PARAMETRIC SURVIVAL ANALYSIS
Survival analysis is the study of time-to-event data. Its terminology traces back to medical studies where the event of interest was death, and to industrial studies where the event of interest was failure, such as burn-out of a motor or bulb. The objective in these seminal studies was to understand the correlates of survival, hence survival analysis.
Survival analysis may be parametric or semi-parametric. This volume of the Statistical Associates "Blue Book" series treats parametric survival analysis. "Parametric" means that an essential parameter, the baseline hazard function, must be specified by the researcher in advance. The baseline hazard function defines the chance of experiencing the event of interest (the "hazard", which traditionally was death or failure) when other predictors in the model are held constant. Positing the correct baseline hazard function is quite challenging, often leading the researcher to rely on semi-parametric survival analysis, which does not require this. Cox regression is the prime example of semi-parametric survival analysis and is treated in a separate volume.
A related term is "event history analysis," which is also called duration analysis, hazard model analysis, failure-time analysis, or transition analysis. Event history analysis is an umbrella term for procedures for analyzing duration-to-event data, where events are discrete occurrences. "Event history" studies have been common in the study of international relations, where events may be wars or civil conflicts. Many of the earlier conflict studies utilized Weibull and other parametric survival analysis models and therefore event history analysis is often seen as a type of parametric survival analysis.
Coleman (1981: 1) defined event history analysis in terms of three attributes: (1) data units (ex., individuals or organizations) move along a finite series of states; (2) at any time point, changes (events) may occur, not just at certain time points; and (3) factors influencing events are of two types, time-constant and time-dependent. Event history models focus on the hazard function, which reflects the instantaneous probability that the event of interest will occur at a given time, given that the unit of analysis has not experienced the event up to that time. While duration until death or failure were the classic examples, duration of peace until the outbreak of war was an example in international relations. In the last few decades, survival analysis has been applied to a wide range of events, including "hazards" which a positive meaning, such as duration until the event of adoption of an innovation in diffusion research. Other applications include study of longevity of trade agreements, strike durations, marriage durations, employment durations, and innumerable other subjects.
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Below is the unformatted table of contents.
PARAMETRIC SURVIVAL ANALYSIS Table of Contents Overview 7 Key Concepts and Terms 8 Types of survival analysis 8 Nonparametric models 8 Semi-parametric models 8 Parametric models 9 Count models 9 States 10 Durations 10 Variables 11 Time variable 11 Covariates 12 The event variable 12 Data 13 Censored data 13 Survival and probability density functions 13 The survival function 13 The cumulative probability function and cumulative hazard plot 14 The probability density function 15 Hazard rates, hazard functions, and hazard ratios 16 Overview 16 The hazard function 16 Parametric survival modeling 17 Duration data 17 Parametric model assumptions 17 Proportional hazard models 17 Accelerated failure time models 18 Parametric models in Stata 18 Overview 18 Types of parametric models using Stata 19 Exponential models 20 Characteristics 20 Exponential models with covariates 20 Weibull models 21 Scale and shape parameters 21 Weibull proportional hazards models 23 Weibull accelerated failure time models 23 Pretesting Weibull models as exponential models 23 Example: Weibull models with Stata 24 Proportional hazards model with Weibull regression (streg command) 29 Overall model significance 29 Significance of predictors 31 Hazard ratios 32 The shape parameter, p 32 AFT (accelerated failure time) version of Weibull regression (time option) in Stata 33 Time ratios (tr option in AFT models) 35 Hazard curve (the Stata stcurve command) 36 The hazard function 36 The cumulative hazard curve 38 The survival curve 39 Parametric survival models in SAS 40 Overview 40 SAS Interface 40 SAS syntax 41 Data setup for SAS 42 PROC LIFEREG in SAS 43 SAS syntax 43 SAS Weibull AFT model output 44 Maximum likelihood parameter estimates 44 Fit statistics 45 The Weibull probability plot 46 Other types of models 47 Generalized gamma models 47 Log-logistic models 48 Log-normal models 48 Logit and probit models 48 Cloglog (discrete time hazard) models 48 Gompertz models 49 Multiple episode and multiple state EHA models 49 Competing risks/Multiple destination models 49 Multiple destinations vs. multiple episodes 49 Analysis 49 Significance 50 Multiple episode models 50 Number of transitions 50 Analysis. 50 Model fit in parametric survival analysis 51 Measures of model fit 51 Log-likelihood and likelihood ratio tests (LR) 51 AIC and BIC 52 Wald tests 52 Analysis of residuals 52 Graphical methods 54 Selecting the best model in Stata 54 Assumptions 55 Parametric models assume a particular shape of the baseline hazard function 55 Frequently Asked Questions 56 Should I use a parametric survival model or a semi-parametric Cox model? 56 Shouldn't censored cases just be dropped from analysis, since for these cases we do not know when the event will occur? 57 How are data organized for event history analysis? 57 I have data on a certain number of events. Is it acceptable to fill in the other periods with zeros, to indicate non-events? 57 Can't I use regular time series methods such as OLS or logistic regression as long as I adjust for autocorrelation? 57 Bibliography 58 Pagecount: 64