## Friedman Test: Non Parametric tests

The Friedman test is a non-parametric statistical test used to compare three or more related groups. It is used when the data does not follow a normal distribution or when the assumptions of a parametric test, such as the repeated-measures ANOVA, are not met.

The Friedman test works by ranking the data in each group and calculating the average rank for each subject across all groups. The test statistic is then calculated based on the difference between the average ranks for each subject and the grand average rank. The test statistic follows a chi-square distribution with k-1 degrees of freedom, where k is the number of groups.

Here are the steps to perform the Friedman test:

- State the null hypothesis and the alternative hypothesis. The null hypothesis is that there is no significant difference between the groups, while the alternative hypothesis is that there is a significant difference between the groups.
- Rank the data in each group from lowest to highest.
- Calculate the average rank for each subject across all groups.
- Calculate the sum of the squared differences between the average ranks for each subject and the grand average rank.
- Calculate the test statistic based on the sum of the squared differences.
- 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 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 groups.

The Friedman test is commonly used in pharmaceutical research to compare the efficacy or safety of three or more different treatments or interventions in a crossover or repeated measures design. For example, it may be used to compare the effectiveness of three different formulations of a drug on blood pressure in patients with hypertension over a period of time.

#### Pharmaceutical example of Friedman Test

Suppose a pharmaceutical company wants to compare the effectiveness of three different formulations (Formulation A, Formulation B, and Formulation C) of a drug on blood pressure in patients with hypertension over a period of time. The company recruits 15 patients with hypertension and administers all three formulations to each patient, with a washout period between each administration.

The company can collect the systolic blood pressure measurements from each patient after each administration of the three formulations. To analyze the data, the company can use the Friedman test to determine if there is a significant difference in blood pressure between the three formulations over time. The null hypothesis is that there is no significant difference in blood pressure between the three formulations, while the alternative hypothesis is that at least one formulation is more effective than the others.

The company can rank the systolic blood pressure measurements for each patient separately for each of the three formulations. They can then calculate the average rank for each patient across all three formulations. The company can then calculate the sum of the squared differences between the average ranks for each patient and the grand average rank.

The Friedman test statistic is then calculated based on the sum of the squared differences. If the p-value is less than the chosen significance level (usually 0.05), the company can conclude that there is a significant difference in blood pressure between the three formulations and perform post-hoc tests to determine which formulations are significantly different from each other. If the p-value is greater than 0.05, the company can conclude that there is no significant difference in blood pressure between the three formulations over time.

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