This is my first post & I'm fairly new to GAMs; apologies.

I ran a series of 16 generalized additive negative binomial models (gam, family=nb, mgcv package) with increasing complexity that modeled the abundances of 12 fish species (independently) as functions of several environmental metrics (and their combinations):

It is a large dataset; a couple examples of models:

m4(s): ccount ~ s(sal, k=6, bs="tp")

m12(ost): ccount ~ s(do, k=6, bs="tp") + s(sal, k=6, bs="tp") + s(temp, k=6, bs="tp")

I conducted a bunch of model checks, then plotted deviance explained, edf, and AIC for each model (in order from least to most complex, approximately) to assist with model selection for each species.

DEVIANCE EXPLAINED

EDF

AIC

It appears that deviance explained is simply the inverse of AIC, and AIC does not seem to be penalized by variation in the model's EDF. I plotted deviance explained vs AIC and was surprised that they were almost perfectly correlated. This seemed counter intuitive.

#AIC vs. Dev.Expl

Question/Commment 1 - I found three different ways to call AIC:

1) model$aic
2) AIC(model)
3) extractAIC(model)

(3) is documented only for parametric models (i.e., not GAM). It's unclear whether (1) or (2) is more appropriate for gam objects based on the mgcv pkg? Both gave practically the same result (see below), but appeared to use different df (edf vs reference df). I assumed (1) would be the best; however, it was unclear what df were used, and there is documentation of logLik used with (2) for gam objects. It was unclear if this is already built into (1)? https://stat.ethz.ch/R-manual/R-devel/library/mgcv/html/logLik.gam.html

Question2 - is it reasonable to expect deviance explained and AIC to be this strongly correlated, regardless of model complexity? I didn't think so. Are improvements in dev.expl large enough that changes in edf are inconsequential? Is deviance explained already penalized for the edf in the model?

Thanks for any help.

Here is a simple reproducable example from mtcars showing the same pattern:

library(mgcv)

# 3 GAM models
b1 <-gam(mpg~s(hp), data=mtcars)
b2 <-gam(mpg~s(wt), data=mtcars)
b3 <-gam(mpg~s(hp)+s(wt), data=mtcars)

# AIC1
a1.1 <- b1$aic
a1.2 <- b2$aic
a1.3 <- b3$aic
a1 <- c(a1.1,a1.2,a1.3)

# AIC2
a2.1 <- AIC(b1)
a2.2 <- AIC(b2)
a2.3 <- AIC(b3)
a2 <- c(a2.1, a2.2, a2.3)

# dev.explained
d1 <- summary(b1)$dev.expl
d2 <- summary(b2)$dev.expl
d3 <- summary(b3)$dev.expl
d <- c(d1,d2,d3)

par(mfrow=c(2,2))
plot(a1~d, type=c("b"), xlab="deviance explained", ylab="model$aic")
plot(a1~a2, col="black", type="b", lwd=1, cex=1, xlab = "AIC(model)", ylab="model$aic")

MTCARS GAM Example: AIC vs Dev.Expl & model$aic vs AIC(model)