GLM instead of Mann-Whitney stats newbie here so please make replies easy to understand!
With the type of data I generate, my colleagues use Mann-Whitneys to generate p values.
However,
I did a stats course recently and we were told there was no longer any excuse to do:
• ANOVA, routinely transforming data to make residuals normal
• non-parametric stats (M-W, K-W, runs, etc) to cope with complex parametric structures
and instead we had to use GLMs.
I have data where my independent variable is a categorical/discrete variable and my dependent variable is continuous variable (non-integer)
I have tried to do a linear model but my data is non normal, and my residuals vs fitted in R looks like this:

I have tried transforming the data using lny 1/y sqrt(y) and it doesn't help at all.
From my stats course the next step would be to try a generalised linear model,
The only GLMS I know are poisson and negative-binomial but my data aren't integers. Any suggestions on what I can do?
 A: First of all, while the advice you received in your course reflects the general sentiment of the stats community, there is nothing wrong with a transformation, and due to simplicity, I would prefer this solution to a GLM in your case. So, before you move to complicated GLM / GLS solutions, try e.g. if https://www.rdocumentation.org/packages/forecast/versions/8.13/topics/BoxCox produces a model with acceptable residuals.
If this is not the case, you can move to either GLM (changing distributions) or GLS (modelling variance), or both.
From eye-balling your data, it seems as if you have a strictly positive response. In this case, you could try a Gamma GLM, and possibly add a variance term (e.g. in glmmTMB) if you still have variance problems (for checking the latter, you can use e.g. https://cran.r-project.org/web/packages/DHARMa/index.html).
Alternatively, to avoid the entire distributional problem, you could just run a quantile regression, e.g. https://cran.r-project.org/web/packages/qgam/index.html
