[R] GLM vs GAM

[Simon Wood]

> I am wondering whether anyone could explain what'd be the difference between running a 'generalized additive regression' versus 'generalized linear regression' with splines.
The smooth terms in mgcv::gam are represented using *penalized* 
regression splines, with the degree of penalization selected as part of 
fitting. Using glm with ns the smooth terms are represented using 
unpenalized regression splines with the smoothness controlled by how 
many knots you chose (none in your example, so the ns terms were 
actually straight lines).

best,
Simon


> Are they same models theoretically? My apologies if this is a silly question. Any comments or direction to references will be highly appreciated.
>
> Thanks in advance,
>
> Ehsan
>
>
> #####################
> set.seed(545)
> require(mgcv)
> n <- 200
> x1 <- c(rnorm(n), 1+rnorm(n))
> x2 <- sqrt(c(rnorm(n,4),rnorm(n,6)))
> y <- c(rep(0,n), rep(1,n))
> #####################
> # GAM version
> #####################
> r1 <- gam(y~s(x1, bs = "cr")+s(x2, bs = "cr"),family=binomial)
> pr1 <- predict(r1, type='response')
> summary(pr1)
> hist(pr1)
> #####################
> # GLM version
> #####################
> r2 <- glm(y~ns(x1)+ns(x2),family=binomial)
> pr2 <- predict(r2, type='response')
> summary(pr2)
> hist(pr2)
> #####################
> # Results
> #####################
>> summary(pr1)
>       Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
> 0.0000394 0.0550200 0.5027000 0.5000000 0.9322000 1.0000000
>> summary(pr2)
>       Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
> 0.0000403 0.0573300 0.5229000 0.5000000 0.9159000 0.9992000 		 	   		
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-- 
Simon Wood, Mathematical Science, University of Bath BA2 7AY UK
+44 (0)1225 386603               http://people.bath.ac.uk/sw283