Supplement

```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)

## Gam in Action

```{r cars}
library(gam)
data(kyphosis)
?kyphosis

General Additive Models allow us to model binary variables. Let us model the response variable Kyphosis which indicates whether or not the condition is present or not. Use Age, Number and Start as predictor variables.

```{r pressure} kyph1.gam = gam(Kyphosis ~ s(Age) + s(Number) + s(Start), family=binomial, data=kyphosis) print(summary(kyph1.gam))

par(mfrow=c(1,3)) plot(kyph1.gam, resid=T, se=T, rug=F, pch=19, col="red")

Perform a likelihood ratio test against the model with no terms. Is incorporating the three variables better than nothing?

```{r}
kyph0.gam = gam(Kyphosis ~ 1,
  family=binomial, data=kyphosis)

print(anova(kyph0.gam,kyph1.gam,test='Chi'))

It appears individual effects are not significant in presence of others. Remove Number and re fit the model

kyph2.gam = gam(Kyphosis ~ s(Age) + s(Start),
  family=binomial, data=kyphosis)
print(summary(kyph2.gam))

par(mfrow=c(1,2))
plot(kyph2.gam, rug=T, se=T)

Visualy, t is clear that the response does not behave linearly in terms of these predictors. Here we fit the corresponding GLM and test the comparison to the GAM.

kyph2.glm = glm(Kyphosis ~ Age + Start,
  family=binomial, data=kyphosis)
print(anova(kyph2.glm,kyph2.gam,test="Chi"))

We can use GAMs for prediction the same as we have done before:

kyph.new = data.frame(Age = c(84,85,86), Start=c(7,8,9))
print(predict(kyph2.gam,kyph.new,type="response"))

Another use of GAM is to transform predictors for use in linear models. Based on the plots for Age and Start it makes sense to include quadratic terms.

kyphosis$Age.sq = (kyphosis$Age - mean(kyphosis$Age))^2
kyphosis$Start.sq = (kyphosis$Start - mean(kyphosis$Start))^2
kyph3.glm = glm(Kyphosis ~ Age + Start + Age.sq + Start.sq,
  family=binomial, data=kyphosis)

print(summary(kyph3.glm))

print(anova(kyph2.glm,kyph3.glm,test="Chi"))