A Statistical Approach to Customer Lifetime Value - Part 3: Duration Dependence
Do retention rates increase over time?
Last time, on CLV, we discussed a very basic approach to modeling customer lifetime value: the linear churn. The basic idea behind linear churn is as follows:
- Every period, a customer flips a weighted coin. If it lands on heads, they churn.
- The coins stay the same over time (i.e. coin flips are independent of each other).
- Everyone has the same coin.
In practice, a subscriber that has been with you for the past 5 years probably has a much lower churn rate than someone who just joined. In this blog post, we’re going to explore the concept of duration dependence, the idea that subscribers may be more or less likely to churn as time goes on.
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A quick review on linear churn
Recall that an exponential model assumes that every subscriber flips the same coin each period, determining whether or not they will churn. The end result is a kind-of-curved line through your data that minimizes the “error” between the actual churn and the expected churn. The equation for is as follows:
Our initial exponential model didn’t fare too well, however part of that was due to some unusual data in the first 6 months. For this blog post, I have removed the first 6 months of data and re-indexed the subscribers to start at 100% in the 7th period. I then re-ran the linear model as before, but this time on all the data (instead of the first 12 months). This is now our new starting point:
So let’s relax the second assumption, that the coin stays the same. After all, a customer that stays with you for 10 years probably has a small probability of churning after so much time. In order to model this phenomenon, we’re going to utilize the Weibull distribution. For our purposes, we’re going to use the discrete Weibull distribution, seeing as our time period (months), is discrete, not continuous.
The Weibull Distribution
While we initially assumed that our \(\theta\) was constant, what happens when we allow it to increase or decrease? The Weibull distribution helps model how a failure rate (or in this case a churn rate) changes over time, making it perfect for our analysis. The Weibull function has two parameters: \(\theta\) and \(\beta\). The \(\theta\) is the same as before: the underlying probability that someone will churn. The \(beta\) is the shape parameter that shows us how the \(\theta\) changes over time. There are three values \(\beta\) can take:
- \(\beta\) = 1. This means that the churn rate stays constant, and we fall back to our earlier exponential model.
- \(\beta\) > 1. This means that churn increases over time, meaning you’re more likely to unsubscribe or “fail” every new period.
- \(\beta\) < 1. This means that churn decreases over time, implying higher loyalty.
The survivor function for the discrete Weibull is as follows:
The only thing that’s changed between the discrete Weibull and the exponential term is that we now raise our retention rate to a \(\beta\) before we raise to the the \(t\) parameter. Solving again, we get this graph:
Pretty neat! The model looks good, and it predicts pretty well what’s happening in the data. Let’s look a little deeper at our calculations:
Variables
| Calculations | Number |
|---|---|
| Churn Rate | 14.2% |
| C | 0.55 |
Results
| Model | % Error | Log-likelihood |
|---|---|---|
| Exponential Churn | 11.73% | -89,167 |
| Discrete Weibull | 0.57% | -84,763 |
In this case, our actual churn rate is 14.2%, but there is a very small “c”, implying that churn significantly decreases each period. For example, churn drops from 14.2% initially to 6%, 4% and then 3.6% in periods 2, 3 and 4. By period 22, churn is steady at 1%. Compared to the exponential churn model, this model has way lower % error (good) and way higher log-likelihood (also good). The story this model tells, an initial high drop off, then high loyalty as time goes on, jives with me and fits the data well.
While it may be tempting to assume that churn is constant across periods, in practice, that model (the exponential churn) does not fit the data at all. In this post, we discussed utilizing the discrete Weibull distribution to allow that churn rate to decrease over time. The output of the model tells a compelling business story and is statistically significant! Next time, we’ll look at an alternative approach, the beta geometric!