A tutorial on time to event analysis for mixed effect modellers
Nick Holford (1) and Marc Lavielle(2)
(1) University of Auckland, New Zealand and (2) INRIA Saclay, Paris, France
The connection between PKPD and clinical outcome events can be made using models for time to event. The time of clinical outcome events is understandable in terms of the event hazard, i.e., the rate of occurrence of a time-related event.
The hazard is similar in principle to a drug elimination rate constant. Therefore pharmacokineticists (who are familiar with factors that can describe drug elimination) can easily appreciate how to build hazard models reflecting the influence of drug exposure, disease progression, age, etc. This tutorial will explain how hazard models can be used to describe clinical event times and develop joint models of hazard with commonly used PKPD and disease progression models.
Events may occur a limited number of times in an individual e.g. death happens just once. Other events may occur several times with no obvious limit e.g. epileptic seizures. In each case, events may be observed at an exact time, or may only be known within an interval, or may not be observed at all e.g. due to censoring at the end of a clinical trial. Thus a wide variety of events can be described using the same fundamental concept of the event hazard.
There are strong links between the hazard, the probability distribution of the number of events in a given interval and the probability distribution of the time between events. It is well known for example that a constant hazard can be used to generate count data with a Poisson distribution and times between events with an exponential distribution.
The hazard is usually not constant and is commonly described with a hazard function of explanatory variables such as time or change in disease state. The likelihood of each of these event types can be simply computed from the hazard function and its integral. Joint models of time to event and PKPD can be used in a population context using software such as NONMEM or MONOLIX.
Recommended reading is the book by Collett (1), a key paper by Hu & Sale (2) and a web presentation of the tutorial materials (3).
. Collett D. Modelling survival data in medical research. 2nd ed. Boca Raton: CRC Press; 2003.
. Hu C, Sale ME. A joint model for nonlinear longitudinal data with informative dropout. J Pharmacokinet Pharmacodyn. 2003;30(1):83-103.
. Holford NHG, Lavielle M. Time to event tutorial. 2011. http://www.page-meeting.org/default.asp?abstract=2281