The Mann-Whitney U-test

The following hypothetical data in Table 11.2 are simulated from two distinct non-normal distributions. The population, from which the observations in group A are taken, has a mean of 1 while the population from which the B group observations were taken has mean equal to 1.5. 

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. 

Example 11.1 Natalizumab in the treatment of relapsing multiple sclerosis 

Miller et al. (2003) report a trial comparing two dose levels of natalizumab (3 mg per kg and 6 mg per km) with placebo. The primary endpoint was the number of new brain lesions during the six month treatment period. Table 11.3 presents the data. 

The distribution of number of new lesions (count data) is clearly not normal within each of the treatment groups. There is a peak at zero in each of the groups with then fewer and fewer patients as the number of lesions increases. A log transformation would not work here because of the presence of the zero values for the endpoint. The authors used the Mann-Whitney U-test to compare each of the natalizumab dose groups with placebo obtaining p 0.001 in each case. Each dose level is significantly better than placebo in reducing the number of new enhancing lesions. 

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The Mann-Whitney U test is employed for the count data, but which test should be employed for the percentage variables should be decided on the same grounds as described later under reproduction studies. 

It may seem strange to see the normal distribution play a part in the p-value calculations in Section 11.5.1 and 11.5.2. The appearance of this distribution is in no sense related to the underlying distribution of the data. For the Mann-Whitney U-test for example it relates to the behaviour of the average of the ranks within each of the individual groups under the assumption of equal treatments where the ranks in those groups of sizes and 2 simply a random split of the numbers 1 through to Hi -b 2-... 

Behavioral data may be analyzed with the Mann-Whitney U test for comparing two groups (parametric Student s t test may be used only if data are normally distributed), or analysis of variance (ANOVA) for multiple groups, followed by an appropriate post hoc test. 

The principal measure taken is the animal s latency to cross to the dark compartment at T2. This score provides an estimate of the animal s retention of the shock received at Tl. The latencies measured at T2 have a 180 second cut-off. The scores in the control group are therefore abnormally distributed because of the presence of numerous ceiling scores. It is therefore essential to apply non-parametric statistics, for example the Mann-Whitney U-test, to analyze the data. 

Parametric data were presented as mean SD. To determine differences in glutamate concentrations, a repeated-measures analysis of variance was performed. The cutaneous sensation, hind-limb motor function, and morphological changes of the spinal cord were analyzed with a non-parametric method (Kruskal-Wallis test) followed by the Mann-Whitney U-test. 

It has been advocated that the area under the ROC curve is a relative measure of a tesfs performance. A Wilcoxon statistic (or equivalently the Mann-Whitney U-Test) statists cally determines which ROC curve has more area under it. Less computationally intensive alternatives, which are no longer necessary, have been described. These methods are particularly helpful when the curves do not intersect. When the ROC curves of two laboratory tests for the same disease intersect, they may offer quite different performances even though the areas under their curves are identical. The performance depends on the region of the curve (i.e., high sensitivity versus high specificity) chosen. Details on how to compare statistically individual points on two curves have been developed elsewhere. ... 

The Mann-Whitney U-test was used to test for differences between two samples that might represent different populations. The frequency distribution of the variables was examined by applying the Kolmogorov-Smimov test. The 95% confidence interval of the median was calculated according to the equation... 

Statistical Analysis. Data are presented as means SEM. Statistical comparisons between groups were performed using ANOVA. Fisher s Protected Least Significant Difference (PLSD) test was used to analyze the difference in lipid levels and the Mann-Whitney U-test was used for differences in atherosclerotic levels between dietary groups. 

N = 20) and the comparison group N = 26) using the Mann-Whitney U test... 

Table 5 Comparisons of the posttest mean TTCT scores in the three dimensions between the treatment group N = 20) and the comparison group (N = 26) using the Mann-Whitney U test...

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. 

MWW Test n short for the Mann-Whitney-Wilcoxon Test, and alternate name for the Mann-Whitney U Test. 

All data are expressed as the mean SD. Differences between groups were examined for statistical significance using the Mann-Whitney U-test. A p value less than 0.05 denoted the presence of a statistically significant difference. 

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