The point here is that the MW-test is generally not testing equality of medians. Generally, it is a test of the equality of stochastic equivalence (a difficult to understand concept), which only in some special cases may be equivalent to a test of a location shift. A "location shift" can only be defined in a meaningful way when the distributions as a whole are only shifted, that ist, when Y = X+d, where X is a random variable with an arbitrary distribution and d is a constant (so that Y is a random variable with the same distribution as X but shifted by d). In such a case, the MW-test test that shift, because this is the only thing that can differ between the distributions. Then, and only then, the test is about the equality of medians (as well as of any other quantile!) and also about the equality of means and the equalty of modes. It is therefore pointless to label the MW-test as a test of medians, because under the special circumstances where this may be correct, the MW-test is also a test of means.

Short advice: use the MW-test only when you are interested in stochastic equivalence. If this is the case, then test it and report it, but do not use this as a proxy to interpret a "location shift".

If you are interested in a location shift and you know the distribution family, you can use the expectation (mean) as the statistic indicating the location. There is only one exact test of a difference in means, which assumes that the distribution of X is normal; however, there are approximate tests available for other distributions. The Likert scale is bounded at both sides, and it is typically convenient and justified to assume that a Likert sclaed variable has a beta-distribution. So it would be advisable to using a GLM of the beta-family (or of the quasi-binomial family, if the absolute likelihood is not of primary interest). The only requirement is to map the Likert-scale onto the interval (0, 1). The mapped vales might be logit-transformed and used with a standard linear model (i.e., a t-test). This is ok to get p-values, but estimates of means and confidence intervals will refer to the transformed values and will be biased when used (after back-transformation) for the original scale.

I am sure that you will find references is you google the relevant terms (beta-distribution, generalized linear models, logit-transformation)

Dear Jochen Wilhelm

Thank you so much for your detailed explanation.

Basically, I am interested in the general differences between the groups, i.e. in the classical sense in the mean differences (which cannot be calculated by t-test due to ordinal scaling, lack of linearity, lack of variance homogeneity and lack of normal distribution). I have already tried a logit transformation of the data, but that didn't help.

The problem is that a dichotomisation (0/1) of my data is out of the question, as the vast majority of my subjects would be in one category (e.g. 397 people in 0 and 4 people in 1). I think the main problem in my data is, as just described, that the results show a very clear tendency in one direction and only very few people are in the middle or high range. Although this is a result, it doesn't help me with the hypothesis testing. It is perhaps worth mentioning that the project is a Master's thesis with limited resources.

I never mentioned dichotomization.

If your Likert variable is reasonable fine-grained, you may approximate its distribution by a continuous random variable. The distribution model must be bounded at both sides. A convenient distribution model for such variables is the beta distribution. The support is the continuous interval (0, 1). Therefore, you'll need to map your Likert values from their range (min, max) to (0, 1).

But if your variable is not fine-grained (which is the case, as I understood now), you may use an appropriate binomial distribution instead and use a GLM of the binomial family.

How about a cumulative probit model, where you assume an underlying normally distributed latent variable for your observed variable. The cumulative probit model would test for a location shift of these latent variables.

Dear Fabianne

the analysis of the likert scale has always been a problem, since sometimes it is assumed that there is a continuity (sorry not to agree with Jochen this time), but it must also be a constant scale, that is, that the distance between 1 and 2 is the same as between 2 and 3, but the answer is based on the "perception" of an individual, this is different from that of others (different psychological continuum, so there could be heterogeneity, among others). And here is my dilemma, I don't know what the objective of the study is, nor what the applied instrument responds to, so I speculate!

And I ask you why analyze the position and not the form, because of what you mention "few people in the upper middle range",

maybe this can help you

Ordinal Methodology in the Analysis of Likert Scales

Goeb, Rainer & McCollin, Chris & Ramalhoto, Maria. (2007). Ordinal Methodology in the Analysis of Likert Scales. Quality and Quantity. 41. 601-626. 10.1007/s11135-007-9089-z.