Mann-Whitney-Wilcoxon test: overview

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Mann-Whitney-Wilcoxon test
Independent variable
One categorical with 2 independent groups
Dependent variable
One categorical of ordinal level
Null hypothesis
Formulation 1:
• The scores in population 1 are not systematically higher or lower than the scores in population 2
Formulation 2:
• The probability that an observation from population 1 exceeds an observation from population 2 is the same as the probability that an observation from population 2 exceeds an observation from population 1
Formulation 3:
• P(observation from population 1 > observation from population 2) = P(observation from population 2 > observation from population 1)
Alternative hypothesis
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: The probability that an observation from population 1 exceeds an observation from population 2 is not the same as the probability that an observation from population 2 exceeds an observation from population 1
• Right sided: The probability that an observation from population 1 exceeds an observation from population 2 is larger than the probability that an observation from population 2 exceeds an observation from population
• Left sided: The probability that an observation from population 1 exceeds an observation from population 2 is smaller than the probability that an observation from population 2 exceeds an observation from population
Formulation 3:
• Two sided: P(observation from population 1 > observation from population 2) $\neq$ P(observation from population 2 > observation from population 1)
• Right sided: P(observation from population 1 > observation from population 2) > P(observation from population 2 > observation from population 1)
• Left sided: P(observation from population 1 > observation from population 2) < P(observation from population 2 > observation from population 1)
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
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 $W$:
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.
Sampling distribution of $W$ and of $U$ 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.
Significant?
For large samples, the table for standard normal probabilities can be used:
Two sided:
Right sided:
Left sided:
Equivalent to
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 context
Do men tend to score higher on social economic status than women?
Pratice questions