Friedman test - overview

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Friedman test
Binomial test for a single proportion$z$ test for a single proportion$z$ test for the difference between two proportionsGoodness of fit testChi-squared test for the relationship between two categorical variablesOne sample $z$ test for the meanOne sample $t$ test for the meanPaired sample $t$ testTwo sample $z$ testTwo sample $t$ test - equal variances not assumedTwo sample $t$ test - equal variances assumedOne way ANOVATwo way ANOVAPearson correlationRegression (OLS)Logistic regressionMann-Whitney-Wilcoxon testKruskal-Wallis testSign testMcNemar's testCochran's Q testMarginal Homogeneity test / Stuart-Maxwell testFriedman testWilcoxon signed-rank testOne sample Wilcoxon signed-rank testSpearman's rho
Independent/grouping variable
One within subject factor ($\geq 2$ related groups)
Dependent variable
One of ordinal level
Null hypothesis
H0: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups

Usually the related groups are the different measurement points. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.
Alternative hypothesis
H1: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups
Assumptions
  • Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
Test statistic
$Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$

Here $N$ is the number of 'blocks' (usually the subjects - so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$.

Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$.

Note: if ties are present in the data, the formula for $Q$ is more complicated.
Sampling distribution of $Q$ if H0 were true
If the number of blocks $N$ is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom.

For small samples, the exact distribution of $Q$ should be used.
Significant?
If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
  • Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
  • Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
Example context
Is there a difference in depression level between measurement point 1 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)?
SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
  • Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables
  • Under Test Type, select the Friedman test
Jamovi
ANOVA > Repeated Measures ANOVA - Friedman
  • Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
Practice questions