Normally distributed errors assumption is violated (multiple regression)

#1
Hi guys,

I am testing the normally distributed errors assumptions for multiple regression with a visual inspection using a P-Plot plot. Now, I have the suspicion that the errors are not normally distributed (see image below). I have two questions about this:
  • Is there any way to test this statistically and not visually?
  • What are my options when the assumption turns out to be violated? What is the best way to proceed?

Schermafbeelding 2021-05-13 om 09.14.29.png
 

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Karabiner

TS Contributor
#2
The question is: are the residuals normally distributed in the population?
Although widely recommended here, descriptive statistics (including such plots)
from the sample IMHO cannot really answer that question.

On the other hand, inferential tests of normality (there are quite many of them)
are mostly useless. Either sample size is small, and the tests are therefore not
very sensitive (small statistical power). Or, the sample size is large, and even
small deviations from normality become "statistically significant", although they
do not practically matter. So the tests often provide no information about
what to do.

Fortunately, all this normality stuff is usually irrelevant, especially if your sample
size is large enough (about n > 30, or n > 50). The central limit theorem then
makes sure that non-normality of the residuals does not affect the realiabilty
of the F-test. Seemingly, your sample size is large enough.

With kind regards

Karabiner
 
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hlsmith

Less is more. Stay pure. Stay poor.
#5
Yeah, I got this new thing. I have been not typing 'not' in my posts by accident. It is a weird middle age thing I suppose. It will get my in trouble yet. I usually have to reread my emails and posts to check for my brain leaving out words or exchanging words with the same first letter. I should add a post script on all of my messages warning people about this, but I would likely mess that up and the results would be the he always tells the truth and he always lies paradox.

I agree that I would 'not' have concerns with the above figure.

P.S., @shannahw - I like your avatar image. Reminds me of of art from Angie Pickman in Kansas.
 

noetsi

Fortran must die
#6
I think the consensus is with enough cases (say 100) normality does not matter. Some worry about outliers, not normality. They can move the regression line in some cases. But qq plots do not get at the issue of outliers, there are other tests like dfbeta that do.

I would be interested to know why Karabiner thinks QQ plots are not a good way to determine normality (they are highly recommended).
 
#7
@Karabiner

Thank you for your elaborate answer. So, let me resume the main message:
  • There is no method available to properly test the assumption of normality of residuals. This assumption only holds as a theoretical statement about the characteristics to which the population should adhere in order for statistical tests on a sample to be validly generalized to the population.
  • So, descriptive tests (like the P-P and Q-Q plot) and inferential tests (like the Kolmogorov-Smirnov and Shapiro-Wilk) are mostly useless.
You basically state that the only assumption that we can make, is that, when the sample size is big enough, the Central Limit Theorem applies. I am performing moderated mediation with one moderator and one mediator. Therefore, I have 3 predictor variables, but I also include 6 covariates in the model. From a rule of thumb, I am using that the sample size should be about 20 observations per predictor. I have 120 observations, but, when I include all predictors and covariates, I theoretize that I need 9 x 20 = 180 observations in order for the Central Limit Theorem to apply. Based on this information, what would be your approach on making a statement about this assumption?

I am also curious on your apply to @noetsi about the Q-Q plots. I have also run these as a further testing of the assumption (see image).

Schermafbeelding 2021-05-17 om 11.14.36.png

@hlsmith - Thank you. It is an interpretation of Watership Down :). What's yours?
 

Karabiner

TS Contributor
#8
I would be interested to know why Karabiner thinks QQ plots are not a good way to determine normality (they are highly recommended).
They are sample descriptions. I cannot simply judge from the difference beteen two sample means whether there is a statisticall significant mean difference in the population. I cannot simply judge from a sample q-q-plot whether the residuals are distributed normally in the population.

With kind regards

Karabiner
 

hlsmith

Less is more. Stay pure. Stay poor.
#9
I haven't read all that is above, but unless I think there are issues I am typically content enough looking at a histogram and qqplot of residuals and any other generated output. I don't get caught up in normality tests, but I acknowledge that I did when I first started out.

Well it looks like everything is still going well in the warren in that WD illustration. Mine is a picture my daughter made in art class of her in a spaceship visiting an alien.
 

hlsmith

Less is more. Stay pure. Stay poor.
#11
Great to hear and welcome to the forum! Side note, if you have ever seen the movie Donnie Darko, they were going to originally use Watership Down as the book covered in his class, but deviated to the fictitious Cellar probably to incorporate the nihism/chaos motif.