Mann-Whitney-Wilcoxon test - overview

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Mann-Whitney-Wilcoxon test
Kruskal-Wallis test
Independent variableIndependent variable
One categorical with 2 independent groupsOne categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent variable
One of ordinal levelOne of ordinal level
Null hypothesisNull 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:
• The median in population 1 is equal to the median in population 2
Else:
Formulation 1:
• The scores in population 1 are not systematically higher or lower than the scores in population 2
Formulation 2:
• P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
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.
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:
• The medians in the $I$ populations are equal
Else:
Formulation 1:
• The scores in any of the $I$ populations are not systematically higher or lower than the scores in any of the other populations
Formulation 2:
• 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.
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.
Alternative hypothesisAlternative 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:
• Two sided: the median in population 1 is not equal to the median in population 2
• Right sided: the median in population 1 is larger than the median in population 2
• Left sided: the median in population 1 is smaller than the median in population 2
Else:
Formulation 1:
• Two sided: The scores in population 1 are systematically higher or lower than the scores in population 2
• Right sided: The scores in population 1 are systematically higher than the scores in population 2
• Left sided: The scores in population 1 are systematically lower than the scores in population 2
Formulation 2:
• 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)
• 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)
• 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 all $I$ populations:
• Not all of the medians in the $I$ populations are equal
Else:
Formulation 1:
• The scores in some populations are systematically higher or lower than the scores in other populations
Formulation 2:
• 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 $g$)
AssumptionsAssumptions
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, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another
Test statisticTest 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 $W$:
The second type of test statistic is the Mann-Whitney $U$ statistic:
• $U = W - \dfrac{n_1(n_1 + 1)}{2}$
where $n_1$ is the sample size of group 1

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.

$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$

Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$.

Note: if ties are present in the data, the formula for $H$ is more complicated.
Sampling distribution of $W$ and of $U$ if H0 were trueSampling distribution of $H$ if H0 were true

Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here \begin{aligned} \mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\ \sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_W = \dfrac{W - \mu_W}{\sigma_W}\\$$ follows approximately a standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.

Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here \begin{aligned} \mu_U &= \dfrac{n_1 n_2}{2}\\ \sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_U = \dfrac{U - \mu_U}{\sigma_U}\\$$ follows approximately a standard normal distribution if the null hypothesis were true.

For small samples, the exact distribution of $W$ or $U$ should be used.

Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.

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.

Significant?Significant?
For large samples, the table for standard normal probabilities can be used:
Two sided:
Right sided:
Left sided:
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 $\alpha$
Equivalent ton.a.
If no ties in the data: two sided Mann-Whitney-Wilcoxon test is equivalent to Kruskal-Wallis test with an independent variable with 2 levels ($I = 2$) -
Example contextExample 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?
SPSSSPSS
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 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
JamoviJamovi
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 > 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
Practice questionsPractice questions