# R: test normality of residuals of linear model - which residuals to use

I would like to do a Shapiro Wilk's W test and Kolmogorov-Smirnov test on the residuals of a linear model to check for normality. I was just wondering what residuals should be used for this - the raw residuals, the Pearson residuals, studentized residuals or standardized residuals? For a Shapiro-Wilk's W test it appears that the results for the raw & Pearson residuals are identical but not for the others.

fit=lm(mpg ~ 1 + hp + wt, data=mtcars)
res1=residuals(fit,type="response")
res2=residuals(fit,type="pearson")
res3=rstudent(fit)
res4=rstandard(fit)
shapiro.test(res1) # W = 0.9279, p-value = 0.03427
shapiro.test(res2) # W = 0.9279, p-value = 0.03427
shapiro.test(res3) # W = 0.9058, p-value = 0.008722
shapiro.test(res4) # W = 0.9205, p-value = 0.02143


Same question for K-S, and also whether the residuals should be tested against a normal distribution (pnorm) as in

ks.test(res1, "pnorm") # D = 0.296, p-value = 0.005563


or a t-student distribution with n-k-2 degrees of freedom, as in

ks.test(res3, "pt",df=nrow(mtcars)-2-2)


Any advice perhaps? Also, what are recommended values for the test statistics W (>0.9?) and D in order for the distribution to be sufficiently close to normality and not affect your inference too much?

Finally, does this approach take into account the uncertainty in the fitted lm coefficients, or would function cumres() in package gof() be better in this respect?

cheers, Tom

• It is rare for such a test to have any point. Ask yourself what specific actions you would take if the residuals turned out to be "significantly" non-normal. Experience teaches you that it depends on how, and how much, they differ from normality. Neither of those is directly (or adequately) measured by SW, KS, or any other formal distribution test. For this work you want to employ exploratory graphics, not formal tests. The question of which residuals might be suitable for plotting still stands, but the remaining questions fall to the wayside as being irrelevant.
– whuber
Commented Oct 7, 2014 at 21:15
• Yes I've noticed that many statisticians advocate this position. But I would still like to check the test statistics of these tests (e.g. check if the value of Shapiro Wilks W is greater than 0.9). And I could always do a Box-Cox transformation or something like that to improve normality in case of large deviations. Plus my question was also partly conceptual - ie what would be the most correct way of doing of this, even if normality is not always that important in practice... Commented Oct 7, 2014 at 21:20

1. For an ordinary regression model (such as would be fitted by lm), there's no distinction between the first two residual types you consider; type="pearson" is relevant for non-Gaussian GLMs, but is the same as response for gaussian models.
4. You could overcome some of the issues in 2. and 3. (dependence in residuals as well as non-normality in standardized residuals) by simulation conditional on your design matrix ($\mathbf{X}$), meaning you could use whichever residuals you like (however you can't deal with the "answering an unhelpful question you already know the answer to" problem that way).