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### Exercise 19 from  Exercises in STK3100/4100 .
##################################

## Looking at the number of people with and without diabetes by gender:
n = matrix(c(377, 17869, 336, 20099), nrow=2)            #obs

#19d)   chi-squared test: 13.3 in Devore & Berk
e = matrix(c(336.3,17906.1,376.8,20059.8), nrow=2)       #Expected, if H0 true = there are no differences between gender
chi_test = sum((n-e)^2/e)
pchisq(chi_test, 1, lower.tail=FALSE)
#conclusion: 0.002058151  < 0.05. Reject H0. There are differences between gender

#19e)   CI for OR:
OR_est    <- n[1,1]*n[2,2]/(n[1,2]*n[2,1])     
varOR_est <- 1/n[1,1] + 1/n[1,2] + 1/n[2,1] + 1/n[2,2]
conf.int  <- OR_est*exp(c(-1,1)*1.96*sqrt(varOR_est))
#conclusion:   CI=(1.087970, 1.463981), and do not contain 1. Reject H0. There are differences between gender


#19g)   GLM:
sich = c(377, 336)
not_sick = c(17869, 20099)
gender <- c(1,0)
mod1 <- glm( cbind(sich,not_sick) ~ 1 + gender, family=binomial)
summary(mod1)
wald_test <-  (0.23274)^2/(0.07573)^2     #est^2/var
pchisq(wald_test, 1, lower.tail=FALSE)    #= 0.00211715
# conclusion: 0.00211715 < 0.05. Reject H0. There are differences between gender




#19h)    Looking at number of people with and without diabetes against BMI over and under 25.
n <- matrix(c(90, 21689, 623, 16274), nrow=2)            #obs
#calculate CI for OR
OR_est    <- n[1,1]*n[2,2]/(n[1,2]*n[2,1])     
varOR_est <- 1/n[1,1] + 1/n[1,2] + 1/n[2,1] + 1/n[2,2]
conf.int  <- OR_est*exp(c(-1,1)*1.96*sqrt(varOR_est))
#conclusion:  CI=(0.08681915,0.13533277), and do not contain 1. Reject H0. There are differences between for people over and under 25.