Jekyll2026-05-21T13:14:48+00:00https://lingying.me/feed.xmlHomepageData Science UndergraduateYing LINGspiritsswin@gmail.comSurvival Analysis Report — Telco Customer Churn2026-04-18T00:00:00+00:002026-04-18T00:00:00+00:00https://lingying.me/posts/2026/04/survival-analysis-telco-churn1. Data Preparation

1.1 Dataset

Item Value
Source IBM Telco Customer Churn
Original rows (Bronze) 7,043
Columns 21

1.2 Filtering for Survival Analysis

Two filters applied:

  1. Contract = “Month-to-month” only
  2. InternetService ≠ “No” (internet subscribers only)
Stage Rows % of Original
Bronze (raw) 7,043 100.0%
Silver (filtered) 3,351 47.6%

1.3 Churn Distribution (Silver Table)

Churn Count Percentage
0 (Retained) 1,795 53.6%
1 (Churned) 1,556 46.4%
Total 3,351  

1.4 Contract Distribution (Full Dataset)

Contract Count Percentage
Month-to-month 3,875 55.0%
Two year 1,695 24.1%
One year 1,473 20.9%

2. Kaplan-Meier Estimator

2.1 What is Kaplan-Meier?

Kaplan-Meier is a non-parametric method that estimates the survival function S(t) — the probability that a customer survives beyond time t. It properly accounts for censored observations.

2.2 Population-Level Survival Curve

Median survival time: 34.0 months

Time Point Survival Probability Interpretation
6 months 0.7803 78.0% survive at least 6 months
12 months 0.6950 69.5% survive at least 1 year
24 months 0.5753 57.5% survive at least 2 years
34 months 0.5000 Median — half have churned
48 months 0.3872 38.7% survive 4 years
60 months 0.2890 28.9% survive 5 years

2.3 Covariate-Level Analysis with Log-Rank Test

The log-rank test determines whether survival curves for different groups are statistically distinguishable. Null hypothesis (H₀): the groups have the same survival distribution.

Results for all 15 categorical variables:

Variable Levels Overall p-value Significant (p < 0.05)?
onlineSecurity 3 < 0.000001 ✅ Yes
onlineBackup 3 < 0.000001 ✅ Yes
deviceProtection 3 < 0.000001 ✅ Yes
techSupport 3 < 0.000001 ✅ Yes
partner 2 < 0.000001 ✅ Yes
dependents 2 < 0.000001 ✅ Yes
internetService 2 0.000001 ✅ Yes
paymentMethod 4 < 0.000001 ✅ Yes
multipleLines 3 < 0.000001 ✅ Yes
streamingMovies 2 0.000023 ✅ Yes
streamingTV 2 0.000322 ✅ Yes
paperlessBilling 2 0.003876 ✅ Yes
gender 2 0.153317 ❌ No
phoneService 2 0.194432 ❌ No
seniorCitizen 2 0.723174 ❌ No

Key findings:

  • 12 out of 15 variables show statistically significant differences.

  • paymentMethod IS significant (overall p < 0.000001).

  • Service-related features are the most significant.

2.4 DSL Subscriber Survival Probabilities

Month Survival Probability
0 1.0000
3 0.8347
6 0.7839
9 0.7508
12 0.7270

3. Cox Proportional Hazards Model

3.1 What is Cox PH?

Cox Proportional Hazards is a semi-parametric regression model:

\[h(t|X) = h_0(t) \times e^{\beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p}\]

  • HR < 1 → protective (reduces churn risk)

  • HR > 1 → risk factor (increases churn risk)

3.2 Feature Encoding

Original Variable Kept Column Dropped (baseline)
dependents dependents_Yes dependents_No
internetService internetService_DSL internetService_Fiber optic
onlineBackup onlineBackup_Yes onlineBackup_No
techSupport techSupport_Yes techSupport_No
paperlessBilling paperlessBilling_Yes paperlessBilling_No

3.3 Model Results

Covariate Coef (β) Hazard Ratio exp(β) p-value 95% CI
onlineBackup_Yes -0.7766 0.4600 < 0.001 [0.4096, 0.5165]
techSupport_Yes -0.6392 0.5277 < 0.001 [0.4553, 0.6117]
dependents_Yes -0.3287 0.7199 < 0.001 [0.6265, 0.8272]
internetService_DSL -0.2173 0.8047 0.0002 [0.7167, 0.9034]
Concordance Index: 0.6409        

3.4 Interpretation

  • Online Backup (HR = 0.460): 54% lower hazard of churning.

  • Tech Support (HR = 0.528): 47.2% lower hazard.

  • DSL Internet (HR = 0.805): 19.5% lower hazard compared to Fiber Optic.

3.5 Proportional Hazards Assumption Check

Variable p-value PH Assumption Violated?
internetService_DSL < 0.0001 ✅ Yes
onlineBackup_Yes < 0.0001 ✅ Yes
techSupport_Yes 0.0002 ✅ Yes
dependents_Yes > 0.05 ❌ No

4. Accelerated Failure Time (AFT) Model

4.1 What is AFT?

AFT models how covariates “accelerate” or “decelerate” the time to event:

\[T = T_0 \times e^{\beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p}\]

  • exp(β) > 1 → time to churn is longer (protective)

  • exp(β) < 1 → time to churn is short (risk factor)

4.2 Feature Encoding

9 covariates: partner, multipleLines, internetService_DSL, onlineSecurity, onlineBackup, deviceProtection, techSupport, paymentMethod_Bank, paymentMethod_Credit.

4.3 Model Results

Median survival time: 135.51 months

Metric Cox PH AFT (Log-Logistic)
Concordance 0.6409 0.7306

4.4 AFT Coefficients

Covariate Coef (β) exp(β) p-value Interpretation
onlineSecurity_Yes 0.8616 2.3669 < 0.001 2.37× longer survival
onlineBackup_Yes 0.8128 2.2542 < 0.001 2.25× longer survival
paymentMethod_Credit 0.7990 2.2234 < 0.001 2.22× longer survival
techSupport_Yes 0.6893 1.9923 < 0.001 1.99× longer survival

5. Customer Lifetime Value (CLV)

5.1 Methodology

\[\text{Expected Profit}_m = S(m) \times \text{Monthly Revenue}\] \[\text{NPV}_m = \frac{\text{Expected Profit}_m}{(1 + \text{Monthly IRR})^m}\]

5.2 Customer Profile

Has dependents, Fiber Optic, has online backup, has tech support.

5.3 CLV Table

Month Survival Prob Expected Profit NPV Cumulative NPV
1 0.9830 $29.49 $29.24 $29.24
6 0.9416 $28.25 $26.90 $168.13
12 0.9073 $27.22 $24.64 $319.76
24 0.8583 $25.75 $21.12 $591.61
36 0.8118 $24.35 $18.06 $824.71

5.4 Profile Comparison (36-Month Cumulative NPV)

Profile 36-Month Cumulative NPV
DSL + TechSupport $1,004.91
Fiber + TechSupport $907.05
Fiber, No TechSupport $596.69

6. Summary & Key Takeaways

6.1 Method Comparison

Method Concordance Key Assumption Met?
Cox PH 0.6409 ❌ 3/4 violated
AFT 0.7306 Partially

6.2 Most Important Findings

  1. Median churn time is 34 months for target segment.

  2. Online Backup is the strongest protective factor (HR = 0.460).

  3. The AFT model outperforms Cox PH (concordance 0.73 vs 0.64).

  4. Gender, phoneService, and seniorCitizen do NOT significantly affect churn.

6.3 Business Recommendations

  • Prioritize Online Backup and Tech Support adoption.

  • Monitor Fiber Optic customers closely.

  • Use AFT model for better prediction accuracy.

Appendix: Corrections

Issue Original Corrected
paymentMethod significance Not significant Significant (p < 0.000001)
AFT interpretation exp(β)>1 = faster churn exp(β)>1 = longer survival
CLV 12m NPV $292.68 $319.76
CLV 36m NPV $799.97 $824.71
]]>Ying LINGspiritsswin@gmail.com