CHAPTER 18

1. CHAPTER 18 SURVIVAL ANALYSIS Damodar Gujarati Econometrics by Example, second edition

2. SURVIVAL ANALYSIS (SA) • The primary goals of survival analysis are to: • (1) Estimate and interpret survivor or hazard functions from survival data • (2) Assess the impact of explanatory variables on survival time • Survival analysis goes by various names, such as: • Duration analysis • Event history analysis • Reliability or failure time analysis • Transition analysis • Hazard rate analysis Damodar Gujarati Econometrics by Example, second edition

3. TERMINOLOGY OF SURVIVAL ANALYSIS • Event: “An event consists of some qualitative change that occurs at a specific point in time . . . The change must consist of a relatively sharp disjunction between what precedes and what follows.” • Duration Spell: The length of time before an event occurs. • Discrete Time Analysis: Some events occur only at discrete times. • Continuous Time Analysis: Continuous time SA analysis treats time as continuous. Damodar Gujarati Econometrics by Example, second edition

4. CUMULATIVE DISTRIBUTION FUNCTION OF TIME • If we treat T, the time until an event occurs, as a continuous variable, the distribution of the T is given by the CDF: • which gives the probability that the event has occurred by duration t. • If F(t) is differentiable, its density function can be expressed as: Damodar Gujarati Econometrics by Example, second edition

5. SURVIVAL AND HAZARD FUNCTIONS • The Survivor Function S(t): is the probability of surviving past time t and is defined as: • The Hazard Function h(t): Consider the following function: where the numerator is the conditional probability of leaving the initial state in the (time) interval {t, t+h}, given survival up to time t. • The hazard function is the ratio of the density function to the survivor function for a random variable: Damodar Gujarati Econometrics by Example, second edition

6. SOME PROBLEMS ASSOCIATED WITH SA • 1. Censoring: A frequently encountered problem in SA is that the data are often censored. • 2. Hazard Function With or Without Covariates: We have to determine if covariates are time-variant or time-invariant. • 3. Duration Dependence: If the hazard function is not constant, there is duration dependence. • 4. Unobserved Heterogeneity: No matter how many covariates we consider, there may be intrinsic heterogeneity among individuals. Damodar Gujarati Econometrics by Example, second edition

7. SOME PROBLEMS ASSOCIATED WITH SA • There are several parametric models that are used in duration analysis. • Each depends on the assumed probability distribution, such as: • Exponential Distribution • Weibull Distribution • Lognormal Distribution • Loglogistic Distribution Damodar Gujarati Econometrics by Example, second edition

8. EXPONENTIAL DISTRIBUTION • Suppose the hazard rate is constant and is equal to h. • A constant hazard implies the following CDF and PDF: • The hazard rate function is a constant, equal to h: Damodar Gujarati Econometrics by Example, second edition

9. WEIBULL DISTRIBUTION • If h(t) is not constant, we have the situation of duration dependence—a positive duration dependence if the hazard rate increases with duration, and a negative duration dependence if this rate decreases with duration. • For this distribution, we have: • and Damodar Gujarati Econometrics by Example, second edition

10. PROPORTIONAL HAZARD MODEL • Originally proposed by Cox • The PH model assumes that the hazard rate for the ith individual can be expressed as: where h0(t) is the baseline hazard • In PH, the ratio of the hazards for any two individuals depends only on the covariates or regressors but does not depend on t, the time. • The hazard rate is proportional to the baseline hazard rate for all individuals: Damodar Gujarati Econometrics by Example, second edition

11. SALIENT FEATURES OF SOME DURATION MODELS Damodar Gujarati Econometrics by Example, second edition