Can someone please explain the difference between 'Central Exposed to risk' and 'Exposed to risk'. I know that the Central Exposed to risk is used in Poisson estimation and the other in Binomial. Also 'Exposed to Risk' = Central Exposed to Risk + 1/2dx But can someone please explain what do these terms mean in words? Thanks
The difference is pretty simple to understand mathematically, See since the INITIAL exposed to risk was developed within the binomial framework, the lives who die have to be counted as exposed to risk until the end of the year of age (not the investigation), whereas the central exposed to risk is developed within the poisson framework, it only measures the total time under observation (the difference being if a life dies, the life is only counted as at risk until the moment of death). So to summarise if we let ai be the latest of the time of entry into observation, the date of attaining age label x, or the beginning of the investigation, and let bi be the earliest of death, date of losing age label x or the end of the investigation, we see that the initial exposed to risk is, Ex = Sum over all lives (1-ai) - Sum over LIVES WHO DO NOT DIE (1-bi) Whereas the central exposed to risk is ExC = Sum over all lives (bi-ai) Hopefully this should make some sense, intuitivley the initial exposed to risk really has no meaning, can only really be understood interms of the above equation. The relationship between the two by assuming that on average deaths occur at age x+1/2. See by making this assumption we know that the length of time we will have to "add on" in respect of each death to the CENTRAL exposed to risk will be 1/2 for each death, since it has to be counted as exposed to risk until the end of the year of age. So this is where the Ex = 1/2dx + ExC comes from.
Cheers DevonMatthews, thats a good explanation. So the key point is that Exposed to risk assumes that if a life is dead during the year, the person dies at the end of the year of age. Where as in Central Exposed to risk the actual time of death is used. Do we use either of these in the two state model to estimate parameters?
Yes, you have the idea, the person doesn't "Die" at the end of the year, they are counted as EXPOSED TO RISK until the end of the year of age, the central exposed to risk only measures the exact time the population is exposed to risk. And for the Two state model, the estimator of the waiting time is E[V]. This is the expected value of the waiting time. It's realisation is ExC. As far as I know Ex can be used in the two state model if it's all that can be calculated from the data available (ie. exact times of death are unknown), but it requires the adjustment we have already discussed.
