# Chapter 4 Joint Models for Longitudinal and Time-to-Event Data

In this Chapter we will see a joint modelling approach in order to analyze two types of outcomes produced usually in longitudinal studies, particularly, a set of longitudinal response measurements and the time to an event of interest, such as default, death, etc.

These two outcomes are usually analyzed separately, using a mixed effects model (Verbeke and Molenberghs 2000) for the longitudinal response and a survival model for the time-to-event. Here, we are going to see how we can analyze them jointly.

#### Why should I use these type of models?

As we mentioned in Chapter 3, the Cox PH hazard model can be extended in order to incorporate time-dependent variables. However, when we focus our interest in the time-to-event and we wish to take into account the effect of the longitudinal variable as a time-dependent covariate, traditional approaches for analyzing time-to-event data (such as the partial likelihood for the Cox proportional hazards models) are not applicable in all situations.

In particular, standard time-to-event models require that time-dependent covariates are external; that is, the value of this covariate at time point $$t$$ is not affected by the occurrence of an event at time point $$u$$, with $$t > u$$ (Kalbfleisch and Prentice 2002, Section 6.3). However, the type of time-dependent covariates that we have in longitudinal studies do not met this condition, this is due to the fact that they are the output of a stochastic process generated by the subject, which is directly related to the failure mechanism. Based on this, in order to produce correct inferences, we need to apply a joint model that takes into account the joint distribution of the longitudinal and survival outcomes.

Another advantage of these models is that they allow to deal with the error measurements in the time dependent variables (longitudinal variable in this case). In a Cox model with time dependent covariates we assume that the variables are measured without error.

When we think in time-dependent covariates, we should first distinguish between two different categories, namely, internal or endogenous covariates or external or exogenous covariates. Internal covariates are generated from the patient herself and therefore require the existence of the patient, for example CD4 cell count and the hazard for death by HIV are stochastic processes generated by the patient herself. On the other hand, air pollution is an external covariate to asthma attacks, since the patient has no influence on air pollution.

### References

Verbeke, Geert, and Geert Molenberghs. 2000. Linear Mixed Models for Longitudinal Data. Springer, New York, NY.

Kalbfleisch, John D, and Ross L Prentice. 2002. The Statistical Analysis of Time Failure Data. The Statistical Analysis of Time Failure Data. John Wiley; Sons New York.