I know the concept of standardized coefficients, which should be applied to continuous variable only and it is used for comparing variable importance among continuous variables. But what about comparison between continuous var and categorical var?
For example, let's assume a dependent variable y to be explained by two continuous variable x_1 and x_2. If we standardize y, x_1 and x_2, and fit the standardized x_1 and x_2 to standardized y, then we could get standardized coefficients beta_1 and beta_2. By comparing the values of beta_1 and beta_2, we could know which independent variable y is more sensitive to.
Now if one switches x_2 with a categorical variable, it isn't that straightforward to compare the sensitivity between x_1 and x_2 anymore since we couldn't standardize the categorical variable x_2. In this case, how should one compare the sensitivity between x_1 and x_2?