Event History Analysis

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Event History Analysis - PowerPoint PPT Presentation

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

PS 791

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
• 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 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
• 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
• 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 is expressed as:
• This expresses the idea that some survival time T is less than or equal to t
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 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
• 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
• 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 hazard rate, survivor function, and distribution and density functions all interrelated.
• Thus the hazard rate can be represented by
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
• 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
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
• 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
• 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
• 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 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
• 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
• Cox Proportional Hazard
• Similar to Weibull
An example
• Events
• Action-reaction Models