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.