Kruskal-Wallis test - overview
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Kruskal-Wallis test | Kruskal-Wallis test | Paired sample $t$ test |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent/grouping variable | Independent variable | |
One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with $I$ independent groups ($I \geqslant 2$) | 2 paired groups | |
Dependent variable | Dependent variable | Dependent variable | |
One of ordinal level | One of ordinal level | One quantitative of interval or ratio level | |
Null hypothesis | 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:
| 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:
| H0: $\mu = \mu_0$
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 | 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:
| 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:
| H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 H0 were true | Sampling distribution of $H$ if H0 were true | Sampling distribution of $t$ if H0 were true | |
For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | For large samples, approximately the chi-squared 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? | Significant? | |
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| Two sided:
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n.a. | 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 $t_{N-1}$ 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. | 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. | n.a. | Visual representation | |
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n.a. | n.a. | Equivalent to | |
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Example context | Example context | Example context | |
Do people from different religions tend to score differently on social economic status? | 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 | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Compare Means > Paired-Samples T Test...
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Jamovi | Jamovi | Jamovi | |
ANOVA > One Way ANOVA - Kruskal-Wallis
| ANOVA > One Way ANOVA - Kruskal-Wallis
| T-Tests > Paired Samples T-Test
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Practice questions | Practice questions | Practice questions | |