offsets_survival.R

# simple simulation of survival data for use in regression analysis
# shows that the effect of a risk factor on an outcome must be estimated using an offset term rather than adjusting for the follow-up duration

# sample size
n <- 1e5

# simulate random binary exposure with 30% prevalence
binary_exposure <- rbinom(n, 1, 0.3)

# simulate total observation duration, associated with binary exposure
observation <- sample(100:400, n, T, prob = (400:100/1000)^4)
observation <- observation * ifelse(binary_exposure == 1, 1.3, 1)

# risk of outcome
baseline_risk <- 1/500 # baseline risk per time increment
risk_ratio <- 2 # risk associated with exposure
risk <- ifelse(binary_exposure == 1, baseline_risk * risk_ratio, baseline_risk) # individual risk

# simulate first event, censor date, and whether event was observed 
first_event <- rgeom(length(risk), risk) + 1
censored_observation <- pmin(observation, first_event) # duration until censor date
event_observed <- as.integer(first_event <= observation)

# correct model offsetting censored observation
m1 <- glm(event_observed ~ binary_exposure + offset(log(censored_observation)), family = 'poisson')
exp(coef(m1)[2]) # should approximate risk ratio

# model offsetting for full observation
m2 <- glm(event_observed ~ binary_exposure + offset(log(observation)), family = 'poisson')
exp(coef(m2)[2]) # does not approximate risk ratio

# model adjusting for censored observation
m3 <- glm(event_observed ~ binary_exposure + censored_observation, family = 'poisson')
exp(coef(m3)[2]) # does not approximate risk ratio