns pertain to the data obtained from masliner runs that combined four Cy5 scans and four Cy3 scans. The highest PMTG values for these scan series were the Cy5 75% and Cy3 85% scans; masliner background SDs are therefore the same as for the "75_85" columns. Numbers and percents of background spots and saturated spots are based on the 6298 of the 6307 spots for which no oligos were spiked in. Background spots are those for which either the Cy5 or Cy3 BSIs is < twice the background SD measured for the fluor. For "65_75" and "75_85", saturated spots are those for which either the Cy5 or Cy3 BSI is > 60000. For masliner, saturated spots are those for which the masliner-computed SATURATION-FLAG is on.
|E2||F2||avg +/- SD||E2||F2||avg +/- SD||E2||F2||avg +/- SD|
|background SD (Cy5, Cy3)||(45,5)||(23,0)||(169,12)||(88,1)||(169,12)||(88,1)|
|background spots (%)||784 (12)||623 (10)||703.5+/-113.8
|648 (10)||657 (10)||652.5+/-6.4
|648 (10)||657 (10)||652.5+/-6.4
|saturated spots (%)||130 (2)||62 (1)||96+/-48.1
|294 (5)||197 (3)||245.5+/-68.6
|0 (0)||0 (0)||0+/-0
Microarrays containing 6,216 PCR-amplified yeast ORFs with a common sequence tag were produced as follows. PCR products were re-amplified from purified PCR products corresponding to the Yeast GenePairs products (Research Genetics). The PCR products were re-amplified using Taq/PfuTurbo (20:1, units) with universal forward (5'-GGAATTCCAGCTGACCACC) and reverse (5'-GATCCCCGGGAATTGCCATG) primers for 30 cycles, annealing at 65 C and extending for 4 minutes. The forward primer also contained 5' acrydite and amino modifications (Operon). These modifications were introduced to facilitate binding to alternative slide chemistries, but are not relevant for attachment to the polylysine slides used in this study. The quality and yield of each PCR reaction was assessed by agarose gel electrophoresis; 475 PCR reactions failed to yield a single band. The list of failed PCR reactions is available here. PCR products were purified by ethanol precipitation and respuspended in 150 mM potassium phosphate (pH 8.0). PCR products were printed onto polylysine slides (CEL Associates, Houston) using a piezoelectric printer (Steffen, Huang, and Church, unpublished data). Slides were UV cross-linked, washed in 0.1% SDS, and boiled in dH2O for 3 min. ORF arrays were pre-hybridized as described previously (Hegde, Qi et al. 1997).
Microarrays containing the Yeast Genome Oligo Set (Operon Technologies) were printed at a concentration of 10 pmols/ml in 150 mM potassium phosphate (pH 8.0) onto 3D-LinkTM slides (Motorola) using an OmniGridTM microarrayer (GeneMachines). Following printing, slides were processed according to the manufacturer's instructions. Oligo arrays were boiled for 3 min. and shaken dry prior to hybridization.
Both types of microarrays were hybridized and washed as follows. Fluorescently labeled probes plus 20 mg salmon sperm DNA (Life Technologies) and 20 mg polyadenylic acid (Sigma) were denatured at 90 C for 3 min. Pre-warmed (42 C) hybridization buffer was added to the probe samples at a final concentration of (25% formamide, 5X SSC, 0.1% SDS) in a final volume of 80 ml. Hybridizations were carried out using LifterSlipTM cover glass (Erie Scientific) and CMTTM hybridization chambers (Corning). Hybridizations were performed at 42 C for approximately 16 hours. Slides were washed at room temperature for 5 min. with shaking in each of the following: 0.2X SSC/0.1% SDS, 0.2X SSC, and 0.1X SSC.
We statistically compared reconstructed expression ratios with actual conventional microarray ratios in two ways based on conventional gal:glu ratios for 6297 spots derived from four independent replications of galactose and glucose sample microarray co-hybridizations, and four sets each of abundance levels derived from independent gal:oligo and glu:oligo microarray hybridizations for the same 6297 spots. All intensities in both the four conventional and 8 calibrated oligo reference experiments were adjusted by masliner.
In our first comparison of reconstructed vs. conventional microarray ratios, we computed 6297 two-sided t-tests for gal:glu ratios from the four conventional microarray hybridizations against four reconstructed ratios formed by taking the individual gal:oligo values and dividing by the average of the four corresponding glu:oligo values. If the distributions are equivalent, 5% of the t-tests for equality should be rejected at P < 0.05. In fact, 346/6297 =~ 5.5% of all t-tests were rejected, a result consistent with the hypothesis that the reconstructed and conventional microarray ratios are statistically equivalent. This method allowed easy computation of a standard deviation for the reconstructed ratios by reconstructing all ratios using a constant denominator, but may be criticized for neglecting (a) the fact that this denominator comes from a distribution and has its own variance, (b) the possible lack normality of the reconstructed ratio distribution, and (c) the small number of values considered by the t-test (four for both conventional and reconstructed ratios). t-tests were computed in Excel 2000 (Microsoft: Seattle).
To overcome these limitations, we performed a second set of statistical tests that compared reconstructed ratios derived from individual gal:oligo and individual glu:oligo values (rather than average glu:oligo values) against the four conventional gal:glu ratios using the Wilcoxon rank sum test, which does not require any particular type of distribution. With four values available for gal:oligo and glu:oligo each, 256 such reconstructed ratios are possible for each of the 6297 spots, but as the Wilcoxon test demands that all values be independent we randomly selected 8 of the 24 possible series of four reconstructed ratios for which each of the gal:oligo and glu:oligo values were used only once. E.g., the first of the 8 sets of reconstructed ratios comprised the four ratios gal:oligo(1)/glu:oligo(1), gal:oligo(3)/glu:oligo(2), gal:oligo(2)/glu:oligo(3), and gal:oligo(4)/glu:oligo(4), where gal:oligo(n) indicates the gal:oligo value for the nth replicate (and similarly for glu:oligo(n)). We calculated these 8 sets of four reconstructed ratios in the same way for each spot, excluding from consideration 8 spots which had at least one glu:oligo value of 0 (leaving N = 6289). For each of these 6289 spots, we performed 8 two-sided Wilcoxon tests each comparing the four conventional microarray gal:glu ratios with one of the 8 sets of four reconstructed ratios, a total of 50312 tests, using SPLUS 6.0 (Insightful: Seattle). All p-values were computed exactly from rank sums except for 202 which involved rank ties (0.4%). We then found the fraction, for each of the 8 comparisons, the fraction of tests for which P <= 0.05:
|comparison||fraction with P<=0.05|
In theory, for equivalent distributions, the percentage of tests for which P<=0.05 should be ~5%. If the reconstructed and conventional microarray ratios had different central locations, we would expect to see fractions > 5%. Here all fractions are ~1%, lower than expected. Part of this may be explicable by the small numbers of values in each series. Given two series of four values, all assumed unequal, one of the series of four can have only 70 possible rank sums and a two-tailed test can only be rejected at P <= 0.05 at the two extreme rank sum values. Thus, the critical region for this test actually comprises only 2/70 =~ 2.9% of the distribution. The fact that the actual fractions are still all less than 2.9% suggests some possible additional bias in the distributions. Nevertheless, these results are also still consistent with hypothesis that the reconstructed and conventional microarray ratio distributions are equivalent.
For Figure 4 in the article, Supplemental table V (on which Figure 4 was based), and our Coefficient of variation analysis, reconstructed ratios were derived as follows: From four independent masliner-adjusted gal:oligo abundance measurements for each microarray spot and four independent glu:oligo abundance measurements for each microarray spot, we computed averages and standard deviations for all 24 possible sets of four gal:oligo/glu:oligo and gal:oligo/(gal:oligo+glu:oligo) ratios, where in each set of four ratios each gal:oligo and glu:oligo measurement was only used once. Reconstructed gal/glu and gal/(gal+glu) ratios were computed as the average of the 24 averages, and the standard deviations of the reconstructed gal/glu and gal/(gal+glu) ratios were computed as the average of the 24 standard deviations. This method of estimating gal/glu and gal/(gal+glu) ratios was used to avoid basing estimations of ratios and errors on non-independent series of measurements; further discussion of these issues and some alternative computations are given above in Reconstructed ratio comparisons.
For Figure 4 in the article and its data source Supplemental table V, errors for reconstructed ratios are given as standard errors of the mean based on the standard deviations given above. Since each of the 24 standard deviations is based on a set of four values, the standard error of the mean is therefore computed as 1/2 of the average standard deviation for each spot.
For Coefficient of variation analysis, coefficients of variation are computed as the reconstructed standard deviation values divided by the reconstructed ratio values (described above), and adjusted for bias as described in Sokal and Rohlf 1995 by multiplying by 1+1/(4n). Here n = 4. This bias adjustment is also applied to the coefficient of the conventional microarray ratios analyzed in this section.
Click here to download experimental data associated with this article. We have provided all unadjusted GenePix 3.0 image analysis files and the masliner-adjusted GenePix files that integrate all the linear ranges for the complete series of scans performed for each array. We have also provided a MIAME-compliant description of these data files.
When assessing whether a gene exhibits an increased or decreased expression level using masliner-adjusted abundance values based on a calibrated oligo reference, an increase in statistical error should be expected compared to assessments based on conventional microarray ratios because assessments based on the abundance values require an extra aggregation of error. For instance, for a conventional (gal/glu) microarray ratio from a galactose (gal) experimental sample against a glucose (glu) reference sample, error in the gal/glu ratio derives from variance in both the numerator and the denominator. For the abundance comparison, the separate gal:oligo and glu:oligo measurements derived from the experimental samples and reference oligo are each associated with effectively the same kind of error as the conventional gal/glu ratio, as, indeed, each of the gal:oligo and glu:oligo measurements are essentially conventional microarray ratios except for their use of a calibrated reference. At this level, the error profiles of conventional ratios and masliner-derived abundance values should be approximately the same. However, the conventional microarray ratio results in an immediate comparison of the galactose and glucose expression levels for a gene, while for the abundance measurements a further computation is required to set up a statistical test for equality, e.g., we may have to compute gal:oligo-glu:oligo and test this difference against 0, or gal:oligo/glu:oligo and test this ratio against 1. This further computation involves an additional aggregation of error that is not required in the conventional case.
We sought to verify the existence of this error by computing the coefficient of variation for 6266 spots on our microarrays for both conventional and reconstructed ratio values for galactose vs. glucose. Details on our computation of coefficient of variation may be found here. We performed this exercise for both gal/glu ratios, the form of conventional microarray ratios, and for gal/(gal+glu) ratios. The rationale for considering the latter form of ratio is that, unlike the former, it is bounded between 0 and 1 and should therefore be subject to less error overall. The results, seen below, verify the expected increase in error for the abundance values based on the calibrated oligo reference for both sorts of ratio.
In these images, light blue dots correspond to reconstructed ratios, and light tan spots to dots to conventional ratios. The eight genes considered in Figure 4 in the article text and in Supplemental table V are highlighted with labeled spots of different shapes; the dark blue spot for each gene corresponds to its reconstructed value and the dark red to its conventional value. Each image is presented as a scatter plot of coefficient of variation against the log10 average of the four glu:oligo values for the gene, representing the degree to which the gene is expressed in glucose. Genes that are not expressed much in glucose are therefore on the left, while genes that are highly expressed in glucose are on the right. Clicking on either of the thumbnail images will bring up a large version of the image.
Left panel: Coefficient of variation of gal/glu conventional and reconstructed ratios against glucose expression level for 6256 genes. Right panel: Coefficient of variation of gal/(gal+glu) conventional and reconstructed ratios against glucose expression level for 6256 genes.
The average coefficient of variation +/- standard deviation for gal/glu ratios over all 6266 spots is 0.83+/-0.38 (reconstructed) vs. 0.42+/-0.20 (conventional), while for gal/(gal+glu) ratios it is 0.33+/-0.14 (reconstructed) vs. 0.18+/-0.10 (conventional). Therefore, on average, reconstructed ratios have a 2.0-fold higher coefficient of variation for gal/glu values compared to conventional ratios, and a 1.8-fold higher coefficient of variation for gal/(gal+glu) values.
Note that despite the overall increase in error for reconstructed vs. conventional ratios, many individual genes have reconstructed errors less than their conventional errors, including some of the genes in Figure 4 of the article and Supplemental table 5.
In the article, we described how the loss of abundance information in conventional microarray ratios may lead to less sensitive analysis of differences in gene behavior compared to microarray abundance data derived from calibrated oligo references. In particular, we showed that the different behavior of a set of galactose-induced and a set of mitochondrial genes could not be seen in a conventional raffinose vs. ethanol (Raff:EtOH) microarray ratios, but can be seen in the microarray abundances derived from raffinose vs. common oligo reference (Raff:oligo) and ethanol vs. common oligo reference (EtOH:oligo) hybridizations. In particular, the Raff:EtOH ratios show genes of both classes as being effectively unchanged between the two conditions (ratios =~ 1), while the Raff:oligo and EtOH:oligo abundances show that this is because the galactose-induced genes are equally unexpressed in either condition, while the mitochondrial genes are equally moderately expressed in both.
To show how the availability of abundance information affects clustering, we clustered these 10 genes in two ways (click on thumbnails to get larger images):
Left panel: Clustering of 10 galactose-induced and mitochondrial genes by conventional Raff:EtOH ratios (data and genes from Table I of article). Right panel: Clustering of these same 10 genes by Raff:oligo and EtOH oligo (data also in Table I of article).
As can be seen, and as expected, clustering of the Raff:EtOH ratios is unable to separate the galactose-induced genes (GAL1, GAL2, GAL3, GAL7, GAL10) from the mitochondrial genes (COX5A, COX7, COX12, QCR6, QCR7) (see left panel), whereas clustering of Raff:oligo and Raff:EtOH abundance values separates them successfully (right panel). We do not take these results to indicate any inherent limitation in clustering conventional microarray ratio data, as the different behavior of these two sets of genes could be revealed by inclusion into the clustering of additional sets of ratios involving other conditions (such as galactose vs. glucose ratios). What we do suggest is that the different behavior of these two sets of genes was already exhibited in the two conditions examined, raffinose and ethanol, without the need for additional conditions, but that use of conventional ratios obscured the ability of clustering to detect it. Clustering was performed in SPLUS 6.0 (Insightful: Seattle) using Ward's algorithm (Everitt 1980).
In theory, using a universal oligo as a microarray control against experimental samples should generate microarray ratios that are proportional to the absolute abundances of mRNA transcripts. How close are they actually? We attempted to address this question by comparing ratios between masliner-adjusted ratios against universal oligo controls of log-phase S. cerevisiae cells grown on glucose with SAGE-derived estimates of mRNA transcript counts/cell grown in comparable conditions (see Velculescu, Zhang et al. 1997).
To perform this comparison, we downloaded SAGE-derived mRNA transcript counts/cell from ExpressDB for 2442 genes for which these counts were based on unambiguous SAGE tags (see Aach, Rindone et al. 2000) and unambiguously associable with ORF names in our log-phase glucose condition microarray experiments. We eliminated from this list 149 genes whose spots on our microarray experiments failed PCR quality tests. (Go here for a list of spots that failed PCR quality tests.) A list of the remaining 2292 genes, their estimated transcripts/cell from Velculescu, Zhang et al. 1997, and the average of their normalized masliner-adjusted intensity values in 4 microarray hybridizations of log-phase cells grown in glucose vs. our universal oligo standard ("Glu/oligo ratio") can be found here.
Though both the SAGE- and microarray-derived values should estimate transcript abundances, we expect only a moderate correlation between them because:
As can be seen, substantial scatter is evident in the SAGE- vs. microarray-derived putative abundance values, and the Pearson correlation coefficient between these two series is a modest but statistically significant (see Note) 0.62. This correlation is due mainly to genes that appear to have high abundance in both data series (see right panel in figure above, 140 genes, correlation = 0.60, also statistically significant -- see Note). Genes were here considered "highly abundant" if they had both a SAGE transcript count and a Glu/oligo ratio >= 90th percentile in their distributions (6 transcripts/cell and a Glu/oligo ratio of 3.01, respectively). By contrast, the Pearson correlation between SAGE and microarray-derived abundance values for the remaining 2152 genes is low at 0.08 (which, however, is still significantly significant -- see Note). The difference between these correlations is presumably due to increased Poisson noise of genes with low SAGE tag counts and the noise associated with genes with low microarray spot intensities.
For purposes of comparison, we also performed a correlation analysis between SAGE-estimated transcript counts and a set of conventional microarray ratios using these same genes. Because absolute abundance information is lost by conventional ratios, we expected that we would find low correlations between SAGE and conventional ratios compared to SAGE and common oligo reference abundances. For the conventional ratio data set, we used the average of the glu:gal ratios from our set of four replicates of a glucose vs. galactose microarray experiment (the averages and their standard deviations are reported here). The Pearson correlation coefficient between the SAGE transcripts/cell and the average glu:gal ratios for the entire set of 2292 genes was a surprisingly high 0.24 vs. 0.62 for the common oligo reference correlation. This 0.24 correlation coefficient is statistically significant compared to 0 (see Note); however it is also statistically signficantly less than the common oligo reference correlation of 0.62 (P < 0.0004; see Note). The Pearson correlation coefficient for the 91 genes that had SAGE and glu:gal ratios >= the 90th percentiles in each series was -0.13 vs. 0.60 for the common oligo reference, indicating that the 0.24 correlation does not reflect the genes with the strongest signal in the two experiments. These results confirm the hypothesis that expression levels based on a common oligo reference retain absolute abundance information significantly better than expression levels based on conventional microarray ratios.
As we noted in the article, we anticipate that use of equimolar oligo reference standards (mixtures of oligos unique to each transcript probe) will improve transcript abundance measurements by reducing the differences between hybridization characteristics of common oligo references and sample target sequences to the probe sequences in microarray spots. Once such equimolar oligo reference controls are available, we hope to repeat this comparison with SAGE or, better yet, to perform a new comparison with SAGE or MPSS (Brenner, Johnson, et al. 2000) data collected on the same RNA sample assayed on a microarray using an equimolar oligo reference standard.
The domination of correlation coefficients by the most abundant gene products has been analyzed in another context in Gygi, Rochon et al. (1999).
The masliner strategy of integrating the linear ranges from a series of scans of the same slide at different scanner sensitivites relies on computation of linear regressions between the linear range BSIs of successive scans. An example of regression-based masliner processing is shown below. Here we discuss a number of observations concerning the analysis of scans of the same slide including scatter plots, regression noise and sensitivity to outliers, mean vs. median pixel intensity, sublinear range bias.
A single array was scanned twice on a ScanArray 5000 at constant laser power (90%) but two PMTG settings (50% and 60%). For each spot, the BSI from the 50% scan (s50) is plotted against its BSI from the 60% scan (s60) (,,). Six spots are saturated in s60 that were not saturated in s50 (). Using masliner, a linear regression was computed for the 176 s50 vs. s60 values within the linear range of the scanner (, 2,000 <= BSI <= 60,000). The regression was used to extrapolate values for the six s60 saturated spots (), effectively generating an extended common linear scale for spots over both scans. One spot saturated in s50 () had been extrapolated to a common linear scale with s50 through an earlier scan (45%) and masliner processing. This spot was carried into a common linear scale by masliner processing of s50 and s60 (). Some spots (N=6,185) were below the linear range in all three of these scans (). 3,288 of these spots were brought into the common linear scale by additional scans at higher gains and executions of masliner. A series of masliner adjustments using data from 4 scans increased the intensity of the brightest spot on this array from 65,535 (saturation) to 3.2 million units with a 0.4% estimated error associated with the linear regression. This data was taken from the Glu:oligo experiment presented in Table 2 of the article. Figure 1 of the article is a slightly-edited version of this figure in which data for the spot saturated in s50 (,) was removed and axes scales and legends were changed. (Click on image for larger view.)
Below are scatter plots of linear range BSIs from one scan against the next higher PMTG scan for the same slide. Spots were considered to be in the linear range if their BSIs were between 2000 and 60000 inclusive. Here BSI measurements are given by median pixel intensity minus background as given in GenePix 3.0 output (as used by masliner). The figures on the left (Slide 11A) are from a PCR-product spotted microarray in which the Cy5-labeled sample was yeast grown in glucose and the Cy3-labeled sample was yeast grown in raffinose. The figures on the right (Slide D2) are from an Operon yeast oligo spotted microarray in which both the Cy5- and the Cy3-labled samples were yeast grown in glucose. Both Cy5 and Cy3 images are given for each slide. In general we find that scatter appears to increase as BSI increases within each scatter plot, and also that more scatter appears at higher PMTGs. This makes sense if noise is amplified along with signal as amplification gain increases. Clicking on any image will bring up a larger version.
|Slide 11A||Slide D2|
Along with noise, outliers also appear in these scatter plots. We considered outliers from the several related perspectives: (a) the degree to which outliers may have affected masliner regressions, and also outlier bias, (b) whether outliers are associated with any specific causes such as spot morphology. Regarding outlier bias in (a), visually it appears that outliers tend to lie to the right of the dense linear body of points approximated by the regression in each graph.
(a) Impact of outliers, outlier bias: We took the Cy5 data from the 1492 spots
corresponding to genes in the linear range of the Slide 11A scans at
PMTG 70 and 85 (lower left scatter plot above) and
performed the following computations: We computed the linear regression
of the PMTG 80 values (Yi) against
the 70 values (Xi), yielding a computed slope
m(0) and intercept b(0) such that the sum of squares of residuals
|Regression slope||Regression intercept|
As can be seen, the regression slope curve increases from a minimum value of ~3.11 at 0 outliers to a local peak. The location of this peak is ~3.48 at 75 outliers removed. Regression slope then drops and tends to rise slowly as additional outliers are removed. Similarly, regression intercept drops from an initial maximum of ~928 to a local minimum of -433 at 75 outliers removed, and then begins to rise. The initial sharp rise in regression slope and increase of regression intercept appears to confirm the visual observation that outliers are biased to the right of the main body of points, as outliers to the right tend to drag down the slope and raise the intercept. This is also seen by the direct observation that 13 or the first 20 outliers in the first regression (resulting in m(0) and b(0)) have negative residuals (their points therefore lie to the right of the main body of points).
However, the degree of influence of outlier removal on slope and intercept is modest. As currently designed, masliner performs no outlier removal and therefore it computes adjustments based on the slope and intercept for the complete set of points (outliers removed = 0). Assuming that outlier removal could be reasonably limited to removal of outliers among the first 200 points (13.4% of the 1492 points), the slope used by masliner is a maximum of ~12% too small (100*(3.48-3.11)/3.11). Similarly, the intercept used by masliner is at most 1361 too large (928-(-433)). As masliner's extrapolation m(0)*Xi + b(0) is performed mainly on larger Xi values (as adjustments are extrapolated only for spots that saturate in the next highest PMTG scan), the impact of the high intercept value used by masliner is likely to be small compared to the impact of the small slope value, so we would expect the presence of outliers on masliner to be about 12%.
These expectations are confirmed by the following two graphs. The graph at the left shows the value of m(j)*X + b(j) for j between 0 and 200, for various values of X within the linear range. The ordinate of this graph is given as the ratio of the extrapolated X to the initial X (see graph legend), rather than extrapolated X alone, in order to accommodate differences of scales of the extrapolated X. It is clear that the effect of regression outliers is the same for all values of X except the very lowest values. This behavior is explained by the fact that only for low X values does the regression intercept have a significant impact on extrapolation, as described above. The graph at the right shows the maximum impact of outlier removal on extrapolation for each of the X values exhibited in the graph at the left. The ordinate of this graph is the maximum possible percent change in the extrapolation from X over all the regressions involving removal of j outliers from j = 0,1,...,200. The maximum possible percent change is 100*(max(m(j)*X+b(j))-min(m(j)*X+b(j))/min(m(j)*X+b(j)) where each max and min is over this range of j. As can be seen, this maximum possible effect is ~ 11%, as anticipated.
As masliner is run iteratively over a series of scans, a larger cumulative effect should be expected. E.g., if masliner-adjusted values are 10% low, and masliner is run 3 times, than BSIs that were adjusted 3 times would be approximately 0.93 =~ 73% what they should be, while BSIs that were adjusted twice would be approximately 80% of what they should be, etc.
The conclusion which we have drawn from this analysis is that regression outliers do affect masliner adjustments, but that their impact appears to be modest. We plan to keep an eye on this effect and will consider implementation of outlier elimination or robust regression logic in masliner as we gather experience with it.
(b) Causes of regression outliers: To understand the possible causes of regression outliers, we selected a number of outliers and non-outliers from Slide 11A and Slide D2 scatter plots and examined the images of the spots using GenePix 3.0. We thought it possible that regression outliers might be associable with irregular spot morphologies. However, when we examined these spots, we found no clear indication that the spot morphologies of outlier spots were different from those of non-outliers. One outlier spot did happen to be a spot that had been marked as an error, but the error concerned the fact that the spot was on the edge of a grid that overlapped another grid; the spots on these overlapping grids did not themselves overlap and appeared to have morphology common to normal spots.
As noted above, outliers appear to be biased towards the right. An outlier to the right of the main body of points in a regression has a BSI in a lower PMTG scan that is too high, compared with most spots, than its BSI in the next highest PMTG scan. This could either be due to (i) pixels in the 'low' scan being too bright compared to the 'high' scan, or (ii) pixels in the 'high' scan being too dim compared to the 'low' scan. On suggestion of Jean O'Malley of Oregon Public Health and Sciences Universally (and whom we gratefully acknowledge with thanks for this suggestion), we explored whether pixel saturation in the 'high' scan could be a cause of (ii). Examination of pixel plots in GenePix 3.0 for selected spots did reveal that some 'high' scan pixels do appear to saturate, but we do not believe that this leads to (ii) because we have measured BSI by median pixel intensity minus background: For pixel saturation to artifactually lower BSI measured by the median would require over 50% of the pixels to become saturated, which we did not observe, and the result would be that the 'high' scan BSI would be saturated and therefore not in the linear range. However, the effect of pixel saturation appears to be visible when BSIs are measured by the mean pixel intensity minus background, as seen below.
Finally, we also note that we have found one case of outliers which appear to be so because they appear to be abnormally bright on a lower PMTG scan (therefore instances of situation (i) above), rather than because they are artifactually dim on the higher PMTG scan: In particular, the point for gene YDL130W appears to be an outlier on the left in the Cy5 scatter plot for Slide 11 for PMTG 60% vs. PMTG 70%, and an outlier on the right in the Cy5 scatter plot for Slide 11 for PMTG 75% vs. PMTG 80% (lowest two scatter plots on the left edge above). This suggests that this spot was artifactually bright in the PMTG 70% scan.
We conclude by noting that at this point, based on the preliminary observations above, we do not have a hypothesis as to any mechanisms that may be associated with regression outliers. However, it does appear that there can be considerable scanner-based noise in BSI measurements, and one interesting but yet unperformed experiment would be to rescan a slide at the same PMTG and laser power to see if the same spots were found to be outliers in multiple scans. To the extent that the presence of scanner-based noise is confirmed, we foresee another potential advantage in the masliner strategy: the use of the multiple scans to estimate scanner noise in BSI measurements for each spot.
As noted above, pixel saturation does not appear to explain the bias of regression outliers to the right. However, the effect of pixel saturation appears clearly visible when BSIs are measured by the mean pixel intensity minus background, as seen in the following Cy5 and Cy5 scatter plots for BSIs measured by mean pixel intensity for Slide 11A. Note that there appears to be a curvilinear distortion in the main body of points in these scatter plots compared to their median pixel intensity counterparts (here). Such a curvilinear distortion would be accounted for by pixel saturation in the 'high' scan, which would tend to lower the mean pixel intensity for points to the right in the graph. By contrast, the in median pixel intensity plots, the main body of points appears to lie along a straight line until saturation (although some curving appears in the highest PMTG scatter plot for Slide D2 Cy3).
This observation supports our use of BSIs based on median pixel intensity by masliner.
|Slide 11A Cy5 Mean-based BSIs||Slide 11A Cy3 Mean-based BSIs|
On examining the scatter plots above, one might think that the linear range of a scanner is bounded only above by saturation, and that there is no reason for masliner to operate with a linear range that is also bounded from below. However, the presence of curvilinear bias in the low range of acquired intensities is clearly visible in scatter plots of log intensities (log-log scatter plots). Below is shown a set of log-log scatter plots for slide E2 for both Cy3 and Cy5, one of the slides used in the spiked oligo experiment described in our article. These plots are based on median pixel intensities and not background-subtracted intensities (BSIs) of spots in the arrays to show that the curvilinear bias is not caused by background subtraction itself. (Log-log scatter plots based on BSIs look very similar.) Also, this spotted oligo-based array contains many empty feature positions that are not spotted with DNA oligos for gene products. The plots below show these positions in light blue vs. the dark blue used for gene-based spots. Dots on the scatter plot for these empty positions are intermixed along with the dots for gene spots, thereby showing that the curvilinear effect comes about wholly through operation of the scanner itself and is not dependent on the contents of the spots on the array. The presence of this curvilinear bias justifies the need to specify and use a lower linear range by masliner. Note that any feature for which the Cy5 or Cy3 median pixel intensity <= 0 does not appear on the scatter plots.
|Slide E2 Cy5 Median feature intensities||Slide E2 Cy3 Median feature intensities|
An interesting way to view masliner operation is through RI plots (Quackenbush, J. 2002; essentially the same as MA plots (Yang, Y.H., Dudoit, S., et. al. 2002)). These are plots of log(Cy5 BSI)-log(Cy3 BSI) against log(Cy5 BSI)+log(Cy3 BSI) over all spots on an array. Shown below are four scatter plots of slide E2, one based on masliner-adjusted data and the other three on pairs of single Cy5 and Cy3 scans of the slide. Slide E2 is one of the arrays used in the spiked oligo experiment described in our article. The hybridization mixture for slide E2 is a self-self hybridization (Cy5-glucose vs. Cy3-glucose) to which a 5-fold dilution series of Cy5-labeled oligos and an equimolar mixture of corresponding Cy3-labeled oligos hybridizing to 9 spots were added (see article). For all other spots Cy5/Cy3 ratios are close to 1 while, for these spots, all ratios but one differ widely from 1 (for ratios based on slide E2 and replicate F2, see supplementary Table I). Thus, spots associated with spiked oligos tend to appear as outliers in these RI plots. As described in the article, masliner-adjusted data for this experiment has comparable or better accuracy, consistency, and completeness compared to data derived from individual scan pairs. The plots below show how masliner extends the signal range for the arrays, avoids saturation, and exploits different scans for the accurate measurement of each spot. To prepare these plots, only array features for which both the Cy5 and Cy3 BSIs were > 0 were used. Cy5 and Cy3 BSIs were then both total-normalized to 20000000. Clicking on an image displays a larger version.
This RI plot is based on a single Cy5 scan at PMTG 65% and a single Cy3 scan at PMTG 75%. As described in the article, data obtained from these scans were closest in accuracy to corresponding masliner-adjusted data. Eight of the 9 spots targeted by spiked oligos can be identified in the plot. The sharp diagonal edge at the right of the plot corresponds to spots saturated in the Cy5 scan.
This RI plot is based on a single Cy5 scan at PMTG 75% and a single Cy3 scan at PMTG 85%. As described in the article, data obtained from these scans were less accurate, consistent, and comprehensive than the corresponding masliner-adjusted data. By virtue of the higher scan PMTG settings compared to those for the plot above, more spots are raised above background levels than above (see article). Here, six of the 9 spots targeted by spiked oligos can be identified in the plot. A larger sharp diagonal edge at the right indicates that more spots saturated in this plot compared to above.
This RI plot is based on masliner-adjusted scan series of four Cy5 and Cy3 scans, including the scans used above. Dots in the RI plot corresponding to array features are represented by different colored symbols depending on which Cy5 and Cy3 scans were used by masliner to pick up BSIs for the features; since these depend on spot brightness in the two channels, the result is that the RI plot is broken up into different regions. For instance, the region with legend "75_85" comprises array features picked up in the Cy5 75% and Cy3 85% scans; other legends are interpreted similarly. The "75_85" region uses the same scans that were processed as a pair above and are represented by the same symbol as in that RI plot; similarly, the "65_75" region uses the same symbol as used for the Cy5 65% and Cy3 75% pair above. The "75_85" region represents the great number of dim spots on the array whose BSI values were picked up by these highest PMTG scans. The masliner plot does not exhibit the saturation noted for the individual scan pairs above (i.e., there is no downward diagonal edge at the right). Instead, the different colors to the right show how saturation has been corrected by incorporation of other scans. The masliner plot also extends to an ordinate of > 10.5 compared to ~9.5 for the individual scan pairs due to the extended signal range computed by masliner. The different symbols for the 8 spots corresponding to the spiked oligos shows how masliner has leveraged different scans towards the accurate computation of these features, whose Cy5 and Cy3 values may differ by orders of magnitude.
This RI plot is based on a pair of lower intensity Cy5 and Cy3 scans. Because some Cy5 or Cy3 values were measured with a BSI = 0, some spiked oligo features do not appear on the plot. The main reason we have included this plot is to indicate that not all RI plots show the upward curve to the left, and that masliner's correction for saturation and construction of a common extended signal range does not correct for this effect. The cause of this upward curving is unknown but may be related to higher background levels on brighter scans. While the masliner plot appears to have less upward curvature to the left than the plot for scan pair Cy5 65% and Cy3 75% above, this is only because masliner points at the left of the masliner plot came from higher power Cy5 75% and Cy3 85% scans that exhibited this effect to a lesser degree.
masliner stands for MicroArray Spot LINEr Regression.
Go here to obtain a copy of the masliner program.
Please cite usage of masliner with
Dudley, A., Aach, J., Steffen, M., and Church, G.M. (2002) Measuring absolute expression with microarrays using a calibrated reference sample and an extended signal intensity range, Proc Natl Acad Sci USA 99(11): 7554-7559
Basic help messages describing parameters may be obtained by executing masliner without any parameters. This section provides a general orientation to masliner use.
General orientation: masliner operates on a pair of GenePix scan analysis files that represent scans at different PMTGs of the same array. The files must be in the same format and describe the same spots in the same order. The two files are specified by the -g1 and -g2 parameters. In general, masliner parameters that relate to the input files come in pairs with a common prefix and a suffix that is either 1 or 2.
Each of the GenePix scan files indicated by -g1 and -g2 contains columns for measurements and calculations reported by the GenePix software. These include measurements of mean and median pixel intensity, background levels, and others. Each execution of masliner operates on a single fluor from each file. To process both fluors therefore requires two executions of masliner each of which produces an output file containing adjusted BSIs for a single fluor.
When processing two GenePix files for a single fluor, masliner
works with the columns for median pixel intensity and median background
intensity. Median pixel intensity is given in a column whose header
begins with "Median F", and median background intensity in a column whose
header begins with "Median B". The suffixes vary from scanner to scanner
and also on GenePix options. For the GSI Lumonics ScanArray 5000 and
GenePix options used in our study, the suffixes corresponding to the two
fluors were simply "1" and "2", so that the column headings masliner
examined were "Median F1" and "Median B1" for fluor "1", and "Median F2" and
"Median B2" for fluor "2". For a collaborating group testing masliner using
an Axon scanner and other GenePix options, the column heading suffixes for
the median pixel intensity and background intensity for the two fluors were
"635" and "532", so that their GenePix files contained the columns
"Median F635" and "Median B635" (and similarly for 532). To run masliner
properly it is necessary to determine what column heading suffixes are used in
the GenePix output files for the options and fluor you wish to process, and
feed it to masliner via the "-f1" and "-f2" parameters. Here "-f1"
is the fluor specification for the "-g1" input file and "-f2" the fluor
specification for the "-g2" input file. The default value for these fields
is simply "1" (suitable for the "Median F1" and "Median B1" columns that were
generated for the files we generated). Thus, a sample command for processing
a file using the Axon scanner and GenePix parameters that our collaborators
used would begin:
Regression modes: As described in the article, masliner adjusts intensity values in the file that corresponds to the higher PMTG that appear to be affected by signal saturation. Adjustments are based on linear regressions of intensities of the higher and lower PMTG scans which are both within the linear range of the scanner. To perform these regressions, masliner must be told or determine the linear range of the scanner. There are two options, selectable by means of the -m (mode) parameter.
Output file: masliner output consists of an adjusted copy of the higher PMTG input file. All adjustments are made in a set of four extra columns for each spot at the end of columns copied from the input file.
Running masliner on a series of scans: To derive a common linear scale for an entire series of scans taken at different sensitivities, process the two lowest PMTG scans through masliner first. Because the lowest power scan measures the most high-intensity spots in the linear range, starting the algorithm with this scan brings as many of the brightest spots on the array as possible into the common linear scale. Spots that are saturated in this lowest PMTG scan cannot be accurately measured and have SATURATION-FLAG set to 1 in the output file. Next, process the adjusted output file generated from the first two scans as an input file against the next highest PMTG scan, and repeat this procedure until all scans have been processed. Once a spot has been adjusted in one regression, masliner will automatically adjust it for each subsequent regression. As each iteration of masliner computes a regression of linear range spots on one scan against the same spots as measured on a higher PMTG scan, which have an increased intensity, the regression slope is always greater than 1. Thus, each application of masliner always results in increased values for all previously adjusted spots, and the common linear scale grows at its upper bound as more scans are processed. masliner increments ADJCOUNT each time it adjusts a value, and aggregates REGERR as described in Technical notes below.
Other usage notes:
Regression statistics are computed based on chapter 14 of (Sokal and Rohlf 1995).
Regression quality: We define regression quality as square
root of the unexplained mean square error of regression, divided by
the average value of the regressed quantity. In the notation of
(Sokal and Rohlf 1995) this is
Regression error: masliner's reported REGERR values
are based on the standard error of linear regression prediction
(here called SELRP) as defined in
(Sokal and Rohlf 1995):
As masliner can take an adjusted value computed from one
linear regression and use it as the Xi of another
linear regression, SELRP values must be
aggregated for each iteration. A question arises as to the
appropriate formula for computing the iterated regression error.
We compute REGERR for an iterated regression as
REGERR is intended to provide an estimate of error introduced by the masliner regression procedure and background noise internal to a single hybridized array. Evaluation of the significance of this error using standard distributions is limited by the facts that (a) linear regression parameters and variates are not necessarily independent as assumed by this formula and (b) the unexplained error of regression does not necessarily follow a normal distribution. Because REGERR deals only with regression error and background for a single hybridized array, it underestimates the error across replicate hybridizations, replicate arrays, and replicate experiments. Therefore error analyses in the article and above have focused on errors empirically estimated across replicates. If an appropriate error model for such replication-based error can be determined, it may be integrated with REGERR by assigning the error predicted by this model to SELRP0 and proceeding with the recursions described above.
Derivation of formula for regression error: We start with
as taken from Sokal and Rohlf 1995, and want to find
Focusing on the last term:
The simplifications in the last step arise because and are both constants so far as the E operator is concerned. ( is constant because we are looking at expectations under the Xi distribution: By assumption, we have assumed that these values are independent of the parameters of the current regression, and is a parameter of the current regression.) If the current value of Xi was derived from the k-1st regression, we then have
If you have additional questions or issues, please contact John Aach or Aimee Dudley.