Mann-Whitney test

The data were statistically analyzed using the SOLO Statistical System (BMDP Statistical Software, Inc., Los Angeles, CA) on a personal computer. Differences between groups were tested by the Mann-Whitney test or a paired t-test in cases where paired data sets were tested. Possible relationships were studied with (multiple) linear regression using least-square estimates. 

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... 

NA, not available p < 0.05 versus control (day 9 sham operation) p < 0.05 versus postis-chemic day 9 Kruskal-Wallis or Mann-Whitney tests... 

Fig. 45 A Statistical analysis of the density of BrdU+ cells in SVZa on postischemic day 9. p < 0.001 versus anterior, caudate, and dorsal SVZa p < 0.05 versus caudate and dorsal SVZa one-way ANOVA followed by Tukey-Kramer post hoc. B Proportions of BrdU+ cells retaining presence in SVZa. Densities of BrdU+ cells on day 44 after sham/ischemia were calculated as a percentage of the densities of day 9 after sham/ischemia. p < 0.05 versus respective sham-operated control Kruskal-Wallis or Mann-Whitney tests...

Proliferating microglia were also observed in olfactory parenchyma as demonstrated by BrdU/Ibal co-staining (Fig. 55C). The double-positive cells constituted 6%-8% of the BrdU+ cells in control monkeys, while in the postischemic monkeys the percentage was considerably higher (15%-20%), a difference which proved to be significant ( p < 0.05, Kruskal-Wallis or Mann-Whitney tests). A few astrocytes and oligodendrocytes also incorporated BrdU (up to 10% of the BrdU+ cells). 

Instead of transforming the data to normality, we could employ one of a range of procedures referred to as non-parametric tests . These partially duplicate the functionality of tests we have already met, but use a method of calculation that does not depend upon a normal distribution. The non-parametric test that is generally substituted for the two sample f-test goes by a variety of names, but is most commonly called the Mann-Whitney test. 

Table 17.4 Generic output from a Mann-Whitney test of the amounts of toxin produced by smokers and non-smokers...

The mean - the process of ranking destroys all information regarding the absolute magnitude of the individual results and consequently it would be very risky to try to claim that a Mann-Whitney test had demonstrated a change in the mean. To justify such a conclusion, you would have to make such extensive assumptions about the distribution of the data that you could probably use a parametric test anyway ... 

When non-parametric methods are applied to data that is normally distributed, they are slightly less powerful than their parametric equivalents, although the difference is not great. For the tests covered in this chapter, the non-parametric test has about 95 per cent of the power of its parametric equivalent. In other words, if our data is normally distributed then a sample of 19 tested by a parametric method would provide about the same power as a sample of 20 tested by the non-parametric equivalent. Since the power of the two types of test is so similar, it is not surprising that the P values generated [0.034 for the f-test (when applied to the transformed data) and 0.036 for the Mann-Whitney test] are barely different. 

An example of dealing with ordinal scale data - applying the Mann-Whitney test to the effectiveness of an analgesic... 

With an unpaired design and measurements on an interval scale, we would have used a two-sample t-test to check for any difference. However, this data are ordinal and not remotely normally distributed, so we will have to move to its non-parametric equivalent - the Mann-Whitney test. 

Non-parametric Mann-Whitney test Wilcoxon paired samples test Kruskal-Wallis test Spearman correlation... 

Mann—Whitney test equivalent to the two-sample /-test ... 

The data are ordinal and extremely positively skewed, with a majority of zero scores and a tail of other scores on one side only. Group 2 appears to have somewhat higher scores (22 positive scores compared with only 11 in group 1). For a formal comparison, we would use the non-parametric Mann-Whitney test. That yields a P value (adjusted for ties) of 0.021, so there is significant evidence of higher scores in group 2. 

In case that both data sets to be compared are normally distributed the F-test is applied. The hypothesis of homogeneity of variance of both test series is eliminated when the significance level for homogeneity of variance is 5 %. The t-test for paired and non-paired data is performed when homogeneity of variance is present. In any case, a paired difference test (for paired data) or the U-test (for non-paired data) is likewise carried out (paired of difference test = Wilcoxon test U-test = Wilcoxon-Mann-Whitney or Mann-Whitney test, respectively). 

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.

The chief non-parametric tests for comparing locations are the Mann-Whitney [/-test and the Kolmogorov—Smirnov test. The former assumes that the frequency distributions of the data sets are similar, whereas the latter makes no such assumption. In the Kolmogorov-Smimov test, significant differences found with the test may be due to differences in location or shape of the distribution, or both. 

The unpaired t-test is an example of a parametric method, which means that it is based on the assumption that the two samples are taken from normal, or approximately normal distributions. Generally, parametric tests should be used where possible because they are more powerful (effectively, more sensitive) than the alternative non-parametric methods [32]. However, significance levels obtained from parametric tests may be inaccurate, and the true power of the test may decrease, if the assumption of normality is poor. The non-parametric alternative to the unpaired t-test is the Mann-Whitney test [32]. In this test, a rank is assigned to each observation (1 = smallest, 2 = next smallest, etc.), and the test statistic is computed from these ranks. Obviously, the test is less sensitive to departures from normality, such as the presence of outliers, since, for example, the rank assigned to the smallest observation will always be 1, no matter how small that observation is. 

Relationships were tested with the non-parametric Spearman rank correlation coefficient, while the translocation capacity was tested with the ratio Ce/Cr, where Ce and Cr are the antimony contents in an epigeal compartment (shoots, leaves, etc.) and in the roots, respectively. The effects of plant traits were tested with the Mann-Whitney /-test in each sampling site for which a sample size no smaller than four observations was possible. The null hypothesis was not rejected when an overlap of observations between small samples occurred. 

The different aspects of each plant trait considered (root morphology, life span, habitus, and systematic rank) have no effect on the Sb contents in the plants nor on the translocation coefficients the U values of the Mann-Whitney test are always nonsignificant. Nevertheless, the species enriched in Sb ([Sb] 100 mg/kg) and best able to accumulate the element in their epigeal parts are all herbaceous, tap-rooted and Dicotyledones, with the exception of P. australis. Most of them are perennials and about a third of them are Compositae (Tables 4 and 5 for antimony content and Appendix A for species attributes). 

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