1. Survival Analysis 1 Always be contented, be grateful, be understanding and be compassionate.

2. Survival Analysis 2 Semiparametric Proportional Hazards Regression (Part I)

3. Survival Analysis 3 Abbreviated Outline The proportional hazards regression model provides a method for Exploring the association of covariates with failure rates, Studying the effect of a primary covariate while adjusting for other variables. The model is neither fully parametric nor fully nonparametric. Inference for the model is based on a likelihood type function that, in censored data, only approximates the probability function of any observed set of values.

4. Survival Analysis 4 Identification of Prognostic Factors Prognosis: the prediction of the future of a patient with respect to duration, course, and outcome of a disease Prognosis plays an important role in medical practice.

5. Survival Analysis 5 Identification of Prognostic Factors Many medical charts contain a large number of patient characteristics: Demographic variables, Physiological variables, Factors associated with lifestyle. A compact summary of the data is useful in revealing the relationship of these characteristics to prognosis.

6. Survival Analysis 6 Example: PBC Primary biliary cirrhosis (PBC) is a rare but fatal chronic liver disease. The bile ducts within the liver become inflamed and are destroyed. A randomized trials conducted by Mayo Clinic during 1974 to 1984 to compare the drug D-penicillamine (DPCA) with a placebo; 312 patients per group.

7. Survival Analysis 7 Example: PBC Data:

8. Survival Analysis 8 Regression Modeling for Survival Data As in the linear regression modeling, regression models for survival data can be used to examine the dependence of survival time on other covariates

9. Survival Analysis 9 Regression Modeling for Survival Data We model the hazard function directly instead of the survival time since our interest centers on the risk of failure. Data:

10. Survival Analysis 10 Cox Regression Model h(y|x): the hazard rate at time y for an individual with risk vector x.

11. Survival Analysis 11 Cox Regression Model

12. Survival Analysis 12 Baseline Hazard Function The baseline hazard function, h 0, characterizes the dependence of the hazard rate on time. It is the hazard function for an individual with x = (0,…,0)’.

13. Survival Analysis 13 Prognostic Index

14. Survival Analysis 14 Cox Regression Model Semiparametric Proportional hazard

15. Survival Analysis 15 Interpretation of Regression Coefficients Single covariate: Categorical Continuous Multiple covariates

16. Survival Analysis 16 Single Categorical Coefficient Consider a single dichotomous covariate x, coded 0 and 1 The hazard ratio, also known as relative risk, is

17. Survival Analysis 17 Example: PBC In the DPCA group: 125 out of 312 patients had died. (right-) censored observations: 11 deaths were not attributable to PBC 8 were lost to follow up 19 had undergone liver transplantation 187 were survived at the end of study

18. Survival Analysis 18 Example: PBC The covariate X is coded as 0 for DPCA and as 1 for placebo. Therefore, the baseline hazard function h 0 is the hazard function for DPCA Interpretation: Estimate of e^  is 0.944 Its 95% C.I. is (0.665, 1.342)

19. Survival Analysis 19 Single Continuous Covariate The interpretation of the coefficient for a continuous covariate is similar. For a continuous covariate, the hazard ratio should be reported for a clinically interesting unit of change.

20. Survival Analysis 20 Example: PBC Consider X = age (in days)  is estimated as 1.095x10^-4 The hazard ratio for a 1-day change in age The hazard ratio for a 5-year change in age

21. Survival Analysis 21 Multiple Covariates Similar to the linear regression model, each coefficient represents the effect of the corresponding covariate after adjusting for other covariate effects. Such adjustment is called Analysis of covariance in statistical applications Control of confounding in clinical and epidemiological investigations

22. Survival Analysis 22 Example: PBC

23. Survival Analysis 23 Example: PBC

24. Survival Analysis 24 Example: PBC