#           ***** Problem S8 *****

birth<-read.table("http://www.uio.no/studier/emner/matnat/math/STK3100/data/Birth.txt",header=T,row.names=NULL)

#models with different numer of covariates:
mod10 <- glm(children~age+I(age^2) ,family=poisson,data=birth)                   # canonical link: log
mod11 <- glm(children~age          ,family=poisson,data=birth)                   # canonical link: log
mod12 <- glm(children~1            ,family=poisson,data=birth)                   # canonical link: log

summary(mod10)      #beta_0, beta_1, beta_2
summary(mod11)      #beta_0, beta_1
summary(mod12)      #beta_0


#a) test H: beta_2=0
# likelihood ratio test
anova.glm(mod11,mod10)          # look under the variable called the "Deviance"
1-pchisq(0.19601,1)             # p-value

# Wald test
b<-mod10$coeff                  # beta_hat
C1<-c(0,0,1)                    # testing for the last beta
W1<-t(C1%*%b)%*%solve(t(C1)%*%vcov(mod10)%*%C1)%*%(C1%*%b) # Wald statistic
1-pchisq(W1,1)                  # p-value


#b) test H: beta_1=beta_2=0

# likelihood ratio test
anova.glm(mod12,mod10)          # look under the variable called the "Deviance"
1-pchisq(29.604,2)              # p-value

C2<-cbind(c(0,0),diag(c(1,1)))  # testing for the last two betas
W2<-t(C2%*%b)%*%solve(C2%*%vcov(mod10)%*%t(C2))%*%(C2%*%b) # Wald statistic
1-pchisq(W2,2)                  # p-value