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ogy come from medical research and failure analysis in manufacturing, the
concepts are tailor made for marketing. Survival tells us when to start worry­
ing about customers doing something important, such as ending their rela­
tionship. It tells us which factors are most correlated with the event. Hazards
and survival curves also provide snapshots of customers and their life cycles,
answering questions such as: “How much should we worry that this customer
is going to leave in the near future?” or “This customer has not made a pur­
chase recently; is it time to start worrying that the customer will not return?”
The survival approach is centered on the most important facet of customer
behavior: tenure. How long customers have been around provides a wealth of
information, especially when tied to particular business problems. How long
customers will remain customers in the future is a mystery, but a mystery that
past customer behavior can help illuminate. Almost every business recognizes
the value of customer loyalty. As we see later in this chapter, a guiding principle

383
384 Chapter 12


of loyalty”that the longer customers stay around, the less likely they are to stop
at any particular point in time”is really a statement about hazards.
The world of marketing is a bit different from the world of medical research.
For one thing, the consequences of our actions are much less dire: a patient
may die from poor treatment, whereas the consequences in marketing are
merely measured in dollars and cents. Another important difference is the vol­
ume of data. The largest medical studies have a few tens of thousands of par­
ticipants, and many draw conclusions from a just a few hundred. When trying
to determine mean time between failure (MTBF) or mean time to failure
(MTTF)”manufacturing lingo for how long to wait until an expensive piece of
machinery breaks down”conclusions are often based on no more than a few
dozen failures.
In the world of customers, tens of thousands is the lower limit, since cus­
tomer databases often contain data on millions of customers and former
customers. Much of the statistical background of survival analysis is focused
on extracting every last bit of information out of a few hundred data points. In
data mining applications, the volumes of data are so large that statistical con­
cerns about confidence and accuracy are replaced by concerns about manag­
ing large volumes of data.
The importance of survival analysis is that it provides a way of understand­
ing time-to-event characteristics, such as:
When a customer is likely to leave
––

The next time a customer is likely to migrate to a new customer segment
––

The next time a customer is likely to broaden or narrow the customer
––

relationship
The factors in the customer relationship that increase or decrease likely
––

tenure
The quantitative effect of various factors on customer tenure
––


These insights into customers feed directly into the marketing process. They
make it possible to understand how long different groups of customers are
likely to be around”and hence how profitable these segments are likely to be.
They make it possible to forecast numbers of customers, taking into account
both new acquisition and the decline of the current base. Survival analysis also
makes it possible to determine which factors, both those at the beginning
of customers™ relationships as well as later experiences, have the biggest effect
on customers™ staying around the longest. And, the analysis can be applied to
things other then the end of the customer tenure, making it possible to deter­
mine when another event”such as a customer returning to a Web site”is no
longer likely to occur.
A good place to start with survival is with visualizing customer retention,
which is a rough approximation of survival. After this discussion, we move
on to hazards, the building blocks of survival. These are in turn combined into
Hazard Functions and Survival Analysis in Marketing 385


survival curves, which are similar to retention curves but more useful. The
chapter ends with a discussion of Cox Proportional Hazard Regression and
other applications of survival analysis. Along the way, the chapter provides
particular applications of survival in the business context. As with all statisti­
cal methods, there is a depth to survival that goes far beyond this introductory
chapter, which is consciously trying to avoid the complex mathematics under­
lying these techniques.


Customer Retention
Customer retention is a concept familiar to most businesses that are concerned
about their customers, so it is a good place to start. Retention is actually a close
approximation to survival, especially when considering a group of customers
who all start at about the same time. Retention provides a familiar framework
to introduce some key concepts of survival analysis such as customer half-life
and average truncated customer tenure.


Calculating Retention
How long do customers stay around? This seemingly simple question
becomes more complicated when applied to the real world. Understanding
customer retention requires two pieces of information:
When each customer started
––

When each customer stopped
––


The difference between these two values is the customer tenure, a good
measurement of customer retention.
Any reasonable database that purports to be about customers should have
this data readily accessible. Of course, marketing databases are rarely simple.
There are two challenges with these concepts. The first challenge is deciding
on what is a start and stop, a decision that often depends on the type of busi­
ness and available data. The second challenge is technical: finding these start
and stop dates in available data may be less obvious than it first appears.
For subscription and account-based businesses, start and stop dates are well
understood. Customers start magazine subscriptions at a particular point in
time and end them when they no longer want to pay for the magazine.
Customers sign up for telephone service, a banking account, ISP service, cable
service, an insurance policy, or electricity service on a particular date and
cancel on another date. In all of these cases, the beginning and end of the rela­
tionship is well defined.
Other businesses do not have such a continuous relationship. This is particu­
larly true of transactional businesses, such as retailing, Web portals, and cata­
logers, where each customer™s purchases (or visits) are spread out over time”or
386 Chapter 12


may be one-time only. The beginning of the relationship is clear”usually the
first purchase or visit to a Web site. The end is more difficult but is sometimes
created through business rules. For instance, a customer who has not made a
purchase in the previous 12 months may be considered lapsed. Customer reten­
tion analysis can produce useful results based on these definitions. A similar
area of application is determining the point in time after which a customer is no
longer likely to return (there is an example of this later in the chapter).
The technical side can be more challenging. Consider magazine subscrip­
tions. Do customers start on the date when they sign up for the subscription?
Do customers start when the magazine first arrives, which may be several
weeks later? Or do they start when the promotional period is over and they
start paying?
Although all three questions are interesting aspects of the customer relation­
ship, the focus is usually on the economic aspects of the relationship. Costs
and/or revenue begin when the account starts being used”that is, on the issue
date of the magazine”and end when the account stops. For understanding
customers, it is definitely interesting to have the original contact date and time,
in addition to the first issue date (are customers who sign up on weekdays dif­
ferent from customers who sign up on weekends?), but this is not the beginning
of the economic relationship. As for the end of the promotional period, this is
really an initial condition or time-zero covariate on the customer relationship.
When the customer signs up, the initial promotional period is known. Survival
analysis can take advantage of such initial conditions for refining models.


What a Retention Curve Reveals
Once tenures can be calculated, they can be plotted on a retention curve, which
shows the proportion of customers that are retained for a particular period of
time. This is actually a cumulative histogram, because customers who have
tenures of 3 months are included in the proportions for 1 month and 2 months.
Hence, a retention curve always starts at 100 percent.
For now, let™s assume that all customers start at the same time. Figure 12.1,
for instance, compares the retention of two groups of customers who started at
about the same point in time 10 years ago. The points on the curve show the
proportion of customers who were retained for 1 year, for 2 years, and so on.
Such a curve starts at 100 percent and gradually slopes downward. When a
retention curve represents customers who all started at about the same time”
as in this case”it is a close approximation to the survival curve.
Differences in retention among different groups are clearly visible in the
chart. These differences can be quantified. The simplest measure is to look at
retention at particular points in time. After 10 years, for instance, 24 percent of
the regular customers are still around, and only about a third of them even
make it to 5 years. Premium customers do much better. Over half make it to 5
years, and 42 percent have a customer lifetime of at least 10 years.
Hazard Functions and Survival Analysis in Marketing 387

100%
90%

80% High End
Percent Survived 70% Regular
60%

50%
40%
30%
20%

10%
0%
0 12 24 36 48 60 72 84 96 108 120
Tenure (Months after Start)
Figure 12.1 Retention curves show that high-end customers stay around longer.


Another way to compare the different groups is by asking how long it takes
for half the customers to leave”the customer half-life (although the statistical
term is the median customer lifetime). The median is a useful measure because
the few customers who have very long or very short lifetimes do not affect it.
In general, medians are not sensitive to a few outliers.
Figure 12.2 illustrates how to find the customer half-life using a retention
curve. This is the point where exactly 50 percent of the customers remain,
which is where the 50 percent horizontal grid line intersects the retention
curve. The customer half-life for the two groups shows a much starker differ­
ence than the 10-year survival”the premium customers have a median life­
time of close to 7 years, whereas the regular customers have a median a bit
under over 2 years.

Finding the Average Tenure from a Retention Curve
The customer half-life is useful for comparisons and easy to calculate, so it is a
valuable tool. It does not, however, answer an important question: “How
much, on average, were customers worth during this period of time?”
Answering this question requires having an average customer worth per time
and an average retention for all the customers. The median cannot provide this
information because the median only describes what happens to the one cus­
tomer in the middle; the customer at exactly the 50 percent rank. A question
about average customer worth requires an estimate of the average remaining
lifetime for all customers.
There is an easy way to find the average remaining lifetime: average cus­
tomer lifetime during the period is the area under the retention curve. There is
a clever way of visualizing this calculation, which Figure 12.3 walks through.
388 Chapter 12

100%
90%

80%
Percent Survived

70% High End
60% Regular
50%
40%
30%
20%

10%
0%
0 12 24 36 48 60 72 84 96 108 120
Tenure (Months after Start)
Figure 12.2 The median customer lifetime is where the retention curve crosses the
50 percent point.


First, imagine that the customers all lie down with their feet lined up on
the left. Their heads represent their tenure, so there are customers of all differ­
ent heights (or widths, because they are horizontal) for customers of all
different tenures. For the sake of visualization, the longer tenured customers
lie at the bottom holding up the shorter tenured ones. The line that connects
their noses counts the number of customers who are retained for a particular
period of time (remember the assumption that all customers started at about
the same point in time). The area under this curve is the sum of all the cus­
tomers™ tenures, since every customer lying horizontally is being counted.
Dividing the vertical axis by the total count produces a retention curve.
Instead of count, there is a percentage. The area under the curve is the total
tenure divided by the count of customers”voilà, the average customer tenure
during the period of time covered by the chart.

T I P The area under the customer retention curve is the average customer
lifetime for the period of time in the curve. For instance, for a retention curve

that has 2 years of data, the area under the curve represents the two-year

average tenure.


This simple observation explains how to obtain an estimate of the average
customer lifetime. There is one caveat when some customers are still active. The
average is really an average for the period of time under the retention curve.
Consider the earlier retention curve in this chapter. These retention curves
were for 10 years, so the area under the curves is an estimate of the average cus­
tomer lifetime during the first 10 years of their relationship. For customers who are still
active at 10 years, there is no way of knowing whether they will all leave at 10
years plus one day; or if they will all stick around for another century. For this rea­
son, it is not possible to determine the real average until all customers have left.

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