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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

- the duration of the events,

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

- 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).

- T is continuous

- T is a positive random variable for survival time – the length of time before a change of state

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
- A dispute already underway

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

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