Survival analysis

Survival Analysis

When to use this skill

Use when the outcome is "how long until X happens?" and some observations haven't experienced X yet (censored). Triggers:

• "Time to churn / cancel / conversion"
• "Survival curve for…"
• "Hazard ratio of feature X"
• "How long do users stay before churning?"
• "Compare retention across segments over time"

If you only care about whether an event happens within a fixed window, use logistic-regression. If you need fixed-time-point retention rates without timing, use cohort-analysis.

Why not just use logistic regression?

Question	Right tool
Did the user churn within 90 days?	logistic
When did the user churn? Distribution?	survival
How does plan tier affect churn timing?	survival
Users still active — do we throw them out?	survival (treats as censored, not missing)

Throwing out censored observations biases logistic regression toward early events.

Required inputs

Input	Why it matters
Subject ID	One row per subject
Duration	Time from start to event OR last observation
Event indicator (1/0)	1 = event happened, 0 = censored (still observed without event)
Covariates	Features that may affect timing
Start time	When observation begins (often signup_at)

Workflow

1. Define event precisely. "Churn" must have an exact definition:

   • Subscription cancellation date
   • No-activity threshold (e.g., 30 consecutive inactive days)
   • Account deletion

2. Define censoring clearly. A subject is censored if:

   • Still active at the end of the observation window (right censoring — most common)
   • Lost to follow-up (e.g., switched accounts, account suspended for unrelated reason)
   • Did not start before observation window (left truncation)

3. Build the duration table: one row per subject with start, duration, event.

   df = pd.DataFrame({
       "subject_id": ...,
       "duration_days": (event_date - start_date).dt.days,
       "event": did_event.astype(int),
       # ... covariates
   })
   

4. Fit Kaplan-Meier (KM) curves overall and by segment.

   • Plot survival function S(t) = P(T > t)
   • 95% CI bands (Greenwood formula)
   • Median survival time (if reached)

5. Compare groups with log-rank test to check if curves differ significantly.

6. Fit Cox Proportional Hazards model for multivariable analysis.

   • Hazard ratios with 95% CIs
   • Check the PH assumption (Schoenfeld residuals, proportional hazards plot). If violated for a feature, stratify or include time-interaction term.

7. Communicate in business language:

   • "Median time-to-churn is 142 days. Users on annual plans have HR = 0.42, meaning ~58% lower instantaneous churn risk at any given time."
   • Visualize KM curves with confidence bands by segment.

Output format

# Survival Analysis: <event>

## Setup
- Event: <e.g., subscription cancellation>
- Start time: <e.g., signup_at>
- Censoring: subjects still active as of <observation_end_date>
- Sample: N = <total>, events = <X (X%)>
- Duration unit: <days | weeks | months>
- Median follow-up: <D days>

## Kaplan-Meier summary
| Segment | N | Events | Median survival | S(30d) | S(90d) | S(180d) |
|---|---|---|---|---|---|---|
| Overall | 24,800 | 6,420 (26%) | 312 days | 92% | 81% | 71% |
| Monthly plan | 12,400 | 4,810 (39%) | 184 days | 88% | 71% | 58% |
| Annual plan | 12,400 | 1,610 (13%) | not reached | 96% | 91% | 84% |

Log-rank test (monthly vs annual): χ² = 2,143, p < 0.0001 — **significantly different**

## Cox proportional hazards model
| Covariate | HR | 95% CI | p | Notes |
|---|---|---|---|---|
| plan = annual (ref: monthly) | 0.42 | [0.38, 0.46] | < 0.001 | annual reduces hazard by 58% |
| tenure_segment = enterprise | 0.31 | [0.25, 0.39] | < 0.001 | |
| support_tickets_30d | 1.18 | [1.14, 1.22] | < 0.001 | each ticket: +18% hazard |
| has_team_admin | 0.67 | [0.61, 0.73] | < 0.001 | |
| age_days (per 30 days) | 0.94 | [0.93, 0.95] | < 0.001 | older accounts more stable |

Concordance index: 0.74

## Proportional hazards assumption check
- Schoenfeld residuals: <p-value per covariate>
- Covariates violating PH: <list, or "none">
- For violators: <stratified by | time-interacted | accepted with caveat>

## Visualization
- KM curves with 95% CI bands (overall + key segments)
- Hazard ratios with CIs as forest plot
- (Optional) Cumulative incidence functions

## Caveats
- <e.g., observation window 18 months; long-term survival beyond ~500 days extrapolated>
- <e.g., Annual-plan users haven't reached median — estimate uncertain>
- <e.g., Right censoring assumption assumes censoring is independent of churn likelihood; verify>

## Decision implications
- <e.g., shifting customers to annual plans should yield ~50% reduction in long-run churn>
- <e.g., support ticket spike is a strong leading indicator — trigger intervention at 3+ tickets in 30 days>

## Next steps
- <e.g., A/B test annual plan upsell campaign>
- <e.g., build early-warning model using ticket count + tenure>

Validation checks

• Event definition is exact (date or threshold)
• Censoring rule documented (and not correlated with event)
• Duration is non-negative
• KM curves include CI bands
• Log-rank test reported when comparing groups
• Cox model: PH assumption checked
• Concordance index reported (Cox's analog of AUC, typically 0.6-0.85)

Edge cases & failure modes

• Informative censoring: if subjects with higher churn risk are more likely to be censored (e.g., paused accounts), Cox model is biased. Hard to detect. Investigate the censoring mechanism.
• Competing risks: a subject can experience multiple mutually exclusive events (churn vs upgrade). Use competing risk models (Fine-Gray), not standard Cox.
• PH violation: a covariate's effect changes over time (e.g., new users have higher early churn). Stratify by that variable or add time-interaction term.
• Time-varying covariates: feature values change during the observation period (e.g., support tickets accumulate). Use time-dependent Cox model with long-format data.
• Left truncation: subjects only enter observation after some time. Use entry parameter in lifelines.
• Heavy ties (many events at same time): use Efron or exact tie-breaking, not Breslow.

Scripts

• scripts/survival_fit.py — Fit KM by segment + Cox PH model, output readout.

python scripts/survival_fit.py \
    --input subjects.csv \
    --duration tenure_days \
    --event churned \
    --segment plan_tier \
    --covariates plan_tier,support_tickets_30d,has_team_admin

Related skills

• cohort-analysis — for fixed-time-point retention without censoring complexity
• logistic-regression — for binary outcome within a fixed window
• data-quality-audit — verify start/end dates and event indicators before fitting
• causal-inference — if you need to estimate intervention effect on survival