36 Friedman Tests and related Post-hoc Tests
The Friedman test (Milton Friedman, 1937) is a non-parametric statistical test used to detect differences in treatments across multiple test attempts. It is the non-parametric alternative to the one-way ANOVA with repeated measures and is suitable for experiments where the data violate the assumption of normality required for ANOVA. This test is commonly used in situations where measurement variables are ordinal or when interval scale measurements fail the assumptions of normality.
36.1 Assumptions
The Friedman test operates under several key assumptions:
- Dependent Samples: The groups are related or matched; typically, measurements are taken from the same subjects at different times or under different conditions.
- Ordinal or Continuous Data: The data should be at least ordinal, though they can also be continuous.
- Same Number of Observations: Each subject is measured the same number of times, ensuring that the data matrix is balanced.
36.2 Hypotheses
The hypotheses for the Friedman test can be framed as:
- Null Hypothesis (H₀): The distributions of the treatments are identical across repeated measures.
- Alternative Hypothesis (H₁): At least one treatment yields a different distribution.
36.3 Formula
The Friedman test statistic ($ ^2_F \() is calculated as follows:\)$ ^2_F = _{j=1}^k R_j^2 - 3n(k+1) $$ Where:
- \(n\) is the number of subjects.
- \(k\) is the number of conditions or treatments.
- \(R_j\) is the sum of ranks for each treatment across all subjects.
36.3.1 Calculation Steps
- Rank each row (or block) of observations separately from 1 to \(k\), treating ties by assigning average ranks.
- Calculate the sum of ranks for each treatment across all subjects.
- Compute the test statistic using the formula above.
36.3.2 Interpretation
A significant $ ^2_F $ statistic indicates that at least one of the treatments significantly differs from the others. The value is compared against critical values from the chi-square distribution with \(k-1\) degrees of freedom. If $ ^2_F $ is greater than the critical value for a given level of significance (\(\alpha\)), the null hypothesis is rejected.
36.4 Example Problem
Imagine a study examining the effect of three different diets on weight loss. Each of four participants tries each diet for one month, and their weight loss is recorded:
- Diet A: 5, 4, 6, 5
- Diet B: 4, 3, 5, 4
- Diet C: 6, 5, 7, 6
Hypotheses:
- Null Hypothesis (H₀): No diet leads to significantly different weight loss.
- Alternative Hypothesis (H₁): At least one diet leads to significantly different weight loss.
36.4.1 Friedman Test using Excel:
36.5 Friedman Test using R and Python
This test is crucial in fields like medicine, psychology, and agronomy where multiple treatments are compared under non-normal conditions and repeated measures are common.
Summary
| Concept | Description |
|---|---|
| Foundations | |
| Friedman Test | A non-parametric test for differences across three or more related conditions on the same subjects |
| Non-parametric Counterpart of RM ANOVA | Stand-in for repeated-measures ANOVA when normality or interval-scale assumptions fail |
| Ranked Within Blocks | Differences are detected by ranking values within each subject rather than across the whole dataset |
| Assumptions | |
| Dependent Samples | Each subject is measured under every condition, creating natural blocks |
| Ordinal or Continuous Outcome | The outcome must be at least ordinal so within-subject ranks can be assigned |
| Balanced Block Design | Each subject contributes exactly one observation per condition, producing a complete data matrix |
| Hypotheses | |
| Null Hypothesis | States that all conditions have the same distribution within blocks |
| Alternative Hypothesis | States that at least one condition tends to produce systematically higher or lower ranks |
| Computation | |
| Within-Block Ranking | Within each subject, observations are ranked from 1 to k across the k conditions |
| Sum of Ranks Per Treatment | Ranks are summed within each treatment column to produce R_j, the rank sum for treatment j |
| Friedman Chi-square Statistic | Chi-square_F is computed from the rank sums, number of subjects and number of conditions |
| Chi-square Approximation | Under H0, the statistic approximately follows a chi-square distribution with k minus 1 degrees of freedom |
| Decision and Follow-up | |
| Decision Rule | Reject H0 when chi-square_F exceeds the critical value or when p is below alpha |
| Post-hoc Tests (Nemenyi, Conover) | Follow a significant Friedman test with pairwise rank-based comparisons such as Nemenyi or Conover |
| In R and Python | |
| R via friedman.test() | Use friedman.test(matrix) where rows are subjects and columns are conditions in R |
| Python via friedmanchisquare() | Use scipy.stats.friedmanchisquare(c1, c2, ..., ck) on equal-length sequences in Python |