Spearman's rho  overview
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Spearman's rho 


Variable 1  
One of ordinal level  
Variable 2  
One of ordinal level  
Null hypothesis  
H_{0}: $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H_{0}: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  
Assumptions  
 
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 H_{0} were true  
Approximately the $t$ 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  