Philip Hougaard

Frailty models for survival data

A frailty model is a random effects model for time variables, where the random effect (the frailty) has a multiplicative effect on the hazard. It can be used for univariate (independent) failure times, i.e. to describe the influence of unobserved covariates in a proportional hazards model. More interesting, however, is to consider multivariate (dependent) failure times generated as conditionally independent times given the frailty. This approach can be used both for survival times for individuals, like twins or family members, and for repeated events for the same individual. The standard assumption is to use a gamma distribution for the frailty, but this is a restriction that implies that the dependence is most important for late events. More generally, the distribution can be stable, inverse Gaussian, or follow a power variance function exponential family. Theoretically, large differences are seen between the choices. In practice, using the largest model makes it possible to allow for more general dependence structures, without making the formulas too complicated.

Multi-state Models: A Review

Multi-state models are models for a process, for example describing a life history of an individual, which at any time occupies one of a few possible states. This can describe several possible events for a single individual, or the dependence between several individuals. The events are the transitions between the states. This class of models allows for an extremely flexible approach that can model almost any kind of longitudinal failure time data. This is particularly relevant for modeling different events, which have an event-related dependence, like occurrence of disease changing the risk of death. It can also model paired data. It is useful for recurrent events, but has limitations. The Markov models stand out as much simpler than other models from a probability point of view, and this simplifies the likelihood evaluation. However, in many cases, the Markov models do not fit satisfactorily, and happily, it is reasonably simple to study non-Markov models, in particular the Markov extension models. This also makes it possible to consider, whether the dependence is of short-term or long-term nature. Applications include the effect of heart transplantation on the mortality and the mortality among Danish twins.

Univariate Survival Data

This chapter gives a description of univariate survival data methods. The topic is also described in many other books. Thus, it is not absolutely necessary for persons experienced in survival analysis to read it, but it does contain notation and key results that will be needed later. Furthermore, some aspects are treated in more detail than elsewhere, in order to create a basis for understanding specific points in later chapters.

Statistical inference for shared frailty models

This chapter considers the statistical inference for the shared frailty models described in the previous chapter. A main part of this is estimation procedures. Estimation difficulties have previously limited the applicability of the shared frailty models. There have, however, been a number of suggestions on how to estimate the parameters. Reasons for the many choices are, of course, that some formulas are complicated and that iteration can be time consuming. One basic direction to take is to integrate out the random frailties, but this is not the only possibility. Alternatively, one can use estimation routines where the frailties are included as unobserved random variables, similar to BLUP (best linear unbiased predictor) methods for normal distribution models. For non-parametric hazard functions, there is one parameter per time point with observed events. This can be specifically included in the model, or one can attempt to remove it from the likelihood, an approach that is inspired by the successful way of doing so in the Cox model. Also for the Nelson-Aalen estimate, it is easy to handle the hazard contributions, because there is a separate equation for each term allowing for an explicit solution. It is, unfortunately, not quite as easy in a frailty model; iteration is necessary as all expressions are non-linear and related to each other.

Shared frailty models

The shared frailty model is a specific kind of the common risks model described in Section 3.1.2. The frailty is the term that describes the common risks, acting as a factor on the hazard function. The approach makes sense both for parallel data and recurrent events data. In this chapter, only parallel data are considered. The results are presented in terms of individuals, which have the same risk in some groups. Recurrent events data will be separately considered in Chapter 9. The shared frailty model is relevant to lifetimes of several individuals, similar organs and repeated measurements. It is not generally relevant for the case of different events. It is a mixture model, because in most cases the common risks are assumed random. The mixture term is the frailty and for this the notation Y will be used. The model assumes that all time observations are independent given the values of the frailties. In other words, it is a conditional independence model. The value of Y is constant over time and common to the individuals in the group and thus is responsible for creating dependence. This is the reason for the word shared, although it would be more correct to call the models of this chapter constant shared frailty models. The interpretation of this model is that the between-groups variability (the random variation in Y) leads to different risks for the groups, which then show up as dependence within groups. The approach is a multivariate version of the mixture calculations of Sections 2.2.7 and 2.4.6.

Shared frailty models for recurrent events

Recurrent events are in several ways more complicated and in other ways more simple to analyze than parallel data of several individuals, and this is why the shared frailty models for such data are described in a separate chapter. This chapter treats the probability model as well as the statistical inference.

Multivariate frailty models

The shared frailty model described in Chapter 7 is very useful for bivariate data with common risk dependence, but in many cases, we do need extensions. In particular, for truly multivariate data, that is, when there are three or more observations, we need more models with varying degrees of dependence. This general frailty approach can be used to create a random treatment by group interaction, or other models with several sources of variation. Secondly, combining subgroups with different degrees of dependence in a single model, for example, monozygotic and dizygotic twins, is difficult in a shared frailty model. Furthermore, this extension can be an improvement for the consideration of effects of covariates in frailty models.

Competing risks models

The term competing risks refers to cause of death models. Competing risks have been introduced and illustrated as a multi-state model in Figure 1.7 and discussed previously, in Sections 1.10, 3.3.9, and Chapter 5; but such data present special challenges, and therefore are considered separately in this chapter. It differs from the rest of this book in that only one event is possible for each individual, and in that sense, the concept of dependence seems irrelevant. However, two problems stand out as particularly important for competing risks data, the possibility of classification error and, more important, the desire to study the effect of modifying the hazards for some causes of death. Therefore, dependence is the key problem for such data. We would like to discuss aspects where dependence is important, but it is impossible to estimate the degree of dependence. In fact, it is impossible even to give a specific interpretation of dependence. The problem is caused by competing risks not being a truly multivariate survival problem. This has been discussed for centuries, since Bernoulli considered what the eradication of smallpox would imply for the mean lifetime.