Wilcoxon Rank-Sum Test

If the two sample populations are not normally distributed, then you can use the nonparametric Wilcoxon rank sum test to compare the population means. The following SAS code compares the ldl change change-from-baseline means for active drug and placebo ... 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

The Wilcoxon Rank-Sum test is commonly used for the comparison of two groups of nonparametric (inteval or not normally distributed) data, such as those which are not measured exactly but rather as falling within certain limits (for example, how many animals died during each hour of an acute study.) The test is also used when there is no variability (variance = 0) within one or more of the groups we wish to compare (Sokal and Rohlf, 1994). 

When k = 2, the Kruskal-Wallis chi-square value has 1 df This test is identical to the normal approximation used for the Wilcoxon Rank-Sum Test. As noted in previous sections, a chi square with 1 df can be represented by the square of a standardized normal random variable. In the case oik = 2, the //-statistic is the square of the Wilcoxon Rank-Sum Z-test (without the continuity correction). 

If every animal were followed until the event occurrence, the event times could be compared between two groups using the Wilcoxon Rank-Sum Test. However, some... 

The Wilcoxon Rank-Sum Test could be used to analyze the event times in the absence of censoring. A Generalized Wilcoxon Test, sometimes called the Gehan Test, based on an approximate chi-square distribution, has been developed for use in the presence of censored observations. 

A basic assumption underlying r-tests and ANOVA (which are parametric tests) is that cost data are normally distributed. Given that the distribution of these data often violates this assumption, a number of analysts have begun using nonparametric tests, such as the Wilcoxon rank-sum test (a test of median costs) and the Kolmogorov-Smirnov test (a test for differences in cost distributions), which make no assumptions about the underlying distribution of costs. The principal problem with these nonparametric approaches is that statistical conclusions about the mean need not translate into statistical conclusions about the median (e.g., the means could differ yet the medians could be identical), nor do conclusions about the median necessarily translate into conclusions about the mean. Similar difficulties arise when - to avoid the problems of nonnormal distribution - one analyzes cost data that have been transformed to be more normal in their distribution (e.g., the log transformation of the square root of costs). The sample mean remains the estimator of choice for the analysis of cost data in economic evaluation. If one is concerned about nonnormal distribution, one should use statistical procedures that do not depend on the assumption of normal distribution of costs (e.g., nonparametric tests of means). 

The Mann-Whitney U-test is equivalent to an alternative test called the Wilcoxon rank sum test. These tests were developed independently, but subsequently shown to be mathematically the same. We will develop the test using the Wilcoxon rank sum methodology. 

Wilcoxon rank-sum test, sign test, winning sites method... 

The choice of an appropriate statistical method is important, and a method suitable for the comparison of two groups in terms of an ordinal outcome measurement is the Mann-Whitney/Wilcoxon rank-sum test (not to be confused with the Wilcoxon matched-pairs signed ranks test, which is appropriate for paired data - see later). It is both inefficient and inappropriate to use a qualitative data test (such as a simple chi-square) for such a measurement, and the application of quantitative data tests (such as one of the f-tests) is also invalid. 

Hypothesis test of the location of two distributions Wilcoxon rank sum test... 

The two-sample t test was based on the assumption that the two samples were drawn from an underlying normal population with the same (assumed) population variance. A rejection of the null hypothesis in the setting of the two-sarnple t test would imply that the two populations from which the samples were drawn were represented by two normal distributions with the same variance (shape), but with different means. The Wilcoxon rank sum test does not... 

Using the Wilcoxon rank sum test, interest is in a location shift between two population distributions so the following null hypothesis is tested ... 

The test statistic for the Wilcoxon rank sum test is the sum of the ranks in group 1 ... 

In a randomized, double-blind, 6-week study, the test treatment (one tablet taken once a day) was compared with placebo. The primary endpoint of the study was the mean change from baseline SBP. Given the small sample size of the study, the primary analysis is based on the Wilcoxon rank sum test with a = 0.05 (two-sided). 

FIGURE 10.9 (a) Representative reversed-phase UPLC-Q-TOF MS TICs of a urine sample analyzed in positive and negative ion modes (b) PLS-DA scores plot (OSC filtered) of healthy, insulin-sensitive subjects (A) and prediabetic, insulin-resistant individuals ( ) (c) corresponding PLS-DA loading plot. The variables are labeled with m/z, and m/z 194.34 is labeled by an arrow and (d) differences in the TIC peak height of m/z between insulin-sensitive subjects (A) and insulin-resistant individuals ( ). p < 0.05 vs. insulin-resistant individuals analyzed by the Wilcoxon rank sum test. (Reprinted from Chen, J. et al., Anal. Chem., 80, 1280, 2008. With permission.)... 

Figure 3. The effect of fluence rate and light fractionation on BPD-mediated PDT. BPD-MA was administered to rats with NBT II tumors implanted into the bladder wall. One hour later tumors were exposed to a total fluence of 30 J/cm of 690 nm irradiation under the following conditions 100 mWcm"- continuous 100 mW/cm - fractionated 15 s on/15 s off 100 mWcm 2 30 s on/30 s off 100 mWcm" 60 s on/60 s off. Tumors were disaggregated 24 h later and tumor cells were plated for colony formation assay. Colonies (50 cells or more) were counted 9 days later after fixing with methanol and staining with crystal violet. The Wilcoxon rank sum test was used to compare the number of clonogenic cells with data at 100 mWcm - and continuous wave irradiations. NS not significant. (Source linuma et al. [32]. Reproduced with permission.)...

Wilcoxon rank sum test was used to examine the difference between pre- and post-intervention samples, UIC were averaged for subjects which had been sampled on more than one occasion in the pre-intervention surveys. 

Quinone conjugates are 4-OHEi(E2)-2-NAcCys, 4-OHEi(E2)-2-Cys, 2-OHEi(E2)-(l+4)-NAcCys, and 2-OHEi(E2)-(l+4)-Cys. Statistically significant differences (compared to controls) were determined using the Wilcoxon rank sum test. 

The Wilcoxon rank-sum test is a nonparametric test for assessing whether two samples of measurements come from the same distribution. That is, as an alternative to the two-sample f-test, this test can be used to discover differentially expressed candidates under two conditions. For example, again consider the measurements of the probe set used for the two-sample t-test. The gene expression values are 12.79, 12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition. Measurement 12.79 has rank 6, measurement 12.53 has 5, and measurement 12.46 has rank 4. The rank sum of the naive condition is 6 -I- 5 -I-4=15. Then after the sum is subtracted by ni(ni-I-l)/2 = 3 x 4/2 = 6, the Wilcoxon rank-sum test statistic becomes 9. Considering all of the combinations of the three measurements, we can compute the probability that the rank sum happens more extremely than 9. The probability becomes its p-value. This is the most extreme among the 20 combinations thus the p-value is 2 x Pr( W > 9) = 2 x = 0.1 for the two-sided test. It is hard to say that the probe set is differentially expressed since the p-value 0.1 > 0.05. This test is also called the Mann - Whitney- Wilcoxon test because this test was proposed initially by Wilcoxon for equal sample sizes and extended to arbitrary sample sizes by Mann and Whitney. As a nonparametric alternative to the paired t-test for the two related samples, the Wilcoxon signed-rank test can be used. The statistic is computed by ordering absolute values of differences of paired samples. For example, consider a peptide in the platelet study data. Their differences for each... 

The Kruskal-Wallis test is a nonparametric alternative to the ANOVA F-test, described above, for multiple conditions. That is, it is an extension of the Wilcoxon rank-sum test to multiple conditions. Expression values are replaced with their ranks to form the test statistic without requiring an assumption of the form of the distribution. For example, the Kruskal-Wallis statistic is 7.2 and the p-value is less than 0.05 for the probe set used for illustration of the ANOVA F-test. 

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