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Models for Multiple Events (01) (Generally, we can think of events as…
Models for Multiple Events (01)
Unordered Events of the Same Type
Repeated Events Models
There are two general approaches.
First, variance-corrected approaches estimate a model and then “fix up” the variance to account for the fact that the observations are not indepen- dent, i.e., they are repeated and therefore correlated
Second, random effects, or frailty, approaches assume that there is stochastic variation across the re- gression parameters that is attributable to unmeasured factors pertaining to the observation.
Variance-Corrected Models for Repeated Events
There are three general modeling frameworks under the variance-corrected approach:
marginal (Wei, Lin, and Weissfeld 1989)
Marginal and conditional models use stratification
conditional (Prentice, Williams, and Peterson 1981)
Marginal and conditional models use stratification
Andersen-Gill (Andersen and Gill 1982)
All three adjust the variance of the parameter estimates by clustering on subject to account for the repeated nature of the data.
The hazard rate for the jth cluster and kth failure in a Cox model is
hk(t)= ho(t)expβ' xkj
If the hazard rate is allowed to vary by the kth failure in a repeated events model by the use of stratification, i.e., the data are stratified according to the kth event, then the hazard rate is given by
hk(t)= hok(t)expβ´xkj
Example 10.1: A Repeated Events Model for Militarized Intervention Data
Frailty Models and Repeated Events Data
The underlying logic of frailty models is that some observations (or groups or clusters) are intrinsically more or less prone to experiencing the event of interest than are others, and that the distribution of these individual-specific effects can be known, or at least approximated.
Competing Risks Models
Event history processes where multiple kinds of events can occur are sometimes referred to as “multi-state” processes because it is assumed that an observation can make a transition into one of several states. This kind of process is also sometimes called a “competing risks” problem because the ob- servation is at risk of experiencing one of several kinds of events.
Latent Survivor Time Approach to Competing Risks
Example 10.2: Competing Risks Model of Congressional Careers
Multinomial Logit Approach to Competing Risks
Example 10.3: MNL Competing Risks Model of Congressional Careers
Generally, we can think of events as being
“ordered”
or “unordered.”
unordered events that are not the same type
unordered events that are the same type
The fundamental problem in an analysis of mul- tiple events data is that the traditional duration model assumes that the event times are independent.