Mann-Whitney-Wilcoxon test - overview
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Mann-Whitney-Wilcoxon test | Kruskal-Wallis test | Two sample $z$ test |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | |
One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with 2 independent 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 both 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: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |
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:
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: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | |
Assumptions | Assumptions | Assumptions | |
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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 $W$:
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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | |
Sampling distribution of $W$ and of $U$ if H0 were true | Sampling distribution of $H$ if H0 were true | Sampling distribution of $z$ if H0 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. | 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. | Standard normal distribution | |
Significant? | Significant? | Significant? | |
For large samples, the table for standard normal probabilities can be used: Two sided:
| 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 $\mu_1 - \mu_2$ | |
- | - | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1 - \mu_2$ can also be used as significance test. | |
n.a. | n.a. | Visual representation | |
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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? | Do people from different religions tend to score differently on social economic status? | Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women. | |
SPSS | SPSS | n.a. | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| - | |
Jamovi | Jamovi | n.a. | |
T-Tests > Independent Samples T-Test
| ANOVA > One Way ANOVA - Kruskal-Wallis
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Practice questions | Practice questions | Practice questions | |