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Reference:
Garson, G. D. (2012). Parametric Survival Analysis. Asheboro, NC: Statistical Associates Publishers.
 

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ASIN number (e-book counterpart to ISBN): ASIN: B008SNZMOU
 
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PARAMETRIC SURVIVAL ANALYSIS

Overview

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
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