## Wilcoxon Rank Sum Test: Non Parametric tests

The Wilcoxon Rank Sum test (also known as the Mann-Whitney U test) is a nonparametric statistical test used to compare two independent samples. It is used when the data does not follow a normal distribution or when the assumptions of a parametric test, such as the t-test, are not met.

The Wilcoxon Rank Sum test works by ranking all the data from both groups combined, then calculating the sum of the ranks for each group. The test statistic is then calculated as the smaller of the two sums, which is compared to the expected distribution of this statistic under the null hypothesis of no difference between the two groups.

The following are the steps to perform the Wilcoxon Rank Sum test:

- State the null hypothesis and the alternative hypothesis. The null hypothesis is that there is no significant difference between the two groups, while the alternative hypothesis is that there is a significant difference between the two groups.
- Combine the data from the two groups and rank them from lowest to highest. Ties are assigned the average rank.
- Calculate the sum of the ranks for each group.
- Calculate the test statistic as the smaller of the two sums.
- Calculate the p-value by comparing the test statistic to the expected distribution of this statistic under the null hypothesis.
- Compare the p-value to the chosen significance level (usually 0.05) to determine if the null hypothesis can be rejected.

If the p-value is less than the chosen significance level, the null hypothesis is rejected, and it is concluded that there is a significant difference between the two groups. If the p-value is greater than the chosen significance level, the null hypothesis is not rejected, and it is concluded that there is no significant difference between the two groups.

#### Pharmaceutical example of Wilcoxon Rank Sum Test

The Wilcoxon Rank Sum test is commonly used in pharmaceutical research to compare the efficacy or safety of two different treatments or interventions. Here is an example of how the Wilcoxon Rank Sum test can be applied in a pharmaceutical study:

Suppose a pharmaceutical company is developing a new drug for the treatment of a particular disease. The company conducts a clinical trial with two groups of patients: one group receiving the new drug, and the other group receiving a placebo. The company wants to compare the efficacy of the new drug to that of the placebo.

The study measures the change in disease severity scores (e.g., based on a validated questionnaire) over a specified period of time for each patient in both groups. The scores are not normally distributed, so the Wilcoxon Rank Sum test is used to compare the two groups.

The steps to perform the Wilcoxon Rank Sum test in this scenario are as follows:

- State the null hypothesis and the alternative hypothesis. The null hypothesis is that there is no significant difference in disease severity scores between the two groups, while the alternative hypothesis is that there is a significant difference between the two groups.
- Combine the disease severity scores from both groups and rank them from lowest to highest. Ties are assigned the average rank.
- Calculate the sum of the ranks for each group.
- Calculate the test statistic as the smaller of the two sums.
- Calculate the p-value by comparing the test statistic to the expected distribution of this statistic under the null hypothesis.
- Compare the p-value to the chosen significance level (usually 0.05) to determine if the null hypothesis can be rejected.

If the p-value is less than the chosen significance level, the null hypothesis is rejected, and it is concluded that there is a significant difference in disease severity scores between the two groups. This indicates that the new drug has a significant effect on disease severity compared to the placebo.

The Wilcoxon Rank Sum test is useful in situations where the data does not meet the assumptions of a parametric test, such as the t-test, and is commonly used in pharmaceutical research to compare the efficacy and safety of different treatments or interventions.

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