Duration models

• Longitudinal data involves time by definition
• Continuous-model observation means we measure duration, and we can model it: what characteristics raise expected duration? which lower it?
• However, with observed durations we usually have significant proportions which are censored, that is spells which are on-going at the time of observation
  • we know the spells are at least `this' long, but we don't know their true duration or outcome.
• As a result simple models such as OLS regression with duration as the $Y$-variable will be seriously biased.
• The alternative is hazard modelling: model the instantaneous or period-by-period hazard of exit
  • instantaneous: continuous time models
  • period-by-period: discrete time models
  • probability conditional on being at risk (i.e., not already experienced exit)
• Conceptually the focus shifts from duration to person-periods:
  • a censored spell of length $n$ contributes $n$ observations of `no-exit', with the last one indicated as censored
  • a terminated spell of length $m$ contributes $m-1$ observations of `no-exit' and one of `exit'
  • Thus we use all the information without suffering censoring bias
• Kaplan-Meier survival estimates use this methodology:
  • Given $N$ cases at the start of period 1, $n$ experiencing a transition and $m$ disappearing from observation
  • The transition rate is estimated as $\frac{n}{N - m/2}$
  • The survival rate between times 1 and 2 is $1 - \frac{n}{N - m/2}$
  • Continuing this curve over time gives the `survival function' which can be considered an estimate of the proportion of $N$ surviving if there were no censoring.

• The key to coping with censoring is thinking in terms of person-time-units
• Parametric approaches exist too: we can model the hazard rate.
• In continuous time:

  
  

  $$h_t = \lim_{\Delta_t\rightarrow 0} \frac{P(T<t+\Delta_t\vert T>t)}{\Delta_t}$$
  
  

  

  Observation window and censoring
• Many types of model exist, in continuous and discrete time, with different parameterisation of the duration dependence
  • the way which hazard changes with time, controlling for covariates
• These include
  • exponential model (constant hazard, therefore an exponential distribution of completed durations)
  • Weibull and Gompertz models
  • Cox proportional hazard model (semi-parametric: model is structured to let duration dependence drop out of the estimation)

• The exponential model is not enormously realistic but very easy to fit, and very flexible.
• Very easy to include time-dependent covariates: explanatory variables which change during the spell
• One strategy is to generate a long-format spell files, with a spell for each period where all the covariates are the same, and a 1 unit spell indicating any transitions that occur: weight by spell duration and fit a logistic regression with transition as the dependent variable.
• Cox PH regression very flexible, very good on TDCs, as long as you have continuous time measurement.
• Cox regression available in SPSS and Stata, though programming the TDCs takes a little time.
• Many more discrete time models available in Stata, including as user-written additions (e.g., pgmhaz and others by Stephen Jenkins)
• TDA was written from scratch to fit duration models and is relatively easy to use and is in some ways a reference implementation of certain models.
• References for hazard modelling:
  • Allison (1984) - good, accessible, clear
  • Blossfeld and Rohwer (2002, 2nd ed) - uses TDA