Duration Data

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Section

• Duration Data

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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

3
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

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• 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?

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• 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

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• 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 youve been unemployed for 12 months
  already

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• PDF, CDF (Failure function), survivor function
  and hazard function are all related
• ?(t) f(t)/S(t) f(t)/(1-F(t))
• We focus on the hazard rate because its
  relationship to time indicates duration
  dependence

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• 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

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• Mathematically
• d ?(t) /dt 0 then there is no duration dep.
• d ?(t) /dt gt 0 there is duration dependence
• the probability the
  spell will end
• increases with time
• d ?(t) /dt lt 0 there is duration dependence
• the probability the
  spell will end
• decreases over time

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• Your choice, is to pick values for f(t) that have
  , - or no duration dependence

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Different Functional Forms

• Exponential
• ?(t) ?
• Hazard is the same over time, a memory less
  process
• Weibull
• F(t) 1 exp(-?ta) where a,? gt 0
• ?(t) a?ta-1
• if agt1, increasing hazard
• if alt1, decreasing hazard
• if a1, exponential

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• Others Lognormal, log-logistic, Gompertz

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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

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• 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

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Key Variables

• max_mths maximum months in the survey.
• Diedin5 respondent died during the 5 years of
  followup
• Note if diedn50, the max_mths60. Diedin5
  identifies whether the data is censored or not.

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Identifying Duration Data in STATA

• Need to identify which is the duration data
• stset length, failure(failvar)
• Lengthduration variable
• Failvar1 when durations end in failure, 0 for
  censored values
• If all data is uncensored, omit failure(failvar)

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• In our case
• Stset max_mths, failure(diedin5)

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Getting Kaplan-Meier Curves

• Tabular presentation of results
• sts list
• Graphical presentation
• sts graph
• Results by subgroup
• sts graph, by(income)