KruskalWallis test  overview
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KruskalWallis test  Spearman's rho 


Independent/grouping variable  Variable 1  
One categorical with $I$ independent groups ($I \geqslant 2$)  One of ordinal level  
Dependent variable  Variable 2  
One of ordinal level  One of ordinal level  
Null hypothesis  Null hypothesis  
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:
 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  Alternative hypothesis  
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:
 H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  
Assumptions  Assumptions  

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