OTHER CENSORING TECHNIQUES
 

 

 

 

 

 

 

 

 

 

 

 

The hazard for Time 1 is 14 percent, since one out of seven customers stop at this time. All seven customers survived to time 1 and all could have stopped. Of these, only one did. At TIME 2, there are five customers left—Customer #7 has already stopped, and Customer 6 has been censored. Of these five, one stops, for a hazard of 20 percent. And so on. This example has shown how to calculate hazard functions, taking into account the fact that some (hopefully many) customers have not yet stopped. This calculation also shows that the hazards are highly erratic—jumping from 25 percent to 50 percent to 0 percent in the last 3 days. Typically, hazards do not vary so much. This erratic behavior arises only because there are so few customers in this simple example. Similarly, lining up customers in a table is useful for didactic purposes to demonstrate the calculation on a manageable set of data. In the real world, such a presentation is not feasible, since there are likely to be thousands or millions of customers going down and hundreds or thousands of days going across.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It is also worth mentioning that this treatment of hazards introduces them as conditional probabilities, which vary between 0 and 1. This is possible because the hazards are using time that is in discrete units, such as days or week, a description of time applicable to customer-related analyses. However, statisticians often work with hazard rates rather than probabilities. The ideas are clearly very related, but the mathematics using rates involves daunting integrals, complicated exponential functions, and difficult to explain adjustments to this or that factor. For our purposes, the simpler hazard probabilities are not only easier to explain, but they also solve the problems that arise when worKing with customer data. Imagine that you are a cancer researcher and have found a medicine that cures cancer. You have to run a study to verify that this fabulous new treat­ ment works. Such studies typically follow a group of patients for several years after the treatment, say 5 years. For the purposes of this example, we only want to know if patients die from cancer during the course of the study (medical researchers have other concerns as well, such as the recurrence of the disease, but that does not concern us in this simplified example).