One common approach is to calculate the area under the ROC curve, which is abbreviated to AUC. It is equivalent to the probability that a randomly chosen positive instance is ranked higher than a randomly chosen negative instance, i.e. it is equivalent to the two sample Wilcoxon rank-sum statistic One good answer is given by the following probabilistic interpretation: The area under ROC curve specifies the probability that, when we draw one positive and one negative example at random, the decision function assigns a higher value to the positive than to the negative example The area under the receiver operator characteristic (ROC) curve is a well established measure for determining the efficacy of tests in correctly classifying diseased and non-diseased individuals

The Area Under Curve (AUC) metric measures the performance of a binary classification . In a regression classification for a two-class problem using a probability algorithm, you will capture the probability threshold changes in an ROC curve. Normally the threshold for two class is 0.5. Above this threshold, the algorithm classifies in one class and. When we need to check or visualize the performance of the multi-class classification problem, we use the AUC (**Area** **Under** The **Curve**) **ROC** (Receiver Operating Characteristics) **curve**. It is one of the most important evaluation metrics for checking any classification model's performance The program lists the results of the individual studies included in the meta-analysis: the area under the ROC curve, its standard error and 95% confidence interval. The pooled Area under the ROC curve with 95% CI is given both for the Fixed effects model and the Random effects model (Zhou et al., 2002) The area under the ROC curve (AUC) is a measure of how well a parameter can distinguish between two diagnostic groups (diseased/normal). Theory summary The diagnostic performance of a test, or the accuracy of a test to discriminate diseased cases from normal cases is evaluated using Receiver Operating Characteristic (ROC) curve analysis (Metz, 1978; Zweig & Campbell, 1993) The area under (a ROC) curve is a measure of the accuracy of a quantitative diagnostic test. A point estimate of the AUC of the empirical ROC curve is the Mann-Whitney U estimator (DeLong et. al., 1988)

Area under the curve. When using normalized units, the area under the curve (often referred to as simply the AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative') The ROC curve. Now let's verify that the AUC is indeed equal to 0.875 in a classical way, by plotting a ROC curve and calculating the estimated AUC using the ROCR package. The ROC curve plots the False Positive Rate (FPR) on the X-axis and the True Postive Rate (TPR) on the Y-axis for all possible thresholds (or cutoff values)

- ation. For risk prediction models these risk distributions can be derived from the population risk distribution so are not independent as in diagnosis
- AUC: Area Under the ROC Curve. AUC stands for Area under the ROC Curve. That is, AUC measures the entire two-dimensional area underneath the entire ROC curve (think integral calculus) from (0,0)..
- Interpretation of ROC curve Total area under ROC curve is a single index for measuring the performance a test. The larger the AUC, the better is overall performance of the medical test to correctly identify diseased and non-diseased subjects. Equal AUCs of two tests represents similar overal
- Naturally, you might want to use the ROC curve to quantify the performance of a classifier, and give a higher score for this classifier than this classifier. That is the purpose of AUC, which stands for Area Under the Curve. AUC is literally just the percentage of this box that is under this curve
- The Genetic Interpretation of Area under the ROC Curve in Genomic Profiling Naomi R. Wray1*, Jian Yang1, Michael E. Goddard2,3, Peter M. Visscher1 1Genetic Epidemiology and Queensland Statistical Genetics, Queensland Institute of Medical Research, Brisbane, Australia, 2Department of Food and Agricultura

- An ROC curve is the most commonly used way to visualize the performance of a binary classifier, and AUC is (arguably) the best way to summarize its performan..
- The Meaning of the Area under an ROC Curve A precise meaning of the area under an ROC curve in terms of the result of a signal detection exPeriment employing the two-alternative forced choice (2AFC) technique has been known for some time. In this system, Green and Swets (6) showed that the area under the curve corresponds to th
- ate between those individuals with the disease and those without the disease. A truly useless test (one no better at identifying true positives than flipping a coin) has an area of 0.5
- The Receiver Operating Characteristic (ROC) curve is a two dimensional measure of classiﬁcation performance. The area under the ROC curve (AUC) is a scalar measure gauging one facet of performance. In this note, ﬁve idealized models are utilized to relate the shape of the ROC curve, and the area under
- In this video I describe how ROC curves are constructed and how to interpret the
- Interpretation of the area under the ROC curve Although it is not obvious from its definition, the area under the ROC curve (AUC) has a somewhat appealing interpretation. It turns out that the AUC is the probability that if you were to take a random pair of observations, one with and one with , the observation with has a higher predicted probability than the other
- Each point in a ROC curve arises from the values in the confusion matrix associated with the application of a specific cutoff on the predictions (scores) of the classifier. To construct a ROC curve, one simply uses each of the classifier estimates as a cutoff for differentiating the positive from the negative class

- As the area under an ROC curve is a measure of the usefulness of a test in general, where a greater area means a more useful test, the areas under ROC curves are used to compare the usefulness of tests. The term ROC stands for Receiver Operating Characteristic
- ative ability of prediction models even though the measure is criticized for being clinically irrelevant and lacking an intuitive interpretation. Most of the criticism is traced back to the fact that the ROC curve was introduced as the discri
- If the base rate is low, it's possible that a model has a high area under the ROC curve but still a low area under the PR curve. For example, Andy Berger notes this is the case for conflict studies, and provides some example graphs

** AUC is an abbrevation for area under the curve**. It is used in classification analysis in order to determine which of the used models predicts the classes best. An example of its application are ROC curves. Here, the true positive rates are plotted against false positive rates. An example is below. The closer AUC for a model comes to 1, the. Step 4: Interpret the ROC curve. The AUC (area under curve) gives us an idea of how well the model is able to distinguish between positive and negative outcomes. The AUC can range from 0 to 1. The higher the AUC, the better the model is at correctly classifying outcomes perform ROC analyses, including estimation of sensitivity and specificity, estimation of an ROC curve and computing the area under the ROC curve. In addition, several macros will be introduced to facilitate graphical presentation and complement existing statistical capabilities of SAS with regard to ROC curves

The receiver operating characteristics (ROC) curve was introduced in medicine [1] to evaluate diagnostic tests. The area under the curve (AUC) is used to measure the separation of patients and nonpatients, i.e. discrimination. Subsequently, the ROC curve AUC has been used to evaluate discrimination of risk prediction models Area Under the ROC curve otherwise known as Area under the curve is the evaluation metric to calculate the performance of a binary classifier. Before getting into details of AUC, lets understand.

The area under the receiver operator characteristic (ROC) curve is a well established measure for determining the efficacy of tests in correctly classifying diseased and non-diseased individuals. We use quantitative genetics theory to provide insight into the genetic interpretation of the area under the ROC curve (AUC) when the test classifier is a predictor of genetic risk ** The area under the curve (AUC) of the receiver operating characteristics curve (ROC) evaluates the separation between patients and nonpatients or discrimination**. For risk prediction models these risk distributions can be derived from the population risk distribution so are not independent as in diagnosis. A ROC curve AUC formula based on the underlying population risk distribution clarifies. Area Under the Curve: The Area Under the Curve gives us an idea of how well the model is able to distinguish between positive and negative outcomes. The AUC can range from 0 to 1. The higher the AUC, the better the model is at correctly classifying outcomes. We can see that the AUC for this particular logistic regression model is .948, which is. AUC: Area Under the ROC Curve AUC stands for 'Area under the ROC Curve.' That is, AUC measures the entire two-dimensional area underneath the entire ROC curve (think integral calculus) from (0,0) to (1,1). Figure 5. AUC (Area under the ROC Curve)

* An incredibly useful tool in evaluating and comparing predictive models is the ROC curve*. Its name is indeed strange. ROC stands for receiver operating characteristic. Its origin is from sonar back in the 1940s; ROCs were used to measure how well a sonar signal (e.g., from a submarine) could be detected from noise (a school of fish). In its current usage, ROC curves are a nice way to see how. One of the most commonly used metrics nowadays is AUC-ROC (Area Under Curve - Receiver Operating Characteristics) curve. ROC curves are pretty easy to understand and evaluate once there is a good understanding of confusion matrix and different kinds of errors. In this article, I will explain the following topics

- ROC curves can be used to evaluate how well these methods perform. Statistics. Area under the ROC curve with confidence interval and coordinate points of the ROC curve. Plots: ROC curve. Methods. The estimate of the area under the ROC curve can be computed either nonparametrically or parametrically using a binegative exponential model. Show m
- Other ROC Curve Examples. Taken from . AUC: Area Under ROC Curve. Area Under ROC Curve Measure for evaluating the performance of a classifier; it's the area under the ROC Curve; total area is 100% so AUC = 1 is for a perfect classifier for which all positive come after all negatives; AUC = 0.5 - randomly ordere
- Therefore, there the AUC score is 0.9 as the area under the ROC curve is large. Whereas, if we see the last model, predictions are completely overlapping each other and we get the AUC score of 0.5
- roc.area: Area under curve (AUC) calculation for Response Operating Characteristic curve. Description This function calculates the area underneath a ROC curve following the process outlined in Mason and Graham (2002). The p-value produced is related to the Mann-Whitney U statistics
- Fig. 1 — Some theoretical ROC curves AUC. While it is useful to visualize a classifier's ROC curve, in many cases we can boil this information down to a single metric — the AUC.. AUC stands for area under the (ROC) curve.Generally, the higher the AUC score, the better a classifier performs for the given task

ROC Area Under Curve (AUC) Score. Although the ROC Curve is a helpful diagnostic tool, it can be challenging to compare two or more classifiers based on their curves. Instead, the area under the curve can be calculated to give a single score for a classifier model across all threshold values. This is called the ROC area under curve or ROC AUC. Some other answers alluded to a simplistic interpretation of the ROC curve: The higher the area under the curve, the better the model is at separating positive and negative groups. The terminology can apply to any two labels, but positive and negative are most commonly used Interpretation. The area under the ROC curve values range from 0.5 to 1. When the binary model can perfectly separate the classes, then the area under the curve is 1. When the binary model cannot separate the classes better than a random assignment, then the area under the curve is 0.5 An ROC curve may be summarized by the area under it (AUC). This area has an additional interpretation. Suppose that a rater is asked to study two subjects, one that is actually disease positive and one that is disease negative. The AUC is equal to the probability that the rater will give the disease positive subject a higher scor

The Area Under the Curve (AUC) is the measure of the ability of a classifier to distinguish between classes and is used as a summary of the ROC curve. The higher the AUC, the better the performance of the model at distinguishing between the positive and negative classes The area under the ROC curve (AUC) is widely recog-nized as the measureof a diagnotic test'sdiscriminatory power.5 The maximum value for the AUC is 1.0, thereby indicating a (theoretically) perfect test (i.e., 100% sensitive Can J Emerg Med 2006;8(1):19-20 Received: Nov. 15, 2005; accepted: Nov. 22, 200 Area under curve (AUC) is directly related to Mann Whitney U test. People from analytics community also call it Wilcoxon rank-sum test. This test assumes that the predicted probability of event and non-event are two independent continuous random variables. Area under the curve = Probability that Event produces a higher probability than Non-Event

Plotting the approach. If the ROC curve were a perfect step function, we could find the area under it by adding a set of vertical bars with widths equal to the spaces between points on the FPR axis, and heights equal to the step height on the TPR axis In this figure, the blue area corresponds to the Area Under the curve of the Receiver Operating Characteristic (AUROC). The dashed line in the diagonal we present the ROC curve of a random predictor: it has an AUROC of 0.5. The random predictor is commonly used as a baseline to see whether the model is useful Upload an image to customize your repository's social media preview. Images should be at least 640×320px (1280×640px for best display)

istic (ROC) curve, which is frequently used in the diagnosis and classi cation literature, and the Gini terminology, which is mainly used in the economic lit-erature, is clari ed. It is shown that the area under the ROC curve is related Key words: Area under the curve (AUC), Gini covariance, Lorenz curve, Classi cation ycorresponding author The ROC curve is also important because the area under the curve (AUC) is a reflection of how good the test is at distinguishing between patients with disease and those without disease. The AUC serves as a single measure, independent of prevalence, that summarizes the discriminative ability of a test across the full range of cut‐offs ( 14 ) ** The AUC is the area under the ROC curve**. It is a number between zero and one, because the ROC curve fits inside a unit square. Any model worth much of anything has an AUC larger than 0.5, as the line segment running between (0, 0) and (1, 1) represents a model that randomly guesses class membership The accuracy of a test is measured by the area under the ROC curve (AUC). AUC is the area between the curve and the x axis. An area of 1 represents a perfect test, while an area of .5 represents a worthless test. The closer the curve follows the left-upper corner of the plot, the more accurate the test

* The area under a receiver operating characteristic (ROC) curve*, abbreviated as AUC, is a single scalar value that measures the overall performance of a binary classifier (Hanley and McNeil 1982) ROC curves are a useful tool in the assessment of the performance of a diagnostic test over the range of possible values of a predictor variable. The area under an ROC curve provides a measure of discrimination and allows investigators to compare the performance of two or more diagnostic tests

Area Under Curve: like the AUC, summarizes the integral or an approximation of the area under the precision-recall curve. In terms of model selection, F-Measure summarizes model skill for a specific probability threshold (e.g. 0.5), whereas the area under curve summarize the skill of a model across thresholds, like ROC AUC Note. This function is used internally in the roc.plot command to calculate areas.. Author(s) Matt Pocernich. References. Mason, S. J. and Graham, N. E. (2002) Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation, Q. J. R. Meteorol.Soc. 128, 2145-2166. See Als **interpretation** of this analysis, such as smooth and empirical **ROC** **curves**, para-metric and nonparametric methods, the **area** **under** the **ROC** **curve** and its 95% confidence interval, the sensitivity at a particular FPR, and the use of a partial **area** **under** the **ROC** **curve** are discussed The receiver operating characteristic (ROC) curve is a popular technique with applications, for example, investigating an accuracy of a biomarker to delineate between disease and non-disease groups. A common measure of accuracy of a given diagnostic marker is the area under the ROC curve (AUC) The ROC curve visualizes the tradeoffs between the true positive rate and false positive rate or sensitivity vs. 1-specificity. Particularly we are usually interested in the area under the ROC curve (AROC or c-statistic). The ROC curve is a measure of a model's discriminatory power

The ROC curve is an often-used performance metric for classification problems. In this article, we attempt to familiarize ourselves with this evaluation method from scratch, beginning with what a curve means, the definition of the ROC curve to the Area Under the ROC curve (AUC), and finally, its variants If you have participated in any online machine learning competition/hackathon then you must have come across Area Under Curve Receiver Operator Characteristic a.k.a AUC-ROC, many of them have it as their evaluation criteria for their classification problems One ROC Curve and Cutoff Analysis Introduction This procedure generates empirical (nonparametric) and Binormal ROC curves. It also gives the area under the ROC curve (AUC), the corresponding confidence interval of AUC, and a statistical test to determine if AUC is greater than a specified value The area under the ROC curve, or AUC, seems like a nice heuristic to evaluate and compare the overall performance of classification models independent of the exact decision threshold chosen. signifies perfect classification accuracy, and is the accuracy of making classification decisions via coin toss (or rather a continuous coin that outputs values in ) This summary index is considered to be more practically relevant than the area under the entire ROC curve (AUC), but because of several perceived limitations, it is not used as often. To improve interpretation, results for pAUC analysis are frequently reported using a rescaled index such as the standardized partial AUC proposed by McClish (1989)

- Help for Area Under ROC curve (AUC) interpretation Reliability of resultant top 200 candidates depends on predictive power of AraNet for the given query gene set as measured by AUC score from ROC analysis
- A measure of the accuracy of the test is the area under the ROC curve (AUC: Area Under Curve). The area can take values between 0.5 and 1, a higher value indicating the better quality. The easiest way to calculate AUC is with the trapezoidal method, which generally estimates the area well
- Area Under the ROC Curve. Besides showing us how thresholds affect test performance, ROC curves can allow us to compare different tests. As we have alluded to earlier, the closer the ROC curve reaches to the top left corner, the better the test. Illustration of 3 different ROC curves from imaginary data
- Sample 64029: Area under the ROC curve measure (AUC) for multinomial models / / / / / / / / >) ® ®.. . The next model is also a generalized logit model but is unrestricted, allowing for separate slopes on the two logits for both predictors. Since the.

- An ROC curve demonstrates several things: It shows the tradeoff between sensitivity and specificity (any increase in sensitivity will be accompanied by a decrease in specificity). The closer the curve follows the left-hand border and then the top border of the ROC space, the more accurate the test
- AUC Area Under the ROC Curve Interpretation Specifies how much better or worse from CIS 375 at Arizona State Universit
- The ROC curve for naive Bayes is generally lower than the other two ROC curves, which indicates worse in-sample performance than the other two classifier methods. Compare the area under the curve for all three classifiers

Interpretation of classification table in stata for a logistic regression and ROC curve 19 Aug 2016, 02:15. Dear Everyone, I performed some tests for my logistic regression: lroc, estat class, cutoff(0.15), and estat gof, group (10). My Your area under the ROC curve is mediocre,. However, -lroc- provides area under ROC curve as point estimate. I wonder if there is a command or a method in STATA that can calculate the point estimate and 95% confidence interval of C-statistics? I did not think that it is necessary to have the CIs until I saw that several articles have reported C-statistics and its 95% confidence intervals A representation and interpretation of the area under a receiver operating characteristic (ROC) curve obtained by the rating method, or by mathematical predictions based on patient characteristic.. The full area under a given ROC curve, or AUC, formulates an important statistic that represents the probability that the prediction will be in the correct order when a test variable is observed (for one subject randomly selected from the case group, and the other randomly selected from the control group) Two areas separated by this ROC curve indicates a simple estimation of the performance level. ROC curves in the area with the top left corner (0.0, 1.0) indicate good performance levels, whereas ROC curves in the other area with the bottom right corner (1.0, 0.0) indicate poor performance levels

The partial area under the ROC curve (pAUC), when focused on the range of practical/clinical relevance, is considered a more relevant summary index than the full AUC. However, several conceptual and analytical difficulties frequently prevent the pAUC from being used. First, i A typical ROC curve looks like this, which shows two ROC curves for Algorithm 1 and Algorithm 2. The goal is to have a model be at the upper left corner, which is basically getting no false positives - a perfect classifier. The receiver operating characteristic area under curve (ROC AUC) is just the area under the ROC curve. The higher it is. Area Under the Curve (AUC) Area under ROC curve is often used as a measure of quality of the classification models. A random classifier has an area under the curve of 0.5, while AUC for a perfect classifier is equal to 1. In practice, most of the classification models have an AUC between 0.5 and 1 After plotting the ROC Curve, the area under it is called Area Under the ROC Curve (AUC), Area Under the Curve (AUC), or AUROC.It's been said that ROC is a probability curve and AUC represents degree or measure of separability.In other words, AUC is a single metric that can be used to quantify how well two classes are separated by a binary classifier..

We have run two different models and have areas under the ROC curve of .5785 and .8330. Next, we will use the two linear predictors with the roccomp command to get a test of the differences in area under the ROC curve ROC curve and Area under the Curve (AUC) ROC - Receiver operating characteristic curve is a curve between true positive rate and false positive rate for various threshold values. ROC curve tells us how good/bad mode * Interpretation of the area under the curve (AUC) We now prove that the area under the ROC curve is the probability that the classification algorithm will rank a randomly chosen data point, , that belongs to class higher than a randomly chosen data point, , that belongs to class (i*.e., that )

ROC curve has a much greater distance from the 45-degree diagonal line. It is recommended that researchers identify whether the scores for the positive and negative groups need to be transformed to more closely follow the Normal distribution before using the Binormal ROC Curve methods. Area under the ROC Curve (AUC * The One ROC Curve and Cutoff Analysis procedure generates empirical (nonparametric) and Binormal ROC curves*. It also gives the area under the ROC curve (AUC), the corresponding confidence interval of AUC, and a statistical test to determine if AUC is greater than a specified value

Area Under (ROC) Curve. The optimal point on the ROC curve is (FPR, TPR) = (0,1). No false positives and all true positives. So the closer we get there the better. The second essential observation is that the curve is by definition monotonically increasing The result of a diagnostic test is in general not binary (positive/negative) but a quantitative parameter (such as a biomarker). If an appropriate threshold for the quan- titative parameter has not yet been defined, the receiver operating characteristic (ROC) curve and in particular the area under this curve, are appropriate for evaluating the overall accuracy of the diagnostic test [1]

The area under the receiver operator characteristic (ROC) curve is a well established measure for determining the efficacy of tests in correctly classifying diseased and nondiseased individuals. We use quantitative genetics theory to provide insight into the genetic interpretation of the area under the ROC curve (AUC) when the test classifier is a predictor of genetic risk Compute the area under the ROC curve. Notes. Since the thresholds are sorted from low to high values, they are reversed upon returning them to ensure they correspond to both fpr and tpr, which are sorted in reversed order during their calculation. References. 1 The objectives are to describe the disadvantages of the area under the receiver operating characteristic curve (ROC AUC) to measure diagnostic test performance and to propose an alternative based on net benefit. We use a narrative review supplemented by data from a study of computer-assisted detection for CT colonography. We identified problems with ROC AUC the ROC curve is a straight line connecting the origin to (1,1). Any improvement over random classiﬁcation results in an ROC curve at least partially above this straight line. The AUC is deﬁned as the area under the ROC curve. Consider a binary classiﬁcation task with m positive examples and n negative examples

A representation and interpretation of the area under a receiver operating characteristic (ROC) curve obtained by the rating method, or by mathematical predictions based on patient characteristics, is presented. It is shown that in such a setting the area represents the probability that a randomly chosen diseased subject is (correctly) rated or ranked with greater suspicion than a randomly. One of the most commonly used summary indices derived from the ROC curve is the area under the curve (AUC). AUC has a convenient interpretation and a close relationship to the well-known Wilcoxon statistic [ 1 ]; as a result, methods for AUC-based analyses are well developed and widely used [ 2 , 3 ] Partial area under the ROC curve (pAUC) is defined as an integration of the ROC curve over only a confined range of FPR. In many situations, pAUC may be a more relevant measure than AUC. For example, the area under the ROC curve over a limited region of very low false-positive rate is of much interest to investigators in cancer screening Good afternoon, I am looking for the AUC value (Area Under the Curve or ROC Index) in SAS Enterprise Miner in SAS 9.3. I have explored all the options from the ROC curve graph (see pic) and from the nodes I am using (logistic regression, decision trees and neural nets) without any success

Abstract. Rather than viewing receiver operating characteristic (ROC) curves directly to compare the performances of diagnostic methods, the whole and the partial areas under the ROC curve (area under the ROC curve [AUC] and partial area under the ROC curve [pAUC]) are 2 of the most popularly used summaries of the curve The ROC Curve measures how accurately the model can distinguish between two things (e.g. determine if the subject of an image is a dog or a cat). AUC measures the entire two-dimensional area underneath the ROC curve. This score gives us a good idea of how well the classifier will perform

Comparative statistical properties of expected utility and area under the ROC curve for laboratory studies of observer performance in screening mammography. Abbey CK(1), Gallas BD(2), Boone JM(3), Niklason LT(4), Hadjiiski LM(5), Sahiner B(2), Samuelson FW(2) **ROC** **curves** that fall **under** the **area** at the top-left corner indicate good performance levels, whereas **ROC** **curves** fall in the other **area** at the bottom-right corner indicate poor performance levels. An **ROC** **curve** of a perfect classifier is a combination of two straight lines both moving away from the baseline towards the top-left corner Sensitivity, specificity, tradeoffs and ROC curves. With a little bit of radar thrown in there for fun

ROCR - 2005. ROCR has been around for almost 14 years, and has be a rock-solid workhorse for drawing ROC curves. I particularly like the way the performance() function has you set up calculation of the curve by entering the true positive rate, tpr, and false positive rate, fpr, parameters.Not only is this reassuringly transparent, it shows the flexibility to calculate nearly every performance. The ROC curve is increasing and invariant under any monotone increasing transformation of the variables X and Y. Several ROC curve summary measures have been proposed in the literature, such as the area under the curve (AUC) or the Youden index (maxc{Se(c)+Sp(c)−1}). They are considered as summaries of the discriminatory accuracy of a test

AZ - Area under the ROC Curve. Looking for abbreviations of AZ? It is Area under the ROC Curve. Area under the ROC Curve listed as AZ. Area under the ROC Curve - How is Area under the ROC the test with the largest area under the ROC curve is the best test, although the interpretation becomes more complex if ROC curves of different tests. Area under the ROC curve is calculated using trapz function. AUC is always in between 0.5 (two classes are statistically identical) and 1.0 (there is a threshold value that can achieve a perfect separation between the classes). Area under ROC Curve (AUC) measure is very similar to Wilcoxon Rank Sum Test (see wilcox.test) and Mann-Whitney U Test