Friedman test - overview
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Friedman test |
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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 | |
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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 $X^2 = Q$:
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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...
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Jamovi | |
ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | |