# Is there a negative impact from imbalance/skew in predictor variables?

I understand that imbalance or skew in the target variable within your training data can negatively impact effectiveness. Does the same apply to the predictor/independent variables?

y ~ B0 + B1*x1 + B2*x2


Consider this simple example. I am trying to predict y, a categorical variable, from two variables x1 and x2 which are also categorical variables. If I have an imbalanced set of y values, this could be a bad thing. What if I have an imbalanced set of x1 or x2? Could the same issue apply?

Lack of balance is not bad for a saturated model. With two categorical variables this means having $AB$ predictors ($A$ is number of categories in the variable, $B$ the second). If you include the interaction term $x_1$ and $x_2$ you should be fine. If this fit is too "noisy" then include it as a random effect in a mixed model - so you get the benefits of partial pooling.
$y = B0 + B1*x1 + B2*x2$? Does it even make sense for categorical variables?
If you look at continuous variables, for example most values of $x$ are clustered together, and there's a couple of outliers, these outliers is going to have high leverage and have a large impact on the regression coefficients. I imagine the same might happen to the strongly imbalanced categorical variables (if you interpret the equation y ~ B0 + B1*x1 + B2*x2 in the sense of, say, logistic regression, with x1, x2 etc being dummy variables).