Logistic regression and other statistical methods for predicting a binary outcome run into problems when the outcome being tested is very rare, even in data sets big enough to insure that the rare outcome occurs hundreds or thousands of times. The problem is that attempts to optimize the model across all of the data will end up looking predominantly at optimizing the negative cases, and could easily ignore and misclassify all or almost all of the positive cases since they consistute such a small percentage of the data. The ROSE package generates artificial balanced samples to allow for better estimation and better evaluation of the accuracy of the model.

Nicola Lunardon, Giovanna Menardi, and Nicola Torelli. ROSE: A Package for Binary Imbalanced Learning. The R Journal 6/1, 79-89, June 2014. Available at https://journal.r-project.org/archive/2014-1/menardi-lunardon-torelli.pdf.

Most nonparametric tests based on ranks (like Kruskal-Wallis and Friedman) do not extend well to additional factors. The reason for this is that the assumption that all rankings are equally likely under the null hypothesis does not work when testing the significance of one factor when another factor might have variation in the outcome measure.

Any extension of these nonparametric tests would therefore no longer be nonparametric. That doesn’t mean that you can’t use them, but it means that most of the motivation for choosing them disappears.

There is an alternative, however. Ordinal logistic regression is able to handle ordinal data like ranks and it is invariant to any montone transformation of your outcome variable. And ordinal logistic regression extends easily to multiple factors. It does have a pretty strong assumption of parallelism, but this is something that you can easily check. It certainly would be less restrictive than the assumption of normality.

You should also consider transformations like the log transform that can sometimes turn data that is highly skewed and heteroscedastic into nice well behaved data amenable to the classic multifactor ANOVA models.

I have three variables: physicians (%), dentists (%), and salary categories. I want to know if there is a difference in the percentage of physicians and dentists in each salary category. What test I need to use? ANOVA is not appropriate because the outcome is not continuous.

You need counts rather than percentages. Can you get them?

Also, can you verify that this is a simple random sample?

If you can get the counts and they are coming from something reasonably
close to a simple random sample, then you are able to proceed. But
lacking either of these means that your data is not amenable to any
credible analysis.

Let’s hope for the best here. If so, then what you are doing is comparing to see if the counts for the physicians and the counts for the dentists are allocated roughly evenly across the salary categories. Salary category is ordinal, and there is a lot of controversy over the best approach here. Probably the best approach would be ordinal logistic regression. Chi square tests have a problem in that they fail to take advantage of the ordinal nature of your outcome variable and lose a lot of power against reasonable alternative hypotheses. Some people will use a rank based test like Mann-Whitney for this data and even though there are a lot of ties, that ends up being the same as ordinal logistic regression.