Rank sums

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

P-values Age = Wilcoxon rank-sum, Gender = Pearson s chi-square, Race = Fisher s exact 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). 

The distribution-free multiple comparison test should be used to compare three or more groups of nonparametric data. These groups are then analyzed two at a time for any significant differences (Hollander and Wolfe, 1973, pp. 124-129). The test can be used for data similar to those compared by the rank-sum test. We often employ this test for reproduction and mutagenicity studies (such as comparing survival rates of offspring of rats fed various amounts of test materials in the diet). 

As with the Wilcoxon Rank-Sum, too many tied ranks inflate the false... 

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. 

Hemoglobin is directly measured and is an independent and continuous variable However, and probably because at any one time a number of forms and conformations (oxyhemoglobin, deoxyhemoglobin, methemoglobin, etc.) of hemoglobin are actually present the distribution seen is not typically a normal one, but rather may be a multimodal one. Here a nonparametric technique such as the Wilcoxon or multiple rank-sum is called for. 

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. 

In general many of the standard parametric tests have non-parametric equivalents. The Mann-Whitney test corresponds to the parametric unpaired t-test. This test is based on rank sums. The combined data are ranked, usually low to high. If the null hypothesis, that the two samples come from identical populations, is true the sum of the ranks assigned to the observations from the two... 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

The statistical test described above is based on a standard confidence interval procedure related to the Wilcoxon Rank Sum/Mann-Whimey rank test, apphed to the log slopes. References to this confidence interval procedure include ... 

In (23.80) and (23.81) the rank sum y+k is an odd number, otherwise these operators are identically equal to zero. We shall separate sets of operators that are scalars in the space of total angular momentum but tensors in isospin space. If we go through a similar procedure for one subshell of equivalent electrons we shall end up with the quasispin classification of its states. It turns out that ten operators l/(00), U 0 vffl, F 0) are generators of a group of five-dimensional quasispin, wnich can be easily verified by comparing their commutation relations with the standard commutation relations for generators of that group. 

Mann-Whitney Rank Sum Test, Sigmastat v 2.0 1 PERE = Potential Effluent Related Effect 3 NF (Nearfield) significantly greater than both Tunnel Bay and Santoy Bay reference areas, 4 t-tests, Mann-Whitney U-tests, Mann-Whitney rank sum tests, a = 0.05,5 National EEM database, unpublished data 6 significantly less at Santoy Bay reference area, Significant difference at Tunnel Bay. 

The Wilcoxon s Rank-Sum Test (WRST) is a non-parametric alternative. The WRST is robust to the normal distribution assumption, but not to the assumption of equal variance. Furthermore, this test requires that the two groups of data under comparison have similarly shaped distributions. Non-parametric tests typically suffer from having less statistical power than their parametric counterparts. Similar to the /-test, the WRST will exhibit false positive rate inflation across a microarray dataset. It is possible to use the Wilcoxon test statistic as the single filtering mechanism however calculation of the false positive rate is challenging (48). 

Fig. 7. Effects of L-PDMP on spatial cognition deficit induced by repeated cerebral ischemia. (A) Protocol for behavioral experiments to evaluate the efficacy of L-PDMP against spatial cognition deficit in rats with cerebral ischemia. (B) Sham (sham-operated rats, n — 11), Group 1-vehicle, l-PDMP (i.p. injections of vehicle or 40 mg/kg L-PDMP twice a day for 6 days from 24 h after ischemia, n = 13), Group 2-vehicle, L-PDMP (i.p. injection of vehicle or 40 mg/kg L-PDMP twice a day for 4 days from 3 days after ischemia, n — 9 and n — 10, respectively). P < 0.001 versus sham-operated rats, P < 0.05 and P < 0.01 versus vehicle-treated rats (Wilcoxon s rank sum test).

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

Quantitative continuous data may be evaluated by standard statistical methods. It is inappropriate to use parametric statistical methods on semiquantitative data (i.e., renal injury light miscroscopic assessment scores), although appropriate non-parametric methods (e.g., Duncan s rank-sum procedure) may be used. 

The alpha spectrometry results were also significantly different at a 99% confidence level from the assigned NPL values (which deviations are 0% by definition). Application of the non-parametric Wilcoxon Signed Rank test, which, like the Rank Sum test, does not assume a normal distribution and does not require the removal of outliers, also resulted in a significant difference at a 99% confidence level between the alpha spectrometry results and the assigned NPL values (the absolute z-value being 3.72). 

Figure 5.2 Comparison of the dose-AUC relationship of R-(-)-apomorphine (11), R-(-)-N- -propylnorapomorphine (80) and R-(-)-l l-OH-N- -propylnoraporphine (12). Data represent mean values S.E.M. of 4 animals. Statistical analysis by t-test p<0.05, p<0.01, p<0.001. For comparison with R-(-)-apomorphine (11) 30 nmol/kg equal variance test failed and than Rank Sum Test followed by Mann-Whitney test was performed.

---