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

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

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

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

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

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

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

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

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

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

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