Bartlett's test is used to test homoscedasticity between two or more independent normal populations (with unknown mean). The actual formula is not important for my question. The important thing is that it compares the adjusted sample variances coming from each iid sample.
In linear regression we know (given the usual assumptions about the linear model) that the residuals are normally distributed with zero mean, and that they are not independent.
Under these assumptions about the model (linearity, homoscedasticity, normal errors), standardized residuals should come from a std normal distribution.
We want to test if these standardized residuals actually have constant variance, so we separate them in groups and we use Bartlett's test.
For each group we calculate the adjusted sample variance, as we would do if we had an iid sample from a population with unknown mean, and then we compare them.
I have two questions (the second one is more important to me):
1)Why is it ignored that the residuals are not independent? Maybe this doesn't change the asymptotic distribution of the test.
2)We know that the residuals distribution has (given the other assumptions that we are not testing) zero mean, but we still use the sample mean of each group to calculate the variances.
Why don't we use, instead of the adjusted sample variance, the sample mean of the squared residuals for each group?