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Event History Analysis. PS 791 Advanced Topics in Data Analysis. Event History Analysis … and its cousins. Event History Analysis is a general term comprising a set of time duration models Survival Analysis Duration analysis Hazard Modeling. Event Duration.

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event history analysis

Event History Analysis

PS 791

Advanced Topics in Data Analysis

event history analysis and its cousins
Event History Analysis … and its cousins
  • Event History Analysis is a general term comprising a set of time duration models
    • Survival Analysis
    • Duration analysis
    • Hazard Modeling
event duration
Event Duration
  • When we look at processes that occur over time, we are often interested in two aspects of the process:
    • the duration of the events,
      • How long a regime or alliance lasts
    • the transition event or state
      • The occurrence of a coup
survival in broader terms
Survival in broader terms
  • Survival analysis is often used to examine the length of time that an entity survives after exposure to a disease or toxin.
  • In toxicity studies this time might be the LC50
    • The concentration of the toxin that will kill 50% of the species during the time of exposure – say 24 hours
    • Used for determining acute toxicity of a chemical compound
survival in a non fatal sense
Survival in a non-fatal sense
  • Other senses of survival
    • Length of time a regime lasts or stays in power
    • Length of a military intervention
    • Duration of wars; or alliances
the mathematics of survival
The Mathematics of Survival
  • Some definitions:
    • T is a positive random variable for survival time – the length of time before a change of state
      • T is continuous
        • Until we assume it isn’t – for later
      • The actual measure of the survival time, or instance of it, is t.
      • The possible values of T have a probability distribution, f(t), and a cumulative distribution function F(t).
the distribution function of t
The distribution function of T
  • The distribution function of T is expressed as:
  • This expresses the idea that some survival time T is less than or equal to t
the unconditional failure rate
The Unconditional Failure Rate
  • If we differentiate F(t), we get the density function
  • We can characterize the distribution of failures by either distribution or density function
the survivor function
The Survivor Function
  • The survivor function denotes the probability a survival time T is equal to or greater that some time T.
  • This is also the proportion of units surviving beyond t.
  • S(t) is a strictly decreasing function since as time passes there are fewer and fewer individuals surviving
the hazard rate
The Hazard Rate
  • Given the survival function and the density of failures, we have a way that “survival” and “death are accounted for in EHA (Event History Analysis)
  • We obtain another important component in EHA when we look at the relationship between the two in the hazard rate.
a conditional failure rate
A Conditional Failure Rate
  • The hazard rate is the rate at which units fail - or durations end – by t given that the unit has survived until t.
  • Thus the hazard rate is a conditional failure rate.
the interrelationships
The Interrelationships
  • The hazard rate, survivor function, and distribution and density functions all interrelated.
  • Thus the hazard rate can be represented by
using ols on durations
Using OLS on Durations
  • If we model the duration of an event using OLS
    • Like the year a regime lasts
  • We regress the duration length on a set of characteristics or exogenous variables
  • Often we will log the duration time because of some extremely durable cases that make the distribution asymmetric.
  • This will cause problems
  • In some cases, a case may not have failed by the end of the observation period.
  • We refer to this as right-censoring.
    • Model adoption of state lottery
    • If a state has not adopted it by the end of the sample time frame, it is right censored
left censoring
  • Left censoring occurs when the history of the event begins prior to the start of the observed period
    • A regime that began before the time frame
    • A dispute already underway
censoring cont
Censoring (cont)
  • Note that both right- and left-censoring is common in many time-series data sets and is not dealt with in regression designs at all.
  • EHA can incorporate censoring in the models.
  • Based on calculating likelihoods
selection bias
Selection Bias
  • Duration Models can give us a tool to look at Selection Bias
    • When we study something like the determinants of regime failure, and we have a data set comprised of regimes, their failure dates, and the exogenous variables we think led to the failure, we have omitted cases that didn’t fail
    • Because they did not fail because of the same factors that those that did fail we have biased our sample.
    • Duration models can account for this bias.
      • Somehow!
time varying covariates
Time Varying Covariates
  • Regression assumes constant relationships (covariates)
  • What if the slope changes over the course of the study?
  • Regression can handle this through Stochastic or Time-Varying Parameter models, but they are usually ignored
distribution of failure times
Distribution of failure times
  • If we can correctly specify the type and shape of the distribution of the failure rate, we can estimate the impact of the covariates on the failure rate.
  • The shape of that failure rate is a function of it’s parameterization
    • The model’s covariates are used to assess that parameterization
the exponential model
The exponential model
  • The exponential model implies a baseline hazard rate that is flat
    • The likelihood of a failure is the same at any given time
  • This implies a constant hazard rate
other distributions
Other distributions
  • Weibell
    • Used if the hazard rate is increasing or decreasing
  • Log-logistic or Log-normal
  • Gompertz
  • How to choose?
    • Theory?
    • Generalized Gamma
proportional hazard models
Proportional Hazard Models
  • Cox Proportional Hazard
    • Similar to Weibull
an example
An example
  • Events
    • Action-reaction Models