How to read a survival curve: how long until a customer goes quiet?

Half of the customers of one UK online shop had gone quiet by month 14 (95 percent interval 13 to 15). But the curve behind that number does something more useful than a median can: it shows that the single biggest drop happens in the first month.

This article walks through a survival curve, the Kaplan-Meier curve, using real purchase records. You will learn what the curve shows, what to do with customers who have not gone quiet yet, and where the curve stops telling you anything.

The question and the data

We asked: how long does a customer keep buying before going quiet? We call a customer lapsed when they go six full months with no purchase. The six months is our rule, not a law of nature. A different rule would move every number below. A customer who lapsed is placed at their last purchase month, the point where the quiet began, even though we only know it six months later. Customers still buying are placed at their last purchase month too.

The records are the Online Retail II data, a UK online shop, with prices in pounds (GBP). We kept the 3,978 customers whose first purchase fell between January 2010 and May 2011, so each one has at least six months in which they could have gone quiet. Of those, 2,017 lapsed. The other 1,961 (49.3 percent) were still buying when the data ends.

Basis for every figure here: one UK online shop, GBP, 3,978 customers, six-month lapse rule.

What a survival curve is

Picture 3,978 bars, one per customer. Each bar starts at the customer's first purchase and runs for as many months as they kept buying. Some bars end because the customer lapsed. Some are cut off because the data ends while the customer is still buying.

The survival curve answers one question at every month: of everyone who started, what share is still buying? It starts at 100 percent and can only step down.

Here are the points on this shop's curve, with the 95 percent interval in brackets:

Time here is counted inclusive: first to last purchase month, plus one. A customer who buys in one month only has a time of one month. We count it this way because the tool drops a duration of zero, and that would have removed all the one-month customers.

The median: where the curve crosses 50 percent

The curve crosses 50 percent between month 12 and month 14, and the tool reports the median as 14 months. The median is the time by which half the group has gone quiet. For durations it is a safer summary than the average, because a few very loyal customers can pull an average up a lot while the median barely moves.

The first month is the cliff

Look at the first row of the table. Only 71.34 percent are still buying after one month. That is because 1,140 of the 3,978 customers (28.7 percent) bought in one month only. After that first drop the curve falls slowly: from month 3 to month 12 it loses about 15 points, less than it lost in the first month alone.

What to do with customers who have not gone quiet yet

This is the part that makes survival analysis different from a simple average. Suppose you computed the average time to lapse using only the 2,017 customers who lapsed. You would be ignoring the 1,961 who are still buying, and those are the loyal ones. The average would come out too short.

Survival analysis keeps those customers in. A customer who is still buying leaves the curve at their last purchase month without pulling it down. We know they lasted at least that long, and we use exactly that much. The word for this is censored: we know the customer made it this far, but not what happens next.

One honest cost: we time a censored customer to their last purchase month, not to the end of the data. That makes the later part of the curve a little short.

What this does not tell you

• It is an association in one shop's data from 2010-01 to 2011-11. It does not say why customers go quiet.
• The six-month rule is a choice (the figures in this bullet come from a separate run on the same data, not from the gated key). With a three-month rule the median drops to 12 months; with a nine-month rule the curve never falls to 50 percent within the data, so there is no median. The population changes with the rule (2,389 lapsed customers under the three-month rule, 2,017 under the six-month rule, 1,594 under the nine-month rule).
• There is one group, so there is no comparison between customer types. A comparison would need a second group and a test.
• If the December 2011 rows are dropped, 32 customers move from still buying to lapsed (2,049 and 1,929).

Run it on your own file

Use the free survival tool at /free/standard_survival_v2. You need three columns: a customer id, how long each customer lasted, and a flag that is 1 when the customer lapsed and 0 when they are still buying. If you have a column that splits customers into types, the tool draws one curve per type and compares them.

Questions people ask

Why not just report an average?

Because half the customers are still buying. Leaving them out makes the average too short; counting them at their last purchase date as if they had lapsed is wrong too. The curve handles them correctly.

Does 49.3 percent censored mean the data is bad?

No. It means the data ends while many customers are still active. That is normal for any study that stops before everyone has had the event.

Can I say customers lapse because of the first purchase?

No. The curve shows when, not why.

Sources

Data: Chen, D. (2012). Online Retail II. UCI Machine Learning Repository. CC BY 4.0. A UK online retailer, prices in GBP; customers grouped by first purchase month, 2010-01 to 2011-05, and lapsed means six full months with no purchase.

Method: Kaplan and Meier, "Nonparametric estimation from incomplete observations", Journal of the American Statistical Association 53:457-481 (1958).