Spearman's rho  overview
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Spearman's rho 


Independent variable  
One of ordinal level  
Dependent variable  
One of ordinal level  
Null hypothesis  
$\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. In words: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  
Two sided: $\rho_s \neq 0$ Right sided: $\rho_s > 0$ Left sided: $\rho_s < 0$  
Assumptions  
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.  
Test statistic  
$t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sampling distribution of $t$ if H0 were true  
Approximately a $t$ distribution with $N  2$ degrees of freedom  
Significant?  
Two sided:
 
Example context  
Is there a monotonic relationship between physical health and mental health?  
SPSS  
Analyze > Correlate > Bivariate...
 
Jamovi  
Regression > Correlation Matrix
 
Practice questions  