Observation model for time-to-event data

Use of time-to-event data

Here, observations are the “times at which events occur”. An event may be one-off (e.g., death, hardware failure) or repeated (e.g., epileptic seizures, mechanical incidents, strikes). Several functions play key roles in time-to-event analysis: the survival, hazard and cumulative hazard functions. We are still working under a population approach here so these functions, detailed below, are thus individual functions, i.e., each subject has its own. As we are using parametric models, this means that these functions depend on individual parameters (\psi_i).

  • The survival function S(t, \psi_i) gives the probability that the event happens to individual i after time t>t_{\text{start}}:

    S(t,\psi_i) = \mathbb{P}(T_i>t;\psi_i)

  • The hazard function h(t,\psi_i) is defined for individual i as the instantaneous rate of the event at time t, given that the event has not already occurred:

    h(t,\psi_i) \ \ = \ \ \lim_{dt\to 0} \frac{S(t,\psi_i) - S(t + dt,\psi_i)}{ S(t,\psi_i) \, dt} .

    This is equivalent to

    h(t,\psi_i) \ \ = \ \ -\frac{d}{dt} \log{S(t,\psi_i)} .

  • Another useful quantity is the cumulative hazard function H(a,b;\psi_i), defined for individual i as

H(a,b;\psi_i) \ \ = \ \ \int_a^b h(t,\psi_i) \, dt .

Note that S(t,\psi_i) = e^{-H(t_{\text{start}},t;\psi_i)}. Then, the hazard function h(t,\psi_i) characterizes the problem, because knowing it is the same as knowing the survival function S(t,\psi_i). The probability distribution of survival data is therefore completely defined by the hazard function.

Observation model syntax

An observation variable for time-to-event or repeated time to event data is defined using the type event. Its additional fields are:

  • eventType: Type of the events. The exact time of the events can be observed, or censored per interval. The respective keywords are exact and intervalCensored. By default, an exact time is assumed.
  • maxEventNumber: Maximum number of events. It is useful for simulation only, and by default the number of simulated events is unlimited.
  • rightCensoringTime: Right censoring time of events. It is useful for simulation only, and by default it is the actual time of the last record.
  • intervalLength: Length of censoring intervals. It is useful for simulation only, and by default it is the tenth part of the global length.
  • hazard: Hazard function.

Example

An example where we define an observation model for this case is proposed here

[LONGITUDINAL]
input={gamma, V, Cl}

EQUATION:
Cc = pkmodel(V,Cl)
haz =  gamma*Cc

DEFINITION:
Seizure = {type = event, eventType = intervalCensored, maxEventNumber = 1, 
rightCensoringTime = 120, intervalLength = 10, hazard = haz}