Key Concepts in Survival Analysis

Survival Analysis Fundamentals

Understanding Survival Analysis

The main objective of survival analysis techniques is to determine differences between two or more treatments applied to a set of individuals. Each individual receives a particular treatment, and the effect (response) is measured by the occurrence of a specific event of interest (e.g., default) and the time elapsed from the start of observation until the aforementioned event occurs. Survival analysis techniques apply to data with the following characteristics:

• The dependent variable (or response variable) is the time that elapses until the individual experiences a specific event of interest, often termed death. Therefore, while the individual does not experience the event of interest, they are considered to be in survival (alive).
• Observations may be censored, meaning they never experience the event of interest during the study period.
• Independent variables (covariates) that characterize individuals may influence the time until the occurrence of the event, and their influence may also be of interest. The use of these variables is optional.

Censoring and Truncation

Censoring (Right Censoring): An observation is censored if it never experiences the event of interest during the study period. Censoring can occur for various reasons:

• Loss to Follow-up: Interruption of observation before the end of the study, without the event of interest occurring.
• Study Completion: The study concludes, and the individual has not experienced the event of interest.

Truncation (Left Censoring): Truncation occurs when an individual begins to be observed at a time after the study's commencement.

Life Expectancy in Survival Analysis

Life expectancy (M) is defined as the average of the variable T. It can be calculated by the following expression:

[Insert Formula 1 Here]

Integrating by parts and using the fact that the derivative of S(t) is -f(t), it can be calculated as follows (proof omitted):

[Insert Formula 2 Here]

This measure is very important when comparing treatments. Two treatments differ in their effects (from the point of view of survival analysis) if and only if they influence the life expectancy measure differently for individuals. Therefore, to differentiate between treatments using distinct samples for each, one calculates the life expectancy for each sample. Conclusions about their effects are then drawn from these values.