Cox Proportional Hazards Model Marlen Roberts & Heyang Liao

1. Cox Proportional Hazards Model Marlen Roberts & Heyang Liao

2. Outline • Introduction to Survival analysis • Comparison with other models • Survival function • Cox regression

3. Survival Analysis • Study time it takes for events to occur • “survival” of new businesses over time • “survival” of machine parts under pressure • Primarily used in biological and medical research • Also widely used in social, economic & engineering research • Censored Data Values • When the study ends some events may not have occurred Source: Wikipedia.org, StatSoft

4. Comparison with Other Methods Source: Intro. to Survival Analysis by Brian F. Gage, Oct 19 2004, Dr. Westfall

5. Methods • Univariate • Parametric models: normal, lognormal, exponential, Weibull, … • Nonparametricmodel: Kaplan-Meier • Regression • Parametric conditional distribution models: normal, lognormal, exponential, Weibull, … • Semiparametric conditional distribution model: Cox proportional hazard

6. Survival function Source: An Introduction to Survival Analysis by Maarten L. Buis

7. Rossi Dataset • Study of recidivism of 432 male prisoners • Observed for 1 year after being released from prison • Information includes: • week, arrest(0/1), fin aid(yes/no), age, race, married etc • Question – how do these variables predict the time of arresting Source: Cox Proportional-Hazards Regression for Survival Data, by John Fox

8. Rossi Dataset • Snippet of the dataset:

9. Survival Curve • Step function • Time interval • Start at 1.0

10. Survival Probability Distribution • 3 more ways to present the time-to-event probability distribution • Cumulative distribution Function (cdf) • Probability Density Function (pdf) • Hazard Function Source: An Introduction to Survival Analysis by Maarten L. Buis

11. Cumulative Probability

12. Probability Density Function

13. Hazard Function

14. Cox regression

15. Assumption #1 - Semi-Parametric

16. Assumption #1 - Semi-Parametric

17. Assumption #1 - Semi-Parametric

18. Assumption #2 – Proportional Hazard

19. Alternate View Coefficient Constant relationship b/w x and y Source: http://data.princeton.edu/wws509/notes/c7.pdf (p. 12)

20. Connection to Survival Function Source: http://data.princeton.edu/wws509/notes/c7.pdf (p. 12)

21. Connection to Survival Function

22. Interpretation of the model 1 year ↑ in age  -0.07161 ↓ in expected log of HR, holding other predictors constant For age: exp(-0.07161) = 0.9309. 1 year ↑ in age  6.91%(=1-0.9309) ↓ in Hazard • 22

23. Interpretation of the model Marginal significant Not significant -coeff. is not significant different from 0 Conf. Interval Incl. 1 Age is significantly negatively related to risk of incarceration! • 23

24. Partial likelihood • Find the value of β that best fit the data • Time factor is irrelevant  partial • Joint density function for subjects’ rank in terms of event order

25. Partial likelihood

26. Partial likelihood

27. Partial likelihood P value is significant - null hypothesis that all of β’s are zero

28. Thank you!