Continuous-time microsimulation in longitudinal analysis

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Continuous-timemicrosimulationin longitudinal
analysis

• Frans Willekens
• Netherlands Interdisciplinary
• Demographic Institute (NIDI)

ESF-QMSS2 Summer School Projection methods for
ethnicity and immigration status, Leeds, 2 9
July 2009
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What is microsimulation?A sample of a virtual
population

• Real population vs virtual population
• Virtual population is generated by a mathematical
  model
• If model is realistic virtual population real
  population
• Population dynamics
• Model describes dynamics of a virtual (model)
  population
• Macrosimulation dynamics at population level
• Microsimulation dynamics at individual level
  (attributes and events transitions)

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Discrete-event simulation

• What is it? the operation of a system is
  represented as a chronological sequence of
  events. Each event occurs at an instant in time
  and marks a change of state in the system.
  (Wikipedia)
• Key concept event queue The set of pending
  events organized as a priority queue, sorted by
  event time.

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Types of observation

• Prospective observation of a real population
  longitudinal observation
• In discrete time panel study
• In continuous time follow-up study (event
  recorded at time occurrence)
• Random sample (survey vs census)
• Cross-sectional
• Longitudinal individual life histories

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Longitudinal datasequences of eventssequences
of states(lifepaths, trajectories, pathways)

• Transition data transition models or multistate
  survival analysis or multistate event history
  analysis
• Discrete time
• Transition probabilities
• Probability models (e.g. logistic regression
• Transition accounts
• Continuous time
• Transition rates
• Rate models (e.g. exponential model Gompertz
  model Cox model)
• Movement accounts
• Sequence analysis Abbott represent trajectory
  as a character string and compare sequences

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Why continuous time? When exact dates are
important

• Some events trigger other events. Dates are
  important to determine causal links.
• Duration analysis duration measured precisely or
  approximately
• Birth intervals
• Employment and unemployment spells
• Poverty spells
• Duration of recovery in studies of health
  intervention
• To resolve problem of interval censoring
• Time to the event of interest is often not
  known exactly but is only known to have occurred
  within a defined interval.

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What is continuous time?

• Precise date (month, day, second)
• Month is often adequate approximation gt discrete
  time converges to continuous time
• Transition models dependent variable
• Probability of event (in time interval)
  transition probabilities
• Time to event (waiting time) transition rates

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Time to event (waiting time) models in
microsimulation

• Examples of simulation models with events in
  continuous time (time to event)
• Socsim (Berkeley)
• Lifepaths (Statistics Canada)
• Pensim ((US Dept. of Labor)

Choice of continuous time is desirable from a
theoretical point of view. (Zaidi and Rake, 2001)
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Time to event (waiting time) models in
microsimulation
Time to event is generated by transition rate
model

• Exponential model (piecewise) constant
  transition (hazard) rate
• Gompertz model transition rate changes
  exponentially with duration
• Weibull model power function of duration
• Cox semiparametric model
• Specialized models, e.g. Coale-McNeil model

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Time to event is generated by transition rate
modelHow?
Inverse distribution function or Quantile function
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Quantile functions

• Exponential distribution (constant hazard rate)
• Distribution function
• Quantile function
• Cox model
• Distribution function
• Quantile function

Parameterize baseline hazard
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Two- or three-stage method

• Stage 1 draw a random number (probability) from
  a uniform distribution
• Stage 2 determine the waiting time from the
  probability using the quantile function
• Stage 3
• in case of multiple (competing) events event
  with lowest waiting time wins
• in case of competing risks (same event, multiple
  destinations) draw a random number from a
  uniform distribution

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Illustration

• If the transition rate is 0.2, what is the median
  waiting time to the event?

The expected waiting time is
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IllustrationExponential model with ?0.2 and
1,000 draws
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Table 1 Number of occurrences, given ?0.2 Random samples of 1000 transitions Table 1 Number of occurrences, given ?0.2 Random samples of 1000 transitions Table 1 Number of occurrences, given ?0.2 Random samples of 1000 transitions Table 1 Number of occurrences, given ?0.2 Random samples of 1000 transitions Table 1 Number of occurrences, given ?0.2 Random samples of 1000 transitions
Number of subjects by number of occurrences within a year Random sample 1 Random sample 2 Random sample 3 Expected values
0 829 797 828 819
1 152 189 153 164
2 18 12 17 16
3 1 2 2 1
4 0 0 0 0
5 0 0 0 0
Total 1000 1000 1000 1000
Total number of occurrences within a year 191 219 193 200
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Table 1 Times to transition Random samples of 1000 transitions and expected values Table 1 Times to transition Random samples of 1000 transitions and expected values Table 1 Times to transition Random samples of 1000 transitions and expected values Table 1 Times to transition Random samples of 1000 transitions and expected values Table 1 Times to transition Random samples of 1000 transitions and expected values
Number of subjects by number of occurrences within a year Random sample 1 Random sample 2 Random sample 3 Expected values

1 0.504 0.478 0.483 0.483
2 0.672 0.705 0.700
3 0.960 0.740 0.596
4 - - -
5 - - -
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Multiple origins and multiple destinationsState
probabilities
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Lifepaths during 10-year periodSample of 1,000
subject ?0.2
Pathway Number Name Mean age at transition Mean age at transition Mean age at transition Mean age at transition Mean age at transition
1 325 HD 4.24D
2 217 H
3 161 H 4.03
4 150 HD 2.68D 5.67
5 84 HDH 3.32D 6.79H
6 40 HDHD 2.36D 4.88H 7.25D
7 11 HDH 1.96D 4.15H 5.77
8 7 HDHDH 1.49D 2.85H 5.74D 7.67H
9 3 HDHD 1.64D 3.86H 4.97D 6.78
10 2 HDHDHD 3.38D 3.92H 6.50D 8.04H 8.17D
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Conclusion

• Microsimulation in continuous time made simple by
  methods of survival analysis / event history
  analysis.
• The main tool is the inverse distribution
  function or quantile function.
• Duration and transition analysis in virtual
  populations not different from that in real
  populations