Spearman's rho  overview
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Spearman's rho  KruskalWallis test 


Variable 1  Independent/grouping variable  
One of ordinal level  One categorical with $I$ independent groups ($I \geqslant 2$)  
Variable 2  Dependent variable  
One of ordinal level  One of ordinal level  
Null hypothesis  Null hypothesis  
H_{0}: $\rho_s = 0$
Here $\rho_s$ is the 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.  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ Here $r_s$ 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.  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $H$ if H_{0} were true  
Approximately the $t$ distribution with $N  2$ degrees of freedom  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  
Significant?  Significant?  
Two sided:
 For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 
Example context  Example context  
Is there a monotonic relationship between physical health and mental health?  Do people from different religions tend to score differently on social economic status?  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 ANOVA > One Way ANOVA  KruskalWallis
 
Practice questions  Practice questions  