# Mann-Whitney-Wilcoxon test - overview

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Mann-Whitney-Wilcoxon test | Friedman test | One sample $z$ test for the mean |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent/grouping variable | Independent variable | |

One categorical with 2 independent groups | One within subject factor ($\geq 2$ related groups) | None | |

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 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)
| H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
Usually the related groups are the different measurement points. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher. | 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 | 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 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)
| H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups
| H_{1} two sided: $\mu \neq \mu_0$H _{1} right sided: $\mu > \mu_0$H _{1} left sided: $\mu < \mu_0$
| |

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. That is, within and between groups, observations are independent of one another
| - Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
| - Scores are normally distributed in the population
- Population standard deviation $\sigma$ is known
- Sample is a simple random sample from the population. That is, observations are independent of one another
| |

Test statistic | Test statistic | Test statistic | |

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. | $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects - so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
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 $W$ and of $U$ if H_{0} were true | Sampling distribution of $Q$ if H_{0} were true | Sampling distribution of $z$ if H_{0} were true | |

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. | If the number of blocks $N$ is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used. | Standard normal distribution | |

Significant? | Significant? | Significant? | |

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 $\alpha$
| If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
- 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 $\alpha$
| 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. | 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. | 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. | n.a. | Visual representation | |

- | - | ||

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

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 men tend to score higher on social economic status than women? | Is there a difference in depression level between measurement point 1 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)? | 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 | SPSS | n.a. | |

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 > K Related Samples...
- Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables
- Under Test Type, select the Friedman test
| - | |

Jamovi | Jamovi | n.a. | |

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
| ANOVA > Repeated Measures ANOVA - Friedman
- Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
| - | |

Practice questions | Practice questions | Practice questions | |