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