Event History Modeling, aka Survival Analysis, aka Duration Models, aka Hazard Analysis

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Event History Modeling,aka Survival
Analysis,aka Duration Models,aka Hazard Analysis
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How Long Until ?

• Given a strike, how long will it last?
• How long will a military intervention or war
  last?
• How likely is a war or intervention?
• What determines the length of a Prime Ministers
  stay in office?
• When will a government liberalize capital
  controls?

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Origins

• Medical Science
• Wanted to know the time of survival
• 0 ALIVE
• 1 DEAD
• Model slightly peculiar once you transition,
  there is no going back.
• Many analogs in Social Sciences

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Disadvantages of Alternatives(Cross Sections)

• Assumes steady state equilibrium
• Individuals may vary but overall probability is
  stable
• Not dynamic
• Cant detect causation.

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Disadvantages of Alternatives(Panel)

• Measurement Effects
• Attrition
• Shape not clear
• Arbitrary lags
• Time periods may miss transitions

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Event History Data

• Know the transition moment
• Allows for greater cohort and temporal
  flexibility
• Takes full advantage of data

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Data Collection Strategy(Retrospective Surveys)

• Ask Respondent for Recollections
• Benefit Can cheaply collect life history data
  with single-shot survey
• Disadvantages
• Only measure survivors
• Retrospective views may be incorrect
• Factors may be unknown to respondent

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Logic of Model

• T Duration Time
• t elapsed time
• Survival Function S(t) P(Tt)

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Logic of Model (2)

• Probability an event occurs at time t
• Cumulative Distribution function of f(t)
• Note S(t) 1 F(t)

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Logic of Model (3)

• Hazard Rate
• Cumulative Hazard Rate

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Logic of Model (4)

• Interrelationships
• so knowing h(t) allows us to derive survival and
  probability densities.

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Censoring and Truncation

• Right truncation
• Dont know when the event will end
• Left truncation
• Dont know when the event began

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Censoring and Truncation (2)
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Discrete vs. Continuous Time

• Texts draw sharp distinction
• Not clear it makes a difference
• Estimates rarely differ
• Need to measure time in some increment
• Big problem comes for Cox Proportional Hazard
  Model it doesnt like ties

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How to Set up Data(Single Record)
Prime Minister Took Office Left Office Days Event
Henry Sewell 7 May 1856 20 May 1856 13 1
William Fox 20 May 1856 2 June 1856 13 1
Edward Stafford 2 June 1856 12 July 1861 1866 1
William Fox 12 July 1861 6 August 1862 390 1
Alfred Domett 6 August 1862 30 October 1863 450 1
Frederick Whitaker 30 October 1863 24 November 1864 391 1
Frederick Weld 24 November 1864 16 October 1865 326 1
Edward Stafford 16 October 1865 28 June 1869 1351 1
William Fox 28 June 1869 10 September 1872 1170 1
Edward Stafford 10 September 1872 11 October 1872 31 1
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Choices / Distributions

• Need to assume a distribution for h(t).
• Decision matters
• Exponential
• Weibull
• Cox
• Many others, but these are most common

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Distributions (Exponential)

• Constant Hazard Rate
• Can be made to accommodate coefficients

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Distributions (Weibull)

• Allows for time dependent hazard rates

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Weibull Survival Functions
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Weibull Hazard Rates
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Distributions (Cox)

• Useful when
• Unsure of shape of time dependence
• Have weak theory supporting model
• Only interested in magnitude and direction
• Parameterizing the base-line hazard rate

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Distributions (Cox 2)
Baseline function of t not X
Involves X but not t
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Distributions (Cox 3)

• Why is it called proportional?

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How to Interpret Output

• Positive coefficients mean observation is at
  increased risk of event.
• Negative coefficients mean observation is at
  decreased risk of event.
• Graphs helpful.

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Unobserved heterogeneity and time dependency

• Thought experiment on with groups
• Each group has a constant hazard rate
• The group with higher hazard rate experience
  event sooner (out of dataset)
• Only people left have lower hazard rate
• Appears hazard drops over time
• Solution akin to random effects

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Extensions

• Time Varying Coefficients
• Multiple Events
• Competing Risk Models