Friedman Test

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Friedman Test

Definition

Friedman Test is a non-parametric alternative to the one-way repeated measures ANOVA. It tests for differences between three or more matched or paired groups, without requiring normality in the data distribution. The test statistic is based on the ranks of the data values within each block.

The Friedman test is used when the same subjects are measured under different conditions or treatments, and the data is ordinal or continuous. The test is sensitive to differences in the central tendency of the data, but not to differences in variance.

If the Friedman test indicates significant differences between groups, post-hoc tests can be used to determine which groups differ from each other.

Test Statistic

Q = \frac{12}{nk(k+1)}\sum_{j=1}^k R_j^2 - 3n(k+1)

Where:

$n$ = number of blocks (subjects)
$k$ = number of treatments
$R_j$ = total rank of treatment $j$
$Q \sim \chi^2_{k-1}$ when $n > 15$ or $k > 4$
$Q \sim F_{k-1, (k-1)(n-1)}$ for better accuracy when $n \leq 15$ and $k \leq 4$

Key Assumptions

Matched Groups: Data must be paired or matched

Ordinal Data: Measurements must be ordinal or continuous

Independent Blocks: Subjects must be independent

Practical Example

Step 1: State the Data

Scores for three treatments across five subjects:

Subject	Treatment A	Treatment B	Treatment C
1	8	6	5
2	7	7	6
3	9	8	6
4	6	5	4
5	7	6	5

Step 2: State Hypotheses

$H_0$ : No difference between treatments
$H_a$ : At least two treatments differ
$\alpha = 0.05$

Step 3: Rank Within Blocks

Subject	A	B	C
1	3	2	1
2	2.5	2.5	1
3	3	2	1
4	3	2	1
5	3	2	1
$R_j$	14.5	10.5	5

Step 4: Calculate Test Statistic

1. Basic Friedman statistic with b = 5 subjects and k = 3 treatments:

Q = \frac{12}{5 \cdot 3(3+1)}(14.5^2 + 10.5^2 + 5^2) - 3 \cdot 5(3+1) = 9.1

2. Correction for ties:

In our data, we have one tie (7,7) in Subject 2's scores.

C = 1 - \frac{\sum T}{n k(k^2-1)}

where $T = (t^3 - t)$ for each group of $t$ tied ranks

One pair of ties gives $T = (2^3 - 2) = 6$

C = 1 - \frac{6}{5 \cdot 3(9-1)} = 0.95

3. Adjusted test statistic:

Q_{\text{adj}} = \frac{Q}{C} = \frac{9.1}{0.95} = 9.58

Step 5: Draw Conclusion

Critical value for $\chi^2$ with $df = 2$ at $\alpha = 0.05$ is $5.991$ . Since $Q = 9.58 \gt 5.991$ , we reject $H_0$ .

There is sufficient evidence to conclude that there are significant differences between at least two treatments.

With $p$ -value = $0.008317 \lt 0.05$ , there is strong evidence to conclude that there are significant differences between at least two treatments.

Effect Size

Kendall's W

Kendall's coefficient of concordance ( $W$ ) ranges from $0$ (no agreement) to $1$ (complete agreement):

W = \frac{Q}{n(k-1)}

For our example:

W = \frac{9.58}{5(3-1)} = 0.958

Interpretation guidelines:

$W \lt 0.3$ : Weak agreement
$0.3 \leq W \leq 0.5$ : Moderate agreement
$W \gt 0.5$ : Strong agreement

Code Examples

1#Example data
2data <- matrix(c(8, 6, 5, 
3                 7, 7, 6, 
4                 9, 8, 6, 
5                 6, 5, 4, 
6                 7, 6, 5), 
7               nrow = 5, byrow = TRUE)
8
9# Perform Friedman test
10friedman.test(data)

Python

1import numpy as np
2from scipy.stats import friedmanchisquare
3
4# Example data
5data = np.array([
6    [8, 6, 5],
7    [7, 7, 6],
8    [9, 8, 6],
9    [6, 5, 4],
10    [7, 6, 5]
11])
12
13# Perform Friedman test
14stat, p = friedmanchisquare(data[:, 0], data[:, 1], data[:, 2])
15
16# Output results
17print("Friedman Test Statistic:", stat)
18print("p-value:", p)

Alternative Tests

Consider these alternatives:

Repeated Measures ANOVA: When normality assumptions hold
Cochran's Q Test: For binary outcomes

Related Calculators

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Friedman Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Friedman Test

Definition

Test Statistic

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Rank Within Blocks

Step 4: Calculate Test Statistic

Step 5: Draw Conclusion

Effect Size

Kendall's W

Code Examples

Alternative Tests

Related Calculators

One-Way ANOVA Calculator

Repeated Measures ANOVA Calculator

Wilcoxon Signed-Rank Test

Chi-Square Test of Independence Calculator

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