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A Nonparametric Test for the General Two-Sample Problem
Abstract
For two independently drawn samples of data, a novel statistical test is proposed for the null hypothesis that both samples originate from the same population. The underlying distribution function does not need to be known but must be continuous, i.e., it is a nonparametric test. It is demonstrated for suitable examples that the test is easy to apply and is at least as powerful as the commonly used nonparametric tests, i.e., the Kolmogorov-Smirnov, the Cramer-von Mises, and the Wilcoxon tests.
... Various distribution-free and nonparametric statistics have been proposed and discussed over the course of years. Baumgartner et al. (1998) defined a novel nonparametric rank statistic, namely B, for the two-sample problem. The power of the B statistic is almost equivalent to that of the well-known Wilcoxon test (Gibbons and Chakraborti, 2003) for shifted location parameters. ...
... Baumgartner et al. asserted that the B statistic could be applied to scale parameters. Baumgartner et al. (1998) and Murakami (2006) showed that the B statistic was more powerful than the Kolmogorov-Smirnov (Hájek et al., 1999), Cramér-von Mises (Hájek et al., 1999), and Anderson-Darling (Pettitt, 1976) statistics for scale parameters. Since its proposal, various modifications to the B statistic have been considered (e.g., Neuhäuser (2001Neuhäuser ( , 2003Neuhäuser ( , 2005, Murakami (2006)). ...
... In Murakami (2007), the exact critical value of the B statistic and the B 1 statistic are calculated when the sample sizes are small. Baumgartner et al. (1998) proposed a critical value for the asymptotic distribution of the B statistic. A much more detailed table with the asymptotic distribution of the B statistic was given by Neuhäuser (2012). ...
ABSTRACT In this paper, we focus on the univariate two-sample Baumgartner statistic and propose a modification of the B statistic for the shifted-scale parameter. A nonparametric rank test based on the Baumgartner statistic was used to test location, scale, and location-scale parameters. Critical values of the test statistics were evaluated; limiting distributions were derived under the null hypothesis. We investigated the power of the proposed statistics by simulation studies. The results of our simulations indicated that the B2 statistic was superior to the B1 statistic for the shifted scale parameters when the sample sizes were equal under symmetric distributions. The differences between these two statistics were small when the sample sizes were unequal. The B2 statistic is more efficient than other nonparametric statistics.
... Toward this end, the Wilcoxon rank sum test is a useful tool when there are reasons to believe that the outcome variables of interest may fail certain distributional assumptions required for parametric methods. However, as discussed in Baumgartner [18], Wilcoxon rank sum test is not suitable for situations where the expected values of the two populations are close to each other. To overcome this problem, they proposed a more powerful nonparametric test to handle the general two-sided two-sample problem [18]. ...
... However, as discussed in Baumgartner [18], Wilcoxon rank sum test is not suitable for situations where the expected values of the two populations are close to each other. To overcome this problem, they proposed a more powerful nonparametric test to handle the general two-sided two-sample problem [18]. Neuhaeuser further extended the two-sided two-sample test to a one-sided test that can detect if one population is stochastically larger than the other [19]. ...
... The permutation results depend on the sample size. But as mentioned in Baumgartner [18], for a two-sided test, the asymptotic distribution can be approximated by the permutation method quite well even with a small sample size (as small as 10). We first derive the empirical formula to fit the asymptotic distribution using the permutation method. ...
DNA methylation profiles differ among disease types and, therefore, can be used in disease diagnosis. In addition, large-scale whole genome DNA methylation data offer tremendous potential in understanding the role of DNA methylation in normal development and function. However, due to the unique feature of the methylation data, powerful and robust statistical methods are very limited in this area.
In this paper, we proposed and examined a new statistical method to detect differentially methylated loci for case control designs that is fully nonparametric and does not depend on any assumption for the underlying distribution of the data. Moreover, the proposed method adjusts for the age effect that has been shown to be highly correlated with DNA methylation profiles. Using simulation studies and a real data application, we have demonstrated the advantages of our method over existing commonly used methods.
Compared to existing methods, our method improved the detection power for differentially methylated loci for case control designs and controlled the type I error well. Its applications are not limited to methylation data; it can be extended to many other case-control studies.
... To accurately estimate the coherence, two sources of errors need to be corrected: 1) removing signal nonstationarity induced by image textures, in which different populations are included during averaging of the coherence estimator [16], [17]; and 2) mitigating bias and variance of the coherence estimator by collecting large samples. To this end, a two-step strategy is developed under the framework of the time-series-based ESD analysis. ...
... where b1 and b2 indicate burst 1 and burst 2, respectively. The similarity of neighboring k with respect to the reference pixel p in the coherence estimate window is tested by the Baumgartner-Wei β-Schindler (BWS) test [16], [17] under a given significance level α ...
... The source to implement this test has been given in http://mijiang.org.cn/index.php/software/. It is worth pointing out that the BWS test places more weight on observations in the tail of the distribution [16], [17]. It is a better measure for the general two-sample problem than its competitors such as Cramér-von Mises and Kolmogorov-Smirnov tests [18], [19]. ...
... The well known test of Wilcoxon (Hollander and Wolfe, 1999) is a test for location as F(x) = G(y -6). Baumgartner, Weif3 and Schindler (1998) presented a new nonparametric two-sample test which has almost same power as the Wilcoxon test for the location parameter. They assert that the Baumgartner statistic can be applied for the scale parameter as F(x) = G(~ ), where a 0. Let R1 < ... < Rn and H1 < ... < Hm denote the combined-samples ranks of the X-values and Y-values in increasing order of magnitude, respectively. ...
... They assert that the Baumgartner statistic can be applied for the scale parameter as F(x) = G(~ ), where a 0. Let R1 < ... < Rn and H1 < ... < Hm denote the combined-samples ranks of the X-values and Y-values in increasing order of magnitude, respectively. The test statistic B proposed by Baumgartner et al. (1998) is where Recently, Neuhauser (2003) researched ties of the Baumgartner statistic. Also, Neuhauser (2000) suggested an exact two-sample test based on the Baumgartner statistic. ...
... The critical values of B statistic and B* statistic are listed in Table 1 including different sample sizes for simulation studies. Baumgartner et al. (1998) derived the asymptotic distribution, and its critical values were Pr(B > 3.880) = 0.010 and Pr(B > 2.493) = 0.050. The asymptotic distribution of the B* statistic is same as the B statistic. ...
The purpose of this paper is to develop a nonparametric k-sample test based on a modified Baumgartner statistic. We define a new modified Baumgartner statistic B* and give some critical values. Then we compare the power of the B* statistic with the t-test, the Wilcoxon test, the Kolmogorov-Smirnov test, the Cramér-von Mises test, the Anderson-Darling test and the original Baumgartner statistic. The B* statistic is more suitable than the Baumgartner statistic for the location parameter when the sample sizes are not equal. Also, the B* statistic has almost the same power as the Wilcoxon test for location parameter. For scale parameter, the power of the B* statistic is more efficient than the Cramér-von Mises test and the Anderson-Darling test when the sizes are equal. The power of the B* statistic is higher than the Kolmogorov-Smirnov test for location and scale parameters. Then the B* statistic is generalized from two-sample to k-sample problems. The B*k statistic denotes a k-sample statistic based on the B* statistic. We compare the power of the B*k statistic with the Kruskal-Wallis test, the k-sample Kolmogorov-Smirnov test, the k-sample Cramér-von Mises test, the k-sample Anderson-Darling test and the k-sample Baumgartner statistic. Finally, we investigate the behavior of power about the B*k statistics by simulation studies. As a result, we obtain that the B*k statistic is more suitable than the other statistics.
... Baumgartner et al. (1998) proposed a novel statistical test for the null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for more than two samples. Whereas the expressions of the exact density and distribution functions of the generalized Baumgartner statistic are not yet found, the characteristic function of its limiting distribution has been obtained. ...
... Introduction Baumgartner et al. (1998) proposed a novel statistical test for a null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for cases with more than two samples. One of the most important procedures for test statistics is to determine their density and distribution functions as well as their percentage points under a null hypothesis. ...
... In addition, the observations are combined and ranked in increasing order of magnitude. We are interested in testing the null hypothesis H 0 : F = G against H A : F G. Further denoting R 1 < · · · < R n and P 1 < · · · < P m the combined-samples ranks of the X-value and Y-value, respectively, Baumgartner et al. (1998) defined a novel nonparametric two-sample test statistic as ...
Baumgartner et al. (1998) proposed a novel statistical test for the null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for more than two samples. Whereas the expressions of the exact density and distribution functions of the generalized Baumgartner statistic are not yet found, the characteristic function of its limiting distribution has been obtained. Due to the development of computational power, the Fourier series approximation can be readily utilized to accurately and efficiently approximate its density function based on its Laplace transform. Numerical examples show that the Fourier series method provides an accurate approximation for statistical quantities of the generalized Baumgartner statistic.
... To evaluate whether two pixels at spatial locations p and q are homogeneous, a twosample nonparametric Baumgartner-Weiβ-Schindler (BWS) test should be applied to cope with various distributions caused by temporal variability. This test is a better measure for the general two-sample problem than its competitors due to smaller type II error (Baumgartner et al., 1998). Given two temporal data vectors I(p) and I(q), the BWS test compares the difference between F p and F q using the following statistic: ...
... where the rank G t (p) (H t (q)) of each element I t (p) (I t (q)) is defined as the number of data in both sets {I(p), I(q)} smaller or equal to I t (p) (I t (q)). The BWS test considers the two data vectors drawn from the same statistical population if B is smaller than a threshold that depends on the significance level α bws and can be found in published tables (Baumgartner et al., 1998). ...
One main challenge in detecting built-up land cover changes using synthetic aperture radar (SAR) instruments is that complicated backscattering behaviours and the superimposition of speckles on rich textures cause a large number of false alarms. Using trajectory-based analyses from time-series SAR imagery can mitigate false alarms since the temporal variability in backscattering during construction improves discrimination capability. This paper presents an approach towards the detection of built-up land change based on a single-channel SAR stack. The proposed methodology includes the generation of a change indicator, the Markov modelling procedure and the delineation of changes over built-up areas. The generation of the change indicator aims to provide a feature with abundant contrast between changed and stable areas, a high signal-to-noise ratio and detail preservation. To this end, all temporal information is converted into a map of the coefficient of variation. After error removal, this change detector is combined with a Markov random field (MRF) criterion function. Rather than MRF modelling by iteration with very complex stochastic models, we propose using SAR temporal trajectory under a hypothesis test framework and interferometric coherence series to establish conditional density for each class. Then, the Graph-cuts theory is applied to delineate the boundary between changed and stable areas, followed by a binary classification procedure based on speckle divergence to exclude natural areas. The technique is tested on both synthetic data and two TerraSAR-X datasets covering representative areas with rich texture. We found that in a complex built environment that is challenging for classical change indicators and state-of-the-art techniques, the presented method can provide smaller overall error with better detail preservation.
... A rank based test for comparing locations of two continuous populations was proposed by Baumgartner, Weiβ, and Schindler (BWS) (Baumgartner et al., 1998). This nonparametric test is based on the squared value of the difference between the two empirical distribution functions weighted by the respective variance. ...
... A nonparametric two-sample test for determining whether the two samples are from the same population was proposed by Baumgartner, Weiβ, and Schindler (Baumgartner et al., 1998). Let the sample corresponding to one population be denoted by Y̰ = (Y 1 , Y 2 , …, Y m 1 )′ and let Z̰ = (Z 1 , Z 2 , …, Z m 2 )′ denote the sample for a second population. ...
We propose a modified nonparametric Baumgartner–Weiß–Schindler test and investigate its use in testing for trends among KK binomial populations. Exact conditional and unconditional approaches to pp-value calculation are explored in conjunction with the statistic in addition to a similar test statistic proposed by Neuhäuser (2006), the unconditional approaches considered including the maximization approach (Basu, 1977), the confidence interval approach (Berger and Boos, 1994), and the E+ME+M approach (Lloyd, 2008). The procedures are compared with regard to actual Type I error and power and examples are provided. The conditional approach and the E+ME+M approach performed well, with the E+ME+M approach having an actual level much closer to the nominal level. The E+ME+M approach and the conditional approach are generally more powerful than the other pp-value calculation approaches in the scenarios considered. The power difference between the conditional approach and the E+ME+M approach is often small in the balance case. However, in the unbalanced case, the power comparison between those two approaches based on our proposed test statistic show that the E+ME+M approach has higher power than the conditional approach.
... We use the Baumgartner-Weiß-Schindler (BWS) statistical test [19] to estimate if the medium utilization is above a given threshold. This is achived by comparing the empirical distribution of CAD values obtained during a live experiment with a known distribution for different medium utilization levels and data rates at which the packets are sent, obtained during the training session on our testbed. ...
... Clearly, such an active probing technique has an inherent tradeoff between the estimation accuracy and the overhead of probe packets in a time interval further adding to congestion. We use the Baumgartner-Weiß-Schindler (BWS) statistical test [19] to estimate if the medium utilization is above a given threshold. This is achived by comparing the empirical distribution of CAD values obtained during a live experiment with a known distribution for different medium utilization levels and data rates at which the packets are sent, obtained during the training session on our testbed. ...
Rate adaptation is a critical component that impacts the performance of IEEE 802.11 wireless networks. In congested networks, traditional rate adaptation algorithms have been shown to choose lower data-rates for packet transmissions, leading to reduced total network throughput and capacity. A primary reason for this behavior is the lack of real-time congestion measurement techniques that can assist in the identification of congestion-related packet losses in a wireless network. In this work, we first propose two real-time congestion measurement techniques, namely an active probe-based method called Channel Access Delay, and a passive method called Channel Busy Time. We evaluate the two techniques in a testbed network and a large WLAN connected to the Internet. We then present the design and evaluation of Wireless cOngestion Optimized Fallback (WOOF), a rate adaptation scheme that uses congestion measurement to identify congestion-related packet losses. Through simulation and testbed implementation we show that, compared to other well-known rate adaptation algorithms, WOOF achieves up to 300 percent throughput improvement in congested networks.
... The performance comparison has been made using t-test, F-test and p-value (in terms of the number of enriched attributes or GO (gene ontology) attributes). In addition, we have used biological and statistical measurements like pi-GSEA [24], Fisher-score [25], KOGS [26], SPEC [27], W-test [28,30], BWS [29] for identifying the biologically and statistically relevant gene set. ...
... Baumgartner-Wei-Schindler non-parametric test (BWS test) makes the same assumptions about the samples as W test does [29]. However, it has probed to be less conservative and to yield better results. ...
In this article, we propose a methodology for selecting genes that may have a role in mediating a disease in general and certain cancers in particular. The methodology, first of all, groups an entire set of genes. Then the important group is determined using two neuro-fuzzy models. Finally, individual genes from the most important group are evaluated in terms of their importance in mediating a cancer, and important genes are selected. A method for multiplying existing data is also proposed to create a data rich environment in which neuro-fuzzy models are effective. The effectiveness of the proposed methodology is demonstrated using five microarray gene expression data sets dealing with human lung, colon, sarcoma, breast and leukemia. Moreover, we have made an extensive comparative analysis with 22 existing methods using biochemical pathways, p-value, t-test, F-test, sensitivity, expression profile plots, pi-GSEA, Fisher-score, KOGS, SPEC, W-test and BWS, for identifying biologically and statistically relevant gene sets. It has been found that the proposed methodology has been able to select genes that are more biologically significant in mediating certain cancers than those obtained by the others.
... These include the Gastwirth test G U and the Wilcoxon test W U for symmetric distributions with short and midlong tails, respectively. For right-skewed distributions, the Baumgartner-Wei β-Schindler test B U is used [33]. The gain of these tests on power actually attributes to the use of the appropriate weight functions in the specified types of distributions. ...
... where U , M , M , and k are identical to the aforementioned case and O 1 < · · · < O M (Q 1 < · · · < Q M ) are the ranks of the combined samples of V (P ) (V (P )) in an increasing order of magnitude. The asymptotic distribution of test statistic B U has been shown in [33]. ...
A novel coherence estimation method for small data sets is presented for interferometric synthetic aperture radar (SAR) (InSAR) data processing and geoscience applications. The method selects homogeneous pixels in both the spatial and temporal spaces by means of local and nonlocal adaptive techniques. Reliable coherence estimation is carried out by using such pixels and by correcting the bias in the estimated coherence caused by the non-Gaussianity in high-resolution SAR scenes. As an example, the proposed method together with coherence decomposition is applied to extract the temporal decorrelation component over an area in Macao. The results show that the proposed algorithms work well over various types of land cover. Moreover, the coherence change with time can be more accurately detected compared to other conventional methods.
... In this paper, we will implement another statistical test: the Baumgartner-Weiß-Schindler (BWS) test (Jiang et al., 2013;Neuhäuser, 2005). The BWS test has been proven to be the most effective in general alternative settings and works with a low complexity (Baumgartner et al., 1998). The test statistic is defined as, ...
... n) values from the first (second) group in increasing order of magnitude. In comparison with standard nonparametric tests, a practical advantage of BWS test is the excellent small sample behavior (Baumgartner et al., 1998) under various distributions. This increases the performance of similarity test algorithm at small stack condition. ...
The Goldstein filter is a well-known filter for interferometric filtering in the frequency domain. The main parameter of this filter, alpha, is set as a power of the filtering function. Depending on it, considered areas are strongly or weakly filtered. Several variants have been developed to adaptively determine alpha using different indicators such as the coherence, and phase standard deviation. The common objective of these methods is to prevent areas with low noise from being over filtered while simultaneously allowing stronger filtering over areas with high noise. However, the estimators of these indicators are biased in the real world and the optimal model to accurately determine the functional relationship between the indicators and alpha is also not clear. As a result, the filter always under- or over-filters and is rarely correct. The study presented in this paper aims to achieve accurate alpha estimation by correcting the biased estimator using homogeneous pixel selection and bootstrapping algorithms, and by developing an optimal nonlinear model to determine alpha. In addition, an iteration is also merged into the filtering procedure to suppress the high noise over incoherent areas. The experimental results from synthetic and real data show that the new filter works well under a variety of conditions and offers better and more reliable performance when compared to existing approaches.
... The expected values of the two populations are required to be different for the Wilcoxon test to be powerful versus the Kolmogorov-Smirnov test that is well suited if the variances are different (Baumgartner et al., 1998). Then which test should be chosen when one does not know in advance whether the mean or variance or both might be different? ...
... Then which test should be chosen when one does not know in advance whether the mean or variance or both might be different? Baumgartner et al. (1998) proposed a test by extending the idea of Anderson and Darling's (1954) for the one-sample case to the two-sample problem. However, one can furnish another methodology by introducing the concept of the simultaneous test for location and scale parameters without further assumptions for the underlying distribution except the continuity. ...
In this study, we propose a simultaneous test procedure based on the individual - values for each sub-null hypothesis with several well-known combining functions. We then compare the efficiency of our procedure with existing tests by obtaining empirical powers through a simulation study. Finally, we discuss some interesting features related to simultaneous test and point out a misconduct for the simulation study published in the previous work.
... Some related works include Pettitt [26] and Baumgartner, Weiß, and Schindler [4]. They considered statistics of Anderson-Darling type that can be viewed as standardized versions of Cramér-von Mises statistics. ...
... The first version is the sample plug-in version of the left side of (7). With τ = m/N and multiplying by mn/N , it is our test statistic defined in (4). H 0 is rejected if the sample version is large, i.e., T > c α (m, n). ...
We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the two-sample Cram\'er-von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the H\'ajek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.
... Convergence was reached after typically 100 iterations. All samples from the Netherlands were compared with the western and eastern samples using a general non parametric two sample test (BWS-test, Baumgartner et al. 1998). This method tests the nullhypothesis H 0 that two independent samples belong to the same underlying distribution against the alternative hypothesis H 1 that the underlying distributions differ significantly. ...
... This was also shown by testing the specimens from The Netherlands against the Western and Eastern F. verna lineages as well as by testing resampled groups comprising mixtures of eastern and western individuals. The BWS-test (Baumgartner et al. 1998) yielded for the specimens of Western Europe (France, Switzerland, Belgium, Great Britain, Ireland, Norway) and The Netherlands a test value of B=6.22 which corresponds to a significance level α<0.001. Thus, the null-hypothesis has to be rejected and, instead, F. verna plants from The Netherlands have to be considered a separate entity and not simply a mixture of Western and Eastern F. verna lineages. ...
The genus Ficaria is now considered to comprize eight Eurasian species. The most widespread European species is the tetraploid F. verna Huds. The present study provides evidence for the existence of two main lineages of F. verna that differ considerably in their genomic size by about 3 pg. A Western F. verna lineage west of river Rhine displays a mean genome size (2C-value) of 34.2 pg and is almost precisely codistributed with the diploid F. ambigua Boreau (20 pg) north of the Mediterranean. The remaining part of Europe appears to be occupied by the Eastern F. verna lineage solely (mean genome size of 31.3 pg) which codistributes in South-Eastern Europe with the diploid F. calthifolia Rchb. (15 pg). There is little overlap at the boundary of Western and Eastern F. verna lineages with the occurrence of a separate intermediate group in the Netherlands (mean genomic size of 33.2 pg) that appears to result from hybridization of both lineages. On the basis of these observations and further considerations we propose development of F. ambigua and F. calthifolia south of the Alps with subsequent divergence to populate their current Western and Eastern European ranges, respectively. The Western F. verna lineage is proposed to originate from autotetraploidization of F. ambigua (precursor) with moderate genomic downsizing and the Eastern F. verna lineage from auto¬tetraploidization of F. calthifolia (precursor).
... the combined-samples ranks of the X-value and Y -value in increasing order of magnitude, respectively. One of the problems is to test the hypothesis H 0 : F = G against H 1 : not H 0 . Baumgartner et al. (1998) ...
A k-sample modified Baumgartner statistic is proposed. For k = 3, the limiting distribution of a k-sample Baumgartner statistic is derived with a procedure similar to Anderson-Darling (1952). For the case of k ≥ 4, a saddlepoint approximation is used to approximate the limiting distribution of the k-sample Baumgartner statistic. The critical values are given for k = 3 to 10, 25, 50 and 100.
... Randomization models are predominantly chosen in biomedical research. Neuhäuser (2006) proposed a modification of the Baumgartner-Weiβ-Schindler (1998) two-sample statistic which utilizes ranks instead of scores. A simulation study is inconclusive and does not reveal a clear winner. ...
A general method is proposed for constructing nonparametric tests of trend for proportions. Such alternatives arise in situations where it is of interest to test for monotonicity in rates of growth. The class of tests is based on the ranks of the observations. The general approach consists of defining two sets of rankings: the first describes the time and the other the binary data itself. The test statistics measures the similarity between the two sets. The asymptotic null distributions are determined for similarity measures due to Spearman, Kendall and Hamming. A limited simulation study shows that the Spearman test has greater power.
... When assuming a symmetric distribution, we have the Lepage-type tests, e.g., the Wilcoxon and Ansari– Bradley (WAB) test, Gastwirh (GA) test, Van der Waerden and Klotz (WK) test, and Long-tail and Mood (LM) test. For asymmetric distributions, the Baumgartner-Weiß-Schindler (BWS) test[53],[54]and KS-and Cramer–von Mises (CM)-type tests with various weight functions are generally useful (see the Appendix). To construct a new test for SAR observations, we compare all the tests mentioned earlier for various distributions. ...
The coherence of radar echoes is a fundamental observable in interferometric synthetic aperture radar (InSAR) measurements. It provides a quantitative measure of the scattering properties of imaged surfaces and therefore is widely applied to study the physical processes of the Earth. However, unfortunately, the estimated coherence values are often biased due to various reasons such as radar signal nonstationarity and the bias in the estimators used. In this paper, we focus on multitemporal InSAR coherence estimation and present a hybrid approach that mitigates effectively the errors in the estimation. The proposed approach is almost completely self-adaptive and workable for both Gaussian and non-Gaussian SAR scenes. Moreover, the bias of the sample coherence can be mitigated with even only several samples included for a given pixel. Therefore, it is a more pragmatic method for accurate coherence estimation and can be applied actually. Different data sets are used to test the proposed method and demonstrate its advantages.
... The difference between the friction angle of sand swimming and non-sand swimming species was highly significant (P < 0.001; one way ANOVA, n = 140). For pair-wise comparison of the different species a rank test, the BWS-test [24,25] was employed. It was found that the friction angles of the sand swimmers do not differ significantly (P > 0.05) when compared pair-wise. ...
The sandfish (Scincidae: Scincus scincus) is a lizard capable of moving through desert sand in a swimming-like fashion. The epidermis of this lizard shows a high resistance against abrasion together with a low friction to sand as an adaption to a subterranean life below the desert's surface, outperforming even steel. The low friction is mainly caused by chemical composition of the scales, which consist of glycosylated β-keratins. In this study, the friction, the micro-structure, the glycosylation of the β-keratin proteins and β-keratin coding DNA of the sandfish in comparison to other reptilian species was investigated, mainly with the closely related Berber skink (Scincidae: Eumeces schneideri) and another sand swimming species, the not closer related Shovel-snouted lizard (Lacertidae: Meroles anchietae). Glycosylated β-keratins of the sandfish, visualized with different lectins resulted in O-linked glycans through PNA employed as carbohydrate marker. Furthermore, the glycosylation of β-keratins in various squamatean species was investigated and all species tested were found positive; however, it seems like both sand swimming species examined have a much stronger glycosylation of their β-keratins. In order to prove this finding through a genetic foundation, DNA of a β-keratin coding gene of the sandfish was sequenced and compared with a homologue gene of Eumeces schneideri. By comparison of the protein sequence, a higher abundance of O-glycosylation sites was found in the sandfish (enabled through the amino acids serine and threonine), giving molecular support for a higher glycosylation of the β-keratins in this species.
... Podgor and Gastwirth (1994) showed that asymptotically the PG test can be recast as a quadratic combination of the Wilcoxon rank test for location and the Mood squared rank test for scale. Neuhäuser (2000) proposed a modification of the Lepage test by replacing the W statistic with the Baumgartner et al. (1998) B statistic. His modified Lepage test statistic is defined as ...
The two-sample location-scale problem arises in many situations like climate dynamics, bioinformatics, medicine, and finance. To address this problem, the nonparametric approach is considered because in practice, the normal assumption is often not fulfilled or the observations are too few to rely on the central limit theorem, and moreover outliers, heavy tails and skewness may be possible. In these situations, a nonparametric test is generally more robust and powerful than a parametric test. Various nonparametric tests have been proposed for the two-sample location-scale problem. In particular, we consider tests due to Lepage, Cucconi, Podgor-Gastwirth, Neuhäuser, Zhang, and Murakami. So far all these tests have not been compared. Moreover, for the Neuhäuser test and the Murakami test, the power has not been studied in detail. It is the aim of the article to review and compare these tests for the jointly detection of location and scale changes by means of a very detailed simulation study. It is shown that both the Podgor–Gastwirth test and the computationally simpler Cucconi test are preferable. Two actual examples within the medical context are discussed.
... This design has many statistical methods such as Hills-Armitage's (1979) method or Koch's (1972) method. In this paper, we propose a nonparametric test for 2×2 Cross-over design based on a two-sample test suggested by Baumgartner et al. (1998). In addition, a Monte Carlo simulation study is adapted to compare the power of the proposed methods with those of previous methods. ...
A 2{\times}2 Cross-over design is widely used in clinical trials for comparison studies of two kinds of drugs or medical treatments. This design has many statistical methods such as Hills-Armitage's (1979) method or Koch's (1972) method. In this paper, we propose a nonparametric test for 2{\times}2 Cross-over design based on a two-sample test suggested by Baumgartner et al. (1998). In addition, a Monte Carlo simulation study is adapted to compare the power of the proposed methods with those of previous methods.
... In addition, for CERNO algorithm the impact to sensitivity and FPR of applied gene ranking metric was tested by using four different ranking metrics that were most suitable for the GSEA algorithm (Zyla et al., 2017b) i.e. Baumgartner-Weiss-Schindler test statistic (BWS; Baumgartner et al., 1998); absolute value from Moderated Welch Test statistics (jMWTj; Demissie et al., 2008); absolute value from signal-to-noise ratio (jS2Nj; Subramanian et al., 2005); MSD (Zyla et al., 2017b). Notably, the BWS metric makes no assumptions about data distribution. ...
Motivation:
Analysis of gene set enrichment is an essential part of functional omics studies. Here, we complement the established evaluation metrics of gene set enrichment algorithms with a novel approach to assess the practical reproducibility of scientific results obtained from gene set enrichment tests when applied to related data from different studies.
Results:
We evaluated eight established and one novel algorithm for reproducibility, sensitivity, prioritization, false positive rate and computational time. In addition to eight established algorithms, we also included CERNO, a flexible and fast algorithm based on modified Fisher p-value integration. Using real-world datasets, we demonstrate that CERNO is robust to ranking metrics, as well as sample and gene set size. CERNO had the highest reproducibility while remaining sensitive, specific and fast. In the overall ranking PADOG, CERNO and ORA performed best, while CERNO and GeneSetTest scored high in terms of reproducibility.
Availability and implementation:
tmod package implementing the CERNO algorithm is available from CRAN (cran.r-project.org/web/packages/tmod/index.html) and an online implementation can be found at http://tmod.online/.
Supplementary information:
Supplementary data are available at Bioinformatics online.
... The new proposed RD test calculates the test statistic by revising the order the the two steps in the WSR test: ranking the observations followed by the difference of the ranks. Specifically, the associated test statistic of the RD test is Recently, Baumgartern, Weiß, and Schindler (BWS) [8] proposed a novel nonparametric test for two independent sample problem, which is based on the squared value of the difference between the two empirical distribution functions weighted by the respective variance. This weighting places more emphasize on the tails of the distribution functions. ...
We propose a new nonparametric test based on the
rank difference between the paired sample for testing the equality of the
marginal distributions from a bivariate distribution. We also consider a
modification of the novel nonparametric test based on the test proposed by Baumgartern,
Weiβ, and Schindler (1998). An
extensive numerical power comparison for various parametric and nonparametric
tests was conducted under a wide range of bivariate distributions for small
sample sizes. The two new nonparametric tests have comparable power to the
paired t test for the data simulated from bivariate normal distributions, and
are generally more powerful than the paired t test and other commonly used
nonparametric tests in several important bivariate distributions.
... Podgor & Gastwirth (1994) devised a test statistic by taking a quadratic combination of the rank test for location and a rank test for scale. Neuhäuser (2000) modified Lepage L-test by replacing the Wilcoxon test for location with a location test proposed by Baumgartner, Weiß & Schindler (1998). Also, Murakami (2007) proposed a modification of the Lepage test. ...
In this article, we propose nonparametric tests for simultaneously testing equality of location and scale parameters of two multivariate distributions by using nonparametric combination theory. Our approach is to combine the data depth based location and scale tests using combining function to construct a new data depth based test for testing both location and scale parameters. Based on this approach, we have proposed several tests. Fisher's permutation principle is used to obtain p-values of the proposed tests. Performance of proposed tests has been evaluated in terms of empirical power for symmetric and skewed multivariate distributions and compared to the existing test based on data depth. The proposed tests are also applied to a real-life data set for illustrative purpose.
... The GSEA method tests if the distribution of the gene ranks in the gene set differs from a uniform distribution by weighted Kolmogorov-Smirnov test. As a ranking metric, the Baumgartner-Weiss-Schindler statistic was used [11], which was highlighted as one of the best metric [9]. The next two methods at first summarize the expression of genes within the same pathway for every sample into single value. ...
Introducing the high-throughput measurement methods into molecular biology was a trigger to develop the algorithms for searching disorders in complex signalling systems, like pathways or gene ontologies. In recent years, there appeared many new solutions, but the results obtained with these techniques are ambiguous. In this work, five different algorithms for pathway enrichment analysis were compared using six microarray datasets covering cases with the same disease. The number of enriched pathways at different significance level and false positive rate of finding enrichment pathways was estimated, and reproducibility of obtained results between datasets was checked. The best performance was obtained for PLAGE method. However, taking into consideration the biological knowledge about analyzed disease condition, many findings may be false positives. Out of the other methods GSVA algorithm gave the most reproducible results across tested datasets, which was also validated in biological repositories. Similarly, good outcomes were given by GSEA method. ORA and PADOG gave poor sensitivity and reproducibility, which stand in contrary to previous research.
... Note that both MH 1 and MH 2 are Lepage-type tests where the multisample Wilcoxon statistic is replaced with the multisample Baumgartner-Weiss-Schindler(Baumgartner et al. 1998) statistic. In MH 1 the multisample Ansari-Bradley statistic is replaced with the multisample Mood(Puri 1965) statistic. ...
The multisample version of the Cucconi rank test for the two-sample location-scale problem is proposed. Even though little known, the Cucconi test is of interest for several reasons. The test is compared with some Lepage-type tests. It is shown that the multisample Cucconi test is slightly more powerful than the multisample Lepage test. Moreover, its test statistic can be computed analytically whereas several others cannot. A practical application example in experimental nutrition is presented. An R function to perform the multisample Cucconi test is given.
... First two metrics are based on Moderated Welch Test statistic (MWT and its absolute value, |MWT|), calculated using weighted pooled and unpooled standard errors in the t-test procedure and adjusted by estimation of the gene-level variance across genes [35]. Next two ranking metrics use non-parametric test statistics: the Sum of Ranks (SoR) and Baumgartner-Weiss-Schindler test statistic (BWS) [36]. Both have been used in GSEA before [31]. ...
Background
There exist many methods for describing the complex relation between changes of gene expression in molecular pathways or gene ontologies under different experimental conditions. Among them, Gene Set Enrichment Analysis seems to be one of the most commonly used (over 10,000 citations). An important parameter, which could affect the final result, is the choice of a metric for the ranking of genes. Applying a default ranking metric may lead to poor results.
Methods and results
In this work 28 benchmark data sets were used to evaluate the sensitivity and false positive rate of gene set analysis for 16 different ranking metrics including new proposals. Furthermore, the robustness of the chosen methods to sample size was tested. Using k-means clustering algorithm a group of four metrics with the highest performance in terms of overall sensitivity, overall false positive rate and computational load was established i.e. absolute value of Moderated Welch Test statistic, Minimum Significant Difference, absolute value of Signal-To-Noise ratio and Baumgartner-Weiss-Schindler test statistic. In case of false positive rate estimation, all selected ranking metrics were robust with respect to sample size. In case of sensitivity, the absolute value of Moderated Welch Test statistic and absolute value of Signal-To-Noise ratio gave stable results, while Baumgartner-Weiss-Schindler and Minimum Significant Difference showed better results for larger sample size. Finally, the Gene Set Enrichment Analysis method with all tested ranking metrics was parallelised and implemented in MATLAB, and is available at https://github.com/ZAEDPolSl/MrGSEA.
Conclusions
Choosing a ranking metric in Gene Set Enrichment Analysis has critical impact on results of pathway enrichment analysis. The absolute value of Moderated Welch Test has the best overall sensitivity and Minimum Significant Difference has the best overall specificity of gene set analysis. When the number of non-normally distributed genes is high, using Baumgartner-Weiss-Schindler test statistic gives better outcomes. Also, it finds more enriched pathways than other tested metrics, which may induce new biological discoveries.
Electronic supplementary material
The online version of this article (doi:10.1186/s12859-017-1674-0) contains supplementary material, which is available to authorized users.
... Comparing two populations in a nonparametric setup, familiar as the goodness of fit (GOF henceforth) test, is an age old problem (Stephens, 1986) that has important applications in many other fields of science like Biology, Physics, Finance, etc., to name a few (Bissantz and Munk, 2001; Baumgartner et al., 1998). In a single-sample GOF problem we test the hypothesis if a set of observations X i , i = 1, . . . ...
A goodness-of-fit test is proposed for the family of exponential polynomial growth curve models (EPGCM; Heinen, 199913.
Heinen , M. ( 1999 ). Analytical growth equations and their Genstat 5 equivalents . Netherlands J. Agricult. Sci. 47 : 67 – 89 . [Web of Science ®]View all references), which has wide applications in different areas of science. The exponential growth curve model (EGCM), the most prominent member of the EPGCM family, is a simple and biologically meaningful growth model. Other members of the EPGCM family also cover many realistic growth processes. Thus, a goodness-of-fit test for the EPGCM class has substantial practical value. The goodness-of-fit test developed here is based on the properties of finite differences. The performance of the theory developed is illustrated through simulation and analysis of real data.
A new approach to constructing nonparametric tests for the general two-sample problem is proposed. This approach not only generates traditional tests (including the two-sample Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling tests), but also produces new powerful tests based on the likelihood ratio. Although conventional two-sample tests are sensitive to the difference in location, most of them lack power to detect changes in scale and shape. The new tests are location-, scale-, and shape-sensitive. so they are robust against variation in distribution.
The Levene test is a widely used test for detecting differences in dispersion. The modified Levene transformation using sample medians is considered in this article. After Levene's transformation the data are not normally distributed, hence, nonparametric tests may be useful. As the Wilcoxon rank sum test applied to the transformed data cannot control the type I error rate for asymmetric distributions, a permutation test based on reallocations of the original observations rather than the absolute deviations was investigated. Levene's transformation is then only an intermediate step to compute the test statistic. Such a Levene test, however, cannot control the type I error rate when the Wilcoxon statistic is used; with the Fisher–Pitman permutation test it can be extremely conservative. The Fisher–Pitman test based on reallocations of the transformed data seems to be the only acceptable nonparametric test. Simulation results indicate that this test is on average more powerful than applying the t test after Levene's transformation, even when the t test is improved by the deletion of structural zeros.
The nonparametric two-sample test recently proposed by Baumgartner,Weiß,and Schindler [1] is not suitable for one-tailed analyses. We propose a modification of the original Baumgartner-Weiß-Schmdler statistic which enables one-sided tests. The test based on the new modified statistic has a less conservative size and is, according to simulation results, more powerful than the one-sided Wilcoxon-Mann-Whitney test. With the modified statistic a Jonckheere-type trend test can be constructed. Simulation studies indicate that this new trend test is superior to the often employed Jonckheere-Terpstra test. Furthermore, the new tests based on the modified statistic are also powerful in case of heteroscedasticity. For all tests the exact permutation distribution was used for inference
This revised book provides a thorough explanation of the foundation of robust methods, incorporating the latest updates on R and S-Plus, robust ANOVA (Analysis of Variance) and regression. It guides advanced students and other professionals through the basic strategies used for developing practical solutions to problems, and provides a brief background on the foundations of modern methods, placing the new methods in historical context. Author Rand Wilcox includes chapter exercises and many real-world examples that illustrate how various methods perform in different situations. Introduction to Robust Estimation and Hypothesis Testing, Second Edition, focuses on the practical applications of modern, robust methods which can greatly enhance our chances of detecting true differences among groups and true associations among variables. * Covers latest developments in robust regression * Covers latest improvements in ANOVA * Includes newest rank-based methods * Describes and illustrated easy to use software.
It is shown that the nonparametric two-saniDle test recently proposed by Baumgartner, WeiB, Schindler (1998, Biometrics, 54, 1129-1135) does not control the type I error rate in case of small sample sizes. We investigate the exact permutation test based on their statistic and demonstrate that this test is almost not conservative. Comparing exact tests, the procedure based on the new statistic has a less conservative size and is, according to simulation results, more powerful than the often employed Wilcoxon test. Furthermore, the new test is also powerful with regard to less restrictive settings than the location-shift model. For example, the test can detect location-scale alternatives. Therefore, we use the test to create a powerful modification of the nonparametric location-scale test according to Lepage (1971, Biometrika, 58, 213-217). Selected critical values for the proposed tests are given.
Nonparametric all-pairs multiple comparisons based on pairwise rankings can be performed in the one-way design with the Steel-Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P-values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner, Weiss, and Schindler (1998, Biometrics54, 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel-Dwass procedure. We illustrate the methods with a practical data set.
Various non-parametric rank tests based on the Baumgartner statistic have been proposed for testing the location, scale and location–scale parameters. The modified Baumgartner statistics are not suitable for the scale shifts for a two-sample problem. Two modified Baumgartner statistics are proposed by changing the weight function. The suggested statistics are extended to the multisample problem. Some exact critical values of the suggested test statistics are evaluated. Simulations are used to investigate the power of the modified Baumgartner statistics.
One-way design nonparametric multiple comparison problems are among the most important topics in nonparametric statistics. Various nonparametric tests have been proposed and considered for a multiple comparison problems by many authors over the years. The analysis of one-way design multiple comparisons are important in biometry. Nonparametric all-pairs multiple comparisons based on pairwise rankings are considered in the one-way layout. The pairwise ranking test based on the Cucconi test for the one-way layout multiple comparison of the location-scale problem is proposed. Simulations are used to investigate the power of the proposed test for jointly location and scale alternatives with various population distributions for small sample sizes. The suggested method is illustrated by the analysis of real data.
In clinical dose-finding trials the identification of the minimum effective dose is important. Recently, a nonparametric step-down procedure based upon Wilcoxon rank sum tests was proposed for identifying the minimum effective dose. However, the recently introduced modified Baumgartner-Weiβ-Schindler statistic is very powerful and, consequently, may be used in the step-down procedure instead of the Wilcoxon rank sum. The two step-down closed testing procedures based upon the Wilcoxon and the modified Baumgartner-Weiβ-Schindler statistic, respectively, were compared in a Monte Carlo study. According to simulation results, the procedure based upon the modified Baumgartner-Weiβ-Schindler statistic is more powerful than that based upon the Mann-Whitney or Wilcoxon test for identifying the minimum effective dose.
The Cramér-von Mises statistic provides a useful goodness of fit test of whether a random sample has been drawn from some given null distribution. Its use in comparing several samples has also been studied, but not systematically. We show that the statistic is capable of significant generalization. In particular, we consider the comparison of the distributions of observations arising from factorial experiments. Provided that observations are replicated, we show that our generalization yields a test statistic capable of decomposition like the sum of squares used in ANOVA. The statistic is calculated using ranked data rather than original observations. We give the asymptotic theory. Unlike ANOVA, the asymptotic distributional properties of the statistic can be obtained without the assumption of normality. Further, the statistic enables differences in distribution other than the mean to be detected. Because it is distribution free, Monte-Carlo sampling can be used to directly generate arbitrarily accurate critical test null values in online analysis irrespective of sample size. The statistic is thus easy to implement in practice. Its use is illustrated with an example based on a man-in-the-loop simulation trial where operators carried out self assessment of the workload that they experienced under different operating conditions.
The non-null limiting distribution of the generalized Baumgartner statistic is approximated by applying the Fourier series approximation. Due to the development of computational power, the Fourier series approximation is readily utilized to approximate its probability density function. The infinite product part for a non-central parameter in the characteristic function is re-formulated by using a formula of the trigonometric function. The non-central parameter of the generalized Baumgartner statistic is formulated by the first moment of the generalized Baumgartner statistic under the alternative hypothesis. The non-central parameter is used to calculate the power of the generalized Baumgartner statistic.
In clinical dose-finding trials the identification of the minimum effective dose is important. Recently, a nonparametric step-down procedure based upon Wilcoxon rank sum tests was proposed for identifying the minimum effective dose. However the recently introduced modified Baumgartner-Weiss-Schindler statistic is very powerful and, consequently, may be used in the step-down procedure instead of the Wilcoxon rank sum. The two step-down closed testing procedures based upon the Wilcoxon and the modified Baumgartner-Weiss-Schindler statistic, respectively, were compared in a Monte Carlo study. According to simulation results, the procedure based upon the modified Baumgartner-Weiss-Schindler statistic is more powerful than that based upon the Mann-Whitney or Wilcoxon test for identifying the minimum effective dose.
The two-sample problem and its extension to the k-sample problem are well known in the statistical literature. However, the discrete version of the k-sample problem is relatively less explored. Here in this work we suggest a k-sample non-parametric test procedure for discrete distributions based on mutual information. A detailed power study with comparison with other alternatives is provided. Finally, a comparison of some English soccer league teams based on their goal-scoring pattern is discussed.
Two-sample problem is frequently discussed problem in statistics. In this paper we consider the hypothese methods for the general two-sample problem and suggest the bootstrap methods. And we show that the modified Kolmogorov-Smirnov test is more efficient than the Kolmogorov-Smirnov test.
The Xiaolangdi Basin of the Yellow River, located between Mengjin County and Jiyuan City in Henan Province, is composed of a reservoir, a barrage dam, flood discharge buildings, and a water diversion and power generation system, among which, the Xiaolangdi project is the only control project in the lower reaches of the Yellow River that has a large storage capacity. In this study, distributed scatterers InSAR (DS-InSAR) deformation analysis was used to process 27 images of ESA Sentinel-1A data for the period between September 10, 2018, and July 19, 2019. Using the generalized likelihood ratio test (GLRT) to select the statistical homogeneous pixels (SHP) and the eigenvalue decomposition method to optimize the interference phase, the time series surface deformation of the study area was obtained. The results showed that the maximum deformation rate in the study area was − 40.71 mm/year, the maximum deformation rate of the Xiaolangdi project was − 10.09 mm/year, and the larger deformation rates in the area around the Xiaolangdi Reservoir were mostly within the range − 35 to − 20 mm/year. Furthermore, small baseline subset (SBAS) InSAR and DS-InSAR are used for verification and analysis, and the monitoring results have high consistency. Therefore, to sum up, the time series deformation of low coherence areas may be monitored using DS-InSAR.
In this paper, we present a novel approach for accurate coherence estimation, based on InSAR data stack. The main advantage of the proposed method is to meet the assumptions that the complex signals are local stationary, and meanwhile take into account the computational efficiency. Therefore, it is possible to obtain a very accurate coherence estimate without loss of resolution. Concretely, two-step is applied to adaptive algorithm: (1) nonparametric hypothesis test is firstly employed to cluster pixels with same statistical distributions; (2) the modified version of maximum likelihood fringe rate estimate is then used to eliminate the non-stationarity of complex signals. The accuracy of such estimation is improved by Cramer-Rao bounds. Experimental results with Envisat ASAR datasets over Los Angeles areas show that the new method performs well under different situations.
In this paper, multivariate two-sample testing problems were examined based on the Jurečková–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.
Automatic road extraction from synthetic aperture radar (SAR) imagery has been studied with success in the past two decades. However, a method that combines full interferometric SAR (InSAR) information is as yet missing. In this paper, we present an algorithm toward robust road extraction by fully exploring the multitemporal InSAR covariance matrix. To improve the detection performance and reduce false alarm ratio, intensity and coherence are first accurately estimated without loss of image resolution by homogeneous pixel selection and robust estimators. After the identification of road candidates from each quantity using multiscale line detectors, novel information fusion rules are applied to integrate the extracted results and generate the final road network. The method is tested and quantitatively evaluated on TerraSAR-X data sets depicting two scenes where complex road features make it hard for standard SAR-based methods. The experimental results show that the new method can achieve satisfactory detection performances.
It is common to test the null hypothesis that two samples were drawn from identical distributions; and the Smirnov (sometimes called Kolmogorov-Smirnov) test is conventionally applied. We present simulation results to compare the performance of this test with three recently-introduced alternatives. We consider both continuous and discrete data. We show that the alternative methods preserve type I error at the nominal level as well as the Smirnov test but offers superior power. We argue for the routine replacement of the Smirnov test with the modified Baumgartner test according to Murakami (2006)7.
Murakami H (2006): A k-sample rank test based on a modified Baumgartner statistic and its power comparison. Journal of the Japanese Society of Computational Statistics 19, 1–13.[CrossRef]View all references, or with the test proposed by Zhang (2006)16.
Zhang J (2006): Powerful two-sample tests based on the likelihood ratio. Technometrics 48, 95–103.[Taylor & Francis Online], [Web of Science ®]View all references.
Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.
In this study we compare the performance of many two-sample tests of significance that might be used to test the equality of means when the effect of the treatment is variable. Of the 19 tests that were compared, the normal scores test is recommended for general use in testing the null hypothesis of no treatment effect against the alternative that the distributions are stochastically ordered when the ratio of the larger standard deviation to the smaller standard deviation does not exceed 1.3. The Baumgartner-Weiß-Schindler tests and an adaptive test also have higher power than the pooled t-test, the unequal variance t-test and the rank-sum test for many distributions. In the simulation studies, data in the first sample are generated from nine distributions, including long-tailed and skewed distributions. Data in the second sample are generated by adding a random treatment effect to a random variable that was generated from the same distribution that was used in the first sample. Because we restricted our power studies to treatment effects that are positive or zero, the population distributions will be stochastically ordered. The results of these studies demonstrate that the normal scores test is often more powerful than the t tests and the rank-sum test. If the ratio of the standard deviations does exceed 1.3 then one of the t tests is recommended.
The development of novel high-throughput experimental techniques makes it possible to comprehensively analyze biological data in health and disease. However, a large amount of data generated results in dramatic data-analytic challenges in discovery of &'signature' molecules, which are specific to different biological conditions (e.g. normal vs. disease, treated vs. untreated). Current statistical methods are effective only in the case their hypothesis can be matched. In this paper, we apply an ensemble statistical method to infer significant molecules. In our approach, four well-done and well-understanding statistical techniques had been used for the analysis to the experimental data, and then the results will be collected into an ensemble framework to find the high confident "significant" molecules which can distinguish the different experimental conditions. We evaluate the performance of our approach on a test dataset which deposited on GEO database with an access number of GSE45114.
Testing Statistical Hypotheses Numerical Recipes in C
- E L W H Lehmann
- S A Teukolsky
- W T Vetterling
- B P Flannery
Lehmann, E. L. (1966). Testing Statistical Hypotheses. New York: John Wiley and Sons. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical Recipes in C. Cambridge, U.K.: Cambridge University Press.
Regulationsmechanismen von nativen und exprimierten L-Typ Ca++-Kanalen
- K Hohenthanner
Hohenthanner, K. (1997). Regulationsmechanismen von nativen und exprimierten L-Typ Ca++-Kanalen. Master's thesis, University of Linz, Linz, Austria.
Numerical Recipes in C
- W H Press
- S A Teukolsky
- W T Vetterling
- B P Flannery
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical Recipes
in C. Cambridge, U.K.: Cambridge University Press.
Sumario: The general decision problem -- The probability background -- Uniformly most powerful tests -- Unbiasedness: theory and first applications -- Unbiasedness: applications to normal distributions; confidence intervals -- Invariance -- Linear hypotheses -- Multivariate linear hypotheses -- The minimax principle -- Conditional inference.
The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given.
The Cramer-von Mises $\omega^2$ criterion for testing that a sample, $x_1, \cdots, x_N$, has been drawn from a specified continuous distribution $F(x)$ is \begin{equation*}\tag{1}\omega^2 = \int^\infty_{-\infty} \lbrack F_N(x) - F(x)\rbrack^2 dF(x),\end{equation*} where $F_N(x)$ is the empirical distribution function of the sample; that is, $F_N(x) = k/N$ if exactly $k$ observations are less than or equal to $x(k = 0, 1, \cdots, N)$. If there is a second sample, $y_1, \cdots, y_M$, a test of the hypothesis that the two samples come from the same (unspecified) continuous distribution can be based on the analogue of $N\omega^2$, namely \begin{equation*}\tag{2} T = \lbrack NM/(N + M)\rbrack \int^\infty_{-\infty} \lbrack F_N(x) - G_M(x)\rbrack^2 dH_{N+M}(x),\end{equation*} where $G_M(x)$ is the empirical distribution function of the second sample and $H_{N+M}(x)$ is the empirical distribution function of the two samples together [that is, $(N + M)H_{N+M}(x) = NF_N(x) + MG_M(x)\rbrack$. The limiting distribution of $N\omega^2$ as $N \rightarrow \infty$ has been tabulated [2], and it has been shown ([3], [4a], and [7]) that $T$ has the same limiting distribution as $N \rightarrow \infty, M \rightarrow \infty$, and $N/M \rightarrow \lambda$, where $\lambda$ is any finite positive constant. In this note we consider the distribution of $T$ for small values of $N$ and $M$ and present tables to permit use of the criterion at some conventional significance levels for small values of $N$ and $M$. The limiting distribution seems a surprisingly good approximation to the exact distribution for moderate sample sizes (corresponding to the same feature for $N\omega^2$ [6]). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges [4].
The patch-clamp technique was used to characterize the mechanism of Ca2+-induced inactivation of cardiac L-type Ca2+ channel alpha(1C-a) + beta3 subunits stably expressed in CHO cells. Single Ca2+ channel activity was monitored with 96 mM Ba2+ as charge carrier in the presence of 2.5 microM (-)BAYK 8644 and calpastatin plus ATP. This enabled stabilization of channel activity in the inside-out patch and allowed for application of steady-state Ca2+ concentrations to the intracellular face of excised membrane patches in an attempt to provoke Ca2+-induced inactivation. Inactivation was found to occur specifically with Ca2+ since it was not observed upon application of Ba2+. Ca2+-dependent inhibition of mean Ca2+ channel activity was characterized by a Hill coefficient close to 1. Ca2+ binding to open and closed states of the channel obtained during depolarization apparently occurred with similar affinity yielding half-maximal inhibition of Ca2+ channel activity at approximately 4 microM. This inhibition manifested predominantly in a reduction of the channel's open probability whereas availability remained almost unchanged. The reduction in open probability was achieved by an increase in first latencies and a decrease in channel opening frequency as well as channel open times. At high (12-28 microM) Ca2+ concentrations, 72% of inhibition occurred due to a stabilization of the closed state and the remaining 28% by a destabilization of the open state. Our results suggest that binding of one calcium ion to a regulatory domain induces a complex alteration in the kinetic properties of the Ca2+ channel and support the idea of a single EF hand motif as the relevant Ca2+ binding site on the alpha1 subunit.