Survival Analysis 1.Survival functions 2.Hazard Function 3.Cumulative hazard 4.Cox Proportional hazard model

Survival curve X r.v with cdf F(x), and pdf f(x) S(x)=Pr(X>x)=1-F(x) = If X is discrete S(x)=1 for x < 0 S(0)=1 if p(0)=0, S(inf) =0 S(x) is nonincreasing Survival functions

The Hazard Function (Conditional failure Rate) Recover S(x) from h(x) h(x)  x :Prob. of experiencing event in next  x. Shapes of the hazard function: 1. Increasing hazard: Natural aging of wear. 2. Decreasing hazard: Early failure, rare. 3. U shape. Infant diseases (early and late hazard) 4. Hump-shaped. After surgery 5. Flat, Constant hazard

Hazard function: Example: Weibull Distr Example: Weibull Distr X discrete: Example: Discrete Distribution

Mean Residua life and Median Life Variance: Median Lifetime: Median Lifetime: 50% percentile of the distribution of X Percentiles: Percentiles: (1-p)th quantile of the survival function Example:

Regression models for survival data: Accelerated Failure Time model X = Failure Time, Z=Vector of Covariates Y = log(X) =  +  ’Z +  W The error distribution W can be normal or extreme value

Cox Proportional hazard model h(x|z) = h 0 (x) exp(  ’z) = h 0 (x) c(  ’z) Also: h(x|z 1 )/h(x|z 2 ) = c(  ’z 1 )/c(  ’z 2 ) The Survival Function can be written in terms of a base survival function S o (x) S(x|z) = S o (x) c(  ’z)

Example: CAD (Coronary Artery Disease) Response: CAD: 0 Patient has no CAD 1 Patient has CAD Predictors: Age, Gender, Race, Smoking, BMI File CAD.sav

SOME DATA FROM CAD

Start End Dead Censored Dead Censored Dead Censored Right Censoring 8 Patients 4 dead 4 Censored

Start End Dead Censored Dead Censored Dead Censored Generalized Right Censoring X1X1 X4X4 X5X5 X7X7 C r2 C r3 C r6 C r8

Start End Dead Censored Dead Censored Dead Censored Generalized Right Censoring X1X1 X4X4 X5X5 X7X7 C r2 C r3 C r6 C r8

Time on Study End Dead Censored Dead Censored Dead Censored Lexis Diagram Generalized Right Censoring X1X1 X4X4 X5X5 X7X7 C r2 C r3 C r6 C r8