This assumption may or may not be true in marketing. Certainly, working with this assumption, survival analysis has proven its worth with customer data. However, it is interesting to consider the possibility that hazards may be changing over time. In particular, if hazards do change, that gives some insight into whether the market place and customers are getting better or worse over time.

One approach to answering this question is to base hazards on customers who are stopping rather than customers who are starting, especially, say, customers who have stopped in each of the past few years. In other words, were the hazards associated with customers who stopped last year significantly different from the hazards associated with customers who stopped the previous year? Earlier, this chapter warned that calculating hazards for a set of customers chosen by their stop date does not produce accurate hazards. How can we overcome this problem?

There is a way to calculate these hazards, although this has not yet appeared in standard statistical tools. This method uses time windows on the customers to estimate the hazard probability. Remember the definition of the empirical hazard probability: the number of customers who stopped at a particular time divided by the number of customers who could have stopped at that time. Up to now, all customers have been included in the calculation. The idea is to restrict the customers only to those who could have stopped during the period in question.

As an example, consider estimating the hazards based on customers who stopped in 2003. Customers who stopped in 2003 were either active on the first day of 2003 or were new customers during the year. In either case, customers only contribute to the population count starting at whatever their tenure was on the first day of 2003 (or 0 for new starts).

Lets consider the calculation of the 1-day hazard probability. What is the population of customers who could have stopped with 1 day of tenure and also have the stop in 2003? Only customers that started between December 31, 2002 and December 30, 2003 could have a 1-day stop in 2003. So, the calculation of the 1-day hazard uses all stops in 2003 where the tenure was 1 day as the total for stops. The population at risk consists of customers who started between December 31, 2002 and December 30, 2003. As another example, the 365-day hazard would be based on a population count of customers who started in 2002.

The result is an estimate of the hazards based on stops during a particular period of time. For comparison purposes, survival proves to be more useful than the hazards themselves. Figure 12.14 provides an example, showing that survival is indeed decreasing over the course of several years. The changes in survival are small. However, the calculations are based on hundreds of thousands of customers and do represent a decline in customer quality.

100% 90%

60% 50%

40% 30%

90 120 150 180 210 240 270 300 2000

Days after Start

2001 2002

Figure 12.14 A time-window technique makes it possible to see changes in survival over time.

30 60

Lessons Learned

Hazards and survival analysis are designed for understanding customers. This chapter introduced hazards as the conditional probability of a customer leaving at a given point in time. This treatment of survival analysis is unorthodox in terms of statistics, which prefers an approach based on continuous rates rather than discrete time probabilities. However, this treatment is more intuitive for analyzing customers.

Hazards are like an x-ray of the customer life cycle. The related idea of survival, which is the proportion of customers who survive up to a particular point in time, makes it possible to compare different groups of customers and to translate results into dollars and cents. When there are enough customers (and usually there are), stratifying the customers by building a separate curve for each group provides a good comparison. It is possible to use other measures, such as the survival at a particular point in time, the customer half-life, and the average tenure, to better understand customers.

One of the key concepts in survival analysis is censoring. This means that some customers are dropped from the analysis. The idea of censoring can be extended to understand competing risks, such as voluntary versus forced attrition. Censoring also makes it possible to discard certain outcomes, such as a one-time boycott, without adversely biasing overall results.

One of the most powerful aspects of hazards is the ability to determine which factors, at the onset, are responsible for increasing or decreasing the hazards. In addition to stratifying customers, there is another technique, Cox proportional hazards regression, which has proven its worth since the 1970s and continues to be extended and improved upon.

Survival analysis has many applications beyond measuring the probability of customers leaving. It has been used for forecasting customer levels, as well as for predicting other types of events during the customer life cycle. It is a very powerful tool, seemingly designed specifically for understanding customers and their life cycles.