Friedman test  overview
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Friedman test  Pearson correlation 


Independent/grouping variable  Variable 1  
One within subject factor ($\geq 2$ related groups)  One quantitative of interval or ratio level  
Dependent variable  Variable 2  
One of ordinal level  One quantitative of interval or ratio level  
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}: $\rho = \rho_0$
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  
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: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  
Assumptions  Assumptions of test for correlation  

 
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.  Test statistic for testing H0: $\rho = 0$:
 
Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $t$ and of $z$ if H_{0} were true  
If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  Sampling distribution of $t$:
 
Significant?  Significant?  
If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 $t$ Test two sided:
 
n.a.  Approximate $C$% confidence interval for $\rho$  
  First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 
n.a.  Properties of the Pearson correlation coefficient  
 
 
n.a.  Equivalent to  
  OLS regression with one independent variable:
 
Example context  Example context  
Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  Is there a linear relationship between physical health and mental health?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  
ANOVA > Repeated Measures ANOVA  Friedman
 Regression > Correlation Matrix
 
Practice questions  Practice questions  