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398 Chapter 12


When the contract is up, customers often rush to leave, and the higher rate
continues for a while because customers have been liberated from the contract.
Once the contract has expired, there may be other reasons, such as the prod­
uct or service no longer being competitively priced, that cause customers to
stop. Markets change and customers respond to these changes. As telephone
charges drop, customers are more likely to churn to a competitor than to nego­
tiate with their current provider for lower rates.

A Real-World Example
Figure 12.6 shows a real-world example of a hazard function, for a company
that sells a subscription-based service (the exact service is unimportant). This
hazard function is measuring the probability of a customer stopping a given
number of weeks after signing on.
There are several interesting characteristics about the curve. First, it starts
high. These are customers who sign on, but are not able to be started for some
technical reason such as their credit card not being approved. In some cases,
customers did not realize that they had signed on”a problem that the authors
encounter most often with outbound telemarketing campaigns.
Next, there is an M-shaped feature, with peaks at about 9 and 11 weeks. The
first of these peaks, at about 2 months, occurs because of nonpayment. Cus­
tomers who never pay a bill, or who cancel their credit card charges, are
stopped for nonpayment after about 2 months. Since a significant number of
customers leave at this time, the hazard probability spikes up.


7%


6%


5%
Weekly Hazard




4%


3%


2%


1%


0%
0




12

16

20
24

28

32

36
40

44

48

52

56

60

64

68

72

76
4

8




Tenure (Weeks after Start)
Figure 12.6 A subscription business has customer hazard probabilities that look like this.
Hazard Functions and Survival Analysis in Marketing 399


The second peak in the “M” is coincident with the end of the initial promo­
tion that offers introductory pricing. This promo typically lasts for about
3 months, and then customers have to start paying full price. Many decide that
they no longer really want the service. It is quite possible that many of these
customers reappear to take advantage of other promotions, an interesting fact
not germane to this discussion on hazards but relevant to the business.
After the first 3 months, the hazard function has no more really high peaks.
There is a small cycle of peaks, about every 4 or 5 weeks. This corresponds to
the monthly billing cycle. Customers are more likely to stop just after they
receive a bill.
The chart also shows that there is a gentle decline in the hazard rate. This
decline is a good thing, since it means that the longer a customers stays around,
the less likely the customer is to leave. Another way of saying this is that cus­
tomers are becoming more loyal the longer they stay with the company.


Censoring
So far, this introduction to hazards has glossed over one of the most important
concepts in survival analysis: censoring. Remember the definition of a hazard
probability, the number of stops at a given time t divided by the population
at that time. Clearly, if a customer has stopped before time t, then that customer
is not included in the population count. This is most basic example of censoring.
Customers who have stopped are not included in calculations after they stop.
There is another example of censoring, although it is a bit subtler. Consider
customers whose tenure is t but who are currently active. These customers are
not included in the population for the hazard for tenure t, because the customers
might still stop before t+1”here today, gone tomorrow. These customers have
been dropped out of the calculation for that particular hazard, although they are
included in calculations of hazards for smaller values of t. Censoring”dropping
some customers from some of the hazard calculations”proves to be a very pow­
erful technique, important to much of survival analysis.
Let™s look at this with a picture. Figure 12.7 shows a set of customers and
what happens at the beginning and end of their relationship. In particular, the
end is shown with a small circle that is either open or closed. When the circle
is open, the customer has already left and their exact tenure is known since the
stop date is known.
A closed circle means that the customer has survived to the analysis date, so
the stop date is not yet known. This customer”or in particular, this cus-
tomer™s tenure”is censored. The tenure is at least the current tenure, but most
likely larger. How much larger is unknown, because that customer™s exact stop
date has not yet happened.
400 Chapter 12




time
Figure 12.7 In this group of customers who all start at different times, some customers
are censored because they are still active.


Let™s walk through the hazard calculation for these customers, paying par­
ticular attention to the role of censoring. When looking at customer data for
hazard calculations, both the tenure and the censoring flag are needed. For the
customers in Figure 12.7, Table 12.2 shows this data.
It is instructive to see what is happening during each time period. At any
point in time, a customer might be in one of three states: ACTIVE, meaning
that the relationship is still ongoing; STOPPED, meaning that the customer
stopped during that time interval; or CENSORED, meaning that the customer
is not included in the calculation. Table 12.3 shows what happens to the cus­
tomers during each time period.

Table 12.2 Tenure Data for Several Customers

CUSTOMER CENSORED TENURE 5

2 N 4

3 N 3

4 Y 3

5 N 2
6 Y 1

7 N 1
Tracking Customers over Several Time Periods
Table 12.3

CUSTOMER CENSORED LIFETIME TIME 0 TIME 1 TIME 2 TIME 3 TIME 4 TIME 5
1 Y 5 ACTIVE ACTIVE ACTIVE ACTIVE ACTIVE ACTIVE
2 N 4 ACTIVE ACTIVE ACTIVE ACTIVE STOPPED CENSORED
3 N 3 ACTIVE ACTIVE ACTIVE STOPPED CENSORED CENSORED
4 Y 3 ACTIVE ACTIVE ACTIVE ACTIVE CENSORED CENSORED
5 N 2 ACTIVE ACTIVE STOPPED CENSORED CENSORED CENSORED
6 Y 1 ACTIVE ACTIVE CENSORED CENSORED CENSORED CENSORED
7 N 1 ACTIVE STOPPED CENSORED CENSORED CENSORED CENSORED
Hazard Functions and Survival Analysis in Marketing
401
402 Chapter 12


Table 12.4 From Times to Hazards

TIME 0 TIME 1 TIME 2 TIME 3 TIME 4 TIME 5

ACTIVE 7 6 4 3 1 1

STOPPED 0 1 1 1 1 0

CENSORED 0 0 2 3 5 5

HAZARD 0% 14% 20% 25% 50% 0%


Notice in Table 12.4 that the censoring takes place one time unit later than
the lifetime. That is, Customer #1 survived to Time 5, what happens after that
is unknown. The hazard at a given time is the number of customers who are




Y
STOPPED divided by the total of the customers who are either ACTIVE or
STOPPED.




FL
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.
AM
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
TE

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, statisti­
cians often work with hazard rates rather than probabilities. The ideas are
clearly very related, but the mathematics using rates involves daunting inte­
grals, 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 work­
ing with customer data.


Other Types of Censoring
The previous section introduced censoring in two cases: hazards for customers
after they have stopped and hazards for customers who are still active. There

Team-Fly®
Hazard Functions and Survival Analysis in Marketing 403


are other useful cases as well. To explain other types of censoring, it is useful
to go back to the medical realm.
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 (med­
ical researchers have other concerns as well, such as the recurrence of the
disease, but that does not concern us in this simplified example).
So you identify 100 patients, give them the treatment, and their cancers
seem to be cured. You follow them for several years. During this time, seven

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