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My question is regarding linear regression and non-constant variance. I've heard that even though a large data set fails a normality test, this does not necessarily mean the data is not normal. This is because the tests for normality can be 'overly' sensitive.' Thus, I've heard that, in practice, people overlook when normality fails.

Are the tests for constant variance like that? Or should I be able to keep transforming the data until it passes tests for constant variance? Also, I read that the constant variance tests like the breusch pagan test are sensitive to normality. If my data set fails the normality test, then will BP also fail?

My data set is over 2500 observations and has many predictor variables. When I take a look at the scatter plots, they look reasonable.

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The assumptions that are required for the linear model to have the least squares estimates optimal is that the residuals are normally distributed with a constant variance. Checking for normality is done formally by goodness of fit tests. It sometimes can happen that whenthe data set is large it is easy to reject normality. That can be because the test is capable of detecting mild departures from normality. It does not mean that the test should be ignored because "even though a large data set fails a normality test, this does not necessarily mean the data is not normal". What it means is that the departure is not large enough to worry about. In practice data do not exactly follow a normal distribution but assuming normality is fine if it holds approximately.
Regarding the F test for equality of variance when comparing samples from two distributions, the test statistics distribution under the null hypothesis has the F distribution only when the two distributions are normal with the same mean and variance. There are two things to note about the test. (1) Unless the sample size is very large the variances will need to be very different to reject equality, and (2) the test is highly dependent on the normality assumption. If the data fail the normality test and the departure is large there is no need to test for constant vriance. There is a difference between saying that a test is invalid (in this case because the normality assumption fails) and saying that the test will likely reject equality of variance because it is invalid. The idea of trying transformation after transformation until you appear to achieve constant variance is never good practice. If the data fails normality because the sample size is large the departure could be mild and the residual qq plots may look nearly linear. In that case failing the normality test is not important.

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