WD5BA.docxRead the section titled "Customer Survival" in the Supplementary Text on pp. 360-367, and answer the following questions (over the course of the week, NOT all in one post):
· How would you explain the meaning of the survival curve shown in Figure 10-3 to a senior manager with no statistical background?
· How would you go about determining the average and median customer lifetimes from the data in Figure 10-3?
· What are some ways a company could use the results of a customer survival analysis like the one described in this section?
Customer Survival Survival is a good way to measure customer retention, a concept familiar to most businesses that are concerned about their customers. Survival curves also provide a well-grounded framework for understanding customer retention, including important measures such as customer half-life and average truncated tenure. What Survival Curves Reveal A survival curve shows the proportion of customers that are expected to survive up to a particular point in tenure, based on the historical information of how long customers have survived in the past. The curve always starts at 100 percent and then descends; a survival curve may flatten out, but it never increases. The curve may descend all the way to zero, but typically does not, indicating that some customers with long tenures remain active. Stopping is a one-time event. After a customer has stopped, the customer cannot come back. Figure 10-1 compares the survival of two groups of customers over a period of ten years. The points on the curve show the proportion of customers who are expected to survive for one year, for two years, and so on. The picture clearly shows that one group is better than the other. How can this difference be quantified? The simplest measure is the survival value at particular points in time. After ten 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. More than half make it to 5 years, and 42 percent have a customer lifetime of at least 10 years. Another way to compare different groups is by asking how long it takes for half the customers to leave—the customer half-life (the statistical term is the median customer lifetime). The half-life 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 10-2 illustrates how to find the customer half-life using a survival curve. This is the point where exactly 50 percent of the customers remain, which is where the 50 percent horizontal gridline intersects the survival curve. The customer half-life for the two groups shows a much starker difference than the ten-year survival—the premium customers have a median lifetime of close to seven years, whereas the regular customers have a median a bit less than two years. Figure 10-1: Survival curves show that high-end customers stay around longer. Figure 10-2: The median customer lifetime is where the retention curve crosses the 50 percent point. Finding the Average Tenure from a Survival Curve The customer half-life is useful for comparisons and easy to calculate. It does not, however, answer an important question: “How much, on average, are customers worth during this period of time?” Answering this question requires having an average customer worth per time and an average survival for all the customers. The median cannot provide this information because the median only describes what happens to the one customer in the middle; the one who leaves when exactly half the original customers have left. The average remaining lifetime is the area under the retention curve. Finding the area under the curve may seem daunting—particularly for readers who may have memories of calculus. In fact, the process is quite easy. Figure 10-3 shows a survival curve with a rectangle holding up each point. The base of each rectangle has a length of 1, measured in the units of the horizontal axis. The height is the survival value. The area under the curve is the sum of the areas of these rectangles. Figure 10-3: Circumscribing each point with a rectangle makes it clear how to approximate the area under the survival curve. The area of a rectangle is base times height, which is 1 multiplied by the survival value. The sum of all the rectangles, then, is just the sum of all the survival values in the curve—an easy calculation in a spreadsheet. Voilà, an easy way to calculate the area and quite an interesting observation as well: The sum of the survival values (as percentages) is the average customer lifetime. Notice also that each rectangle has a width of one time unit, in whatever the units are of the horizontal axis. So, the units of the average are also in the units of the horizontal axis. TIP The area under the survival curve is the average customer lifetime for the period of time in the curve. For instance, for a survival curve that has two 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 small clarification. The average is really an average for the period of time under the survival curve. Consider the pair of survival curves in the previous figure. These survival curves were for ten years, so the area under them is an estimate of the average customer lifetime during the first 10 years of the customer relationship. For customers who are still active at ten years, there is no way of knowing whether they will all leave at ten years plus one day; or if they will all stick around for another century. For this reason, determining the real average is not possible until all customers have left. This value, called truncated mean lifetime by statisticians, is very useful. As shown in Figure 10-4, the better customers have an average 10-year lifetime of 6.1 years; the other group has an average of 3.7 years. If, on average, a customer is worth, say, $1,000 per year, then the premium customers are worth $6,100 – $3,700 = $2,400 more than the regular customers during the 10 years after they start, or about $240 per year. This $240 might represent the return on a retention program designed specifically for the premium customers, or it might give an upper limit of how much to budget for such retention programs. Figure 10-4: Average customer lifetime for different groups of customers can be compared using the areas under the survival curve. Customer Retention Using Survival How long do customers stay around? This seemingly simple question becomes more complicated when applied to the real world. Understanding customer survival requires two pieces of information: ▪ When each customer starts ▪ When each customer stops The difference between these two values is the customer tenure. Any reasonable database that purports to be about customers should have this data readily accessible. There are two challenges. The first is deciding on the business definition of a start and a stop. The second is technical: finding 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, Internet 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 relationship is well defined. Even when the business definition is well understood, translating it to the technical data can be challenging. Consider magazine subscriptions. 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? In making decisions about what data to use, 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 interesting to have the original contact date and time in addition to the first issue date (are customers who sign up on weekdays different 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. Many businesses do not have continuous, account-based relationships with their customers. Examples are retailing, web portals, and hotels. Each customer's purchases (or visits) are spread out over time—or may be one-time only. The beginning of the relationship is clear—usually the first purchase or visit. 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. Looking at Survival as Decay Although the authors don't generally advocate comparing customers to radioactive materials, the comparison is useful for understanding survival. Think of customers as a lump of uranium that is slowly, radioactively decaying into lead. The “good” customers are the uranium; the ones who have left are the lead. Over time, the amount of uranium left in the lump looks something like the survival curves seen earlier, with the perhaps subtle difference that the timeframe for uranium is measured in billions of years, as opposed to smaller time scales for human activities. One very useful characteristic of uranium is that scientists have determined how to calculate exactly how much uranium will survive after a certain amount of time. They are able to do this because they have built mathematical models that describe radioactive decay, and these have been verified experimentally. Radioactive materials have a process of decay described as exponential decay. This means that the same proportion of uranium turns into lead during a given amount of time, regardless of how much has already done so. The most common form of uranium has a half-life of about 4.5 billion years. So, about half the lump of uranium has turned into lead after this time. After another 4.5 billion years, half the remaining uranium will decay, leaving only a quarter of the original lump as uranium and three-quarters as lead. WARNING Exponential decay has many useful properties for predicting beyond the range of observations. Unfortunately, time-to-event problems in business hardly ever exhibit exponential decay. What makes exponential decay so nice is that the decay fits a nice simple equation that describes how much uranium is around at any given point in time. Wouldn't it be nice to have such an equation for customer survival? It would be very nice, but it is unlikely, as shown in the example in the sidebar “Parametric Approaches Do Not Work for Survival of Customers.” To shed some light on the issue, imagine a world where customers did exhibit exponential decay. For the purposes of discussion, these customers have a half-life of one year. Of 100 customers starting on a particular date, exactly 50 would still be active one year later. After two years, 25 would be active and 75 stopped. Exponential decay would make forecasting the number of customers in the future easy. The problem with this scenario is that the customers who have been around for one year are behaving 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 customers. In the real world, 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 approximation, because it assumes the opposite: that the tenure of the customer relationship has no effect on the rate that customers are leaving (an even worse scenario would have longer term customers leaving at consistently higher rates than newer customers—the “familiarity breeds contempt” scenario). Parametric Approaches Do Not Work for Survival of Customers Trying to fit some known function to a survival curve is tempting. This approach is an example of regression, where a few parameters describe the shape of the survival curve. The power of a parametric approach is that it can be used to estimate survival values in the future, beyond the data used to generate the curves. A line is the most common shape for a regression function. A line has two parameters, the slope of the line and where it intersects the y-axis. Another common shape is a parabola, which has an additional X2 term, so a parabola has three parameters. The exponential that describes radioactive decay actually has only one parameter, the half-life. The following figure shows part of a survival curve for seven out of thirteen years of data. The figure also shows three best-fit curves. The statistical measure of fit is R2, which varies from 0 to 1. Values over 0.9 are quite good, so by standard statistical measures, all these curves fit very well. Fitting parametric curves to a survival curve is easy. The real question, though, is not how well these curves fit the data in the range used to define it, but rather how well do these curves work beyond the original range? The next figure answers this question. It extrapolates the curves ahead another five years. Quickly, the curves diverge from the actual values, and the difference seems to get larger farther out. 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 does not work in all cases. Perhaps some function is out there that, with the right parameters, would fit the observed survival curve for customers 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 the author's experience even when using more data points. What looks like a very good fit to the survival curve in one range turns out to diverge pretty quickly outside that range.
