Conduct and Interpret a Mann-Whitney U-Test
What is the Mann-Whitney U-Test?
The Mann-Whitney U-test compares differences between two independent groups on a continuous or ordinal scale. It doesn’t assume a specific data distribution. This makes it useful for data that don’t meet the normal distribution requirements of t-tests or ANOVA.
Essentially, the Mann-Whitney U-test evaluates whether the ranks of two independent samples differ significantly. Researchers rank all observations together, regardless of group, then sum the ranks for each group. The U statistic, calculated from rank sums, assesses the likelihood that two samples come from the same population.
Unlike the t-test, which compares means, the Mann-Whitney U-test compares medians. It is a robust alternative for non-normal or ordinal data. The Mann-Whitney U-test also serves as the foundation for the Kruskal-Wallis H-test. The Kruskal-Wallis H-test extends the comparison to more than two groups through multiple pairwise U-tests. Its flexibility and lack of stringent distributional assumptions make it a preferred choice for researchers. This is especially true when dealing with non-normal data distributions or ordinal measurements.
Overview of the Mann-Whitney U-Test
The Mann-Whitney U-test utilizes a unified approach to ranking all observations across groups, distinguishing it from parametric counterparts like the t-test and F-test, which compare mean values. Its primary focus is on medians rather than means, enhancing its resilience against outliers and distributions with heavy tails. The non-parametric nature of it means that it does not assume a specific data distribution, making it ideal for non-normally distributed ordinal data.
Robustness of the Mann-Whitney U-Test
The robustness of it stems from its ability to provide reliable comparisons without the strict distributional requirements needed for parametric tests. It’s ideal for evaluating median differences when dealing with non-normal distributions or ordinal data. For significance testing, the U-test assumes that with sample sizes greater than 80, or when each sample size exceeds 30, the distribution of the U statistic approximates a normal distribution. This allows researchers to assess the U statistic, derived from sample data, against a normal distribution to determine confidence levels.
Evaluating Differences Between Groups
The Mann-Whitney U-test detects differences in medians influenced by an independent variable. The test assesses whether one sample stochastically dominates another, with the U-value quantifying how often one group’s observations rank higher than the other’s. This is based on the probability concept that one sample is likely to yield higher values than the other. In some cases, it helps to determine if two samples come from the same population by comparing their distributions.
Alternative Non-Parametric Tests
Other non-parametric tests for comparing distributions include the Kolmogorov-Smirnov Z-test and the Wilcoxon signed-rank test, each offering unique approaches to analyze data that does not fit the assumptions required by parametric methods.
The Mann-Whitney U-Test in SPSS
The research question for our U-Test is as follows: Do the students that passed the exam achieve a higher grade on the standardized reading test? The question shows that the independent variable is whether students passed or failed the final exam, while the dependent variable is their grade on the standardized reading test (A to F). You can find the it in Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples.
In the dialog box for the nonparacontinuous-level two independent samples test, we select the ordinal test variable ‘mid-term exam 1’, which contains the pooled ranks, and our nominal grouping variable ‘Exam‘. With a click on ‘Define Groups…‘ we need to specify the valid values for the grouping variable Exam, which in this case are 0 = fail and 1 = pass. We also need to select the Test Type. It is marked by default. Like the Mann-Whitney U-Test the Kolmogorov-Smirnov Z-Test and the Wald-Wolfowitz runs-test have the null hypothesis that both samples are from the same population. Moses extreme reactions test has a different null hypothesis: the range of both samples is the same. The U-test compares the ranking, Z-test compares the differences in distributions, Wald-Wolfowitz compares sequences in ranking, and Moses compares the ranges of the two samples.
Mann-Whitney U-Test in SPSS: Data Requirements and Output Options
The Kolmogorov-Smirnov Z-Test requires continuous-level data (interval or ratio scale), the Mann-Whitney U-Test, Wald-Wolfowitz runs, and Moses extreme reactions require ordinal data. If we select it, SPSS will calculate the U-value and Wilcoxon’s W, which the sum of the ranks for the smaller sample. If the values in the sample are not already ranked, SPSS will sort the observations according to the test variable and assign ranks to each observation. The dialog box Exact… allows us to specify an exact non-para continuous-level test of significance and the dialog box Options… defines how missing values are managed and if SPSS should output additional descriptive statistics.