# Friedman test - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Friedman test | Binomial test for a single proportion | McNemar's test |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent variable | Independent variable | |

One within subject factor ($\geq 2$ related groups) | None | 2 paired groups | |

Dependent variable | Dependent variable | Dependent variable | |

One of ordinal level | One categorical with 2 independent groups | One categorical with 2 independent groups | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: 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. | H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options: - First score of pair is 0, second score of pair is 0
- First score of pair is 0, second score of pair is 1 (switched)
- First score of pair is 1, second score of pair is 0 (switched)
- First score of pair is 1, second score of pair is 1
_{0} is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is the same as the probability that a pair of scores switches from 1 to 0.
Other formulations of the null hypothesis are: - H
_{0}: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group - H
_{0}: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups
| H_{1} two sided: $\pi \neq \pi_0$H _{1} right sided: $\pi > \pi_0$H _{1} left sided: $\pi < \pi_0$
| The alternative hypothesis H Other formulations of the alternative hypothesis are: - H
_{1}: $\pi_1 \neq \pi_2$ - H
_{1}: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
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Assumptions | Assumptions | Assumptions | |

- Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
| - Sample is a simple random sample from the population. That is, observations are independent of one another
| - Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
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Test statistic | Test statistic | 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. | $X$ = number of successes in the sample | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | |

Sampling distribution of $Q$ if H_{0} were true | Sampling distribution of $X$ if H0 were true | Sampling distribution of $X^2$ if H_{0} 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. | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | |

Significant? | Significant? | 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 $\alpha$
| Two sided:
- Check if $X$ observed in sample is in the rejection region or
- Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
- Check if $X$ observed in sample is in the rejection region or
- Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
- Check if $X$ observed in sample is in the rejection region or
- Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
| For test statistic $X^2$:
- 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 $\alpha$
- Check if $b$ observed in sample is in the rejection region or
- Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$
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n.a. | n.a. | Equivalent to | |

- | - | - Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups
- Cochran's Q test, with two related groups
- Two sided sign test: $b = W$, and $X^2 = z^2$
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Example context | Example context | 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)? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | |

SPSS | SPSS | 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
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
- Put your dichotomous variable in the box below Test Variable List
- Fill in the value for $\pi_0$ in the box next to Test Proportion
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
- Put the two paired variables in the boxes below Variable 1 and Variable 2
- Under Test Type, select the McNemar test
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Jamovi | Jamovi | Jamovi | |

ANOVA > Repeated Measures ANOVA - Friedman
- Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
| Frequencies > 2 Outcomes - Binomial test
- Put your dichotomous variable in the white box at the right
- Fill in the value for $\pi_0$ in the box next to Test value
- Under Hypothesis, select your alternative hypothesis
| Frequencies > Paired Samples - McNemar test
- Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns
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Practice questions | Practice questions | Practice questions | |