# Time-to-event observation model

Related resources on modeling time-to-event data in Monolix:

Columns used to define observations: formatting of time-to-event data in the MonolixSuite.

Time-to-event model library : detailed description of the library of time-to-event models integrated within Monolix.

Time-to-event data models: examples of time-to-event data models from the Monolix demos.

On the current page, we explain the different models for time-to-event data that can be used in Monolix and their syntax.

**Time-to-event observation model**

## 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 ().

The

gives the probability that the event happens to individual**survival function***i*after time :

The

is defined for individual**hazard function***i*as the instantaneous rate of the event at time**t**, given that the event has not already occurred:

This is equivalent to

Another useful quantity is the

, defined for individual**cumulative hazard function***i*as

Note that . Then, the hazard function characterizes the problem, because knowing it is the same as knowing the survival function . The probability distribution of survival data is therefore completely defined by the hazard function.

### Repeated events

Sometimes, an event can potentially happen again and again, e.g., epileptic seizures, heart attacks. For any given hazard function **h**, the survival function **S** for individual *i* now represents the survival since the previous event at , given here in terms of the cumulative hazard from to :

## 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 (integer). By default the number of simulated events is unlimited. If the event is one-off (as death for instance), it is important to indicate

`maxEventNumber=1`

to speed up simulations (including simulations for the prediction interval of the TTE plot in Monolix).rightCensoringTime: Right censoring time of events (number). It is useful for simulation only, and by default it is the actual time of the last record.

intervalLength: Length of censoring intervals (number). 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}
```

## TTE model library

A library of typical parametric models is provided in Monolix: Complete description of the TTE model library.