Extending Kaplan-Meier for Business: A Value-Based Survival Analysis

# Extending Kaplan-Meier for Business: A Value-Based Survival Analysis

## Introduction

Survival Analysis is a powerful statistical tool traditionally used to answer the question: “How long will something last?” While it originated in biology to study lifespans, its applications have expanded into business, marketing, and engineering. The Kaplan-Meier estimator is one of the most widely used methods in survival analysis, but its binary nature (alive/dead) limits its effectiveness in business contexts where outcomes are more nuanced.

In this article, we’ll explore:
– Why traditional Kaplan-Meier falls short in business
– How to extend it to handle continuous economic value
– A practical approach to Value Kaplan-Meier
– Real-world applications and benefits

## The Problem with Traditional Kaplan-Meier in Business

Kaplan-Meier was designed for biological studies where outcomes are clear-cut: a patient either survives or dies. However, in business:

### 1. “Death” Is Ambiguous
– In e-commerce, does “churn” mean account deletion, inactivity, or reduced spending?
– Defining a threshold (e.g., “2 months of no purchases”) is arbitrary and discards valuable data.

### 2. Resurrections Are Common
– Customers unsubscribe and resubscribe (e.g., streaming services).
– Users return after periods of inactivity (e.g., retail loyalty programs).

### 3. Binary Outcomes Ignore Value
– A customer spending $10 vs. $100 is treated the same if both are “alive.”
– Economic impact is lost in a yes/no framework.

## Introducing Value Kaplan-Meier

To address these limitations, we introduce Value Kaplan-Meier (VKM), which:
– Preserves continuous data (e.g., revenue, engagement scores).
– Accounts for resurrections (value can fluctuate over time).
– Scales values relative to baseline (individual contributions are weighted fairly).

### How It Works

The traditional Kaplan-Meier formula:
“`python
survival_rate_t = surv_t[obs_t].sum() / surv_t_minus_1[obs_t].sum()
kaplan_meier[t] = kaplan_meier[t-1] * survival_rate_t
“`

VKM modifies this by:
1. Replacing binary survival indicators (`surv_t`) with continuous values (`val_t`).
2. Normalizing by baseline value (`val_t_0`) to ensure fair comparisons.

“`python
value_rate_t = (val_t[obs_t] / val_t_0[obs_t]).sum() / (val_t_minus_1[obs_t] / val_t_0[obs_t]).sum()
value_kaplan_meier[t] = value_kaplan_meier[t-1] * value_rate_t
“`

### Example: Customer Spending Over Time

Consider three e-commerce users:

| Month | User A | User B | User C |
|——-|——–|——–|——–|
| 0 | $100 | $200 | $150 |
| 1 | $80 | $180 | $120 |
| 2 | $0 | $160 | $0 |
| 3 | $0 | $140 | $0 |
| 4 | $50 | $120 | $100 |

Traditional Kaplan-Meier (assuming “death” = 2 months of $0):
– Declares Users A and C “dead” at Month 3.
– Ignores User A’s revival in Month 4.

Value Kaplan-Meier:
– Tracks relative value retention (e.g., User A’s Month 4 spending is 50% of baseline).
– Accounts for rebounds in spending.


## Practical Applications

### 1. Experiment Evaluation
– Compare treatment vs. control groups in A/B tests.
– Measure not just “did they stay?” but “how much value did they retain?”

### 2. Customer Lifetime Value (CLV) Forecasting
– Predict revenue retention more accurately by incorporating spending fluctuations.

### 3. Subscription Services
– Track resubscriptions without penalizing temporary cancellations.

## Advantages Over Traditional Kaplan-Meier

1. Higher Precision
– Reflects actual economic impact, not just binary survival.

2. Flexibility
– Works with any continuous metric (revenue, engagement, etc.).

3. Real-World Alignment
– Captures the messy, non-binary nature of customer behavior.

## Implementing Value Kaplan-Meier in Python

Using the `lifelines` library as a base, we can adapt it for VKM:

“`python
import numpy as np

def value_kaplan_meier(durations, values, baseline_values, observed):
km = np.ones(len(durations))
for t in range(1, len(durations)):
mask = observed[t]
value_ratio = (values[t][mask] / baseline_values[mask]).sum()
prev_value_ratio = (values[t-1][mask] / baseline_values[mask]).sum()
km[t] = km[t-1] * (value_ratio / prev_value_ratio)
return km
“`

## Conclusion

Value Kaplan-Meier bridges the gap between traditional survival analysis and business needs by:
– Eliminating arbitrary binary thresholds.
– Incorporating economic value directly into survival curves.
– Allowing for customer “resurrections.”

For businesses, this means:
– More accurate retention metrics.
– Better experiment insights.
– Deeper understanding of customer value dynamics.

By moving beyond “alive vs. dead,” VKM unlocks a richer, more actionable view of how value evolves over time.

Want to try it yourself? Check out the Python implementation on GitHub.

Further Reading:
– Survival Analysis in Data Science
– Lifelines Documentation

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