100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

60 90 120 150 180 210 240 270 300 330 360

Days after Deactivation

Figure 12.12 Survival curve (upper curve) and hazards (lower curve) for reactivation of mobile telephone customers.

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After 90 days, the hazards are practically zero-customers do not reactivate. Once again, the business processes provide guidance. Telephone numbers are reserved for 90 days after customers leave. Normally, when customers reactivate, they want to keep the same telephone number. After 90 days, the number may have been reassigned, and the customer would have to get a new telephone number.

This discussion has glossed over the question of how new (reactivated) customers were associated with the expired accounts. In this case, the analysis used the telephone numbers in conjunction with an account ID. This pretty much guaranteed that the match was accurate, since reactivated customers retained their telephone numbers and billing information. This is very conservative but works for finding reactivations. It does not work for finding other types of winback, such as customers who are willing to cycle through telephone numbers in order to get introductory discounts.

Another approach is to try to identify individuals over time, even when they are on different accounts. For businesses that collect Social Security numbers or drivers license numbers as a regular part of their business, such identifying numbers can connect accounts together over time. (Be aware that not everyone who is asked to supply this kind of identifying information does so accurately.) Sometimes matching names, addresses, telephone numbers, and/or credit cards is sufficient for matching purposes. More often, this task is outsourced to a company that assigns individual and household IDs, which then provide the information needed to identify which new customers are really former customers who have been won back.

Studying initial covariates adds even more information. In this case, initial means whatever is known about the customer at the point of deactiva-tion. This includes not only information such as initial product and promotion,

but also customer behavior before deactivating. Are customers who complain a lot more or less likely to reactivate? Customers who roam? Customers who pay their bills late?

This example shows the use of hazards to understand a classic time-to-event question. There are other questions of this genre amenable to survival analysis:

When customers start on a minimum pricing plan, how long will it be before they upgrade to a premium plan?

When customers upgrade to a premium plan, how long will it be before they downgrade?

What is the expected length of time between purchases for customers, given past customer behavior and the fact that different customers have different purchase periods?

One nice aspect of using survival analysis is that it is easy to ask about the effects of different initial conditions-such as the number of times that a customer has visited in the past. Using proportional hazards, it is possible to determine which covariates have the most effect on the desired outcome, including which interventions are most and least likely to work.

Forecasting

Another interesting application of survival analysis is forecasting the number of customers into the future, or equivalently, the number of stops on a given day in the future. In the aggregate, survival does a good job of estimating how many customers will stick around for a given length of time.

There are two components to any such forecast. The first is a model of existing customers, which can take into account various covariates during the customers life cycle. Such a model works by applying one or more survival models to all customers. If a customer has survived for 100 days, then the probability of stopping tomorrow is the hazard at day 100. To calculate the chance of stopping the day after tomorrow, first assume that the customer does not stop tomorrow and then does stop on day 101. This is the conditional survival (one minus the hazard-the probability of not stopping) at day 100 times the hazard for day 101. Applying this to all customer tenures, it is possible to forecast stops of existing customers in the future.

Figure 12.13 shows such a forecast for stops for 1 month, developed by survival expert Will Potts. Also shown are the actual values observed during this period. The survival-based forecast proves to be quite close to what is actually happening. By the way, this particular survival estimate used a parametric model on the hazards rather than empirical hazard rates; the model was able to take into account the day of the week. This results in the weekly cycle of stops evident in the graph.

Actual ш m m Predicted

l-i-i-i-i-i-I i--I--I i-i-i-i-i-i-i-i-i-i-i i-i-i-i-i-r

1 3 5 7 9 111315171921 23252729 31

Day of Month

Figure 12.13 Survival analysis can also be used for forecasting customer stops.

The second component of a customer-level forecast is a bit more difficult to calculate. This component is the effect of new customers on the forecast, and the difficulty is not technical. The challenge is getting estimates for new starts. Fortunately, there are often budget forecasts that contain new starts, sometimes broken down by product, channel, or geography. It is possible to refine the survival models to take into account these effects. Of course, the forecast is only as accurate as the budget. The upside, though, is that the forecast, based on survival techniques, can be incorporated into the process of managing actual levels against budgeted levels.

The combination of these components-stop forecasts for existing customers and stop forecasts for new customers-makes it possible to develop estimates of customer levels into the future. The authors have worked with clients who have taken these forecasts forward years. Because the models for new customers included the acquisition channel, the forecasting model made it possible to optimize the future acquisition channel mix.

Hazards Changing over Time

One of the more difficult issues in survival analysis is whether the hazards themselves are constant or whether they change over time. The assumption in scientific studies is that hazards do not change. The goal of scientific survival analysis is to obtain estimates of the real hazard in various situations.