Survival Analysis Notes

Survival Analysis Notes - Notes on Survival analysis...

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Notes on Survival analysis Develop models for survival times Y . f ( y ) is its pdf and F ( y ) its distribution function. Survivor function : probability of survival beyond time y or S ( y ) = P [ Y > y ] = 1 - F ( y ) . Hazard function : probability of death in an infinitesimal interval given survival to time y We showed that the hazard h ( y ) satisfies h ( y ) = f ( y ) S ( y ) = - d dy [ log ( S ( y ))] May think of the survival function in terms of cummulative hazard H ( y ) . UNM

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S ( y ) = exp ( - H ( y )) where H ( y ) = Z y 0 h ( t ) dt . Inference could be related to quantiles of F ( y ) (or S ( y ) ) q-quantile y ( q ) solution to the equation F ( y ( q )) = q or S ( y ( q )) = 1 - q Simplest parametric model is f ( y | θ ) = exp ( - θ y ); θ > 0 , y > 0 In particular its hazard function is simply θ (does not depend on y ). UNM
Another one is the Weibull distribution. f ( y | λ, φ ) = λφ y λ - 1 exp ( - φ y λ ); y , λ, φ > 0 More flexible than exponential. In particular, its hazard function is h ( y | λ, φ ) = λφ y λ - 1 which is an ”accelerated failure time model”. It can be shown that for this distribution log ( - log ( S ( y | λ, φ ))) = log ( φ ) + λ log ( y ) A plot of log ( y ) values vs. empirical values of log ( - log ( S ( y | λ, φ ))) may suggest if the Weibull dist. is adequate. UNM

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Non-parametric estimation Empirical survivor function (without censoring). ˜ S ( y ) = number of subjects with time y total no. of subjects Step/decreasing function that starts at 1.
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