# 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.

**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.