1. 6/1/2015 1 Survival Analysis

2. Introduction

3. Abbreviated Outline Descriptive overview of survival analysis Terminology and notation Goals of survival analysis 6/1/2015 3 Survival Analysis

4. Survival analysis is a collection of statistical procedures for data analysis for which the outcome variable of interest is time until an event occurs. The event of interest is typically referred to as a failure. The time to the event is referred to as survival time (or failure time in reliability analysis) When more than one event is considered, the statistical problem is characterized as a competing risk problem. 6/1/2015 4 Survival Analysis

5. Examples Engineering: the lifetime of a light bulb; the time from start to burned out Social science: the survival time of a marriage ; the time from start to the end of marriage due to annulment, divorce or death Public health: the time to local or regional recurrence since the first radiation course for the breast cancer patient the time to death due to AIDS or ALDS-related complications after a confirmed diagnosis of HIV 6/1/2015Survival Analysis 5

6. 3 elements to determine the survival time 1. Time origin or starting point 2. Ending event of interest 3. Measurement scale It is usually in time such as days or months; but can be otherwise such as in dollars: the cost from admission to discharge. 6/1/2015 6 Survival Analysis

7. 6/1/20157Survival Analysis

8. Survival Distributions Y: the actual survival time of an individual; Y > 0 4 functions characterize the distributions of Y: the cumulative distribution function, the survival function, the hazard function, and the mean residual life function 6/1/2015 8 Survival Analysis

9. CDF of Y 6/1/2015 9 Survival Analysis

10. Survival Function of Y 6/1/2015 10 Survival Analysis

11. About Survival Function The survival function S is monotone nonincreasing over its support [0,infinity). S(0)=1 and S(infinity)=0. A steep curve represents shorter survival, while a gradual curve represents longer survival. 6/1/2015 11 Survival Analysis

12. Hazard Function of Y 6/1/2015 12 Survival Analysis

13. About Hazard Function The hazard function, h(y), gives the instantaneous rate at time y for experiencing the event of interest, given that the event has not occurred prior to time y. At time y, h(y) > 0 and has no upper bound. An analogy: the concept of velocity What does it mean that your car’s speedometer shows 70 mph? Would you really travel exactly 70 miles in the next hour? 6/1/2015 13 Survival Analysis

14. About Hazard Function Increasing hazard rates may arise when there is natural aging or wear. Decreasing hazard rates are less common but find occasional use when there is an early likelihood of failure (such as after surgery). 6/1/2015 14 Survival Analysis

15. The Mean Residual Life Function 6/1/2015 15 Survival Analysis

16. Goals of Survival Analysis To estimate and interpret survival and/or hazard functions from survival data. To compare survival and /or hazard functions. The hazard function is usually more informative about the underlying mechanism of failure than the survival function. To assess the relationship of explanatory variables to survival time. 6/1/2015 16 Survival Analysis