# Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry

Edited by Zvi Kam, Weizmann Institute of Science, Rehovot, Israel, and accepted by the Editorial Board January 15, 2014 (received for review October 24, 2013)

## Significance

Much of what is currently known about how cells work has been derived through visual interpretation of microscopy images. Computational methods for image analysis have emerged as quantitative alternatives to visual interpretation. We describe an analysis pipeline for cell image databases that combines statistical pattern recognition with the mathematics of optimal mass transport. The approach is fully automated and does not require the use of ad hoc numerical features. It enables the identification of discriminant phenotypic variations, or biomarkers, between sets of cells (e.g., normal vs. diseased) while at the same time allowing for the visualization of meaningful differences. The approach can be used for fully automated high content screening with a variety of microscopic image modalities.

## Abstract

Modern microscopic imaging devices are able to extract more information regarding the subcellular organization of different molecules and proteins than can be obtained by visual inspection. Predetermined numerical features (descriptors) often used to quantify cells extracted from these images have long been shown useful for discriminating cell populations (e.g., normal vs. diseased). Direct visual or biological interpretation of results obtained, however, is often not a trivial task. We describe an approach for detecting and visualizing phenotypic differences between classes of cells based on the theory of optimal mass transport. The method is completely automated, does not require the use of predefined numerical features, and at the same time allows for easily interpretable visualizations of the most significant differences. Using this method, we demonstrate that the distribution pattern of peripheral chromatin in the nuclei of cells extracted from liver and thyroid specimens is associated with malignancy. We also show the method can correctly recover biologically interpretable and statistically significant differences in translocation imaging assays in a completely automated fashion.

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Quantitative analysis of cell images is extensively used in several health sciences applications (1). Scientists wishing to quantify the effects of certain drugs, genes, and other perturbations (e.g., benign vs. malignant cancer cells) routinely make use of numerical software that are capable of evaluating statistical differences between two populations of cells captured under the microscope (2). Beyond simple automation facilitating the analysis of thousands of cells, the purpose of such software is to attempt to extract information that the human visual system is unable to cope with. A well-known drawback of existing methods is that the visual interpretation of any differences found is usually hidden from the user. The popular numerical features used to quantify and compare cells, such as form factor, Gabor and Haralick texture features, color histograms, etc. (3–5), usually do not have a direct biological interpretation. The situation is even more complicated when multiple features are needed simultaneously to characterize differences between cells, given that the physical interpretation of a combination of features with different units is a nontrivial task. Consequently, statistical tests are limited to determining whether or not two or more cell populations are different. Visual interpretation of any obtained result is usually nonintuitive and difficult.

Here we describe a method, which we call transport-based morphometry (TBM), that takes as input a database of presegmented cell images and outputs a representation for the same data, which can be used for simultaneous visualization and quantitative evaluation in commonplace biological domains. An a priori set of numerical features is not needed as all calculations for comparing cells are done using the entire information present in each cell image. Our approach is based on combining a framework for image analysis based on the theory of optimal mass transport (6, 7) together with a modified version of the linear discriminant analysis technique (8), as well as other modifications and extensions as explained below. Mass, in this case, refers to the intensity associated with each particular pixel, which is often linear with respect to number of molecules (e.g., proteins) present in that location. The idea is demonstrated in Fig. 1. The segmented images are first normalized to eliminate the effect of translation and rotation and are spatially morphed to a precomputed reference image. A weighted Euclidean distance computed between two transformations approximates the optimal transport of mass (corresponding to image intensity) between each image and thus defines a linear embedding for the data. The main phenotype variations in a given dataset can then be given by the standard principal component analysis (PCA) technique. PCA, however, can only be used for visualizing variations as a whole. Statistically significant differences between different classes of cells are computed through a modified version of the penalized linear discriminant analysis (

*p*LDA) (8) applied to the linear optimal transport (LOT) coordinates computed by the method described in ref. 6. Given that the used embedding technique is invertible, the result of any statistical analysis (e.g., PCA,*p*LDA) can be directly visualized in image space.Fig. 1.

With respect to the work in refs. 6 and 8, we describe the following extensions. We use the LOT framework, in combination with a modified version of the

*p*LDA method, to derive a linear discriminant subspace for different cell populations by selecting only the dimensions that contain statistically meaningful distances, as described in*Materials and Methods*and*SI Text*. We also describe a method that allows one to obtain a visual and quantitative interpretation of the phenotype variations in a cell population with respect to a chosen independent variable (e.g., time and drug concentration). Here we also focus on describing how these advances can play a role in eliciting previously unavailable information regarding cell morphology in several problems relevant to cell biology and pathology. Finally, we combine all these advances into a freely available concise software package that can be used by scientists to perform high content screening tasks (9).In

*Results*, we describe the application of TBM to discover information in several high content screening applications. The first application is concerned with discovering the principal differences in nuclear chromatin arrangement between normal, benign, and malignant cells extracted from the liver and thyroid of pediatric patients (10). Nuclear structure has long been a highly used biomarker in image-based pathology. For many malignancies, however, nuclear structure is not used given the absence of knowledge related to discriminating structural information. We use TBM to uncover statistically meaningful and at the same time easy to interpret differences in nuclear structure in these malignancies.The second application is concerned with automated screening for cell phenotype changes in imaging-based assays. Such assays are routinely used for a wide variety of applications, including drug discovery, functional genomics, and chemical probe discovery. In this paper, we detail the application of TBM to quantifying translocation of the Forkhead fusion protein as a function of Wortmannin dosage in stably transfected human osteosarcoma cells (U2OS) (11). We show that TBM is able to correctly identify the underlying trend of cytoplasm-to-nucleus translocation in a manner that is both statistically significant and intuitive to understand. We note that in contrast to currently available methods (11), the trend does not have to be assumed a priori. Rather it is automatically discovered without any human intervention.

## Results

### Visualizing Variations of Chromatin Patterns in Normal and Cancerous Cells.

Exploratory visual analysis is an important part of coming to a comprehensive understanding of the phenotype variability in a given set of cells derived from a particular experiment. It can be used to obtain an understanding of the main trends regarding shape, structure, and texture variation in a given experiment. We applied TBM to visualize the most significant nuclear structure variations present in the thyroid and liver specimens, as well as in the Forkhead fusion protein in the cytoplasm-to-nucleus translocation imaging assay (

*Materials and Methods*). Using the PCA technique (12), in conjunction with the transport approach described in ref. 6, we are able to conclude that the main modes of variation (in order of decreasing corresponding variance) are as follows: nuclear size, elongation, shifts in chromatin concentration, and shifts in chromatin concentration accompanied by nuclear envelope protrusions. The first 19 components of the TBM-enabled PCA analysis of the liver data are shown in Fig. 2. These modes correspond to roughly 90% of the variance in the dataset. In a similar manner (*SI Text*), the top three modes of variation in the thyroid dataset (preserving 90% of the total variation) were the cell size, cell shape (elongated vertically vs. elongated horizontally), and shifts in chromatin concentration (*Supporting Information*). The top six PCA directions preserving 90% of the variations in the U2OS dataset is also shown in*Supporting Information*.Fig. 2.

### Peripheral Migration of Nuclear Chromatin Is Predominantly Responsible for Fetal-Type Hepatoblastoma Cancer in Liver Cells.

We used the TBM approach to discover the most discriminant, and visually interpretable, differences between normal liver and fetal-type hepatoblastoma (FHB) specimens. Although the PCA technique is useful for visualizing overall morphology trends in a given population of cells, by itself, it contains no information regarding which morphology changes are responsible for discriminating two subpopulations. To that end, we applied the

*p*LDA-based method, described in detail in*Materials and Methods*. Fig. 2, for example, contains no information regarding which modes could be used for differentiating normal vs. cancerous liver cells. Fig. 3 summarizes the visual information uncovered by TBM when investigating FHB cancer in liver cells. We note this result substantially differs from the result obtained by simply applying our earlier work on discriminant subspace selection (8) to the LOT embedding described in ref. 6, in that it provides a description of differences that are more statistically significant (*SI Text*). The horizontal axis is plotted in units of SD of the chromatin spread along the most statistically significant discriminant direction [in transport space (6)] between benign and FHB cells. In this visualization, each bar in both histograms shown corresponds to the relative number of nuclei that most closely resembled the nuclear structure shown right below. The representative images corresponding to each histogram coordinate are shown below the horizontal axis. The*P*value of the histogram separation (computed using cross-validation) is zero within numerical precision, and therefore, the separation of the normal and cancer subpopulations is highly significant. In this case, TBM discovers that as the axis of chromatin discrimination slides from left to right, typical nuclear chromatin migrates from peripheral bands in the nucleus to being more and more concentrated at the center of the nucleus. In other words, significant concentration of nuclear chromatin in the center of the nucleus, as opposed to its concentration around the nuclear boundary, can suggest possible FHB condition in the liver cells.Fig. 3.

### Patterns of Circumferential Bands in Chromatin Identifies Progression of Cancer from Normal to Follicular Adenoma of the Thyroid Through Follicular Carcinoma of the Thyroid in the Thyroid.

Interesting insights regarding visual differences between the normal, follicular adenoma of the thyroid (FA), and follicular carcinoma of the thyroid (FTC) populations were discovered (Fig. 4) when the thyroid dataset consisting of three subpopulations of normal, FA, and FTC cells were input to the TBM pipeline. Fig. 4 demonstrates the difference between the normal, FA, and FTC subpopulations, computed using the methods described below. The horizontal axis represents the highest level of visual difference inside the subpopulations, i.e., is directed along the most significant direction of difference between normal, FA, and FTC cells. The representative images are generated between every unit of SD from the mean image in the dataset along the discriminant direction. Similar to the previous experiment, the positions of alternate circumferential bands of chromatin concentration are revealed to be a possible biomarker for identifying and distinguishing FA and FTC from the normal case. Whereas one can detect the existence of peripheral and central chromatin concentrations in the benign case, the FTC case seems to have a more uniform chromatin spread across the nucleus, and the FA subpopulations are distinguished by a single circumferential concentration band approximately halfway between the periphery and the center. To facilitate a clearer inspection of the pairwise differences between the three classes (normal, FA, and FTC),

*Supporting Information*shows the pairwise histogram projections on the most discriminant direction found in Fig. 4.Fig. 4.

### Gradual Translocation of the Forkhead Protein in the Nucleus of U2OS Cells with Variation of Dosage of Wortmannin Is Visually Verified by TBM.

The third dataset containing Wortmannin-injected assays of U2OS cells to affect translocation of the Forkhead protein in the nucleus serves as a verification tool for the statistical and representational veracity of TBM. As shown in Fig. 5

*A*, the Forkhead protein (FKHR-EGFP) gradually translocates from the cytoplasm toward the nucleus of U2OS cells (left to right) with increasing dosage of the drug Wortmannin (13). Although there are four realizations of 12 assays in the dataset (13), with the 1st assay being the negative control (no Wortmannin) and the 12th and last assay being the positive control (maximum Wortmannin of 250 nM added), we only show six equispaced assays in Fig. 5*A*from the first realization (13).Fig. 5.

TBM can be used to automatically recover the pattern of translocation of FKHR-EGFP in the U2OS cells through a representational axis signifying the most significant pattern of variation of FKHR-EGFP in the cytoplasm. Fig. 5

*B*bundles all of the realizations of the first six assays into a positive control and the last six assays into a negative control, and the discrimination visualization step of the TBM is applied to the two subpopulations to verify the FKHR-EGFP translocation. It can be seen from Fig. 5*B*that the projections of the positive control (0.00–7.81 nM) in cyan are clearly separated from the projections of the negative control (15.63–250 nM) in red along the most visually discriminant direction estimated by*p*LDA. Moreover, the horizontal axis (again plotted in units of SD of the FKHR-EGFP variation) was tagged with synthesized images that statistically express the average FKHR-EGFP translocation along the dosage increase. It can be observed from the synthesized images in Fig. 5*B*that the FKHR-EGFP translocation was accurately captured.In addition, the projections of all U2OS cells in 12 individual assays along the discriminating direction in Fig. 5

*B*are shown in Fig. 5*C*. All cells in a particular assay were given a unique color. It can be observed from Fig. 5*C*that the projection histograms of individual assays shifts from the negative control toward the positive control with increase in Wortmannin dosage, with a rather abrupt change occurring between 7.8 and 15.6 nM. This figure serves to verify that the negative to positive discrimination direction in Fig. 5*B*estimated by TBM in fact gives reasonable progression of the translocation.### Maximally Correlated Images with Respect to Wortmannin Dosage Shows Quantifiable Translocation of the FKHR-EGFP from the Nucleus Toward the Periphery of the Average Cell Boundary.

TBM also provides considerable insight into the response of the U2OS cells to Wortmannin dosage variation with respect to FKHR-EGFP translocation. Using the TBM pipeline, we computed the direction in LOT space that is most correlated with dosage values (

*Materials and Methods*). Fig. 6 demonstrates the dosage-response curve. The horizontal axis represents the logarithm of the Wortmannin dosage values [log(0.977 nM)–log(250 nM)]; note that 0.00 nM is not included in this experiment. The vertical axis represents the normalized projected value of the data on the maximally correlated direction, as described above. These normalized projected values serve as a measure of Wortmannin activity, where 0 corresponds to the negative control and 1 corresponds to the positive control. The images corresponding to 0−100% activity of the Wortmannin are shown along the vertical axis. The presented curve matches with the one reported in the product specification of FKHR redistribution assay provided by Thermo Fisher Scientific in 2008 (14).Fig. 6.

## Discussion

We described TBM (see ref. 9 for download) as a method for decoding morphology differences in cell populations. The method builds on previous work related to image processing using optimal transport (6), as well as

*p*LDA (8), by adding the capability to construct a quantitative, discriminant, linear subspace of cell phenotypes that can be directly visualized. We note the LOT-based discriminant subspace described by our TBM approach is substantially different from our previous work in ref. 8 in that it provides a more reliable description of statistically significant differences between two cell populations (*SI Text*). In addition, we also describe how to use the framework for visualizing the morphological variations most correlated with a given independent variable (e.g., drug dosage). This visual analysis of structural differences can lead to improved understanding of interrelationships between cellular structure and functions. We believe TBM is a systematic approach, which serves as a totally automated visual exploratory tool in cell image analysis that offers both the benefits of observation and involved statistical tests on cell image databases containing one or more phenotypes.We applied TBM to discover statistically significant and visually interpretable differences of nuclear chromatin configuration in normal vs. cancerous cells. The analysis shows that, on average, the most discriminative information in these was how much chromatin is present in the center vs. periphery of these. Malignant cancerous cells were shown, on average, to have more chromatin concentrated and packed at the center of the nuclear envelope, a finding consistent with the biology of cancer cells (15). We believe TBM could be used in numerous pathology and cytology applications to recover visually interpretable differences between normal, benign, and malignant cells.

We note here that there are few methods for direct application of statistical analysis tools, such as PCA and LDA, to the image pixel intensities directly. For example, a direct and naive application of the PCA to the pixel intensities following ref. 16 can lead to the discovery of biologically meaningless artifacts as evidenced in

*SI Text*. Pixel intensity variations inside real images are highly nonlinear, and any linear interpolation of pixel intensities in the image space leads to presence of aliasing artifacts inconsistent with real images.In addition, we also used TBM to blindly recover known information regarding nuclear to cytoplasm protein translocation in a screening assay. In the U2OS dataset, TBM not only confirms the translocation of FKHR-EGFP visually, establishing the veracity of the functionality of TBM, but it also outputs a visual variant of the dosage response curve of the drug Wortmanin that clarifies the representative effect of the dosage increase on the average U2OS cell.

In conclusion, we anticipate important use of TBM in cell phenotype analysis in part due to the visual exploration component that provides intuitive insight into the structural modes of cell construction. In addition, TBM is fully automated and relieves the end user of manual and tedious definition, as well as selection of arbitrary image features, potentially leading to more accuracy and promises of generative modeling of cells. It can be applied to analyze segmented cell images where the cell content (intensity or texture) can be viewed as a distribution of free mass. In this work, we applied TBM to 2D fluorescence and transmitted light microscopy images, but other microscopy imaging modalities (e.g., coherent anti-Stokes Raman scattering) could also benefit. The framework is also amenable to 3D images (albeit at an increase in computation cost). Finally, the TBM framework was presented here in the context of scalar images. It can also be used to analyze the relationship between multiple protein distributions, obtained through multiple fluorescence labels or spectral imaging modalities, in cell populations. This topic will be the subject of future work.

## Materials and Methods

### Datasets.

To demonstrate the ability of TBM to discover visual information hitherto impossible with the standard feature-based approaches, we identified three cell image datasets that include cell texture variation either due to functional difference (cancer vs. noncancer) or drug infusion (drugs inhibiting protein translocation in cells). The first two datasets were obtained from the archives of the University of Pittsburgh Medical Center. The first dataset contains microscopy images of liver tissue samples obtained from 10 different subjects including five cancer patients suffering from FHB, with the remaining images from the liver of five healthy individuals. The second dataset contains microscopy images of resection specimens of thyroid from 20 different subjects. The first 10 subjects provide images of normal thyroid tissue, patients diagnosed with FA provide the next 5 cases, and the last 5 cases belong to patients diagnosed with FTC. The acquisition of these datasets is described in detail in ref. 7. Briefly, images were stained with the Feulgen technique to tag DNA content and scanned at 0.074 μm/pixel resolution.

The third image dataset demonstrates cytoplasm to nucleus translocation of the Forkhead (FKHR-EGFP) fusion protein in U2OS cells provided by Ilya Ravkin from the Broad Bioimage Benchmark Collection (Broad Institute Imaging Platform, Cambridge, MA) (13). In this assay, the images are obtained from 48-well plates of cells incubated with 12 different dosages of Wortmannin. The images have a resolution of 0.6 μm/pixel.

### Image Segmentation.

Before being analyzed with our TBM approach described below, each morphological exemplar (DNA pertaining to one nucleus or protein distribution from one cell) was first segmented using standard approaches. The nuclear datasets were segmented as described in ref. 10. The liver dataset were segmented to have 500 nuclei with an average of 50 nuclei per patient. The thyroid dataset consisted of 2,053 cell images with an average of 102 images per patient.

The U2OS cell dataset was segmented using Cellprofiler (11) with the exact pipeline described in ref. 11. The DNA channel is used to locate the nuclei and consequently mark the seed points of a growth step where every seed point grows into a closed curve that encircles its respective cell boundary. On average, 730 images of individual cells were extracted from images of the wells incubated with the same dosage of Wortmannin, leading to 8,756 images of cells with 12 different classes (12 different dosages of Wortmannin). A detailed discussion of the segmentation procedure is presented in ref. 11.

### Transport-Based Cell Morphometry.

The aim of our TBM approach is to take as input segmented morphological exemplars and output coordinates for each exemplar that can be used for both visualizing the main modes of variation of a dataset and the main ways in which two or more groups of morphological exemplars differ from one another. To that end, the images are normalized following a similar approach as described in ref. 17, the linear optimal transportation embedding is calculated from the normalized images (6), and a modified version of the

*p*LDA method (8) is used to capture the most statistically significant discriminant direction. Finally, we used the well-known Kolmogorov–Smirnoff test (18) for assessing significance when comparing distributions over a chosen linear subspace.### Preprocessing.

Each segmented structure is first normalized so that the sum of its intensities equals 1, to remove differences in staining procedures during imaging. This normalization limits us to investigating relative changes in overall mass distribution while all information regarding absolute amounts is lost. In addition, due to computational complexity considerations, each segmented structure is also approximated with point masses using the algorithm described in ref. 6. Briefly, a weighted

*K*-means clustering algorithm is used, in conjunction with the available image intensities, to best approximate the input image. The number of point masses is chosen to keep the computation time within a reasonable range (e.g., 30 s per image pair). Details of the algorithm used are available in ref. 6. When it comes to visualizing images from particle approximations (processed as explained below), bilinear interpolation is used to distribute the masses onto the image grid, and a Gaussian function of small variance is used to render a more realistic visualization of the discrete particles. In the end, each morphological exemplar is represented by , where*N*is the number of masses being used, are the 2D Cartesian coordinates of the*j*th particle, and*p*_{j}is its mass. Here, is an unit impulse function placed at the location . In this study, we used masses per structure. Following this procedure, each point mass approximation is translated so that its center of mass is at the center of the field of view, and its main axis of orientation, whose computation includes its mass distribution, is aligned with the vertical axis. For cells whose mass distribution is perfectly circular and uniform, such an axis is impossible to define, as they are identical under all possible rotations.### Optimal Transport and Linear Embedding.

The main idea in TBM is to quantify similarities between morphological structures in each dataset (all three datasets are processed independently) by measuring the amount of effort (quantified as mass times distance that it must be transported) that would have to be spent to rearrange the particle approximation of one structure onto another (7). Here we use the linearized version of this metric constructed based on a tangent space approximation of the underlying Riemannian manifold (6). The idea is to first compute a reference structure and then compute the optimal transport between each image in the database and the reference structure. As in ref. 6, we compute an average structure by running the particle approximation algorithm on the Euclidean average of the input digital images (computed after normalization for rotation, translation, and intensity, as described above). Let be the representation of the reference structure for the given dataset. Let be a sample structure from the dataset. The optimal transport between

*μ*and*σ*is computed bysubject to . Here is the family of all transport plans from

*σ*to*μ*, and*f*denotes the optimal transport plan. Thus, the transport plan is simply an assignment function that states what proportion of the mass (intensity) of the particle at has moved to the particle at (6). Because these can take fractional values between 0 and 1, splitting and joining of particles is possible. Particle splitting and merging are relatively rare occurrences as in most cases, whole (either 0 or 1) assignments are made. Similarly, let*g*be the optimal transport plan between structure*σ*and . A linear embedding for structures*μ*and*ν*can then be computed bywhereas the linear optimal transport between

*μ*and*ν*is given byThus, after preprocessing, the linear embedding for each morphological structure in a database of images is computed from Eq.

**2**. The approximate transport distance between any pair of images in the database is computed from Eq.**3**. It can be noted that the resultant dimensionality of the linear embedding is .### Visualizing Principal Phenotypic Variations.

Given the LOT embedding computed from Eq.

**2**, we use the standard PCA (12) technique for data visualization. The covariance matrix for the LOT embedded image set is , with , where**x**_{m}is the*m*th vectorized LOT-embedded image. The principal components are given by the eigenvectors of and can be used to explain, and in this case visualize, the main modes of phenotypic variation in a dataset. As customary, we retained the top*k*PCA directions that preserve 90% of the variation of the original LOT embeddings. We plotted the texture variation within cell populations by simply reconstructing intermediate LOT embeddings along the PCA axes.### Detecting and Visualizing Phenotype Differences.

The LOT embeddings can also be used for visualizing the discriminating modes of cell texture variation between cell subpopulations. To that end, we apply the

*p*LDA technique described in ref. 8 to compute the most discriminant components explaining the differences between two subsets (classes) of a given dataset. The*p*LDA direction that denotes the direction along which the projections of the*C*classes are maximally separated (in the LDA sense) is given by the solution to the optimization problemwhere represents the “within-class scatter matrix.” The penalty weight

*α*signifies a tradeoff between the traditional LDA direction and the topmost PCA directions of the same dataset (see ref. 8 for a motivation). The precise methodology for the determination of*α*for this work is based on fitting an exponential decay model to a metric that measures how far two consequent subspaces are from one another (*SI Text*). As in the case of PCA, several mutually orthogonal*p*LDA directions can be found from an augmented version of Eq.**4**, which give consecutive directions that show maximal residual discrimination between the populations. Consistent with existing literature, we retained the topmost*p*LDA direction that shows significant statistical difference (*P*≤ 0.05 in a Kolmogorov–Smirnov test between distributions) between the projected histograms of the cell subpopulations along the direction. An important distinction here is that in our method we choose to report only the statistically meaningful directions computed using cross-validation (using a portion of the data held out from the training process). We make note that, as a whole, the method described in this subsection yields a linear discriminant subspace that significantly differs from the procedure described in our earlier work (8). Comparisons between the method described here and our earlier method (8) are available in*SI Text*and show that this method can provide more reliable information regarding difference between cell populations.### Computing Morphology Variations Most Correlated with an Independent Variable.

Define vector such that

*v*_{i}is a scalar attribute of the*i*th image (i.e., dosage of Wortmannin). We are able to search in the LOT space to find a direction, which is most correlated with**v**. Hence, we are able to visualize the statistical effect of that specific attribute in the dataset. The most correlated direction,**w**_{corr}, is found as follows:where is a matrix that contains the vectorized and mean subtracted LOT images as its columns. The given direction can then be visualized by plotting , with

*λ*as a chosen length along the projection (in units of SD of the projected data along**w**_{corr}).### Note on Numerical Implementation.

All computations of TBM were performed in a highly parallel distributed computing cluster in the Electrical and Computer Engineering Department at Carnegie Mellon, and average computation time for generation of results in each dataset extended to a couple of hours. Computer code in the MatLab language is available (9).

## Acknowledgments

We thank Drs. Wei Wang, John A. Ozolek, and Dejan Slepcev for informative discussions. This work was supported by National Institutes of Health Grant GM090033.

## Supporting Information

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*Nonâ Parametric Statistics, Statistical Methods in Practice: For Scientists and Technologists*(Wiley, New York), pp 129–138.## Information & Authors

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**Published online**: February 18, 2014

**Published in issue**: March 4, 2014

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#### Acknowledgments

We thank Drs. Wei Wang, John A. Ozolek, and Dejan Slepcev for informative discussions. This work was supported by National Institutes of Health Grant GM090033.

#### Notes

This article is a PNAS Direct Submission. Z.K. is a guest editor invited by the Editorial Board.

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The authors declare no conflict of interest.

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