0%

1 2 3 4 5 6 7 8 9 10 11 12 13

Tenure (Years)

The parametric curves that fit a retention curve do not fit well beyond the range where

they are defined.

Of course, this illustration does not prove that a parametric approach will

not work. Perhaps there is some function out there that, with the right

parameters, would fit the observed retention curve very well and continue

working beyond the range used to define the parameters. However, this

example does illustrate the challenges of using a parametric approach for

approximating survival curves directly, and it is consistent with our experience

even when using more data points. Functions that provide a good fit to the

retention curve turn out to diverge pretty quickly.

Another way of describing this is that the customers who have been around

for 1 year are going to behave just like new customers. Consider a group of 100

customers of various tenures, 50 leave in the following year, regardless of the

tenure of the customers at the beginning of the year”exponential decay says

that half are going to leave regardless of their initial tenure. That means that

customers who have been around for a while are no more loyal then newer cus

tomers. However, it is often the case that customers who have been around for

a while are actually better customers than new customers. For whatever reason,

longer tenured customers have stuck around in the past and are probably a bit

less likely than new customers to leave in the future. Exponential decay is a bad

situation, because it assumes the opposite: that the tenure of the customer rela

tionship has no effect on the rate that customers are leaving (the worst-case sce

nario would have longer term customers leaving at consistently higher rates

than newer customers, the “familiarity breeds contempt” scenario).

394 Chapter 12

Hazards

The preceding discussion on retention curves serves to show how useful reten

tion curves are. These curves are quite simple to understand, but only in terms

of their data. There is no general shape, no parametric form, no grand theory

of customer decay. The data is the message.

Hazard probabilities extend this idea. As discussed here, they are an exam

ple of a nonparametric statistical approach”letting the data speak instead of

finding a special function to speak for it. Empirical hazard probabilities simply

let the historical data determine what is likely to happen, without trying to fit

data to some preconceived form. They also provide insight into customer

retention and make it possible to produce a refinement of retention curves

called survival curves.

The Basic Idea

A hazard probability answers the following question:

Assume that a customer has survived for a certain length of time, so the cus-

tomer™s tenure is t. What is the probability that the customer leaves before t+1?

Another way to phrase this is: the hazard at time t is the risk of losing

customers between time t and time t+1. As we discuss hazards in more detail,

it may sometimes be useful to refer to this definition. As with many seemingly

simple ideas, hazards have significant consequences.

To provide an example of hazards, let™s step outside the world of business

for a moment and consider life tables, which describe the probability of

someone dying at a particular age. Table 12.1 shows this data, for the U.S. pop

ulation in 2000:

Table 12.1 Hazards for Mortality in the United States in 2000, Shown as a Life Table

AGE PERCENT OF POPULATION THAT

DIES IN EACH AGE RANGE

0“1 yrs 0.73%

1“4 yrs 0.03%

5“9 yrs 0.02%

10“14 yrs 0.02%

15“19 yrs 0.07%

20“24 yrs 0.10%

25“29 yrs 0.10%

30“34 yrs 0.12%

Hazard Functions and Survival Analysis in Marketing 395

Table 12.1 (continued)

AGE PERCENT OF POPULATION THAT

DIES IN EACH AGE RANGE

35“39 yrs 0.16%

40“44 yrs 0.24%

45“49 yrs 0.36%

50“54 yrs 0.52%

55“59 yrs 0.80%

60“64 yrs 1.26%

65“69 yrs 1.93%

70“74 yrs 2.97%

75“79 yrs 4.56%

80“84 yrs 7.40%

85+ yrs 15.32%

A life table is a good example of hazards. Infants have about a 1 in 137

chance of dying before their first birthday. (This is actually a very good rate; in

less-developed countries the rate can be many times higher.) The mortality

rate then plummets, but eventually it climbs steadily higher. Not until some

one is about 55 years old does the risk rise as high as it is during the first year.

This is a characteristic shape of some hazard functions and is called the bathtub

shape. The hazards start high, remain low for a long time, and then gradually

increase again. Figure 12.5 illustrates the bathtub shape using this data.

3.0%

2.5%

2.0%

Hazard

1.5%

1.0%

0.5%

0.0%

0-1 yrs

1-4 yrs

5-9 yrs

10-14 yrs

15-19 yrs

20-24 yrs

25-29 yrs

30-34 yrs

35-39 yrs

40-44 yrs

45-49 yrs

50-54 yrs

55-59 yrs

60-64 yrs

65-69 yrs

70-74 yrs

Age (Years)

Figure 12.5 The shape of a bathtub-shaped hazard function starts high, plummets, and then

gradually increases again.

396 Chapter 12

The same idea can be applied to customer tenure, although customer haz

ards are more typically calculated by day, week, or month instead of by year.

Calculating a hazard for a given tenure t requires only two pieces of data. The

first is the number of customers who stopped at time t (or between t and t+1).

The second is the total number of customers who could have stopped during

this period, also called the population at risk. This consists of all customers

whose tenure is greater than or equal to t, including those who stopped at time

t. The hazard probability is the ratio of these two numbers, and being a proba

bility, the hazard is always between 0 and 1. These hazard calculations are pro

vided by life table functions in statistical software such as SAS and SPSS. It is

also possible to do the calculations in a spreadsheet using data directly from a

customer database.

One caveat: In order for the calculation to be accurate, every customer

included in the population count must have the opportunity to stop at that par

ticular time. This is a property of the data used to calculate the hazards, rather

than the method of calculation. In most cases, this is not a problem, because haz

ards are calculated from all customers or from some subset based on initial con

ditions (such as initial product or campaign). There is no problem when a

customer is included in the population count up to that customer™s tenure, and

the customer could have stopped on any day before then and still be in the data set.

An example of what not to do is to take a subset of customers who have

stopped during some period of time, say in the past year. What is the problem?

Consider a customer who stopped yesterday with 2 years of tenure. This cus

tomer is included in all the population counts for the first year of hazards.

However, the customer could not have stopped during the first year of tenure.

The stop would have been more than a year in the past and precluded the

customer from being in the data set. Because customers who could not have

stopped are included in the population counts, the population counts are too

big making the initial hazards too low. Later in the chapter, an alternative

method is explained to address this issue.

WA R N I N G To get accurate hazards and survival curves, use groups of

customers who are defined only based on initial conditions. In particular, do

not define the group based on how or when the members left.

When populations are large, there is no need to worry about statistical

ideas such as confidence and standard error. However, when the populations

are small”as they are in medical research studies or in some business

applications”then the confidence interval may become an issue. What this

means is that a hazard of say 5 percent might really be somewhere between 4

percent and 6 percent. When working with smallish populations (say less than

a few thousand), it might be a good idea to use statistical methods that provide

Hazard Functions and Survival Analysis in Marketing 397

information about standard errors. For most applications, though, this is not

an important concern.

Examples of Hazard Functions

At this point, it is worth stopping and looking at some examples of hazards.

These examples are intended to help in understanding what is happening, by

looking at the hazard probabilities. The first two examples are basic, and, in

fact, we have already seen examples of them in this chapter. The third is from

real-world data, and it gives a good flavor of how hazards can be used to

provide an x-ray of customers™ lifetimes.

Constant Hazard

The constant hazard hardly needs a picture to explain it. What it says is that

the hazard of customers leaving is exactly the same, no matter how long the

customers have been around. This looks like a horizontal line on a graph.

Say the hazard is being measured by days, and it is a constant 0.1 percent.

That is, one customer out of every thousand leaves every day. After a year (365

days), this means that about 30.6 percent of the customers have left. It takes

about 692 days for half the customers to leave. It will take another 692 days for

half of them to leave. And so on, and so on.

The constant hazard means the chance of a customer leaving does not vary

with the length of time the customer has been around. This sounds a lot like

the exponential retention curve, the one that looks like the decay of radioactive

elements. In fact, a constant retention hazard would conform to an exponential

form for the retention curve. We say “would” simply because, although this

does happen in physics, it does not happen much in marketing.

Bathtub Hazard

The life table for the U.S. population provided an example of the bathtub-

shaped hazard function. This is common in the life sciences, although bathtub

shaped curves turn up in other domains. As mentioned earlier, the bathtub haz

ard initially starts out quite high, then it goes down and flattens out for a long

time, and finally, the hazards increase again.

One phenomenon that causes this is when customers are on contracts (for

instance, for cell phones or ISP services), typically for 1 year or longer. Early in

the contract, customers stop because the service is not appropriate or because

they do not pay. During the period of the contract, customers are dissuaded

from canceling, either because of the threat of financial penalties or perhaps

only because of a feeling of obligation to honor the terms of the initial contract.