tbl <- cbind(expand.grid(Ulcer=0:1,Blood=0:1,Town=c("London","Manchester","Newcastle")),
             Freq = c(911 ,4578 ,579 ,4219 ,361 ,4532 ,246 ,3775 ,396 ,6598 ,219 ,5261)) 

attach(tbl)
 
fit1 <- glm(Freq ~ 1, family="poisson",data=tbl)   
fitU.B.T <- glm(Freq ~ Ulcer + Blood + Town, family="poisson",data=tbl)  
fitUB.T   <- glm(Freq ~Ulcer*Blood + Town , family="poisson",data=tbl)  
fitUB.UT   <- glm(Freq ~Ulcer*Blood + Ulcer*Town , family="poisson",data=tbl)  
fitUB.UT.BT   <- glm(Freq ~Ulcer*Blood + Ulcer*Town+ Blood*Town  , family="poisson",data=tbl)  
fitUBT   <- glm(Freq ~Ulcer*Blood*Town  , family="poisson",data=tbl)  
anova(fit1, fitU.B.T , fitUB.T , fitUB.UT, fitUB.UT.BT,fitUBT)

cbind(tbl,fitUB.UT.BT$fit,fitUBT$fit)

summary(fitUBT)
summary(fitUB.UT.BT)
exp(model.matrix(fitUB.UT.BT) %*% fitUB.UT.BT$coef)


# Use log-linear model to compute Blood OR on Ulcer in London

# OR = log(P(U = 1 | B = 1, London)) -  log(P(U = 0 | B = 1, London)) -  log(P(U = 1 | B = 0, London)) + log(P(U = 0 | B = 0, London)) 
#    = log(P(U = 1,  B = 1, London)) -  log(P(U = 0,  B = 1, London)) -  log(P(U = 1,  B = 0, London)) + log(P(U = 0,  B = 0, London)) 
#    = lambda^{UB}_{Ulcers=1, B=1}

#  Fit logistic regression to data

fit.logistic.1   <- glm(Ulcer ~ Blood + Town , family="binomial",weight = Freq)  
fit.logistic.2   <- glm(Ulcer ~ Blood * Town , family="binomial",weight = Freq)  

anova(fit.logistic.2)
summary(fit.logistic.1)
summary(fit.logistic.2)

model.matrix(fit.logistic.2)
cbind(tbl,fit.logistic.2$fit)

4578 / (4578 + 911)
5261 / (5261 + 219)

(lp.2 <- model.matrix(fit.logistic.2) %*% fit.logistic.2$coef)
exp(lp.2) / (1 + exp(lp.2))

lp.2[3] - lp.2[1]
log((911*4219)/(579*4578))
model.matrix(fit.logistic.2)
sqrt(summary(fit.logistic.2)$cov.unscaled[2,2])