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