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

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 Minister’s stay in office? When will a government liberalize capital controls?

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

Disadvantages of Alternatives (Cross Sections) Assumes steady state equilibrium Individuals may vary but overall probability is stable Not dynamic Can’t detect causation.

Disadvantages of Alternatives (Panel) Measurement Effects Attrition Shape not clear Arbitrary lags Time periods may miss transitions

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

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

Logic of Model T = Duration Time t = elapsed time Survival Function = S(t) = P(T≥t)

Logic of Model (2) Probability an event occurs at time t Cumulative Distribution function of f(t) Note: S(t) = 1 – F(t)=

Logic of Model (3) Hazard Rate Cumulative Hazard Rate

Logic of Model (4) Interrelationships so knowing h(t) allows us to derive survival and probability densities.

Censoring and Truncation Right truncation Don’t know when the event will end Left truncation Don’t know when the event began

Censoring and Truncation (2)

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 doesn’t like ties

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 2 June 1856 Edward Stafford 12 July 1861 1866 6 August 1862 390 Alfred Domett 30 October 1863 450 Frederick Whitaker 24 November 1864 391 Frederick Weld 16 October 1865 326 28 June 1869 1351 10 September 1872 1170 11 October 1872 31

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

Distributions (Exponential) Constant Hazard Rate Can be made to accommodate coefficients

Distributions (Weibull) Allows for time dependent hazard rates

Weibull Survival Functions

Weibull Hazard Rates

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

Distributions (Cox – 2) Baseline function of “t” not “X” Involves “X” but not “t”

Distributions (Cox –3) Why is it called proportional?

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.

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

Extensions Time Varying Coefficients Multiple Events Competing Risk Models