How to see if a categorical variable or interactions between variables affects a continuous outcome variable

Question

(1) Generally, if I have a categorical variable (X) that is non-ordinal, many unique values, and an outcome variable that is continuous, highly skewed (not normally distributed at all), then what kind of testing or technique to use to see if there is any relationship or not?

(2) I am trying to do an inference study and currently in the stage of testing each individual independent variable's impact on the outcome continuous variable (then I will try some methods on the interaction between some variables). My next question is, how can we combine one categorical (say, already encoded) and one continous variable (not normally distributed), and use this interaction or newly created feature to see how it affects the outcome variable?

For the first case, I encounter Kruskal-Wallis test but am still trying to understand carefully. But what are your experiences dealing with such cases?

Answer 1

(1) I think Kruskal-Wallis as you mentioned here is a good start since the parametric alternative (ANOVA) will probably have violated assumptions (as you have tested maybe?). But this depends a bit on the amount of skewness, sometimes simple transformations can still do the trick and enable you to perform a classical ANOVA/t-test, given sufficient sample size.

Extra: watch out with the 'sparsity' of your variable: you say you have many unique categories, this might make it hard to find any association at all. If this problem occurs, try to combine some categories maybe, if possible.

Further, note that Kruskal-Wallis will just tell you 'there is some difference somewhere', but it is not going to tell you how big/strong is that difference and in what direction. If this is not what you need, then don't worry about this.

(2) You can just incorporate the interaction between your nominal variable and skewed continuous variable as you would do in other cases. Your independent variables have no distributional restrictions. But there is no need to 'combine' variables or define a new variable here.

To see the effect of this interaction on an outcome variable, it might be good to look into (nonparametric) regression techniques, depending on your dependent variable.