1. Review of survival models:
exponential (constant hazard ):
Weibull (power hazard):
Consider the following “cure model”, in which S(Y) does not approach 0 at y gets larger, but instead approaches p , the proportion “cured” (or at least “not relapsed” or …). There are three ways to look at this model:
piecewise model
stepwise exponential model
Gompertz model

2. Piecewise model - this assumes we can have a “mixture” of two groups of patients: one that follows a regular survival curve (like exponential) and a second one in which a cure occurs (survival probability 0≤p ≤1 after a particular time).
Exercise: Use R to sketch the above survival curve with p=.25 and beta=5 for example.
Note that any survival curve could be substituted for the exponential part of the piecewise function. We’ll learn later how to estimate the parameters in this model.
Stepwise exponential model - this assumes different exponential survival models on different time intervals. Another way to say this is that the hazard function takes on different constants over different time intervals; e.g.,

3. Use R to sketch various Gompertz models
Exercise: Use R to sketch this survival curve - take beta=5 and t0=12. As with the piecewise model, we’ll show later how to estimate the parameters of this model too.
The Gompertz model looks a little complex at first, since it’s a two parameter model, but it has similarities with models we’ve alread considered.
Notes:
As when  < 0
As so you can see that the exponential is a special case of this Gompertz model with large beta.
Use R to sketch various Gompertz models
We’ll show how to estimate these parameters later…