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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

- 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

- 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

- 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?

- Survivor function, what is the chance you live past (t) ‘t’.
- S(t) = 1 – F(t)
- If 10% of a cohort dies by their 65th birthday, 90% will die sometime after their 65th birthday

- Hazard function, h(t) ‘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

- 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’

- 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

- Mathematically lower the chance they will exit unemployment – ‘damaged goods’
- 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

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

Different Functional Forms duration dependence

- 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

- Others: Lognormal, log-logistic, Gompertz duration dependence

NHIS Multiple Cause of Death duration dependence

- 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

- Our sample duration dependence
- 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 duration dependence

- 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 duration dependence If all data is uncensored, omit failure(failvar)

- 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

- In our case duration dependence
- Stset max_mths, failure(diedin5)

Getting Kaplan-Meier Curves duration dependence

- Tabular presentation of results
sts list

- Graphical presentation
sts graph

- Results by subgroup
sts graph, by(income)

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