Discrete time approximation

• Although it is nice to have information on the exact timing of events, we are often not so lucky. it is very common in longitudinal datasets to have information on an individual's status at regular intervals of time, but not information on when status changed in the interval for cases where it changed. Examples:
  • labor force status
  • school status
  • marital change
• These are cases of what is often called discrete time. In actuality, the data are discrete time approximations of a continuous time process. That is, we believe that there is some continuous hazard of a change in labor force status between our recording intervals, but all we know is whether the change did or did not occur during this interval.
• Because we don't have information on exact timing, we cannot directly calculate hazard rates, but we can calculate the probability of an event occurring in the interval.
• What is the relationship between the probability and the rate? If we assume that the hazard rate is constant over the interval, then the probability of an event for individual $i$ in time period $t$ is given by:
  $$p_{it}=1-e^{\lambda t_{i}}$$
  Where $t_{i}$ is the length of the time period, and $\lambda$ is the constant hazard rate. This follows directly from the survival curve:
  $$p_{it}=1-S(t_{i})==1-e^{-\lambda t_{i}}$$

• If we relate our covariate $x_{it}$ linearly to the log of $\lambda$ as before, then
  $$\lambda=e^{\beta_{0}+\beta_{1}x_{it}}$$
  Then plugging into our previous equation:
  $$p_{it}=1-e^{e^{\beta_{0}+\beta_{1}x_{it}} (t_{i)}}$$
  This is ugly. What we want to is to get a linear expression on the right. Let's log both sides:
  $$\log(p_{it})=-e^{\beta_{0}+\beta_{1}x_{it}} (t_{i)}$$
  Getting better. Let's multiply both sides by $-1$ and log again.
  $$\log(-\log(p_{it}))=\log(t_{i})+\beta_{0}+\beta_{1}x_{it}$$
  On the left we have a link function for the probability known as the complementary log-log function. On the right we have a linear function of the independen variables, plus an offset of the log of the length of the time period.

  So the discrete time approximation can be run directly on the dichotomous status variable for each wave using a GLM with a cloglog link and an offset of $\log(t_{i})$. In many cases, the data are collected once a year, so that the offset will be $\log(1)=0$ and need not be explicitly added.

• Duration dependence can be incorporated directly into this type of model be treating time (measured as the wave of data) as a covariate. It can be treated as a set of dummies or it can be modeled using a more parametric approach.
• True discrete time data
  • There are a few (rare) cases where the researcher may believe that the process is truly governed by a discrete time process. The classical example would be the passage of state legislation. Whether a bill gets passed early or late in legislative session may not really give us any information on the hazard of the bill being passed overall.
  • When the process is a true discrete time process, then the appropriate way to model it is using a logit or probit transformation rather than a complementary log-log transformation. Because, we want to model the probability of transition directly, rather than the rate.
  • Many textbooks do not make a distinction between true discrete time processes and continuous time processes approximated with discrete time data, and many researchers use the logit or probit transformation when they should be using the cloglog transformation - but you know better,
  • When the intervals are short and even for everyone the difference between cloglog and logit is fairly trivial, but cloglog will always give closer estimates to what the continuous time process would have produced than logit.
  • Let's take our example. Let's say we didn't know the exact time students dropped out, but just whether they had dropped out by the end of the school year. First, we make our split-episode file

    Student	Race	Length of Inverval	Dropped out
    1	White	1	N
    2	White	1	N
    2	White	1	N
    3	White	1	N
    3	White	1	N
    4	White	1	N
    4	White	1	Y
    5	Black	1	N
    5	Black	1	N
    6	Black	1	Y
    7	Black	1	N
    7	Black	1	N
    8	Black	1	N
    8	Black	1	Y

  • Now let's run a discrete time approximation using first the logit and the cloglog. How do the relative risks compare to our exponential model?

    Model	Constant	log(RR)
    Exponential	-1.86	0.77
    Discrete logit	-1.79	0.88
    Discrete cloglog	-1.87	0.78