🐛Biostatistics Unit 10 – Survival Analysis: Kaplan-Meier in Biology

What's Survival Analysis?

• Branch of statistics focused on analyzing the expected duration of time until one or more events happen
• Commonly used in medical research to measure the fraction of patients living for a certain amount of time after treatment
• Incorporates data from a cohort of individuals, some of whom remain event-free for the duration of the study (right-censored observations)
• Survival analysis methods can accommodate censoring and provide a survival function that estimates the probability of an event occurring beyond a certain time
• Kaplan-Meier estimator is a non-parametric statistic used to estimate the survival function from lifetime data
  • Non-parametric means it makes no assumptions about the underlying distribution of the survival times
• Useful for analyzing the distribution of time between an initial event (diagnosis, treatment) and a terminal event (death, relapse)
• Can compare survival curves between groups using statistical tests (log-rank test) to determine if differences are significant

Key Concepts in Kaplan-Meier

• Survival function $S(t)$ gives the probability that an individual survives longer than some specified time $t$
• Hazard function $h(t)$ represents the instantaneous event rate at time $t$ conditional on survival until time $t$ or later
• Censoring occurs when the survival time for some individuals is unknown due to loss to follow-up or study termination before the event occurs
  • Right-censoring is most common where the event occurs after the observed survival time
• Kaplan-Meier curve is a series of horizontal steps of declining magnitude that approaches the true survival function for the population
• Median survival time is the time at which $S(t) = 0.5$, representing when 50% of the individuals have experienced the event
• Confidence intervals can be calculated for the survival function to quantify the uncertainty in the estimates
• Log-rank test compares the survival distributions of two or more groups to determine if they are statistically equivalent

Setting Up Your Data

• Data should be structured with one row per individual and columns for the survival time, censoring indicator, and any covariates of interest
• Survival time is the duration from the initial event (start of follow-up) to the terminal event (failure) or censoring
• Censoring indicator is a binary variable (0 for censored, 1 for event) that distinguishes between complete and incomplete observations
  • Censored observations contribute to the survival function only up to their observed survival time
• Time scale should be chosen based on the research question and the granularity of the available data (days, months, years)
• Data should be checked for inconsistencies, such as negative survival times or missing values, and cleaned accordingly
• Covariates can be included to explore their association with the survival outcome and to adjust for potential confounding factors
• Stratification can be used to estimate separate survival curves for different subgroups (treatment arms, risk categories) within the same model

Calculating Survival Probabilities

• Kaplan-Meier estimator calculates the survival probability at each distinct event time $t_i$ as the product of the conditional probabilities of surviving to each event time up to $t_i$
• Conditional probability of surviving beyond time $t_i$ given survival to $t_i$ is estimated as $(n_i - d_i) / n_i$, where:
  • $n_i$ is the number of individuals at risk (not censored and still event-free) just prior to time $t_i$
  • $d_i$ is the number of events (failures) at time $t_i$
• Survival probability at time $t_i$ is the product of the conditional probabilities up to and including $t_i$: $\hat{S}(t_i) = \prod_{j=1}^i \frac{n_j - d_j}{n_j}$
• Standard error of the survival probability can be estimated using Greenwood's formula to construct confidence intervals
• Calculations are typically performed using statistical software (R, SAS, STATA) that can handle tied event times and produce the necessary outputs

Plotting the Kaplan-Meier Curve

• Kaplan-Meier curve is a graphical representation of the survival function over time
• X-axis represents the survival time, and the Y-axis represents the estimated survival probability
• Curve starts at a survival probability of 1 (100% of individuals are event-free at the beginning of follow-up)
• At each distinct event time, the curve drops vertically by an amount proportional to the number of events at that time
• Censored observations are typically marked with a tick or cross on the curve at their observed survival time
• 95% confidence intervals can be plotted as dashed lines around the survival curve to show the uncertainty in the estimates
• When comparing multiple groups, separate curves are plotted on the same graph, often with different colors or line types
• Median survival time for each group can be marked on the x-axis or provided in a legend

Interpreting the Results

• Kaplan-Meier curve provides a visual summary of the survival experience over time
• Steeper drops in the curve indicate time periods with a higher rate of events
• Flatter sections of the curve suggest time periods with a lower rate of events or a higher proportion of censored observations
• Median survival time represents the time at which half of the individuals have experienced the event
  • Useful summary measure, especially when the maximum follow-up time is insufficient for all individuals to experience the event
• Confidence intervals that do not overlap between groups suggest statistically significant differences in survival
• Log-rank test provides a formal comparison of the survival curves, with a small p-value indicating that the curves are significantly different
• Hazard ratios can be estimated using Cox proportional hazards regression to quantify the relative risk of an event between groups while adjusting for covariates
• Results should be interpreted in the context of the study design, population, and potential limitations (selection bias, confounding, limited follow-up)

Real-World Applications in Biology

• Cancer research: Comparing survival outcomes between different treatment regimens or risk groups
  • Example: Kaplan-Meier curves for overall survival in patients with advanced lung cancer receiving chemotherapy versus immunotherapy
• Epidemiology: Analyzing time to infection or disease onset in exposed and unexposed populations
  • Example: Estimating the incubation period distribution for a novel infectious disease using data from contact tracing studies
• Ecology: Studying factors affecting animal lifespan or time to specific events (migration, reproduction)
  • Example: Comparing survival curves for different populations of a threatened species in habitats with varying levels of human disturbance
• Genetics: Investigating the effect of genetic variants on age-related phenotypes or disease progression
  • Example: Assessing the impact of a particular gene mutation on the time to onset of Alzheimer's disease symptoms
• Biomarkers: Evaluating the prognostic value of biological markers in predicting survival outcomes
  • Example: Using Kaplan-Meier curves to demonstrate the association between high levels of a circulating protein and reduced progression-free survival in cancer patients

Common Pitfalls and How to Avoid Them

• Violating the assumption of non-informative censoring, which requires that censored individuals have the same survival prospects as those who remain under observation
  • Ensure that censoring is not related to the outcome of interest and that follow-up is as complete as possible
• Failing to account for competing risks, which occur when an individual experiences an event that precludes the occurrence of the primary event of interest
  • Use specialized methods (cumulative incidence function, cause-specific hazard function) to properly analyze competing risks data
• Misinterpreting the survival probability as the probability of being event-free at a specific time, rather than the probability of surviving beyond that time
  • Emphasize that the survival function represents the cumulative probability of surviving beyond each time point
• Overinterpreting small differences in survival curves, especially when confidence intervals are wide or overlapping
  • Focus on clinically meaningful differences and consider the uncertainty in the estimates when drawing conclusions
• Extrapolating survival estimates beyond the observed follow-up time, which can lead to unrealistic predictions
  • Restrict interpretations to the time period covered by the data and avoid making predictions far beyond the last observed event time
• Failing to report key information (median survival time, confidence intervals, p-values) needed to fully interpret the results
  • Follow reporting guidelines (CONSORT, STROBE) and include all relevant statistics and graphical displays to ensure transparency and reproducibility
• Ignoring the impact of covariates or confounding factors on the survival outcomes
  • Use multivariate regression methods (Cox proportional hazards model) to adjust for potential confounders and explore the effects of covariates on survival