Review of survival models exponential constant hazard Weibull

• 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

• 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. ,

• Exercise: Use R to sketch this survival curve - take beta=5 and t 0=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 to estimate these parameters later…