Decision trees - hypothesis test for quality of split

Question

I was just introduced to the concept of decision tree. I read that hypothesis testing can be used to asses the quality of each split

$$H_0: \text{split was bad}$$ $$H_a: \text{split was good}$$

I read that chi-squared goodness of fit test or t-test can be used. But those two tests are parametric tests, and require normality assumption. What if target & feature data are not normal?

I also learned that another way of assessing split is using the Gini impurity score and entropy. They are non-parametric methods and do not require normality of data.

Question 1: What technique (ex: Gini impurity, entropy, hypothesis test) should I use to assess quality of split, under what circumstances each?

Question 2: what is the implication of the data not being normal?

Answer 1

A nice thing about decision trees is that they exploit high-dimensional interactions among the features. A seemingly useless split at level 1 may lead to ground-breaking splits at levels, 2, 3 and further. No simple statistical test will be able to "see the big picture" if applied at level 1. This will hold no matter whether there is approximate normality in the data or not.

In general, goodness-of-fit chi-square test does not require normality, rather the number of data points in each bin must be big. In general, in most cases, non-normality in small samples can be fought successfully with a form of bootstrap.