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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


Censoring
Censoring

  • 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

  • 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


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