总结outline： 🔗
- model time? e.g., log T = Z/linear predictor –> censoring/ missing data problem. the out come time-to-event is not always observed due to censoring/competing risk.
- Model rate: the aggregated data: known the total number of years and the total number of events
- why rate works? the rate param captures the instantaneous occurrence of the outcome at any given time.（count of success = 每个瞬时时间点上的发生数量，乘上观测总时间）
- while risk param is the probability of an event occurring within a specific time period.
- risk-rate connection:
- In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate(?) and independently of the time since the last event.
- Poisson model
- The likelihood: D ~ Poisson(lambda*Y), where log(lambda) = something (the log-linear model), then parametrize log(lambda) in terms of alpha and beta. substitute the model into the PDF to get the likelihood.
2024-06-09更新 🔗
- 傻傻分不清到底哪个是constant –> rate is constant（instantaneous occurrence of the outcome at any given time). 我们把总时间Y划分为N个小bin，每个小bin为一个单位unit h, that is, Y = Nh. h足够小的时候，每一个小bin只有一个事件发生，则每个小bin都是一个Bern(pi),–> 每个小bin的expected number = pi(Bern dist property). mu = pi(risk) (1),
- 在pois dist,我们有 mu = lambda(rate)*h(2)
- 这样结合(1)和 (2)，我们有了rate-risk connection: mu = pi(risk)=lambda(rate)h
- 然后我们就可以算survival probability啦：survival up to time T = (1 - risk)^N [T时间内一共有N个bin]= (1-lambda*h)^N –> approx. exp(-lambda*T)其中，lambda*T 是cumulative hazard. surv prob(T) = (1-lambda*h)^N = Aexp(-lambda*T)
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