Survival analysis marketing

Survival Analysis for Marketing

Time-to-event modeling with proper censored data handling. Three-tier approach: Kaplan–Meier (nonparametric) → Cox PH (semiparametric) → Time-to-Event prediction.

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Core Equations (Katsov §2.6.2)

Survival Function (eq. 2.107–2.109)

S(t) = Pr(T > t) = 1 − F(t)           # probability of surviving past t

# Kaplan-Meier estimator (eq. 2.109):
Ŝ(t) = ∏_{i ≤ t}  (1 − d_i / n_i)    # product over observed event times

Hazard Function (eq. 2.112–2.116)

h(t) = lim_{dt→0} Pr(t < T ≤ t+dt | T > t) / dt
     = f(t) / S(t)   =   −d/dt [log S(t)]

H(t) = ∫₀ᵗ h(τ)dτ    [cumulative hazard]
S(t) = exp(−H(t))

Cox Proportional Hazards (eq. 2.119–2.133)

h(t | x) = h₀(t) · exp(wᵀx)            # baseline × risk ratio

# Partial likelihood (eq. 2.127–2.128):
L_i(w) = exp(wᵀx_i) / Σ_{j ∈ R(t_i)} exp(wᵀx_j)

# Personalized survival function (eq. 2.133):
S(t | x) = S₀(t)^{exp(wᵀx)}

# Breslow baseline cumulative hazard (eq. 2.131):
Ĥ₀(t) = Σ_{t_i ≤ t}  d̂_i / Σ_{j ∈ R(t_i)} exp(wᵀx_j)

Key marketing interpretations:

• exp(w_k) = hazard ratio for feature k → how much it accelerates/decelerates the event
• HR > 1 → accelerates event (e.g., discount → faster purchase)
• HR < 1 → decelerates event (e.g., retention offer → slower churn)

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Input Data Format (eq. 2.104, 2.118)

Each row: (customer_id, t_i, δ_i, x_i...)

Column	Description
t	Survival/censoring time (days, weeks, months from treatment)
delta	1 = event observed; 0 = censored (customer still active at analysis date)
x_1..N	Feature vector: recency, frequency, discount_depth, n_emails, etc.

Censoring rule: Customer is censored if event not observed by analysis date. Never drop censored records — they carry information (§2.6.2, p. 58–59).

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Workflow

Step 1 — Prepare Data

python scripts/censored_data_prep.py \
    --transactions data/txns.csv \
    --analysis-date 2024-12-31 \
    --event purchase \
    --output data/survival_ready.csv

Step 2 — Kaplan-Meier (nonparametric baseline)

python scripts/kaplan_meier.py \
    --data data/survival_ready.csv \
    --group-col treatment \
    --output results/km_output.json

Output: survival curve per group + log-rank test p-value.

Step 3 — Cox PH Model (covariate effects)

python scripts/cox_model.py \
    --data data/survival_ready.csv \
    --features recency frequency discount_depth n_emails \
    --output results/cox_output.json

Output: hazard ratios, 95% CI, p-values, concordance index.

Step 4 — Predict Time-to-Event per Customer

python scripts/time_to_event_pred.py \
    --data data/survival_ready.csv \
    --cox-model results/cox_output.json \
    --output results/tte_predictions.csv

Output: median survival time, p25/p75 per customer → use as campaign trigger.

Step 5 — Visualize Hazard Ratios

python scripts/hazard_ratio_viz.py \
    --cox-results results/cox_output.json \
    --output results/hr_forest_data.json

Step 6 — Export Dashboard JSON + Render

python scripts/export_survival_dashboard_json.py \
    --km results/km_output.json \
    --cox results/cox_output.json \
    --tte results/tte_predictions.csv \
    --output dashboard_data.json

Then call show_widget with references/survival_dashboard_template.html.

Output sequence:

1. [bash_tool] censored_data_prep.py
2. [bash_tool] kaplan_meier.py + cox_model.py + time_to_event_pred.py
3. [bash_tool] export_survival_dashboard_json.py → JSON
4. [show_widget] Render dashboard
5. [text] Exec recommendation: HR interpretation + campaign trigger timing
6. [text] PH assumption caveat if relevant

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Scripts Reference

Script	Key Inputs	Key Outputs
censored_data_prep.py	raw txns CSV, analysis date, event type	survival-ready CSV with (t, delta, features)
kaplan_meier.py	survival CSV, optional group col	KM curve JSON + log-rank p-value
cox_model.py	survival CSV + feature cols	HR table, concordance, baseline cumhaz
time_to_event_pred.py	survival CSV + cox JSON	per-customer median TTE + p25/p75
hazard_ratio_viz.py	cox JSON	forest plot data JSON
export_survival_dashboard_json.py	km + cox + tte JSONs	unified dashboard JSON

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Output Format — Visualization First

Primary output: inline HTML dashboard. Always render before text.

Dashboard panels (see references/survival_dashboard_template.html):

1. KPI bar — median survival time, concordance index, n events, n censored, % censored
2. Survival curve(s) — KM estimator, one line per group, with 95% CI band
3. Hazard ratio forest plot — one row per feature, dot + CI bar, HR=1 reference line
4. Cumulative hazard H(t) — log scale, shows proportional hazard structure
5. TTE distribution — histogram of predicted median time-to-event across customers
6. Log-rank test summary — p-value and interpretation per group comparison

Fallback: Katsov Example 2.1 format — t | n_at_risk | events | S(t) | CI_lower | CI_upper

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Marketing Applications (§3.5.6)

Application	Event	Censoring rule	Campaign action
Replenishment	Next purchase	Analysis date	Send notification N days before predicted purchase
Churn detection	Subscription cancel	Still active	Retention offer at t = median × 0.7
Acquisition	First purchase	Analysis date	Retarget users nearing median TTE
Promotion redemption	Coupon use	Expiry date	Adjust offer timing to HR of discount_depth
Reactivation	Return after lapse	End of observation	Winback trigger at predicted re-engagement

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Key Caveats

• Proportional hazards assumption. Cox model requires h_i(t)/h_j(t) = constant across time. Validate with Schoenfeld residuals or log-log plot. Violation → use time-varying coefficients or stratified Cox.
• Censoring must be non-informative. Censored customers must not differ systematically from uncensored ones (e.g., don't censor high-value customers selectively).
• Event definition determines the model. Purchase ≠ redemption ≠ churn. Build separate models per event type.
• Repeated events. KM/Cox assume single event per customer. For repeat purchases, model each purchase interval separately (gap time model).
• Marketing interpretation of HR. HR = exp(w_k): a 1-unit increase in feature k multiplies the event rate by HR. For binary features (treatment=1/0), HR is the treatment effect directly.

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Integration with Agency Growth OS

Skill	Handoff
customer-lifetime-value	S_u(t) feeds LTV survival model (eq. 3.31)
response-uplift-modeling	Survival curves compare treated vs control groups
crm-journey-architect	Median TTE → message timing in replenishment journeys
audience-segmentation-brief	TTE segments → urgency tiers for targeting
measurement-incrementality	KM group comparison = A/B test measurement

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Reference Files

• references/katsov_survival_excerpts.md — Equations 2.104–2.133, Example 2.1, §3.5.6
• references/model_selection_survival.md — KM vs Cox vs parametric decision guide
• references/survival_dashboard_template.html — Reusable HTML dashboard; inject SKILL_DATA_JSON