Pearson correlation  overview
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Pearson correlation  Friedman test 


Variable 1  Independent/grouping variable  
One quantitative of interval or ratio level  One within subject factor ($\geq 2$ related groups)  
Variable 2  Dependent variable  
One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  
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.  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.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  
Assumptions of test for correlation  Assumptions  

 
Test statistic  Test statistic  
Test statistic for testing H0: $\rho = 0$:
 $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
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 $t$ and of $z$ if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  
Sampling distribution of $t$:
 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.  
Significant?  Significant?  
$t$ Test two sided:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
Approximate $C$% confidence interval for $\rho$  n.a.  
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
   
Properties of the Pearson correlation coefficient  n.a.  
   
Equivalent to  n.a.  
OLS regression with one independent variable:
   
Example context  Example context  
Is there a linear relationship between physical health and mental health?  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)?  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  