# Kruskal-Wallis test - overview

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Kruskal-Wallis test | Mann-Whitney-Wilcoxon test | Mann-Whitney-Wilcoxon test |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | |

One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with 2 independent groups | One categorical with 2 independent groups | |

Dependent variable | Dependent variable | Dependent variable | |

One of ordinal level | One of ordinal level | One of ordinal 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:
- H
_{0}: the population medians for the $I$ groups are equal
Formulation 1: - H
_{0}: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups
- H
_{0}: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups.
| If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
- H
_{0}: the population median for group 1 is equal to the population median for group 2
Formulation 1: - H
_{0}: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
- H
_{0}: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 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 both populations:
- H
_{0}: the population median for group 1 is equal to the population median for group 2
Formulation 1: - H
_{0}: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
- H
_{0}: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
| |

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:
- H
_{1}: not all of the population medians for the $I$ groups are equal
Formulation 1: - H
_{1}: the poplation scores in some groups are systematically higher or lower than the population scores in other groups
- H
_{1}: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)
| If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
- H
_{1}two sided: the population median for group 1 is not equal to the population median for group 2 - H
_{1}right sided: the population median for group 1 is larger than the population median for group 2 - H
_{1}left sided: the population median for group 1 is smaller than the population median for group 2
Formulation 1: - H
_{1}two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2 - H
_{1}right sided: the population scores in group 1 are systematically higher than the population scores in group 2 - H
_{1}left sided: the population scores in group 1 are systematically lower than the population scores in group 2
- H
_{1}two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1) - H
_{1}right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1) - H
_{1}left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 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 both populations:
- H
_{1}two sided: the population median for group 1 is not equal to the population median for group 2 - H
_{1}right sided: the population median for group 1 is larger than the population median for group 2 - H
_{1}left sided: the population median for group 1 is smaller than the population median for group 2
Formulation 1: - H
_{1}two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2 - H
_{1}right sided: the population scores in group 1 are systematically higher than the population scores in group 2 - H
_{1}left sided: the population scores in group 1 are systematically lower than the population scores in group 2
- H
_{1}two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1) - H
_{1}right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1) - H
_{1}left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
| |

Assumptions | Assumptions | Assumptions | |

- Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another
| - Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
| - Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
| |

Test statistic | Test statistic | Test statistic | |

$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
- $W$ = sum of ranks in group 1
- $U = W - \dfrac{n_1(n_1 + 1)}{2}$
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
- $W$ = sum of ranks in group 1
- $U = W - \dfrac{n_1(n_1 + 1)}{2}$
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | |

Sampling distribution of $H$ if H_{0} were true | Sampling distribution of $W$ and of $U$ if H_{0} were true | Sampling distribution of $W$ and of $U$ if H_{0} 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. | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | |

Significant? | Significant? | Significant? | |

For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
- Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
- Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
| For large samples, the table for standard normal probabilities can be used: Two sided: - Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
| For large samples, the table for standard normal probabilities can be used: Two sided: - Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
| |

n.a. | Equivalent to | Equivalent to | |

- | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | |

Example context | Example context | Example context | |

Do people from different religions tend to score differently on social economic status? | Do men tend to score higher on social economic status than women? | Do men tend to score higher on social economic status than women? | |

SPSS | SPSS | SPSS | |

Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum
- Continue and click OK
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
- Continue and click OK
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
- Continue and click OK
| |

Jamovi | Jamovi | Jamovi | |

ANOVA > One Way ANOVA - Kruskal-Wallis
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
| T-Tests > Independent Samples T-Test
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
- Under Tests, select Mann-Whitney U
- Under Hypothesis, select your alternative hypothesis
| T-Tests > Independent Samples T-Test
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
- Under Tests, select Mann-Whitney U
- Under Hypothesis, select your alternative hypothesis
| |

Practice questions | Practice questions | Practice questions | |