Pearson correlation  overview
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Pearson correlation 


Variable 1  
One quantitative of interval or ratio level  
Variable 2  
One quantitative of interval or ratio level  
Null hypothesis  
H_{0}: $\rho = \rho_0$
$\rho$ is the unknown Pearson correlation in the population, $\rho_0$ is the 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  
H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  
Assumptions of test for correlation  
 
Test statistic  
Test statistic for testing H0: $\rho = 0$:
 
Sampling distribution of $t$ and of $z$ if H_{0} were true  
Sampling distribution of $t$:
 
Significant?  
$t$ Test two sided:
 
Approximate $C$% confidence interval for $\rho$  
First compute approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get approximate $C$% confidence interval for $\rho$:
 
Properties of the Pearson correlation coefficient  
 
Equivalent to  
OLS regression with one independent variable:
 
Example context  
Is there a linear relationship between physical health and mental health?  
SPSS  
Analyze > Correlate > Bivariate...
 
Jamovi  
Regression > Correlation Matrix
 
Practice questions  