# DepthTools: an R package for a robust analysis of gene expression data

- Aurora Torrente
^{1, 2}Email author, - Sara López-Pintado
^{3, 4}and - Juan Romo
^{5}

**14**:237

https://doi.org/10.1186/1471-2105-14-237

© Torrente et al.; licensee BioMed Central Ltd. 2013

**Received: **24 April 2013

**Accepted: **17 July 2013

**Published: **25 July 2013

## Abstract

### Background

The use of DNA microarrays and oligonucleotide chips of high density in modern biomedical research provides complex, high dimensional data which have been proven to convey crucial information about gene expression levels and to play an important role in disease diagnosis. Therefore, there is a need for developing new, robust statistical techniques to analyze these data.

### Results

depthTools is an R package for a robust statistical analysis of gene expression data, based on an efficient implementation of a feasible notion of depth, the Modified Band Depth. This software includes several visualization and inference tools successfully applied to high dimensional gene expression data. A user-friendly interface is also provided via an R-commander plugin.

### Conclusion

We illustrate the utility of the depthTools package, that could be used, for instance, to achieve a better understanding of genome-level variation between tumors and to facilitate the development of personalized treatments.

## Keywords

## Background

The DNA microarrays and the oligonucleotide chips of high density are broadly used in modern biomedical research and in the study of numerous diseases like cancer or diabetes; also, in the last few years, the validity of this technology has become common in differentiation and development studies and in prenatal diagnostic testing for syndromes that involve small changes on chromosomes which are not seen through a microscope [1-6]. Microarrays allow gene expression profiling-based diagnosis and clinical risk stratification and facilitate the development of specific therapeutic treatments. They have made a great impact in the field of genomics, provide prognostic information for potential patients and play an important role in diagnostics and drug development. Additionally, a few recent studies have shown the application of this technique for revealing novel pathways or responses of important genes which were unstudied previously (see e.g. [7, 8]).

Microarray gene expression data are complex and high dimensional (with usually small sample size) and suggest numerous statistical problems; to fully take advantage of the information conveyed by this technology and its impact in the understanding of living processes sound analyses of these data are needed. In this direction, a cohort of techniques, like for instance classifier algorithms, have been developed for gene expression data. A particularly intuitive, computationally inexpensive, and effective collection of methods suitable for the analysis of microarray data is the one proposed by [9]. This consists of a set of robust nonparametric tools, based on the concept of data depth, which generalizes unidimensional order statistics, ranks and medians to high dimensional data. A data depth notion measures the centrality of an observation within a sample and allows the definition of a natural ordering in a multidimensional space from center outwards and of other robust statistics such as trimmed means. Several depth definitions for multivariate data have been proposed and analyzed by [10-14], among others. However, most of these definitions of depth are intractable for dimensions larger than 3 or 4. The Modified Band Depth (MBD) proposed by [14] is computationally feasible for very high dimensions, what makes it specially appropriate for analyzing gene expression data. With this depth notion, it is possible, for instance, to define the most representative (or deepest) sample within a collection of observations which measure the expression (level) of a large set of genes in a group of individuals affected by a particular tumor type. This concept provides the basis for the statistical methods studied in [9]. In particular, classification techniques based on new similarity measures are proposed. The basic idea in these methods is to classify a new sample to the group having the representative (deepest sample) that is the most similar to the new observation.

More precisely, the nonparametric techniques described in [9] include: 1) a scale curve for measuring and visualizing the variability or dispersion of a set of tissue samples in a multidimensional (gene) space; 2) a rank test for deciding if two groups of samples come from the same population, e.g, for deciding whether they correspond to the same type of cancer or not; and 3) two classification techniques for assigning a new sample to one of *G* given groups (multi-class classification). This can translate into a more reliable diagnosis, based on sample profiles. These methods have been successfully applied to real microarray data and have been proven to be robust, efficient, and competitive with other procedures proposed in the literature, outperforming them in several situations [9].

The depthTools package implements these methods with an improved computational cost, allowing the visualization and analysis of gene expression data in a simple framework. Note that there are other packages implementing depth notions (such as the depth package), but they are not applicable for gene expression data, as they become computationally intractable for dimensions larger than 3 or 4. Thus, the depthTools package appears as a suitable choice to analyze gene expression data and should ultimately be useful for improving the characterization of tumor types, and for providing a clinical tool for early diagnosis of cancer and other diseases, or for abnormalities detection.

## Implementation

The statistical tools described in [9] and implemented in the depthTools package are based on the computation of the MBD of a high dimensional observation **y** within a collection **y**_{1},…,**y**_{
n
}. The MBD of **y** with respect to **y**_{1},…,**y**_{
n
} represents the mean, over all possible pairs of distinct observations from **y**_{1},…,**y**_{
n
}, of the proportion of coordinates of **y** that are between the corresponding components of two elements in the set **y**_{1},…,**y**_{
n
}. The deepest sample has the largest of such average proportions.

In this section, we describe the functions implemented in the depthTools package, that also includes, for testing purposes, the prostate data, a subset of the data published by [15], normalized as described in the Prostate dataset subsection, and which contains both normal and tumor samples. The efficiency of the depthTools package stems from an alternative implementation of the MBD, described in the Methods subsection. Finally, in the R-commander support subsection, we describe briefly the implementation of a second package, the RcmdrPlugin.depthTools, that provides a user-friendly interface to make use of the depthTools package without the command line.

### Functions in the depthTools package

Function MBD:

The basic function in the package is MBD, which computes the depth of each element in a given data set and assigns a rank to it from center outwards. Additionally, this function allows the user to decide whether a plot of the data in parallel coordinates [16] will be returned. The most basic usage of the function is: MBD(x), where the mandatory argument x is an *n* × *d* data matrix containing the observations (samples) by rows and the variables (genes) by columns. In addition, several optional arguments can be provided. plotting is a logical value indicating whether the observations should be plotted (set to TRUE by default). In many situations, for instance in the context of classifying new data, the user will be interested only in knowing or envisaging the deepest sample of a group, which is, as mentioned before, the most representative gene expression profile within that group. For this reason, the default implementation of the MBD represents the dataset in a single colour, except for the deepest sample, which is distinctly drawn in a different one. In addition, it is also possible to depict each sample in grayscale, with intensities according to the order provided by the MBD, from deepest (light gray) to most external (dark gray).

Nevertheless, when the gene expression data set contains many samples, which are typically very irregular, such plots might become little informative or noisy, especially if the data set contains samples from different tissues or disease statuses. Therefore, an alternative is to picture the depth structure of the data, instead of by drawing all the curves, by plotting convex regions or bands, each containing a given proportion of the most central curves. To depict these bands in parallel coordinates, the minimum expression level of the samples that determine the band is computed for each gene, and the corresponding points are connected by straight lines, and analogously for the maximum expression levels. Representing these bands for different proportions helps understand how the data varies from center outwards. The logical parameters grayscale and band allow controlling, respectively, the use of gray intensities to reflect each sample position in the MBD ranking, and the representation of the bands. It is possible to draw different bands simultaneously through the argument band.limits. Further standard graphical parameters can be used to modify the aspect of the plot. Finally, we can pass to the parameter xRef an alternative data matrix containing a second collection of samples with respect to which the MBD is computed; this is useful when the user is interested in comparing the depth of a sample with respect to two different groups, as for instance in the rank test.

Panel (a) shows 25 normal samples, measured in 100 genes and represented in parallel coordinates, with the deepest gene expression profile depicted in red. Panel (b) corresponds to the computation of the MBD of these normal samples with respect to the 25 prostate cancer samples. The normal set is described in terms of the convex regions containing the percentages 25%, 50%, 75% and 100% of the most central curves, whereas the reference set is depicted in blue. The (normal) most representative sample with respect to the (cancer) reference set is shown by a red line. Note that in this case, the representative sample, that is, the one lying deepest in the prostate cancer set, is different from the most representative sample in the normal group, as one would expect.

In addition to possibly plotting the sample, the function returns a list containing two components: $ordering, a vector giving the ordering of the samples according to their depths, and $MBD, a vector with the computed depths.

Function tmean:

*n*×

*d*data matrix and alpha is the proportion of observations that are trimmed out when computing the mean (0.2 by default), will provide an R list with two components: $tm.x, which is a matrix containing the deepest points of x after removing the proportion alpha of less deep samples, and $tm, a vector of length

*d*with the alpha-trimmed mean (i.e., the ordinary mean of $tm.x). However, by setting the logical parameter plotting equal to TRUE, as in the following code:

computes the alpha-trimmed mean, for alpha in 0,0.1,…,0.9, for the normal and cancer samples separately, and leads to Figure 2(b), where we focus on a few genes (from the 33-rd to the 42-nd) to illustrate different situations. For instance, gene #35 shows very little variation across both normal (blue lines) and tumor (red lines) samples, and has similar expression values between both types. In contrast, gene #38 has considerably varying expression levels, that in addition depend on the disease status. Finally, gene #40 does not show a large variation across normal samples, but has a more volatile behaviour in cancer samples.

Function centralPlot:

*p*of the most central curves (according to the MBD ordering), which is represented distinctly from the rest of samples, using different colours and/or line types (by default, red solid lines for the central curves and gray dashed lines for the external ones). Additionally it is also possible to incorporate to the plot the information about the depth ordering by setting the logical parameter gradient equal to TRUE and by defining a colour palette with the argument gradient.ramp, a vector with two components corresponding to the first and the last colours in the palette (red-yellow, by default). The first of such colors is used to draw the deepest sample. The following code:

In addition, depthTools includes the following functions, which make use of the MBD and the tmean functions:

Function scalecurve:

Given the ordering defined by the function MBD, it is possible to determine, for each *p* ∈ [0,1], the band containing the proportion *p* of most central samples. The area of this band for each *p* defines the scale curve of the collection of samples, and the slope of the scale curve shows how the dispersion varies in the data.

The function scalecurve computes the scale curve of the *n* × *d* data matrix x and draws the corresponding plot by typing scalecurve(x). To allow the comparison of scale curves from different collections of samples (e.g. normal vs disease) there is an optional argument y, which takes as value a vector defining the class of each observation in x.

Function R.test:

In addition to visually comparing the dispersion of two collections of samples with the function scalecurve, it is also possible to decide whether two sets of samples come from the same population using the statistical rank test (see [9] for details), implemented in the function R.test. This allows the user, for instance, to decide if two groups of samples correspond to the same type of cancer or not. To use the function, we need to determine the four following arguments: an *n*_{1} × *d* data matrix x containing the observations from the first population, an *n*_{2} × *d* data matrix y with the observations from the second population, and two integers, n and m, representing, respectively, the size of the subsets randomly chosen from the first and second populations that will be used in the test. Additionally, due to this random component in the rank test, the user can initialize the random number generation with the optional parameter seed, which is set to 0 by default.

As we can see, R.test rejects that the normal and tumor samples come from the same population (p-value < 0.01).

Functions classDS and classTAD:

These two functions implement the depth-based classification techniques DS and TAD, described in [9]. As any classification procedure, they both use a learning set, xl, in which the class of each sample, yl, is known, and a test set, xt, whose samples are to be classified into one of *G* known classes. In the DS procedure, which computes the trimmed mean of each group and classifies a new observation **y** from the test set in the class having the trimmed mean which is closest to **y**, we also need to specify the proportion alpha of observations that are trimmed out when computing this mean. In the TAD procedure, to determine the class of the new sample **y**, a weighted average distance from **y** to the most central points in each class is computed. The proportion alpha of most external points that are not considered in this weighted distance has to be specified as well. The output in both cases is a vector containing the class predicted for each observation in the test set.

Both methods classify correctly 9 out of 10 prostate samples. Notice that these techniques have been previously validated, and have been proven to be robust and very appropriate to analyze the complicated structures of gene expression data.

### Prostate dataset

This is a normalized subset of the real data published by [15]. The raw data comprise the expression of 52 tumor and 50 non-tumor prostate samples, obtained using the Affymetrix technology. The data were preprocessed by setting thresholds at 10 and 16000 units, excluding genes whose expression varied less than 5-fold relatively or less than 500 units absolutely between the sample, applying a base 10 logarithmic transformation, and standardizing each experiment to zero mean and unit variance across the genes. The 100 most variable genes were selected following the B/W criterion [17] and a random selection of 25 normal samples and 25 tumor samples was performed. The data are included in a matrix, where the 100 first columns correspond to the gene expression levels, and the last one contains the sample type: 0 for normal and 1 for tumor.

### Methods

*d*-dimensional point

**y**

_{ i }= (

*y*

_{i,1},...,

*y*

_{i,d}) in the sample

**y**

_{1},…,

**y**

_{ n }is computed as

**y**

_{ i }inside the bands defined by every two different multidimensional points from the sample [14]. The implementation of this formula leads to nested for loops, which are known to be very inefficient in the R environment. Thus, to improve the computational cost of the MBD we used an alternative expression, developed in [18]. If we store the data in an

*n*×

*d*matrix

**Y**, we can calculate the multiplicity of each value

*y*

_{i,k}in the corresponding

*k*-th column of

**Y**, rather than exhaustively search for all pairs of samples, and use this multiplicity to obtain the MBD of

**y**

_{ i }. It can be easily shown that this efficient expression can be computed by rewriting the MBD as follows. Given a data set

*Y*= {

**y**

_{1},...,

**y**

_{ n }} and a point

**y**

_{ i }∈

*Y*, consider matrices

**Y**has been increasingly ordered. Let

*l*

_{ k }be the smallest index in the

*k*-th column of $\stackrel{~}{\mathbf{Y}}$ that verifies ${y}_{i,k}={y}_{({l}_{k}),k}$, and

*η*

_{ k }be the multiplicity of

*y*

_{i,k}within the same column. Then, the MBD of

**y**

_{ i }is given by

assuming that $\left(\genfrac{}{}{0ex}{}{{\eta}_{k}}{2}\right)=0$ whenever *η*_{
k
} = 1.

This is an alternative to the computation recently published by [19], in which they use a rank matrix to obtain the MBD; however, in formula (2), the cases in which there are variables (genes) with repeated values across different individuals (samples) are taken into account. Implementing expression (2) for MBD in R avoids the use of inefficient nested for loops, and reduces the computational time, for *n* data points in *d* dimension, from *O*(*d* × *n*^{2}) (to compute the proportion for all pairs) to *O*(*n*^{3/4}) (to compute the expression above), plus *O*(*d*) to reorder the data, using a variant of that of [20]. This is particularly important in the common situation where the number of dimensions (i.e. number of genes) is in the order of thousands.

**x**= (

*x*

_{1},...,

*x*

_{ d }),

**x**∉

**Y**, that is, the depth of sample

**x**with respect to a data set which does not contain it. This is given by

where *l*_{1,k} = |{*y*_{(i),k} < *x*_{
k
}, 1 ≤ *i* ≤ *n*}| and *l*_{2,k} = |{*y*_{(i),k} ≤ *x*_{
k
}, 1 ≤ *i* ≤ *n*}| represent the number of *k*-th coordinates in **Y** that are, respectively, smaller than *x*_{
k
} and not larger than *x*_{
k
}, whereas *η*_{
k
} = *l*_{2,k}−*l*_{1,k} provides the multiplicity of *x*_{
k
}, assuming that $\left(\genfrac{}{}{0ex}{}{{\eta}_{k}}{2}\right)=0$ whenever *η*_{
k
} = 0, 1.

**Computational cost of both MBD implementations**

Computational cost | ||
---|---|---|

Dataset size | Nested implementation. Time (s) | Matrix reordering. Time (s) |

10 × 500 | 0.1663 | 0.0151 |

25 × 500 | 1.1386 | 0.0148 |

10 × 1000 | 0.3570 | 0.0318 |

25 × 1000 | 2.2870 | 0.0277 |

### R-commander support

## Discussion

We have developed depthTools, an R package that implements different robust statistical tools for the visualization and analysis of high dimensional gene expression data as illustrated with the prostate dataset. These tools make use of the MBD, a depth notion which is feasible for data with several hundreds or thousands of variables (genes), and of the ordering from center outwards that is derived from this depth. The computational cost of the MBD is drastically improved if we use an implementation based on reordering the columns of the data matrix rather than on exhaustively searching for all possible pairs of samples. An additional plugin for the R-commander has been designed to provide a friendly interface to users who are not familiar with the command line code. These methods can be easily applied to achieve a better understanding of genome-level variation between tumors and to facilitate the development of personalized treatments.

## Conclusions

The depthTools package implements the statistical tools based on the MBD in a very efficient way, greatly improving the computational cost of the original definition. It allows users to order from center outwards the samples in a high dimensional dataset like gene expression data, and to visualize the change in the dispersion through the scale curve or the central regions defined by different percentages of the most central samples. It is also possible to use the rank test to decide whether two sets of biological samples have the same disease status, and to identify the status of a new sample by means of the classification techniques DS and TAD. These tools will ultimately be useful for rapid diagnosis and efficient treatments. The R-commander plugin facilitates the use of the package avoiding the command line, and allows a wider use of these statistical methods.

## Availability and requirements

The package and the R-commander plugin have been developed for the statistical R environment (http://www.R-project.org) and are freely available at http://cran.r-project.org/. The packages are accompanied by documentation files to facilitate their use.

**Project name:** depthTools

**Project home page:** http://cran.r-project.org/web/packages/depthTools/index.html and http://cran.r-project.org/web/packages/RcmdrPlugin.depthTools/index.html

**Operating system(s):** Platform independent.

**Programming language:** R platform.

**Other requirements:** No.

**License:** GPL (≥ 2)

**Any restrictions to use:** It is available for free download.

## Declarations

### Acknowledgements

We would like to thank Alvis Brazma and Gabriella Rustici in European Bionformatics Institute for their useful comments. This work was partially supported by research project ECO2011-25706 from Ministerio de Economía y Competitividad, Spain. Conflict of Interest: None declared.

## Authors’ Affiliations

## References

- Golub TR, Slonim DK, Tamayo P, Huard C, Gaasenbeek M, Mesirov JP, Coller H, Loh ML, Downing JR, Caligiuri MA, Bloomfield CD, Lander ES: Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science. 1999, 286: 531-537. 10.1126/science.286.5439.531.View ArticlePubMedGoogle Scholar
- Ross DT, Scherf U, Eisen MB, Perou CM, Rees C, Spellman P, Iyer V, Jeffrey SS, van de Rijn M, Waltham M, Pergamenschikov A, Lee JC, Lashkari D, Shalon D, Myers TG, Weinstein JN, Botstein D, Brown PO: Systematic variation in gene expression patterns in human cancer cell lines. Nat Genet. 2000, 24 (3): 227-235. 10.1038/73432.View ArticlePubMedGoogle Scholar
- Kote-Jarai Z, Matthews L, Osorio A, Shanley S, Giddings I, Moreews F, Locke I, Evans DG, Eccles D, Williams RD, Girolami M, Campbell C, Eeles R: Accurate prediction of BRCA1 and BRCA2 heterozygous genotype using expression profiling after induced DNA damage. Clin Cancer Res. 2006, 12: 3896-3901. 10.1158/1078-0432.CCR-05-2805.View ArticlePubMedGoogle Scholar
- Jing L, Ng MK, Zeng T: On gene selection and classification for cancer microarray data using multi-step clustering and sparse representation. Adv Adaptative Data Anal. 2011, 3 (1-2): 127-148.View ArticleGoogle Scholar
- Chuang LY, Yang CH, Li JC, Yang CH: A hybrid BPSO-CGA approach for gene selection and classification of microarray data. J Comput Biol. 2012, 19 (1): 92D10-View ArticleGoogle Scholar
- Hillman SC, Mcmullan DJ, Williams D, Maher ER, Kilby MD: Microarray comparative genomic hybridization in prenatal diagnosis: a review. Ultrasound Obstet Gynecol. 2012, 40: 385-391. 10.1002/uog.11180.View ArticlePubMedGoogle Scholar
- Takayama K, Kaneshiro K, Tsutsumi S, Horie-Inoue K, Ikeda K, Urano T, Ijichi N, Ouchi Y, Shirahige K, Aburatani H, Inoue S: Identification of novel androgen response genes in prostate cancer cells by coupling chromatin immunoprecipitation and genomic microarray analysis. Oncogene. 2007, 26 (30): 4453-4463. 10.1038/sj.onc.1210229.View ArticlePubMedGoogle Scholar
- Shi T, Mazumdar T, DeVecchio J, Duan ZH, Agyeman A, Aziz M, Houghton J: cDNA Microarray gene expression profiling of hedgehog signaling pathway inhibition in human colon cancer cells. PLoS ONE. 2010, 5 (10): e13054-10.1371/journal.pone.0013054.PubMed CentralView ArticlePubMedGoogle Scholar
- López-Pintado S, Romo J, Torrente A: Robust depth-based tools for the analysis of gene expression data. Biostatistics. 2010, 11 (2): 254-264. 10.1093/biostatistics/kxp056.View ArticlePubMedGoogle Scholar
- Mahalanobis PC: On the generalized distance in statistics. Proc Nat Acad Sci India. 1936, 12: 49-55.Google Scholar
- Tukey JW: Mathematics and picturing data. Proceedings of the International Congress of Mathematic: 1974; Vancouver. Edited by: Ralph D. James. 1975, Vancouver, 2:523-531.Google Scholar
- Liu R: General notions of statistical depth functions. Annal Stat. 2000, 28: 461-482. 10.1214/aos/1016218226.View ArticleGoogle Scholar
- Zuo Y, Serfling R: General notions of statistical depth functions. Annal Stat. 2000, 28: 461-482. 10.1214/aos/1016218226.View ArticleGoogle Scholar
- López-Pintado S, Romo J: On the concept of depth for functional data. J Ame Stat Assoc. 2009, 104: 486-503. 10.1198/jasa.2009.0015.View ArticleGoogle Scholar
- Singh D, Febbo PG, Ross K, Jackson DG, Manola J, Ladd C, Tamayo P, Renshaw AA, D’Amico AV, Richie JP, Lander ES, Loda M, Kantoff PW, Sellers WR, R G T: Gene expression correlates of clinical prostate cancer behavior. Cancer Cell. 2002, 1: 203-209. 10.1016/S1535-6108(02)00030-2.View ArticlePubMedGoogle Scholar
- Inselberg A: The plane parallel coordinates. Vis Comput. 1985, 1: 69-91. 10.1007/BF01898350.View ArticleGoogle Scholar
- Dudoit S, Fridlyand J, Speed TP: Comparison of discrimination methods for the classification of tumors using gene expression data. J Am Stat Assoc. 2002, 97 (457): 77-87. 10.1198/016214502753479248.View ArticleGoogle Scholar
- Torrente A: Clustering methods for gene expression data. PhD thesis Universidad Carlos III de Madrid, Department of Statistics; 2007Google Scholar
- Sun Y, Genton M, Nychka D: Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?. Stat. 2012, 1: 68-74. 10.1002/sta4.8.View ArticleGoogle Scholar
- Sedgewick R: A new upper bound for Shell sort. J Algorithms. 1986, 7: 159-173. 10.1016/0196-6774(86)90001-5.View ArticleGoogle Scholar
- Fox J: The R commander: a basic-statistics graphical user interface to R. J Stat Softw. 2005, 14 (9): 1-42.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.