Abstract
Survival analysis is almost always reserved for an endpoint of mortality or recurrence. (Mantel, 1966) However, it can be used for many different types of endpoints as the survival distribution is defined as the time to an event. That event can be any endpoint of interest. For patients with chronic diseases, there are many endpoints to examine. For example, patients with diabetes want to avoid organ failure as well as death, and treatments that can prolong the time to organ failure will be beneficial. For patients with resistant infections, treatments that prevent one or multiple recurrences should be explored. Survival data mining differs from survival analysis in that multiple patient events can occur in sequence. The first step in survival data mining is to define an episode of treatment so that the patient events can be found for analysis. It can be thought of as a sequence of survival functions. In this chapter, we will look at the time to a switch in medications, and contrast how prescriptions are given to patients, either following or disregarding treatment guidelines.
TopLog Rank Statistics
The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. (Peto & Peto, 1972) It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all time points where there is an event.
The hazard rate is defined from the failure rate, λ(t), which can be thought of as the probability that a failure occurs in a specified interval, given that there is no failure before time t. It can be defined with the aid of the survival function (also called the reliability function), R(t), the probability of no failure before time t, such that
where Δt=t
2-t
1. This is a conditional probability. By taking the limit, we define the hazard function, which is the instantaneous failure rate at any point in time,