What are the myths associated with linear regression, data transformations? I have been encountering many assumptions associated with linear regression (especially ordinary least squares regression) which are untrue or unnecessary. For example:

*

*independent variables must have a Gaussian distribution

*outliers are the points either above or below the upper or lower whiskers correspondingly (employing Boxplot terminology)

*and that the sole purpose of transformations is to bring a distribution close to normal in order to suit the model.

I would like to know what are the myths that are generally taken for facts/assumptions about linear regression, especially concerning associated nonlinear transformations and distributional assumptions.  How did such myths come about?

Myths:

*

*The normality of residuals (and possibly other assumptions of the model) should be tested with a formal hypothesis test, such as the Shapiro-Wilk test.

*A small $p$-value of such tests indicates that the model is invalid.

Facts:

*

*Formal test of normality (and of other assumptions such as homoskedasticity) do not answer the relevant questions and if they are used to guide subsequent actions, can distort the operating characteristic of the models (e.g. inflating type 1 errors, change distribution of $p$-values under the null etc.).

*A "significant" Shapiro-Wilk test of the residuals just indicates some degree of incompatibiltiy with a normal distribution. It does not say that the (inevitable) deviation from a normal distribution is meaningful or impactful concerning the operating characteristics of the model. Some aspects - e.g. prediction intervals - are more sensitive with regards to the distribution of the errors than others. The $t$-test of the coefficients are reasonably robust (with regards to type 1 errors), for example. Whether or not the deviation of the residuals from a normal distribution is worrysome depends on a number of things: the goal of the analysis, the sample size, the degree of deviation, and more.

