Friedman test

The Friedman test is a non-parametric statistical test developed by the American economist Milton Friedman. It is used to detect differences in treatments across multiple test attempts. The test is an extension of the Kruskal-Wallis test, which is used when the data are measured at least at the ordinal level and the samples are independent. The Friedman test, however, is used for one-way repeated measures analysis of variance by ranks. In essence, it is used to test for differences between groups when the dependent variable being measured is ordinal.

Overview

The Friedman test is particularly useful when the assumptions of one-way ANOVA (analysis of variance) are not met. Specifically, it does not require the assumption of normality, making it suitable for data that is not normally distributed. It is also robust against outliers, which can significantly affect the results of parametric tests.

Application

The test is commonly applied in situations where data are collected from the same subjects at different times or under different conditions. This makes it particularly useful in medical, psychological, and behavioral science research where measurements are often taken from the same subjects under different conditions. For example, it can be used to compare the effects of different medications on the same group of patients over time.

Procedure

The Friedman test ranks each row (or block) together, then considers the values of ranks by columns. The null hypothesis of the test is that the treatments have the same effect. Rejection of the null hypothesis indicates that at least one of the treatments has a different effect.

1. **Rank the data**: For each block, rank the observations from 1 to k (number of treatments). 2. **Sum the ranks for each treatment**: Calculate the sum of ranks for each treatment across all blocks. 3. **Compute the test statistic**: The test statistic is based on the differences between the sum of ranks.

Formula

The Friedman test statistic, \( \chi^2_F \), is given by: \[ \chi^2_F = \frac{12}{n \cdot k \cdot (k+1)} \left( \sum_{j=1}^k R_j^2 - \frac{k \cdot (n \cdot k + 1)^2}{4} \right) \] where \( n \) is the number of blocks, \( k \) is the number of treatments, and \( R_j \) is the sum of ranks for treatment \( j \).

Assumptions

The Friedman test assumes that:

• The observations within each block are ranked, and the ranks are the data analyzed.
• Each block is a matched set of subjects, receiving each treatment once and only once (i.e., there is no replication within a block).
• The observations are independent across blocks.

Limitations

While the Friedman test is a powerful non-parametric alternative to the one-way ANOVA, it does have limitations:

• It is less powerful than ANOVA when the normality assumption holds.
• It can only handle one blocking factor or repeated measures factor, limiting its use in more complex experimental designs.

See Also

• Kruskal-Wallis test
• ANOVA
• Non-parametric statistics

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