Size and power estimation for the Wilcoxon-Mann-Whitney test for ordered categorical data

The Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is a nonparametric test used to determine if there is a significant difference between the distributions of two independent samples. This test is designed specifically for two independent populations and has the following key requirements:

The Wilcoxon signed-rank test for the median of a single population is a nonparametric test used to evaluate whether the median of a population differs from a specified value. Unlike parametric tests, it does not require data to follow a normal distribution, making it suitable for non-normal or small samples. The test begins by calculating the difference (d) between each observation and the hypothesized median. The absolute values of these differences are ranked in ascending order, with ties...

The Wilcoxon signed-rank test for matched pairs evaluates the null hypothesis by combining the ranks of differences with their signs. It essentially tests whether the median of the differences in a population of matched pairs is zero. Since the test incorporates more information than the sign test, it generally yields more trustable conclusions. This test also does not require the data to follow a normal distribution, but two conditions must be met for it to be applicable: (1) the data must...

The Mantel-Cox log-rank test is a widely used statistical method for comparing the survival distributions of two groups. It tests whether a statistically significant difference exists in survival times between the groups without assuming a specific distribution for the survival data, making it a non-parametric test. This flexibility makes the log-rank test particularly valuable in medical research and other fields where the timing of an event, such as death or disease recurrence, is of interest.

Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures from...

The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can be stated as...