BALANCE HIDRICO DE LAS OPERACIONES DE SMCV

So far the methods for survival analysis we have discussed are applicable to the situation when there is one single (possibly censored) failure time of the same type on each study subject, and the different subjects are assumed to be independent. This is what we call uni- variate survival data. In some applications, the survival data observed might be of more complex structures, such as the so-called multivariate survival data (Hougaard, 2000). The fundamental characteristic of multivariate survival data is that independence between sur- vival times cannot be assumed, which adds complexity when modeling and analyzing the data.

One structure of multivariate survival data involves recurrent events. That is, a single individual might experience the same event multiple times during the study period. The failure times observed on the same individual apparently cannot be assumed to be inde- pendent. One simple example arises for times to tumor recurrence among patients of a certain type of cancer. The shared frailty model presented in Chapter 9 of Hougaard (2000) is exclusively for analyzing recurrent events data. Chapter 9 of Kalbfleisch and Prentice (2002) also provides methods which directly model the intensity process corresponding to the recurrent events. The rationale of these methods is quite intuitive with some knowledge about counting process theory.

Another situation when multivariate survival data arise is multi-state data. Similar to the recurrent events data, multiple failure times are observed on a single individual, however, these times correspond to the occurrence of events of distinct types. Generally the life history of an individual under study may involve multiple types of failures that happen longitudinally. For example, a patient in a study might be first observed to have a certain disease, and the patient is followed until death. The patient therefore experiences

two events: the disease and the death. He or she is transitioned from the state of being free of the disease, to the state of being diseased, and then to the state of being dead. Chapter 5 and 6 of Hougaard (2000) elaborates different scenarios of multi-state data and the corresponding modeling techniques. Kalbfleisch and Prentice (2002) also provides models based on Markov process in Chapter 8.

In some studies, although each individual is only observed with one event, the sur- vival times of different individuals may not be independent, such as the times recorded on members from the same family, or the data generated from multi-center studies. The frailty model has the flexibility for modeling this kind of correlated (or clustered) data and is presented in Chapter 7 and 8 of Hougaard (2000). The model of jointly modeling the correlated survival times within a cluster is described in Chapter 10 of Kalbfleisch and Prentice (2002).

One more type of multivariate survival data is competing risks. Each individual is observed with one failure time, however, the failure may be one of several distinct failure types; or the failure happens because of one of multiple causes, that is, the different causes are competing to be the final reason for the failure to happen on the patient. With competing risks data, three problems might be of interest (Kalbfleisch and Prentice, 2002): 1. To estimate the relationship between some explanatory variables and the rate of occurrence of failures of specific types (or causes). 2. To study the interrelation between failure types. 3. To estimate failure rates for certain types of failure given the removal of some or all other failure types. “Strictly speaking, however, competing risks data is not multivariate survival data, as only one time is observed on each subject, and thus it is likely impossible to study the dependence between failure types” (Hougaard, 2000). This fact determines that only under some specific study conditions, the three problems of interest, especially the last two problems, can be answered. The reader can refer to page 249 of Chapter 8 in Kalbfleisch and Prentice (2002) for a more detailed discussion.

study the association between explanatory variables and the failures caused by different reasons. In the next chapter, we will introduce more about competing risks, including the mathematical notation and some modeling techniques.

More About Competing Risks

In Chapter 1, we introduced the topic of survival analysis and reviewed some popular mod- eling techniques for univariate survival data. Some topics where multivariate survival data arise were also introduced at the end of Chapter 1. In this chapter, we will discuss in detail the specific topic of competing risks. We start with Section 2.1 to introduce the probabilis- tic framework for the description of competing risks survival data. The hazard functions for competing risks are defined in Section 2.2. The modeling techniques are introduced in Section 2.3 and Section 2.4, including the traditional latent time variable approach and its limitation, and the hazard based approach - the proportional hazards model for competing risks. Two examples, one derived from a prostatic cancer clinical trial and the other from an ongoing NIH (National Institutes of Health) funded project studying hepatitis C virus (HCV) infected patients diagnosed with hepatocellular carcinoma (HCC), are presented in Section 2.5 to illustrate the situation where competing risks can arise in real-world research.