[R] Theta in Negative binomial GAM

[Eva Maria Leunissen]

Hi, I'm fitting a negative binomial GAM (using mgcv) to my data using
family=nb() so theta is estimated during the fitting process. When I then
extract this theta from the model and refit the same model with
family=negbin(theta) it gives a much lower AIC. I know using AIC to compare
negative binomial models should be done with caution (
http://r.789695.n4.nabble.com/How-to-compare-GLM-and-GAM-models-tt827923.html#a827926)
but approximately it is ok. My questions are:

Is it better to do model selection with nb() or with negbin and a known
theta. if the latter how do you know what theta is?

can you compare models using AIC if the theta is different for each model?

Thanks in advance, any help is much appreciated

Kind regards,

Eva Leunissen

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[Simon Wood]

nb() will default to "REML" smoothing parameter estimation, while
negbin() will default to "UBRE" unless you use the 'method="REML"'
option to 'gam'. Using UBRE in place of REML may lead to differences in
model fit, and will also mean that the AIC is not corrected for
smoothing parameter uncertainty (this correction increases the AIC). You
also expect AIC to drop by 2 or so, because negbin is treating theta as
fixed.

So...

m <- gam(...,family=nb())

should give similar AIC values about 2 higher than

th1 <- m$family$getTheta(TRUE)
m1 <- gam(...,family=negbin(th1),method="REML")

If you don't know theta it is better to estimate it.

best,
Simon

ps. Tedious details of AIC smoothing uncertainty correction are in
sections 4 and 5 of this:
http://amstat.tandfonline.com/doi/pdf/10.1080/01621459.2016.1180986

--
Simon Wood, School of Mathematics, University of Bristol BS8 1TW UK
+44 (0)117 33 18273 http://www.maths.bris.ac.uk/~sw15190