Intro to Business Statistics: Nonparametric Tests Friedman Test Repeated Measures

What This Is

The Friedman Test, also known as the Repeated Measures ANOVA on Ranks, is a non-parametric test used to compare more than two related samples or repeated measurements on a single sample to assess if there are statistically significant differences between them. A retail chain wants to know if the average daily sales of its three different store locations (A, B, and C) exceed $10,000. The Friedman Test can help determine if there are significant differences in the daily sales among the three locations.

Key Formulas & Symbols

• Friedman Test Statistic (χ²): χ² = (12/n) * [ΣRi² - (n+1)(n+1)/4] where n = number of treatments, Ri² = sum of ranks for each treatment.
• Ranks (Ri): The ranks of the observations within each treatment.
• n: The number of treatments.
• k: The number of observations per treatment.
• χ²: The test statistic.
• df: The degrees of freedom, which is n-1.
• α: The significance level, typically set at 0.05.
• p-value: The probability of observing the data (or more extreme) if the null hypothesis is true.

Step-by-Step Procedure

1. State hypotheses: Formulate the null and alternative hypotheses. For example, H₀: μA = μB = μC and H₁: Not all means are equal.
2. Choose test: Select the Friedman Test as the appropriate non-parametric test for comparing related samples.
3. Compute test statistic: Calculate the χ² test statistic using the formula χ² = (12/n) * [ΣRi² - (n+1)(n+1)/4].
4. Find p-value or critical value: Determine the p-value associated with the χ² test statistic or find the critical value from the χ² distribution table with n-1 degrees of freedom and α = 0.05.
5. Compare to α: Compare the p-value to α (0.05) or the test statistic to the critical value. If the p-value is less than α or the test statistic is greater than the critical value, reject the null hypothesis.
6. Conclude: If the null hypothesis is rejected, conclude that there are statistically significant differences among the means of the treatments.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check the assumptions of the Friedman Test, such as the data being ranked and the number of observations per treatment being equal.
• Correction: The Friedman Test assumes that the data are ranked and that the number of observations per treatment is equal. If these assumptions are not met, the test may not be valid.
• Mistake: Using the Friedman Test with a small sample size (n < 5).
• Correction: The Friedman Test is sensitive to sample size, and small sample sizes may lead to inaccurate results. It is recommended to use a larger sample size or a different test.

Quick Practice Problems

1. A company wants to compare the average response times of three different customer service representatives. The response times are as follows:

Calculate the χ² test statistic.

Answer: χ² = 3.00. The ranks were assigned based on the response times, and the χ² test statistic was calculated using the formula χ² = (12/n) * [ΣRi² - (n+1)(n+1)/4].

1. A researcher wants to compare the average scores of three different teaching methods. The scores are as follows:

What is the p-value associated with the χ² test statistic?

Answer: The p-value is 0.001. The χ² test statistic was calculated using the formula χ² = (12/n) * [ΣRi² - (n+1)(n+1)/4], and the p-value was determined using a χ² distribution table with n-1 = 2 degrees of freedom and α = 0.05.

1. A company wants to compare the average sales of three different products. The sales are as follows:

What is the conclusion if the p-value is less than α = 0.05?

Answer: There are statistically significant differences among the means of the products. The p-value is less than α = 0.05, indicating that the null hypothesis can be rejected.

Last-Minute Cram Sheet

1. Friedman Test Statistic (χ²): χ² = (12/n) * [ΣRi² - (n+1)(n+1)/4].
2. Ranks (Ri): The ranks of the observations within each treatment.
3. n: The number of treatments.
4. k: The number of observations per treatment.
5. χ²: The test statistic.
6. df: The degrees of freedom, which is n-1.
7. α: The significance level, typically set at 0.05.
8. p-value: The probability of observing the data (or more extreme) if the null hypothesis is true.
9. ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
10. Assumptions of the Friedman Test: Ranked data and equal number of observations per treatment.