Question

• For each of the following nonparametric tests––Wilcoxon signed-rank test, Mann–Whitney U test, and Spearman’s rho––discuss the following points:
  • Give an overview of the test.
  • State what parametric test it replaces.
  • Provide an example of when you would choose to use that test.

Answer 1

Wilcoxon signed rank test

Overview :

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it is a paired difference test).

Replaces : Paired t test

Example :

research team wants to test whether a new teaching method increases the literacy of children. Therefore the researchers take measure the literacy of 20 children before and after the teaching method has been applied. The literacy is measured on a scale from 0 to 10, with 10 indicating high literacy. The initial baseline shows an average literacy score of 5.9 and after the method has been used the average increases to 7.6.

A dependent samples t-test can not be used, as the distribution does not approximate a normal distribution. Also both measurements are not independent from each other and therefore the Mann-Whitney U-test can not be used.Hence Wilcoxon signed dank test is used.

Mann Whitney U test

Overview:

the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less than or greater than a randomly selected value from a second sample.

This test can be used to determine whether two independent samples were selected from populations having the same distribution

Replaces: t test

Example :

Consider a Phase II clinical trial designed to investigate the effectiveness of a new drug to reduce symptoms of asthma in children. A total of n=10 participants are randomized to receive either the new drug or a placebo. Participants are asked to record the number of episodes of shortness of breath over a 1 week period following receipt of the assigned treatment. The data are shown below.

Placebo	7	5	6	4	12
New Drug	3	6	4	2	1

Is there a difference in the number of episodes of shortness of breath over a 1 week period in participants receiving the new drug as compared to those receiving the placebo? By inspection, it appears that participants receiving the placebo have more episodes of shortness of breath, but is this statistically significant?

Spearman rho test :

Overview:

Spearman Rank Correlation Coefficient is a non-parametric measure of correlation, using ranks to calculate the correlation. The Spearman Rank Correlation Coefficient is its analogue when the data is in terms of ranks. One can therefore also call it correlation coefficient between the ranks. The correlation coefficient is sometimes denoted by r_s.

Replaces : Pearson product moment correlation coefficient

Example :

Suppose that scores of the judges (out of 10 were as follows):

Contestant No.	1	2	3	4	5	6	7	8	9	10
Score by Judge A	5	9	3	8	6	7	4	8	4	6
Score by Judge B	7	8	6	7	8	5	10	6	5	8

Is there any significant differnce in the judging ? Test at 0.05 level of significance.