KruskalWallis test  overview
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KruskalWallis test  One sample $z$ test for the mean 


Independent/grouping variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  None  
Dependent variable  Dependent variable  
One of ordinal level  One quantitative of interval or ratio 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}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  
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: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.  
Sampling distribution of $H$ if H_{0} were true  Sampling distribution of $z$ 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.  Standard normal distribution  
Significant?  Significant?  
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  
  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu$ can also be used as significance test.  
n.a.  Effect size  
  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$  
n.a.  Visual representation  
  
Example context  Example context  
Do people from different religions tend to score differently on social economic status?  Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$  
SPSS  n.a.  
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
   
Jamovi  n.a.  
ANOVA > One Way ANOVA  KruskalWallis
   
Practice questions  Practice questions  