BACKGROUND
genome-wide expression analysis is an increasingly important tool for identifying gene function, disease-related genes and transcriptional patterns related to drug treatments.
microarrays enable the simultaneous measurement of the expression levels of tens of thousands of genes and have found widespread application in biological and biomedical research.
increasing numbers of multi-class microarray studies are performed, but the vast majority continues to be two class  studies, for example when both control and a treatment are examined
<cit> .
the objective of the study in most of them, is to determine the genes that are differentially expressed between the two classes.
differentially expressed genes are usually detected using statistics based on means or medians.
however, if there are genes differentially expressed on different subclasses, those techniques do not select them because either mean or median values tend to be similar between the considered groups.
genes with a bimodal or a multimodal distribution within a class  may indicate the presence of unknown subclasses with different expression values
<cit> , meaning that there are two separate peaks in the distribution; one peak due to a subclass clustered around a low expression level, and a second peak due to a subclass clustered around a higher expression level.
as a consequence, the identification of such subclasses may provide useful insights on biological mechanisms underlying physiologic or pathologic conditions.
in cancer research, a common approach for prioritizing cancer-related genes is to compare gene expression profiles between cancer and normal samples, selecting genes with consistently higher expression levels in cancer samples.
such an approach ignores tumor heterogeneity and is not suitable for finding cancer genes that are overexpressed in only a subgroup of a patient population.
as a result, important genes differentially expressed in a subset of samples can be missed by gene selection criteria based on the difference of sample means
<cit> .
the particular application that motivated our work concerns the development of a methodology which could simultaneously identify up- and down-regulated genes and differentially expressed with bimodal or multimodal distributions with similar means on both groups.
for convenience, the latter case is referred to as special genes.
different statistical tests have been proposed to select differentially expressed genes
<cit> .
among them, is the receiver operating characteristic  analysis, which is widely used to evaluate a diagnostic system but can be interpreted as a measure of separation between two distributions.
a roc curve displays the relationship between the proportion of true positive  and false positive  classifications resulting from each possible decision threshold value in a two class classification task
<cit> .
these proportions depend on the classification rule and in general higher values of the marker are associated with the case group.
however, if roc analysis is blindly applied to select genes differentially expressed, i.e., keeping the same classification rule for all genes in an experiment, not proper roc curves
<cit>  can be produced because genes with positive and negative regulation have opposite classification rules.
nproc curves are obtained when they cross or are below of the reference line .
genes can be ranked using the area under the roc curve
<cit> , a common measure of discrimination, which should range between  <dig>  and  <dig>  but for nproc curves auc can have values below  <dig> .
nevertheless, different scenarios can lead to nproc curves, for instance, when the means of the two groups are similar and one of the groups has a bimodal distribution  , or when both distributions are unimodal with similar means and significant different variances .
on both cases the corresponding roc curve will have a sigmoidal-shape with an auc close to  <dig> .
proper binormal model
<cit>  and contaminated binormal model
<cit>  are methods that force the roc curves to be set above the reference line when they are not proper and consequently the auc will be higher than  <dig> .
however in the context of this work, not proper roc curves have an important role in the selection of different kinds of differentially expressed genes.
since it is not possible to decide beforehand the direction of the classification rule, we considered the same classification rule for all of the genes, i.e., values of expression levels above the threshold correspond to up-regulation.
in that sense, auc values near  <dig> will correspond to up-regulated genes, auc values near  <dig> will correspond to down-regulated genes, and special genes  will have an auc around  <dig> .
however, regardless of the type of distributions, if means are similar , auc will be near  <dig> .
so, using auc is not sufficient to select special genes.
we used the overlapping coefficient  to further separate these different situations which produce values of auc near  <dig> .
bradley
<cit>  and inman and bradley
<cit>  promote the use of ovl as an intuitive measure of the similarity between two probability distributions.
graphically, ovl is the area where the densities of the two distributions overlap when plotted on the same axes.
we propose using auc and ovl simultaneously to select different types of differentially expressed genes and plotting ovl against auc we get a picture which we named as arrow plot.
if we consider that groups have different variances, special genes can be mixed with genes which are not differentially expressed as illustrated on figure
<dig>  that is, genes with unimodal densities, with similar means but significantly different variances.
these genes will have auc values around  <dig>  and low ovl values.
with the purpose of identifying genes under these conditions, allowing their separation from the special genes, we developed an algorithm based on finding bimodality  using kernel densities estimates.
nonparametric techniques are used to estimate auc and ovl.
to estimate auc, we used the mann-whitney u statistic
<cit> , which is equivalent to the trapezoidal rule for integration.
for the ovl, we developed an algorithm where a naive kernel density estimator
<cit>  is used to construct a nonparametric estimator of ovl.
we first describe the algorithm and later we evaluate the performance of our method by comparing the gene expression profiles in two different classes using data from a publicly dataset
<cit>  and from a simulated dataset.
the first dataset consists of  <dig> different samples of normal circulating b cells  and  <dig> heterogeneous lymphomas
<cit> .
the gene expression data were obtained on  <dig> genes.
the simulated dataset consists of  <dig> genes generated from a lognormal distribution, where each group sample has  <dig> arrays.
using publicly available data, we compared our results with those obtained by parodi et al.
<cit>  using as methods, the area between the roc curve and rising diagonal  and a test for not-proper roc curves .
we used both data sets to assess the relative performance of our proposed method as compared to the most common different statistical gene ranking measures.
all the analysis were performed using the open-source r software
<cit>  and packages from bioconductor
<cit> .
RESULTS
algorithm description
for illustrative purposes, we divided the algorithm in two parts .
the first part describes the ovl estimation  and the second part describes the selection of different kinds of differentially expressed genes .
the ovl estimation was based on a non-parametric form with densities estimated using kernel functions.
figure
<dig> shows the pseudo-code which implements the ovl estimation and tables
<dig> and
<dig> describe the notation and functions used there.
the ovl values are computed by finding the points that belong to the area of intersection of the two densities  and the jump points between densities, which are estimated by interpolation .
the points are combined into one set and sorted in ascending order .
finally ovl is estimated using a trapezoidal rule considering a non-uniform grid-spacing .
symbols are listed in order of appearance in the algorithm.
functions are listed in order of appearance.
the selection of differentially expressed genes is based on simultaneous analysis of ovl and auc.
the arrow plot is obtained by plotting ovl on abscissas and auc on ordinates.
figure
<dig> shows the pseudo-code which implements the algorithm to select differentially expressed genes based on these two measures and tables
<dig> and
<dig> present the notation and functions used there.
symbols are listed in order of appearance.
functions are listed in order of appearance.
selection of differentially expressed genes with positive regulation  and negative regulation , is made according to arbitrarily selected cutoff points for auc and ovl.
however, auc values are expected to be close to  <dig> for up-regulated and close to  <dig> for down-regulated genes and ovl will have low values on both situations.
selection of special genes is performed in two steps.
the first step consists on the selection of genes with auc values near  <dig>  and low values of ovl .
since the variances on both groups can be different, it is possible to find genes with no-differential expression mixed with the special ones.
accordingly, the second step aims at removing the genes without differential expression, through the bimodality analysis.
bimodality  is analyzed based on the behavior of the ordinates of the kernel based estimated densities of both groups, considering only the gene list that is selected in the first step mentioned above .
the points of both densities are ordered in increasing order of abscissas .
if an ordinate is equal or less than the ordinate immediately after, it is assigned a label  <dig> and  <dig> otherwise .
this allows us to analyze the variation of the density over the observed range.
considering only the points where the function is increasing, if the differences between the ranks of adjacent ordinates is  <dig>  the distribution is expected to be unimodal, otherwise the distribution will be bimodal or multimodal .
to declare a gene to be special it is enough to find bimodality in one of the groups , yet it is of interest to analyze in which group bimodality is observed, and this is possible using different color labels on the arrow plot.
performance and implementation
the running time of the algorithm in a dataset with  <dig> genes, takes less than  <dig> minutes on a  <dig> mhz pentium.
r source code for the implementation of this algorithm is available in additional file
<dig>
lymphoma data
from a total of  <dig> genes, our method selected  <dig> differentially expressed genes, where  <dig> corresponded to up-regulated genes,  <dig> to down-regulated and  <dig> corresponded to special genes.
we used auc≥ <dig>  and ovl< <dig>  to select up-regulated genes, auc≤ <dig>  and ovl< <dig>  to select down-regulated genes and ovl< <dig>  and  <dig> <auc< <dig>  to select special genes.
thresholds were chosen arbitrarily, although an analysis of the the arrow plot  could help on deciding which thresholds to use.
table
<dig> shows the  <dig> selected special genes.
genes are listed in ascending order of ovl, which ranged between  <dig>  and  <dig> .
auc values ranged between  <dig>  and  <dig> .
bimodality was tested on the  <dig> special genes;  <dig> genes have bimodality in the control group,  <dig> genes on the experimental group and  <dig> genes on both.
for the  <dig> special genes, kernel densities and their corresponding empirical roc curves can be analyzed in figure
<dig>  all the selected genes had a sigmoidal-shaped roc curve.
special genes were selected using ovl< <dig>  and  <dig> <auc< <dig> .
e: bimodality in experimental group, c: bimodality in control group and b: bimodality in both groups.
genes are ordered by ascending order of ovl.
among the  <dig> special genes selected list ,  <dig> have an unknown regulatory function.
all the remaining  <dig> genes are related with proteins encoding.
gene3323x  and the gene3388x  are presented in other clones in the same dataset, gene3322x and gene3389x respectively.
alizadeh et al.
<cit>  observed that bcl7a gene can be altered by translocation in lymphoid malignancies.
the biological properties of the  <dig> selected genes are described in the additional file
<dig>
we compared our results with those obtained by parodi et al.
<cit> , where abcr and tnrc statistics were used.
according to the highest tnrc value, a total of  <dig> differentially expressed genes were considered, and  <dig> of them were special genes.
eight of them are considered to be special according to our methodology.
the remaining  <dig> genes of their list have auc and ovl values slightly higher than the considered cutoff points on our study.
however, if we choose threshold values for auc and ovl to catch those genes, we will select  <dig> more special genes.
nine feature selection methods were applied to the full dataset, namely welch t-statistic, fold change , rank products
<cit> , average difference
<cit> , weighted average difference
<cit> , moderated t-statistic
<cit> , intensity-based moderated t-statistic
<cit> , significance analysis of microarrays
<cit>  and area under the roc curve .
we assessed the overlap between gene lists produced by different feature selection methods and ranked lists of differentially expressed genes were produced.
we examined the top  <dig> mostly highly ranked genes, and for all methods the  <dig> special genes selected by our methodology are missed.
simulated data
we simulated ten thousand genes , among which  <dig> were non-differentially expressed,  <dig> were up-regulated,  <dig> were down-regulated and  <dig> were special genes.
analyzing the arrow plot , we considered  <dig>  as threshold value for the ovl.
as for the auc, we classified as up-regulated those genes with auc above  <dig> , as down-regulated genes those with auc below  <dig>  and special genes those with auc between  <dig>  and  <dig> .
in the arrow plot we can observe the distribution of the truly  <dig> differentially expressed genes, and we can conclude that 95% of them were selected by our methodology.
in the first step of the algorithm used to select special genes ,  <dig> genes which were candidate to special genes were selected.
through the second step we found that all of the genes had bimodality in at least one of the groups.
we can conclude that our algorithm for detection of bimodality performed with 100% of accuracy on that list.
roc analysis was conducted to evaluate and compare the performance of the above methods.
we analyzed the performance of these methods regarding the discrimination between differentially expressed genes and non-differentially expressed genes considering two scenarios.
first we studied the performance of the methods concerning the capacity to differentiate among up-regulated, down-regulated and special genes; secondly we studied the performance concerning only the capacity to identify special genes.
the construction of the roc curves were based on the absolute values of the following statistics: fc, ad, wad, rp, welch-t, sam, samroc, ibmt, modt and shrinkt, where high values are related to de genes.
the roc curve for the auc method was constructed considering auc values ranging from  <dig>  to 1; in this way, any auc value below  <dig>  was substituted by its complementary value, i.e., by 1−auc.
high auc values are related to de genes.
when analyzing the arrow plot, we verified that only the ovl statistic is needed since lower values of the ovl correspond to de genes.
the empirical roc curves, under the first scenario are represented in figure
<dig>  and the respective empirical auc values are displayed on table
<dig>
comparison of auc values where the goal is to select up- and down-regulated genes and special genes.
the auc values are sorted by decreasing order.
the ovl with an estimated auc value near of the unit showed to be the one with a better performance followed by the rank products method.
the method with lowest performance was samroc, however all methods showed high values of performance.
considering the scenario where the goal is to select only special genes, the empirical roc curves  and the empirical auc values  showed that ovl was the method with better performance followed by the fc method, however with an auc value considerably low.
wad and shrinkt were the methods with the lowest performance.
comparison of auc values where the goal is to select special genes.
the auc values are sorted by decreasing order.
CONCLUSIONS
we have presented a graphical and computational method for microarray experiments which allow the identification of genes that express differently under two conditions even if the behavior in average is similar.
the main objective of this work was to select differentially expressed genes due to the presence of different subclasses, which could give important information about their inherent biological functions, and that are usually missed by usual methods.
auc and ovl statistics were used to achieve this goal.
both statistics are invariant when a suitable common transformation is made on variables
<cit> , and on microarray data analysis log transformations are widely used.
arrow plot is obtained by plotting ovl against auc.
this plot is easily interpreted because both statistics range between  <dig> and  <dig>  and in addition to detecting genes with up- or down-regulation, arrow plot is also able to detect special genes, however for the latter genes a bimodality analysis needs to be added.
the approach used by the arrow plot is similar to the volcano plot, in the sense that two selection criteria are needed to select genes.
using double filtering criterion will obtain a more robust result.
yet, the cost we pay is that some true differentially expressed genes might be missed.
however, arrow plot allows us to pick some genes from the single filtering region for further examination.
non-parametric techniques were used because they eliminate the need to specify parametric models.
the non-parametric kernel density method has few assumptions about the form of the distributions.
this is attractive because it can be used on thousands of genes on an automatic way.
the disadvantage of non-parametric techniques is that it results in a loss of efficiency.
yet, the loss of efficiency is balanced by the reduction of the risk of misinterpreting the results by incorrectly specifying a parametric form for the distribution.
the proposed algorithm is particularly useful in situations where bimodality exists in the gene expression data.
the proposed methodology outperforms other well known methods for detecting different kinds of differentially expressed genes.
future work includes further evaluation of this methodology on other real datasets.
we recognize that selecting de genes through an arrow plot has shortcomings.
for instance, using arbitrary cut-off points for auc and ovl will require the user to have some experience and results are sensitive to the cut-off choice.
nevertheless, the analysis of the arrow plot will help the user to select the cut-off points for auc and ovl.
this plot has to be seen as a statistical exploratory tool rather than an inference tool.
the objective of the plot is the visual identification of genes which can play a special role.
no other plot is able to achieve this goal.
arrow plot is an exploratory graphical tool for microarray experiments, useful in the identification of different kinds of differentially expressed genes, particularly in the identification of genes with a special behavior which are not detected by usual methods and yet can bring relevant biological information.
this methodology can be used in all platforms.

 genome-wide expression analysis is an increasingly important tool for identifying gene function, disease-related genes and transcriptional patterns related to drug treatments. microarrays enable the simultaneous measurement of the expression levels of tens of thousands of genes and have found widespread application in biological and biomedical research. increasing numbers of multi-class microarray studies are performed, but the vast majority continues to be two class studies, for example when both control and a treatment are examined. the objective of the study in most of them, is to determine the genes that are differentially expressed between the two classes. differentially expressed genes are usually detected using statistics based on means or medians. however, if there are genes differentially expressed on different subclasses, those techniques do not select them because either mean or median values tend to be similar between the considered groups. genes with a bimodal or a multimodal distribution within a class may indicate the presence of unknown subclasses with different expression values, meaning that there are two separate peaks in the distribution; one peak due to a subclass clustered around a low expression level, and a second peak due to a subclass clustered around a higher expression level. as a consequence, the identification of such subclasses may provide useful insights on biological mechanisms underlying physiologic or pathologic conditions. in cancer research, a common approach for prioritizing cancer-related genes is to compare gene expression profiles between cancer and normal samples, selecting genes with consistently higher expression levels in cancer samples. such an approach ignores tumor heterogeneity and is not suitable for finding cancer genes that are overexpressed in only a subgroup of a patient population. as a result, important genes differentially expressed in a subset of samples can be missed by gene selection criteria based on the difference of sample means. the particular application that motivated our work concerns the development of a methodology which could simultaneously identify up- and down-regulated genes and differentially expressed with bimodal or multimodal distributions with similar means on both groups. for convenience, the latter case is referred to as special genes. different statistical tests have been proposed to select differentially expressed genes. among them, is the receiver operating characteristic analysis, which is widely used to evaluate a diagnostic system but can be interpreted as a measure of separation between two distributions. a roc curve displays the relationship between the proportion of true positive and false positive classifications resulting from each possible decision threshold value in a two class classification task. these proportions depend on the classification rule and in general higher values of the marker are associated with the case group. however, if roc analysis is blindly applied to select genes differentially expressed, i.e., keeping the same classification rule for all genes in an experiment, not proper roc curves can be produced because genes with positive and negative regulation have opposite classification rules. nproc curves are obtained when they cross or are below of the reference line. genes can be ranked using the area under the roc curve, a common measure of discrimination, which should range between and but for nproc curves auc can have values below. nevertheless, different scenarios can lead to nproc curves, for instance, when the means of the two groups are similar and one of the groups has a bimodal distribution, or when both distributions are unimodal with similar means and significant different variances. on both cases the corresponding roc curve will have a sigmoidal-shape with an auc close to. proper binormal model and contaminated binormal model are methods that force the roc curves to be set above the reference line when they are not proper and consequently the auc will be higher than. however in the context of this work, not proper roc curves have an important role in the selection of different kinds of differentially expressed genes. since it is not possible to decide beforehand the direction of the classification rule, we considered the same classification rule for all of the genes, i.e., values of expression levels above the threshold correspond to up-regulation. in that sense, auc values near will correspond to up-regulated genes, auc values near will correspond to down-regulated genes, and special genes will have an auc around. however, regardless of the type of distributions, if means are similar, auc will be near. so, using auc is not sufficient to select special genes. we used the overlapping coefficient to further separate these different situations which produce values of auc near. bradley and inman and bradley promote the use of ovl as an intuitive measure of the similarity between two probability distributions. graphically, ovl is the area where the densities of the two distributions overlap when plotted on the same axes. we propose using auc and ovl simultaneously to select different types of differentially expressed genes and plotting ovl against auc we get a picture which we named as arrow plot. if we consider that groups have different variances, special genes can be mixed with genes which are not differentially expressed as illustrated on figure that is, genes with unimodal densities, with similar means but significantly different variances. these genes will have auc values around and low ovl values. with the purpose of identifying genes under these conditions, allowing their separation from the special genes, we developed an algorithm based on finding bimodality using kernel densities estimates. nonparametric techniques are used to estimate auc and ovl. to estimate auc, we used the mann-whitney u statistic, which is equivalent to the trapezoidal rule for integration. for the ovl, we developed an algorithm where a naive kernel density estimator is used to construct a nonparametric estimator of ovl. we first describe the algorithm and later we evaluate the performance of our method by comparing the gene expression profiles in two different classes using data from a publicly dataset and from a simulated dataset. the first dataset consists of different samples of normal circulating b cells and heterogeneous lymphomas. the gene expression data were obtained on genes. the simulated dataset consists of genes generated from a lognormal distribution, where each group sample has arrays. using publicly available data, we compared our results with those obtained by parodi et al. using as methods, the area between the roc curve and rising diagonal and a test for not-proper roc curves. we used both data sets to assess the relative performance of our proposed method as compared to the most common different statistical gene ranking measures. all the analysis were performed using the open-source r software and packages from bioconductor.  algorithm description for illustrative purposes, we divided the algorithm in two parts. the first part describes the ovl estimation and the second part describes the selection of different kinds of differentially expressed genes. the ovl estimation was based on a non-parametric form with densities estimated using kernel functions. figure shows the pseudo-code which implements the ovl estimation and tables and describe the notation and functions used there. the ovl values are computed by finding the points that belong to the area of intersection of the two densities and the jump points between densities, which are estimated by interpolation. the points are combined into one set and sorted in ascending order. finally ovl is estimated using a trapezoidal rule considering a non-uniform grid-spacing. symbols are listed in order of appearance in the algorithm. functions are listed in order of appearance. the selection of differentially expressed genes is based on simultaneous analysis of ovl and auc. the arrow plot is obtained by plotting ovl on abscissas and auc on ordinates. figure shows the pseudo-code which implements the algorithm to select differentially expressed genes based on these two measures and tables and present the notation and functions used there. symbols are listed in order of appearance. functions are listed in order of appearance. selection of differentially expressed genes with positive regulation and negative regulation, is made according to arbitrarily selected cutoff points for auc and ovl. however, auc values are expected to be close to for up-regulated and close to for down-regulated genes and ovl will have low values on both situations. selection of special genes is performed in two steps. the first step consists on the selection of genes with auc values near and low values of ovl. since the variances on both groups can be different, it is possible to find genes with no-differential expression mixed with the special ones. accordingly, the second step aims at removing the genes without differential expression, through the bimodality analysis. bimodality is analyzed based on the behavior of the ordinates of the kernel based estimated densities of both groups, considering only the gene list that is selected in the first step mentioned above. the points of both densities are ordered in increasing order of abscissas. if an ordinate is equal or less than the ordinate immediately after, it is assigned a label and otherwise. this allows us to analyze the variation of the density over the observed range. considering only the points where the function is increasing, if the differences between the ranks of adjacent ordinates is the distribution is expected to be unimodal, otherwise the distribution will be bimodal or multimodal. to declare a gene to be special it is enough to find bimodality in one of the groups, yet it is of interest to analyze in which group bimodality is observed, and this is possible using different color labels on the arrow plot. performance and implementation the running time of the algorithm in a dataset with genes, takes less than minutes on a mhz pentium. r source code for the implementation of this algorithm is available in additional file lymphoma data from a total of genes, our method selected differentially expressed genes, where corresponded to up-regulated genes, to down-regulated and corresponded to special genes. we used auc≥ and ovl< to select up-regulated genes, auc≤ and ovl< to select down-regulated genes and ovl< and <auc< to select special genes. thresholds were chosen arbitrarily, although an analysis of the the arrow plot could help on deciding which thresholds to use. table shows the selected special genes. genes are listed in ascending order of ovl, which ranged between and. auc values ranged between and. bimodality was tested on the special genes; genes have bimodality in the control group, genes on the experimental group and genes on both. for the special genes, kernel densities and their corresponding empirical roc curves can be analyzed in figure all the selected genes had a sigmoidal-shaped roc curve. special genes were selected using ovl< and <auc<. e: bimodality in experimental group, c: bimodality in control group and b: bimodality in both groups. genes are ordered by ascending order of ovl. among the special genes selected list, have an unknown regulatory function. all the remaining genes are related with proteins encoding. gene3323x and the gene3388x are presented in other clones in the same dataset, gene3322x and gene3389x respectively. alizadeh et al. observed that bcl7a gene can be altered by translocation in lymphoid malignancies. the biological properties of the selected genes are described in the additional file we compared our results with those obtained by parodi et al., where abcr and tnrc statistics were used. according to the highest tnrc value, a total of differentially expressed genes were considered, and of them were special genes. eight of them are considered to be special according to our methodology. the remaining genes of their list have auc and ovl values slightly higher than the considered cutoff points on our study. however, if we choose threshold values for auc and ovl to catch those genes, we will select more special genes. nine feature selection methods were applied to the full dataset, namely welch t-statistic, fold change, rank products, average difference, weighted average difference, moderated t-statistic, intensity-based moderated t-statistic, significance analysis of microarrays and area under the roc curve. we assessed the overlap between gene lists produced by different feature selection methods and ranked lists of differentially expressed genes were produced. we examined the top mostly highly ranked genes, and for all methods the special genes selected by our methodology are missed. simulated data we simulated ten thousand genes, among which were non-differentially expressed, were up-regulated, were down-regulated and were special genes. analyzing the arrow plot, we considered as threshold value for the ovl. as for the auc, we classified as up-regulated those genes with auc above, as down-regulated genes those with auc below and special genes those with auc between and. in the arrow plot we can observe the distribution of the truly differentially expressed genes, and we can conclude that 95% of them were selected by our methodology. in the first step of the algorithm used to select special genes, genes which were candidate to special genes were selected. through the second step we found that all of the genes had bimodality in at least one of the groups. we can conclude that our algorithm for detection of bimodality performed with 100% of accuracy on that list. roc analysis was conducted to evaluate and compare the performance of the above methods. we analyzed the performance of these methods regarding the discrimination between differentially expressed genes and non-differentially expressed genes considering two scenarios. first we studied the performance of the methods concerning the capacity to differentiate among up-regulated, down-regulated and special genes; secondly we studied the performance concerning only the capacity to identify special genes. the construction of the roc curves were based on the absolute values of the following statistics: fc, ad, wad, rp, welch-t, sam, samroc, ibmt, modt and shrinkt, where high values are related to de genes. the roc curve for the auc method was constructed considering auc values ranging from to 1; in this way, any auc value below was substituted by its complementary value, i.e., by 1−auc. high auc values are related to de genes. when analyzing the arrow plot, we verified that only the ovl statistic is needed since lower values of the ovl correspond to de genes. the empirical roc curves, under the first scenario are represented in figure and the respective empirical auc values are displayed on table comparison of auc values where the goal is to select up- and down-regulated genes and special genes. the auc values are sorted by decreasing order. the ovl with an estimated auc value near of the unit showed to be the one with a better performance followed by the rank products method. the method with lowest performance was samroc, however all methods showed high values of performance. considering the scenario where the goal is to select only special genes, the empirical roc curves and the empirical auc values showed that ovl was the method with better performance followed by the fc method, however with an auc value considerably low. wad and shrinkt were the methods with the lowest performance. comparison of auc values where the goal is to select special genes. the auc values are sorted by decreasing order.  we have presented a graphical and computational method for microarray experiments which allow the identification of genes that express differently under two conditions even if the behavior in average is similar. the main objective of this work was to select differentially expressed genes due to the presence of different subclasses, which could give important information about their inherent biological functions, and that are usually missed by usual methods. auc and ovl statistics were used to achieve this goal. both statistics are invariant when a suitable common transformation is made on variables, and on microarray data analysis log transformations are widely used. arrow plot is obtained by plotting ovl against auc. this plot is easily interpreted because both statistics range between and and in addition to detecting genes with up- or down-regulation, arrow plot is also able to detect special genes, however for the latter genes a bimodality analysis needs to be added. the approach used by the arrow plot is similar to the volcano plot, in the sense that two selection criteria are needed to select genes. using double filtering criterion will obtain a more robust result. yet, the cost we pay is that some true differentially expressed genes might be missed. however, arrow plot allows us to pick some genes from the single filtering region for further examination. non-parametric techniques were used because they eliminate the need to specify parametric models. the non-parametric kernel density method has few assumptions about the form of the distributions. this is attractive because it can be used on thousands of genes on an automatic way. the disadvantage of non-parametric techniques is that it results in a loss of efficiency. yet, the loss of efficiency is balanced by the reduction of the risk of misinterpreting the results by incorrectly specifying a parametric form for the distribution. the proposed algorithm is particularly useful in situations where bimodality exists in the gene expression data. the proposed methodology outperforms other well known methods for detecting different kinds of differentially expressed genes. future work includes further evaluation of this methodology on other real datasets. we recognize that selecting de genes through an arrow plot has shortcomings. for instance, using arbitrary cut-off points for auc and ovl will require the user to have some experience and results are sensitive to the cut-off choice. nevertheless, the analysis of the arrow plot will help the user to select the cut-off points for auc and ovl. this plot has to be seen as a statistical exploratory tool rather than an inference tool. the objective of the plot is the visual identification of genes which can play a special role. no other plot is able to achieve this goal. arrow plot is an exploratory graphical tool for microarray experiments, useful in the identification of different kinds of differentially expressed genes, particularly in the identification of genes with a special behavior which are not detected by usual methods and yet can bring relevant biological information. this methodology can be used in all platforms. 
in this paper we propose a new graphical tool which not only identifies genes with up and down regulations, but also genes with differential expression in different subclasses, that are usually missed if current statistical methods are used. we compared our results with the ones obtained using some of the standard methods for detecting differentially expressed genes, namely welch t-statistic, fold change, rank products, average difference, weighted average difference, moderated t-statistic, intensity-based moderated t-statistic, significance analysis of microarrays and area under the roc curve. we also compared our results with area between roc curve and rising area and the test for not proper roc curves. we found our methodology more comprehensive, because it detects both bimodal and multimodal distributions and different variances can be considered on both samples. our results indicate that the arrow plot represents a new flexible and useful tool for the analysis of gene expression profiles from microarrays. it is well known that biological samples are heterogeneous because of factors such as molecular subtypes or genetic background that are often unknown to the experimenter. bimodal or multimodal distributions often reflect the presence of subsamples mixtures. on both datasets all differentially expressed genes with bimodal or multimodal distributions were not selected by all standard selection procedures. another advantage of our method is that we can analyze graphically the behavior of different kinds of differentially expressed genes.
 a common task in analyzing microarray data is to determine which genes are differentially expressed across two kind of tissue samples or samples submitted under experimental conditions. several statistical methods have been proposed to accomplish this goal, generally based on measures of distance between classes. it is well known that biological samples are heterogeneous because of factors such as molecular subtypes or genetic background that are often unknown to the experimenter. for instance, in experiments which involve molecular classification of tumors it is important to identify significant subtypes of cancer. bimodal or multimodal distributions often reflect the presence of subsamples mixtures. consequently, there can be genes differentially expressed on sample subgroups which are missed if usual statistical approaches are used. in this paper we propose a new graphical tool which not only identifies genes with up and down regulations, but also genes with differential expression in different subclasses, that are usually missed if current statistical methods are used. this tool is based on two measures of distance between samples, namely the overlapping coefficient between two densities and the area under the receiver operating characteristic curve. the methodology proposed here was implemented in the open-source r software.  this method was applied to a publicly available dataset, as well as to a simulated dataset. we compared our results with the ones obtained using some of the standard methods for detecting differentially expressed genes, namely welch t-statistic, fold change, rank products, average difference, weighted average difference, moderated t-statistic, intensity-based moderated t-statistic, significance analysis of microarrays and area under the roc curve. on both datasets all differentially expressed genes with bimodal or multimodal distributions were not selected by all standard selection procedures. we also compared our results with area between roc curve and rising area and the test for not proper roc curves. we found our methodology more comprehensive, because it detects both bimodal and multimodal distributions and different variances can be considered on both samples. another advantage of our method is that we can analyze graphically the behavior of different kinds of differentially expressed genes.  our results indicate that the arrow plot represents a new flexible and useful tool for the analysis of gene expression profiles from microarrays. 