Introduction to Survival Analysis

Why Do We Analyze Time?

Many real-world problems involve time:

• Monthly sales
  
• Machine lifetime
  
• Customer retention
  
• Time to graduation

Question:
Do all time-related data use the same analysis method?

Two Fundamental Time Questions

1. How does a value change over time?
   
2. How long does it take until an event occurs?

➡ Different questions → different statistical methods

Time Series Analysis or Time to Event?

What Is Time Series Analysis?

Time Series Analysis focuses on:

• Observations recorded at regular time intervals
• The value of a variable at each time point

Examples:

• Monthly revenue
  
• Daily temperature
  
• Stock prices

Characteristics of Time Series Data

• Each time point has a value
• No concept of event
• No censoring

Typical outputs:

• Trend
  
• Seasonality
  
• Forecasting

Example of Time Series Data

Focus: values, not events

Limitations of Time Series

Time Series is not appropriate when:

• The interest is waiting time
• Not all subjects experience the event
• Observation ends before the event occurs

Shift in Perspective

New question:

“How long does it take until an event happens?”

➡ This leads to Time-to-Event (TTE) analysis

What Is Time-to-Event Data?

Time-to-Event data measure:

The duration from a defined start time
until a specific event occurs

Examples:

• Enrollment → graduation
  
• Contract start → churn
  
• Machine installation → failure

Core Elements of TTE Data

Every Time-to-Event dataset must include:

1. Start time
   
2. Event definition
   
3. Time duration

Start Time

Examples:

• Date of enrollment
• Date of diagnosis
• Date of installation

Must be clearly defined and consistent

Event

An event is the outcome of interest:

• Graduation
  
• Failure
  
• Death
  
• Churn

Usually one event per subject

When the Event Has Not Occurred

• Some subjects do not experience the event
• Observation ends before the event occurs

➡ This situation is called censoring

What Is Censoring?

Censoring occurs when:

• The event has not happened yet
• We only know that survival time exceeds a certain value

Censoring ≠ missing data

Example of Time-to-Event Data

Event = 1 → occurred
Event = 0 → censored

Why Survival Analysis?

Time-to-Event data:

• Contain censored observations
• Focus on duration
• Cannot be analyzed using ordinary statistics

➡️ Require Survival Analysis

Definition of Survival Analysis

Survival Analysis is a statistical framework to:

• Analyze time until an event
• Handle censored data
• Estimate survival probability and risk

Logical Flow of Analysis