Post on 20-Dec-2015
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 = ALIVE1 = 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)= ( )t
f u du
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)
( ) ( )t R
t
S t f u du
( )
( )
( )
t t
t
t
f u du
h t
f u du
( ) ( )
t L
S t f u du
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 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
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
( )
( )
( )
t
t
t
t
f t e
S t e
eh t
e
0 1X
Distributions (Weibull)
Allows for time dependent hazard rates
1
1
1
Weibull Survival Functions
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Alpha = 1 (Exponential)
Alpha = 0.5
Alpha = 1.5
Weibull Hazard Rates
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time
Haz
ard
Rat
eAlpha = 1 (Exponential)
Alpha = 0.5
Alpha = 1.5
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)
0( | ) ( ) Xh t X h t e
Baseline function of “t” not “X”
Involves “X” but not “t”
Distributions (Cox –3)
Why is it called proportional?0( | ) ( ) Xh t X h t e
0
( 1)0 0
0
( | ) ( )
( | 1) ( ) ( )
( | 1) ( ) ( | )
x
x x
x
h t X x h t e
h t X x h t e h t e e
h t X x e h t e e h t X x
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