Friedman test overview

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Friedman test	Binomial test for a single proportion	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent/grouping variable	Independent variable
One within subject factor ($\geq 2$ related groups)	None
Dependent variable	Dependent variable
One of ordinal level	One categorical with 2 independent groups
Null hypothesis	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.	H₀: $\pi = \pi_0$ Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
Alternative hypothesis	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	H₁ two sided: $\pi \neq \pi_0$ H₁ right sided: $\pi > \pi_0$ H₁ left sided: $\pi < \pi_0$
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
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
Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $X$ 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.	Binomial($n$, $P$) distribution. Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).
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 $\alpha$ Right sided: 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 $\alpha$ Left sided: 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 $\alpha$
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 $\pi_0 = 0.2$?
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
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
Practice questions	Practice questions

Friedman test - overview