Lecture 8 Notes — Survival Analysis
Learning outcomes: LO3, LO4
Companion files: slides L08, notebook Week8_Survival_Analysis.ipynb, data member_tenure.csv
1. Time-to-event questions
How long until a member cancels, a guest rebooks, a machine fails? Our synthetic data: 4,000 members observed for up to 24 months; 1,711 cancelled (the event) and 57.2% were still members when observation ended (censored).
2. Censoring
For a censored member we know only that tenure is at least the observed time. Ignoring this biases every average:
| Method | Estimate (months) |
|---|---|
| Mean tenure, everyone | 8.9 (treats censored time as final) |
| Mean tenure, cancelled only | 7.2 (drops the long-stayers) |
| Kaplan–Meier median | 13.6 |
3. Kaplan–Meier estimator
At each event time t_i with d_i events and n_i members at risk:
S(t) = Π (1 − d_i / n_i) over event times up to t.
Censored members leave the risk set without causing a drop. Our hand-built estimator matches the lifelines library. Survival at 6, 12, 18 and 24 months: 77%, 54%, 40%, 27%; median 13.6 months.
4. Comparing groups
Premium members: median 19.5 months, 12-month retention 68%. Free members: median 11.2, retention 47%. The log-rank test gives p < 0.001 for plan and for acquisition channel.
5. Hazard
The hazard h(t) is the instantaneous risk of cancelling given you are still a member. Cumulative hazard H(t) and survival are linked by S(t) = exp(−H(t)). The cumulative hazard bends upward, so in this data risk rises with tenure (Weibull shape greater than 1).
6. Cox proportional-hazards model
h(t | x) = h₀(t) · exp(β₁x₁ + …). exp(β) is a hazard ratio.
| Variable | HR | 95% CI |
|---|---|---|
| Premium | 0.57 | 0.51–0.64 |
| Social channel | 1.46 | 1.32–1.62 |
| Average rating given (per star) | 0.62 | 0.59–0.66 |
| Bookings in year 1 (each) | 0.93 | 0.91–0.96 |
The concordance index is 0.64. The estimates recover the effects built into the synthetic data. The model assumes proportional hazards; a Schoenfeld-residual test gave a smallest p-value of 0.66, so there is no evidence against it. When violated, stratify or add time interactions.
7. Prediction and what-if
Predicted 12-month retention: at-risk free member 18%, typical free member 54%, engaged premium member 83%. Upgrading a typical member to premium raises 12-month retention from 54% to 71%. This assumes the effect is causal: confirm with an experiment before budgeting a campaign.
8. Spark
A life table needs only counts of events and at-risk members per period, which are aggregations. The Spark life table gave S(12) = 0.545 versus 0.544 from Kaplan–Meier (a small difference from handling of ties). Cox fitting is normally done on a sample or per segment.
9. Discussion questions
- Why is dropping censored members worse than keeping them with their observed time?
- A retention offer costs 5 units per member. How would you use the Cox model to decide whom to target?
- What would a crossing of two survival curves imply for the Cox model?
Next lecture: text and sentiment analytics.