Survival Analysis When No One Dies: A Value-Based Approach

 

A generalized version of Kaplan-Meier allows to model a continuous value (like money) instead of a binary signal (like survival)

Survival analysis is a statistical approach used to answer the question: “How long will something last?” That “something” could range from a patient’s lifespan to the durability of a machine component or the duration of a user’s subscription.

One of the most widely used tools in this area is the Kaplan-Meier estimator.

Born in the world of biology, Kaplan-Meier made its debut tracking life and death. But like any true celebrity algorithm, it didn’t stay in its lane. These days, it’s showing up in business dashboards, marketing teams, and churn analyses everywhere.

But here’s the catch: business isn’t biology. It’s messy, unpredictable, and full of plot twists. This is why there are a couple of issues that make our lives more difficult when we try to use survival analysis in the business world.

First of all, we are typically not just interested in whether a customer has “survived” (whatever survival could mean in this context), but rather in how much of that individual’s economic value has survived.

Secondly, contrary to biology, it’s very possible for customers to “die” and “resuscitate” multiple times (think of when you unsubscribe/resubscribe to an online service).

In this article, we will see how to extend the classical Kaplan-Meier approach so that it better suits our needs: modeling a continuous (economic) value instead of a binary one (life/death) and allowing “resurrections”.

A refresher on the Kaplan-Meier estimator

Let’s pause and rewind for a second. Before we start customizing Kaplan-Meier to fit our business needs, we need a quick refresher on how the classic version works.

Suppose you had 3 subjects (let’s say lab mice) and you gave them a medicine you need to test. The medicine was given at different moments in time: subject a received it in January, subject b in April, and subject c in May.

Then, you measure how long they survive. Subject a died after 6 months, subject c after 4 months, and subject b is still alive at the time of the analysis (November).

Now, even if we wanted to measure a simple metric, like average survival, we would face a problem. In fact, we don’t know how long subject b will survive, as it is still alive today.

This is a classical problem in statistics, and it’s called “right censoring“.

Right censoring is stats-speak for “we don’t know what happened after a certain point” and it’s a big deal in survival analysis. So big that it led to the development of one of the most iconic estimators in statistical history: the Kaplan-Meier estimator, named after the duo who introduced it back in the 1950s.

So, how does Kaplan-Meier handle our problem?

First, we align the clocks. Even if our mice were treated at different times, what matters is time since treatment. So we reset the x-axis to zero for everyone — day zero is the day they got the drug.

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