The Friedman Test is a non-parametric method used for two-way Analysis of Variance (Two Way Anova) on ordinal data. This test can be considered an extension of the Sign Test for dependent or paired samples. If there are only two dependent sample groups, then the results of the Friedman Test will be equivalent to the Sign Test. The Friedman Test becomes a suitable alternative for two-way variance analysis (such as Two Way Anova or Randomized Block Design/RBD) when the assumptions of variance analysis are not met and an appropriate data transformation cannot be found. If the results of the Friedman Test indicate a significant treatment effect, then one can proceed with Post hoc Tests. In SmartstatXL, some available options for Post hoc Tests include the Dunn and Nemenyi tests.
Case Example
Sourced from: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition by David J. Sheskin
Example 25.1
A psychologist conducts a study to determine whether noise can inhibit learning. Each of the six subjects is tested under three experimental conditions. In each experimental condition, subjects are given 20 minutes to memorize a list of 10 nonsensical syllables, and they are informed that they will be tested the following day. The three experimental conditions each subject undergoes are as follows: Condition 1, the no-noise condition, requires the subject to study the list of nonsensical syllables in a quiet room. Condition 2, the moderate noise condition, requires the subject to study the list of nonsensical syllables while listening to classical music. Condition 3, the extreme noise condition, requires the subject to study the list of nonsensical syllables while listening to rock music. Although each subject is given a different list of nonsensical syllables in each experimental condition, the three lists are comparable in variables known to affect a person's ability to learn nonsensical syllables. To control for order effects, the presentation order of the three experimental conditions is fully balanced. The number of nonsensical syllables correctly recalled by the six subjects under the three experimental conditions is as follows. (Subjects' scores are listed in the order of Condition 1, Condition 2, Condition 3.) Subject 1: 9, 7, 4; Subject 2: 10, 8, 7; Subject 3: 7, 5, 3; Subject 4: 10, 8, 7; Subject 5: 7, 5, 2; Subject 6: 8, 6, 6. Does the data suggest that noise affects the subjects' performance?
Analysis Steps
The following are the steps for the Friedman Test and Post hoc Tests using SmartstatXL, an Excel Add-in:
- Activate the worksheet (Sheet) that will be analyzed.
- Place the cursor on the dataset (for creating a dataset, see the Data Preparation method). The dataset can be arranged in two layouts:
- Grouped by level/tier (comparison between tiers)
- Grouped by variable (comparison between variables)
- If the active cell is not on the dataset, SmartstatXL will automatically try to determine the dataset.
- Activate the SmartstatXL Tab
- Click on the Non Parametric Menu. SmartstatXL will display a dialog box to confirm whether the dataset is correct or not (usually the cell address of the dataset is automatically selected correctly).
- If it is correct, Click the Next Button
- The following Non Parametric Test Dialog Box will appear:
- If the data layout used is based on a comparison between variables, the following dialog box will appear:
- Next, Press the "OK" button
Analysis Results
The following is the Output from the Friedman Test Analysis (Two Way Anova):
Statistical Summary
Based on the analysis:
- Hypotheses Proposed:
- H0 (Null Hypothesis): All experimental conditions have the same median. This means there is no difference between the three experimental conditions based on noise levels.
- H1 (Alternative Hypothesis): At least one experimental condition has a median different from the other conditions.
- Statistical Results:
- Ties correction is 0.958.
- The Chi Square value (after correction for ties) is 11.565.
- The critical value for this test (with df = 2) is 5.991.
- The p-value (probability) is 0.003.
- Interpretation:
- Since the Chi Square value (11.565) is greater than the critical value (5.991), and the p-value (0.003) is less than 0.05, we reject the null hypothesis (H0). This means there is a significant difference between the three experimental conditions based on noise levels in terms of their performance in recalling nonsensical syllables.
Conclusion:
Based on the Friedman Test analysis, there is sufficient evidence to state that noise levels affect the subjects' performance in recalling nonsensical syllables. In other words, the noise level has an effect on a person's ability to remember information.
Average Comparison
Table of average values, Mean Rank, Sum Rank, and Post hoc Tests
From the table, we can see the comparison between experimental conditions based on average values, mean ranks, and sum ranks. Additionally, there are also results from several Post hoc Tests (Dunn, Nemenyi) used to determine which differences are significant after the Friedman Test showed a difference.
Let's interpret these comparison results:
- Data Description:
- Condition 1: With an average of 8.50, a mean rank of 3.00, and a sum rank of 18.00.
- Condition 2: With an average of 6.50, a mean rank of 1.92, and a sum rank of 11.50.
- Condition 3: With an average of 4.83, a mean rank of 1.08, and a sum rank of 6.50.
- Post-Hoc Test Interpretation:
- Dunn: Condition 1 (b) is significantly different from Condition 3 (a), but not significantly different from Condition 2 (ab). Conditions 2 and 3 are not significantly different from each other as they both have label "a".
- Nemenyi: The results are the same as the Dunn test.
- Critical Distance (CD-F):
- CD-F provides the critical distance between the sum ranks of two conditions to determine whether their difference is significant. If the difference between the sum ranks of two conditions is greater than CD-F, then the difference is significant.
- From the given data, CD-F is 8.29. If we compare the differences in sum ranks between conditions, we can see that the difference between Condition 1 and Condition 3 is 11.5, which is greater than 8.29, making the difference significant. However, the differences between Condition 1 and Condition 2, and between Condition 2 and Condition 3, are both smaller than 8.29, making them not significant.
Conclusion:
Based on the post-hoc tests (Dunn, Nemenyi), Condition 1 performs significantly better than Condition 3 in recalling nonsensical syllables. However, there is no significant difference between Conditions 2 and 3, as well as between Conditions 1 and 2. This suggests that extreme noise (rock music) may have a greater impact on memory ability compared to no-noise or moderate noise conditions (classical music).
