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Section Duration Data Introduction Sometimes we have data on length of time of a particular event or ‘spells’ Time until death Time on unemployment Time to complete a PhD

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section

Section

Duration Data

introduction
Introduction
  • Sometimes we have data on length of time of a particular event or ‘spells’
    • Time until death
    • Time on unemployment
    • Time to complete a PhD
  • Techniques we will discuss were originally used to examine lifespan of objects like light bulbs or machines. These models are often referred to as “time to failure”
notation
Notation
  • T is a random variable that indicates duration (time til death, find a new job, etc)
  • t is the realization of that variable
  • f(t) is a PDF that describes the process that determines the time to failure
  • CDF is F(t) represents the probability an event will happen by time t
slide4
F(t) represents the probability that the event happens by ‘t’.
  • What is the probability a person will die on or before the 65th birthday?
slide5
Survivor function, what is the chance you live past (t)
  • S(t) = 1 – F(t)
  • If 10% of a cohort dies by their 65th birthday, 90% will die sometime after their 65th birthday
slide6
Hazard function, h(t)
  • What is the probability the spell will end at time t, given that it has already lasted t
  • What is the chance you find a new job in month 12 given that you’ve been unemployed for 12 months already
slide7
PDF, CDF (Failure function), survivor function and hazard function are all related
  • λ(t) = f(t)/S(t) = f(t)/(1-F(t))
  • We focus on the ‘hazard’ rate because its relationship to time indicates ‘duration dependence’
slide8
Example: suppose the longer someone is out of work, the lower the chance they will exit unemployment – ‘damaged goods’
  • This is an example of duration dependence, the probability of exiting a state of the world is a function of the length
slide9
Mathematically
      • d λ(t) /dt = 0 then there is no duration dep.
      • d λ(t) /dt > 0 there is + duration dependence

the probability the spell will end

increases with time

      • d λ(t) /dt < 0 there is – duration dependence

the probability the spell will end

decreases over time

different functional forms
Different Functional Forms
  • Exponential
    • λ(t)= λ
    • Hazard is the same over time, a ‘memory less’ process
  • Weibull
    • F(t) = 1 – exp(-γtα) where α,γ > 0
    • λ(t) = αγtα-1
    • if α>1, increasing hazard
    • if α<1, decreasing hazard
    • if α=1, exponential
nhis multiple cause of death
NHIS Multiple Cause of Death
  • NHIS
    • annual survey of 60K households
    • Data on individuals
    • Self-reported healthm DR visits, lost workdays, etc.
  • MCOD
    • Linked NHIS respondents from 1986-1994 to National Death Index through Dec 31, 1995
    • Identified whether respondent died and of what cause
slide14
Our sample
    • Males, 50-70, who were married at the time of the survey
    • 1987-1989 surveys
    • Give everyone 5 years (60 months) of followup
key variables
Key Variables
  • max_mths maximum months in the survey.
  • Diedin5 respondent died during the 5 years of followup
  • Note if diedn5=0, the max_mths=60. Diedin5 identifies whether the data is censored or not.
identifying duration data in stata
Identifying Duration Data in STATA
  • Need to identify which is the duration data

stset length, failure(failvar)

      • Length=duration variable
      • Failvar=1 when durations end in failure, =0 for censored values
  • If all data is uncensored, omit failure(failvar)
slide17
In our case
  • Stset max_mths, failure(diedin5)
getting kaplan meier curves
Getting Kaplan-Meier Curves
  • Tabular presentation of results

sts list

  • Graphical presentation

sts graph

  • Results by subgroup

sts graph, by(income)