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Continuous-time microsimulation in longitudinal analysis

Explore the concept of microsimulation in continuous time through the lens of survival analysis and event history analysis. Learn about the generation of virtual populations, transition modeling, and the importance of precise event timing in simulating real-world scenarios accurately. Discover various models such as exponential, Gompertz, and Cox, and the application of quantile functions to determine waiting times to events. Dive into the complexities of duration and transition analysis and see how continuous-time microsimulation simplifies studying virtual populations.

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Continuous-time microsimulation in longitudinal analysis

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  1. 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

  2. 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)

  3. 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.

  4. 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

  5. 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

  6. 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.

  7. What is continuous time? • Precise date (month, day, second) • Month is often adequate approximation => 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

  8. 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)

  9. 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

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

  11. Quantile functions • Exponential distribution (constant hazard rate) • Distribution function • Quantile function • Cox model • Distribution function • Quantile function Parameterize baseline hazard

  12. 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

  13. Illustration • If the transition rate is 0.2, what is the median waiting time to the event? The expected waiting time is

  14. IllustrationExponential model with =0.2 and 1,000 draws

  15. Multiple origins and multiple destinationsState probabilities

  16. Lifepaths during 10-year periodSample of 1,000 subject; =0.2

  17. 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

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