[R] Nonnormal Residuals and GAMs

[Simon Wood]

If you use GCV smoothness selection then, in the Gaussian case, the key
assumptions are constant variance and independence. As with linear
modelling, the normality assumption only comes in when you want to find
confidence intervals or p-values. (The GM Thm does not require normality 
btw. but I don't know if it has a penalized analogue).

With REML smoothness selection it's less clear (at least to me).

Beyond Gaussian the situation is much as it is with GLMs. The key
assumptions are independence and that the mean variance relationship is
correct. The theory of quasi-likelihood tells you that you can make
valid inference based only on specifying the mean-variance relationship
for the response, rather than the whole distribution, with the price
being a small loss of efficiency. It follows that getting the
distribution exactly right is of secondary importance.

It's also quite easy to be misled by normal qq plots of the deviance
residuals when you have low count data. For example, section 4 of
   http://opus.bath.ac.uk/27091/1/qq_gam_resub.pdf
shows a real example where the usual qq plots look awful, suggesting
massive zero inflation, but if you compute the correct reference
quantiles for the qq plot you find that there is nothing wrong and no
evidence of zero inflation.

best,
Simon

ps. in response to the follow up discussion: The default link depends on 
the family, rather than being a  gam (or glm) default. Eg the default is 
log for the Poisson, but identity for the Gaussian.


On 06/11/13 21:46, Collin Lynch wrote:
> Greetings, My question is more algorithmic than prectical.  What I am
> trying to determine is, are the GAM algorithms used in the mgcv package
> affected by nonnormally-distributed residuals?
>
> As I understand the theory of linear models the Gauss-Markov theorem
> guarantees that least-squares regression is optimal over all unbiased
> estimators iff the data meet the conditions linearity, homoscedasticity,
> independence, and normally-distributed residuals.  Absent the last
> requirement it is optimal but only over unbiased linear estimators.
>
> What I am trying to determine is whether or not it is necessary to check
> for normally-distributed errors in a GAM from mgcv.  I know that the
> unsmoothed terms, if any, will be fitted by ordinary least-squares but I
> am unsure whether the default Penalized Iteratively Reweighted Least
> Squares method used in the package is also based upon this assumption or
> falls under any analogue to the Gauss-Markov Theorem.
>
> Thank you in advance for any help.
>
> 	Sincrely,
> 	Collin Lynch.
>
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-- 
Simon Wood, Mathematical Science, University of Bath BA2 7AY UK
+44 (0)1225 386603               http://people.bath.ac.uk/sw283