Friedman test overview

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Friedman test	Binomial test for a single proportion	Pearson correlation
Independent/grouping variable	Independent variable	Variable 1
One within subject factor ($\geq 2$ related groups)	None	One quantitative of interval or ratio level
Dependent variable	Dependent variable	Variable 2
One of ordinal level	One categorical with 2 independent groups	One quantitative of interval or ratio level
Null hypothesis	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.	H₀: $\rho = \rho_0$ Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.
Alternative 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: $\pi < \pi_0$	H₁ two sided: $\rho \neq \rho_0$ H₁ right sided: $\rho > \rho_0$ H₁ left sided: $\rho < \rho_0$
Assumptions	Assumptions	Assumptions of test for correlation
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	In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions) Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
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	Test statistic for testing H0: $\rho = 0$: $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}} $ where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size Test statistic for testing values for $\rho$ other than $\rho = 0$: $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$ $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $X$ if H0 were true	Sampling distribution of $t$ and of $z$ 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).	Sampling distribution of $t$: $t$ distribution with $N - 2$ degrees of freedom Sampling distribution of $z$: Approximately the standard normal distribution
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 $\alpha$ 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 $\alpha$	$t$ Test two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $t$ Test right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $t$ Test left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $z$ Test two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ $z$ Test right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ $z$ Test left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Approximate $C$% confidence interval for $\rho$
-	-	First compute the approximate $C$% confidence interval for $\rho_{Fisher}$: $lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}}$ $upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}}$ where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$ and the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval). Then transform back to get the approximate $C$% confidence interval for $\rho$: lower bound = $\dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1}$ upper bound = $\dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1}$
n.a.	n.a.	Properties of the Pearson correlation coefficient
-	-	The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables. The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable. The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship. The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example: the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y - 6$. the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $-3x + 5$ and $2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $-3$. The Pearson correlation coefficient does not say anything about causality. The Pearson correlation coefficient is sensitive to outliers.
n.a.	n.a.	Equivalent to
-	-	OLS regression with one independent variable: $b_1 = r \times \frac{s_y}{s_x}$ Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$
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 $\pi_0 = 0.2$?	Is there a linear relationship between physical health and mental health?
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 > Correlate > Bivariate... Put your two variables in the box below Variables
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	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Pearson (selected by default) Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions	Practice questions

Friedman test - overview