Transcription of Parametric Survival Models - Princeton University
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Parametric Survival Models - Princeton University
Parametric Survival ModelsGerm an Rodr 2001; revised Spring 2005, Summer 2010We consider briefly the analysis of Survival data when one is willing toassume a Parametric form for the distribution of Survival Survival NotationLetTdenote a continuous non-negative random variable representing sur-vival time, with probability density function (pdf)f(t) and cumulative dis-tribution function (cdf)F(t) = Pr{T t}. We focus on thesurvival func-tionS(t) = Pr{T > t}, the probability of being alive att, and the hazardfunction (t) =f(t)/S(t). Let (t) = t0 (u)dudenote the cumulative (orintegrated) hazard and recall thatS(t) = exp{ (t)}.Any distribution defined fort [0, ) can serve as a Survival can also draft into service distributions defined fory ( , ) byconsideringt= exp{y}, so thaty= logt.]
The cumulative hazard is ( t) = ( t)p, the survivor function is S(t) = expf ( t)pg, and the hazard is (t) = pptp 1: The log of the Weibull hazard is a linear function of log time with constant plog + logpand slope p 1. Thus, the hazard is rising if p>1, constant if p= 1, and declining if p<1.
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