_2025-03-25 _{#statistical-analysis #kaplan-meier #survival-analysis #r-programming} _{[[maps-of-content]]} # Understanding Life Tables: Tracking Who's still in the Game Survival tables help us track what happens to people in a study over time, especially when some participants disappear before the study ends. This is a key component of [[survival-analysis]]. ## The Big Picture: What We're Tracking Imagine we're following 8 patients after they receive a new treatment. We want to know: 1. How many are still alive at each time point? 2. What's the probability of surviving beyond each time point? 3. How do we handle it when people drop out of the study? ## How the Calculations Work Each column in this table means: ### Time Column This shows when something happened - either a death or someone leaving the study (marked with a "+"). ### Number of Patients Alive This starts with all 8 patients and decreases whenever someone dies or leaves the study. ### Number Who Died This simply counts the deaths at each time point. ### Proportion Surviving Past Each Time Point This is calculated as: (Number alive - Number died) ÷ Number alive Think of it like this: If 8 people are climbing a mountain and 2 turn back, the proportion continuing is (8-2)/8 = 0.75 or 75%. ### The Key Column: Probability of Survival This is where the magic happens. The probability of surviving beyond any time point equals: - Previous probability × Current proportion For example: - Day 0: Everyone starts at 100% (1.0) - Day 1: 2 people die, so 1.0 × 0.75 = 0.75 - Day 4: 1 more person dies, so 0.75 × 0.83 = 0.625 - Day 5: 1 more person dies, so 0.625 × 0.8 = 0.5 ## Handling People Who Leave (Censoring) Now here's where it gets interesting. Sometimes people leave the study for reasons other than death: When someone is "censored" (leaves the study): - They still count in the "Number alive" column up until they leave - They don't affect the survival probability calculation for that time point - But we reduce the count of people we're tracking afterward For example: - At day 6, we have 4 people, and 0 die but 1 leaves the study - The proportion surviving is (4-0)/4 = 1 (100%) - The probability remains at 0.5 (unchanged because no deaths) - But going forward, we only have 3 people to track | Time (t) in days | Number of patients alive at time t | Number of patients who died at time t | Proportion of patients surviving past time t | Probability of survival _past_ time t | | ---------------- | ---------------------------------- | ------------------------------------- | -------------------------------------------- | ------------------------------------- | | 0 (study start) | 8 | 0 | 1 | 1 | | 1 | 8 | 2 | 0.75 | 0.75 | | 4 | 6 | 1 | 0.83 | 0.625 | | 5 | 5 | 1 | 0.8 | 0.5 | | 6+ | 4 | 0 | (4-0)/4 = 1 | 0.5*1 = 0.5 | | 9 | 3 | 1 | (3-1)/3 = 0.67 | 0.5*0.67 = 0.33 | | 9+ | 2 | 0 | (2-0)/2 = 1 | 0.33*1 = 0.33 | | 22 | 1 | 1 | (1-1)/1 = 0 | 0.33*0 = 0 | ## Visualizing the Survival Curve When-axis shows time - The y-axis shows the survival probability - The line forms a "stair-step" pattern, dropping when deaths occur - Each "+" mark shows when someone was censored (left the study) The resulting graph starts at 100% on day 0, drops to 75% on day 1, then to 62.5% on day 4, then to 50% on day 5, stays at 50% through day 6 and 9 (though some people dropped out), and finally reaches 0% on day 22. ![[life-tables-1742984818647.webp]] ## Key Insights to Remember A survival table tracks: 1. When deaths or drop-outs occur 2. How many people remain at each time point 3. The proportion surviving each specific time period 4. The cumulative probability of survival over time People who leave the study ("censored observations") don't affect the survival probability at that moment, but they do reduce the number of people we continue to follow. ## Most Important Takeaway **We can calculate meaningful survival probabilities even when people drop out of studies.** This is crucial because in real research, people rarely all stay until the end. By properly accounting for these "censored" observations, we get a much more accurate picture of survival patterns over time. These calculations can be performed in [[rstudio-r|R]] using specialized packages for statistical analysis. -- Reference: Statistical Analysis with R for Public Health, Imperial College London