Multiple linear regression, backward selection : Normality of the residuals?

Question

I need to create a Multiple Linear regression model on those data explaining max03 T9 T12 T15 Ne9 Ne12 Ne15 Vx9 Vx12 Vx15 maxO3v

!My data 1

My first intuition was to make a backward selection :

attach(ozone)
res <- lm(maxO3~T9+T12+T15+Ne9+Ne12+Ne15+Vx9+Vx12+Vx15+maxO3v)
shapiro.test(res$residuals)

data: res$residuals

W = 0.9682, p-value = 0.008945

But the first full model return non-normal residuals.

Is it okay to continue doing a backward selection (AIC criterion)?

I don't think it [non-normal residuals] has an impact on that sort of selection, but I can't find a definite answer to that question.

If I keep doing the backward selection process

[...]
res <- lm(maxO3~T12+Ne9+Vx15+maxO3v)
drop1(res)

summary(res)
shapiro.test(res$residuals)

Shapiro-Wilk normality test

data: res$residuals W = 0.9622, p-value = 0.002946

My residuals aren't normal at the end ...

Answer 1

Normality is relevant only when you do inference. Using AIC as a criteria does not require normality. Saying that I think it is better you transform the dependent variable using e.g. BoxCox transformations to get normality before proceeding with any kind of variable selection methods

Answer 2

Be careful of using only the Shapiro test for determining if the error follow a normal distribution. I led to bad results. Here is a full explanation of this point.

When I see errors, I always plot an histogram and perform a Q-Q plot. Basically Q-Q plot shows how a distribution would be against what you have, if the theoretical dist is far away from what you have, it don't follow the distribution you are testing.

You can use Q-Q plot for testing several distributions, one of them could be the "normal"

More points are around the line, more likely is your data normal (in this case)

Here is the R code for plotting this curve:

qqplot(res$residuals) # Following your example

Produces a plot like this one: