# Mann-Whitney-Wilcoxon test - overview

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Mann-Whitney-Wilcoxon test | Friedman test | Logistic regression |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent/grouping variable | Independent variables | |

One categorical with 2 independent groups | One within subject factor ($\geq 2$ related groups) | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | |

Dependent variable | Dependent variable | Dependent variable | |

One of ordinal level | One of ordinal level | One categorical with 2 independent groups | |

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. | Model chi-squared test for the complete regression model:
- H
_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
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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
| Model chi-squared test for the complete regression model:
- H
_{1}: not all population regression coefficients are 0
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested: - H
_{1}right sided: $\beta_k > 0$ - H
_{1}left sided: $\beta_k < 0$
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$
| |

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. 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
| - In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1 - \pi_{y=1}})$ is linear
- The residuals are independent of one another
- Variables are measured without error
- Multicollinearity
- Outliers
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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. | Model chi-squared test for the complete regression model:
- $X^2 = D_{null} - D_K = \mbox{null deviance} - \mbox{model deviance} $
$D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis.
The wald statistic can be defined in two ways: - Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$
- Wald $ = \dfrac{b_k}{SE_{b_k}}$
Likelihood ratio chi-squared test for individual $\beta_k$: - $X^2 = D_{K-1} - D_K$
$D_{K-1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.
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Sampling distribution of $W$ and of $U$ if H_{0} were true | Sampling distribution of $Q$ if H_{0} were true | Sampling distribution of $X^2$ and of the Wald statistic 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. | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
- chi-squared distribution with $K$ (number of independent variables) degrees of freedom
- If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chi-squared distribution with 1 degree of freedom
- If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution
- chi-squared distribution with 1 degree of freedom
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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 $\alpha$
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
- 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 $\alpha$
- If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chi-squared tests. Wald can be interpret as $X^2$
- If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.
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n.a. | n.a. | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | |

- | - | $b_k \pm z^* \times SE_{b_k}$ 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). | |

n.a. | n.a. | Goodness of fit measure $R^2_L$ | |

- | - | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | |

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)? | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | |

SPSS | SPSS | SPSS | |

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
| Analyze > Regression > Binary Logistic...
- Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
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Jamovi | Jamovi | Jamovi | |

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
| Regression > 2 Outcomes - Binomial
- Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates
- If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well
- Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'
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Practice questions | Practice questions | Practice questions | |