Survival analysis

Survival analysis is the statistical study of time until an event occurs (often called time-to-event analysis). It is widely used in medical research where patients are followed over time to observe outcomes such as:

• Death
• Disease recurrence
• Time to recovery
• Time to hospital readmission

Used in:

• Longitudinal cohort studies
• Randomised controlled trials (RCTs)

Challenges in time-to-event Data

• Unequal follow-up: Patients may enter the study at different times.
• Non-normal distributions: Time-to-event data are rarely normally distributed.
• Censoring: Patients may leave the study, be lost to follow-up, or the study ends before they experience the event.

Kaplan-Meier (KM) survival curve

A non-parametric method to estimate survival over time.

• Y-axis: Proportion of patients surviving (event-free)
• X-axis: Time
• Steps: Occur at each event (e.g., death)
• Vertical ticks: Represent censored data
• Does not change at time of censoring, only at events

Kaplan-Meier Outputs:

• Median survival time: Time by which 50% have experienced the event
• Survival probability: Chance of being event-free at a given time
• Comparative plots: Useful to compare survival in treatment vs. control groups
• Use the log-rank test to statistically compare survival curves between groups.

Hazard and hazard ratio (HR)
Hazard: The instantaneous risk of the event at a given time.

Hazard Ratio (HR) interpretation:

• HR = 1 → No difference
• HR < 1 → Reduced risk (e.g., treatment helps)
• HR > 1 → Increased risk (e.g., harmful exposure)

Cox proportional hazards regression
A semi-parametric model used to:

• Adjust for multiple covariates
• Provide adjusted hazard ratios
• Assumes the proportional hazards assumption (i.e., relative risk is constant over time)

Summary table of key methods