KruskalWallis test  overview
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KruskalWallis test  Paired sample $t$ test 


Independent/grouping variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  2 paired groups  
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 of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  
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  
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  
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.  $t$ distribution with $N  1$ 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:
 
n.a.  $C\%$ confidence interval for $\mu$  
  $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). 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 of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$  
n.a.  Visual representation  
  
n.a.  Equivalent to  
 
 
Example context  Example context  
Do people from different religions tend to score differently on social economic status?  Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 Analyze > Compare Means > PairedSamples T Test...
 
Jamovi  Jamovi  
ANOVA > One Way ANOVA  KruskalWallis
 TTests > Paired Samples TTest
 
Practice questions  Practice questions  