Paired alternate ranking

The Siegel-Tukey test pools the two data samples with identification, ranks them, applies paired alternate ranking to generate rank sums, and allows for the sample sizes, to provide a test statistic that can be evaluated using the same tables as for the Mann-Whitney U-test. 

Paired comparisons Each employee is compared with every other employee one at a time on each criterion May appear to be less subjective than alternation ranking Limited feedback... 

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... 

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. 

For any object (5,X, A) of Ag,d the kernel of A is a finite flat group scheme of rank d2 over S. We will construct a nondegenerate alternating pairing... 

The ELECTRE family has been extended frequently to improve the original method and develop methods for different applications. ELECTRE II aims at ranking alternatives. It constructs a downward and an upward ranking in a multi-step procedure based on a strong and a weak outranking relationship and consequently requires two pairs of concordance and discor-... 

On a molar basis, most organic compounds contain similar amounts of hydrogen and carbon, and processes involving transfer of hydrogen between covalently bound sites rank in importance in organic chemistry second only to those involving the carbon-carbon bond itself. Most commonly, hydrogen is transferred as a proton between atoms with available electron pairs (l), i.e. Bronsted acid/base reactions. The alternative closed shell process, hydride transfer or shift, involves motion of a proton with a pair of electrons between electron deficient sites (2). These processes have four and two electrons respectively to distribute over the three atomic centres in their transition structures. It is the latter process, particularly when the heavy atoms are both first row elements, which is the subject of this review. The terms transfer and shift are used here only to differentiate intermolecu-lar and intramolecular reactions. 

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... 

An important advantage of the signed rank test is that it can also be used on paired data, because they can be transformed into the type of data given in the previous example. The signed rank method can thus be used as a non-parametric alternative to the paired t-test (Section 3.4). 

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