survival-analysis-guide

Survival Analysis Guide

A skill for conducting time-to-event analyses including Kaplan-Meier estimation, log-rank tests, and Cox proportional hazards regression. Covers censoring concepts, assumption checking, and reporting standards for clinical and social science research.

Core Concepts

What Is Survival Analysis?

Survival analysis studies the time until an event of interest occurs. Despite the name, the "event" need not be death -- it can be any well-defined transition:

Medical:      Time to disease recurrence, death, or recovery
Engineering:  Time to equipment failure
Social:       Time to job termination, divorce, or graduation
Business:     Time to customer churn or first purchase
Ecology:      Time to species extinction in a habitat

Censoring

Right censoring (most common):
  The event has not occurred by the end of the study period.
  Example: Patient is still alive at study end.
  The survival time is "at least T" -- we know T but not the true event time.

Left censoring:
  The event occurred before the observation period began.
  Example: HIV infection detected, but seroconversion happened before testing.

Interval censoring:
  The event occurred between two observation times.
  Example: A patient tests negative at visit 3 and positive at visit 4.

Kaplan-Meier Estimation

Computing the Survival Curve

import numpy as np


def kaplan_meier(times: list[float], events: list[int]) -> dict:
    """
    Compute Kaplan-Meier survival estimates.

    Args:
        times: Observed times (event or censoring time)
        events: Event indicator (1 = event occurred, 0 = censored)

    Returns:
        Dict with time points and survival probabilities
    """
    data = sorted(zip(times, events), key=lambda x: x[0])
    n = len(data)

    unique_event_times = sorted(set(t for t, e in data if e == 1))
    survival = 1.0
    results = {"time": [0], "survival": [1.0]}

    at_risk = n
    idx = 0

    for t_event in unique_event_times:
        # Count censored before this event time
        while idx < n and data[idx][0] < t_event:
            if data[idx][1] == 0:
                at_risk -= 1
            idx += 1

        # Count events at this time
        d = sum(1 for t, e in data if t == t_event and e == 1)
        c = sum(1 for t, e in data if t == t_event and e == 0)

        survival *= (at_risk - d) / at_risk
        results["time"].append(t_event)
        results["survival"].append(survival)

        at_risk -= (d + c)
        idx = max(idx, sum(1 for t, _ in data if t <= t_event))

    return results