Re: [R] Testing for normality of residuals in a regression model

[Spencer Graves]

     OK, I'll expose myself: 

     I tend to do normal probability plots of residuals (usely deletion 
/ studentized residuals as described by Venables and Ripley in Modern 
Applied Statistics with S, 4th ed, MASS4).  If the plots look strange, I 
do something.  I'll check apparent outliers for coding and data entry 
errors, and I often delete those points from the analysis even if I 
can't find a reason why.  Robust regression will usually handle this 
type of problem, and I am gradually migrating to increasing use of 
robust regression, especially the procedures recommended by MASS4.  . 

     However, I recently encountered a situation that would be masked 
by standard use of robust regression without examining residual plots:  
A normal probability plot looked like three parallel straight lines with 
gaps, suggesting a mixture of 3 normal distributions with different 
means and a common standard deviation.  Further investigation revealed 
that an important 3-level explanatory variable that had been miscoded.  
When this was corrected, that variable entered the model and the gaps in 
the normal plot disappeared. 

     I tend NOT to use tests of normality for the reasons Andy 
mentioned.  Instead, I do various kinds of diagnostic plots and modify 
my model or investigate the data in response to what I see. 

     Comments?
     hope this helps.  spencer graves
Liaw, Andy wrote:
Let's see if I can get my stat 101 straight:
We learned that linear regression has a set of assumptions:
1. Linearity of the relationship between X and y.
2. Independence of errors.
3. Homoscedasticity (equal error variance).
4. Normality of errors.
Now, we should ask:  Why are they needed?  Can we get away with less?  What
if some of them are not met?
It should be clear why we need #1.
Without #2, I believe the least squares estimator is still unbias, but the
usual estimate of SEs for the coefficients are wrong, so the t-tests are
wrong.
Without #3, the coefficients are, again, still unbiased, but not as
efficient as can be.  Interval estimates for the prediction will surely be
wrong.
Without #4, well, it depends.  If the residual DF is sufficiently large, the
t-tests are still valid because of CLT.  You do need normality if you have
small residual DF.
The problem with normality tests, I believe, is that they usually have
fairly low power at small sample sizes, so that doesn't quite help.  There's
no free lunch:  A normality test with good power will usually have good
power against a fairly narrow class of alternatives, and almost no power
against others (directional test).  How do you decide what to use?
Has anyone seen a data set where the normality test on the residuals is
crucial in coming up with appriate analysis?
Cheers,
Andy
 

From: Federico Gherardini
Berton Gunter wrote:
   

Exactly! My point is that normality tests are useless for 
         

this purpose for
   

reasons that are beyond what I can take up here. 

         

Thanks for your suggestions, I undesrtand that! Could you 
possibly give 
me some (not too complicated!)
links so that I can investigate this matter further?

Cheers,
Federico
   

Hints: Balanced designs are
robust to non-normality; independence (especially 
         

"clustering" of subjects
   

due to systematic effects), not normality is usually the 
         

biggest real
   

statistical problem; hypothesis tests will always reject 
         

when samples are
   

large -- so what!; "trust" refers to prediction validity 
         

which has to do
   

with study design and the validity/representativeness of 
         

the current data to
   

future. 

I know that all the stats 101 tests say to test for 
         

normality, but they're
   

full of baloney!
Of course, this is "free" advice -- so caveat emptor!
Cheers,
Bert
    

         

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Spencer Graves, PhD, Senior Development Engineer
O:  (408)938-4420;  mobile:  (408)655-4567
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