🫁Intro to Biostatistics Unit 9 – Survival Analysis

What's Survival Analysis?

• Branch of statistics focused on analyzing time-to-event data where the outcome variable is the time until an event of interest occurs
• Commonly used in medical research to study the effectiveness of treatments, risk factors for disease, and prognostic factors
• Allows for the inclusion of censored data, which occurs when the event of interest has not been observed for a subject during the study period
• Differs from other statistical methods as it accounts for the fact that the event of interest may not have occurred for all subjects by the end of the study
• Provides insights into the probability of an event occurring over time and the factors that influence this probability
• Enables researchers to compare survival patterns between different groups (treatment vs. control) and identify risk factors associated with the event of interest
• Offers a flexible framework for handling various types of censoring (right, left, or interval) and accommodating time-dependent covariates

Key Concepts and Terms

• Event: The specific occurrence of interest, such as death, disease recurrence, or equipment failure
• Survival time: The duration from the starting point (e.g., diagnosis or treatment initiation) until the event occurs or the subject is censored
• Censoring: Occurs when the event of interest has not been observed for a subject during the study period
  • Right censoring: The most common type, where the subject is still event-free at the end of the study or is lost to follow-up
  • Left censoring: When the event of interest has already occurred before the subject is included in the study
  • Interval censoring: When the event is known to have occurred within a specific time interval, but the exact time is unknown
• Survival function $S(t)$: The probability that an individual survives beyond time $t$
• Hazard function $h(t)$: The instantaneous rate of experiencing the event at time $t$, given that the individual has survived up to that point
• Kaplan-Meier estimator: A non-parametric method for estimating the survival function from observed survival times, accounting for censoring
• Cox proportional hazards model: A semi-parametric regression model that relates the hazard function to a set of covariates, assuming that the hazard ratios between groups remain constant over time

Types of Survival Data

• Right-censored data: The most common type, where the event of interest has not occurred by the end of the study period or the subject is lost to follow-up
• Left-censored data: When the event of interest has already occurred before the subject is included in the study
• Interval-censored data: When the event is known to have occurred within a specific time interval, but the exact time is unknown
• Truncated data: When subjects are not included in the study until they have reached a certain point in their survival time
  • Left truncation: Subjects are only included if they have survived up to a specific time point
  • Right truncation: Subjects are only included if the event occurs before a specific time point
• Competing risks data: When subjects are at risk of experiencing multiple, mutually exclusive events (e.g., death from different causes)
• Recurrent event data: When the event of interest can occur multiple times for the same subject (e.g., asthma attacks or hospital readmissions)

Survival and Hazard Functions

• Survival function $S(t)$ represents the probability that an individual survives beyond time $t$
  • Defined as $S(t) = P(T > t)$, where $T$ is the survival time
  • Ranges from 1 at the start of the study (when $t = 0$) to 0 as $t$ approaches infinity
  • Can be estimated non-parametrically using the Kaplan-Meier method or parametrically by assuming a specific distribution for the survival times (e.g., exponential, Weibull, or log-normal)
• Hazard function $h(t)$ represents the instantaneous rate of experiencing the event at time $t$, given that the individual has survived up to that point
  • Defined as $h(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t + \Delta t | T \geq t)}{\Delta t}$
  • Provides insights into how the risk of the event changes over time
  • Can be modeled using the Cox proportional hazards model or parametric models (e.g., exponential, Weibull, or log-normal)
• The survival and hazard functions are related through the cumulative hazard function $H(t)$, which is defined as the integral of the hazard function from 0 to $t$
  • The relationship between $S(t)$ and $H(t)$ is given by $S(t) = \exp(-H(t))$

Kaplan-Meier Method

• Non-parametric method for estimating the survival function from observed survival times, accounting for censoring
• Calculates the probability of surviving beyond each observed event time, conditional on having survived up to that point
• Estimates the survival function as a step function, with drops at each observed event time
• Formula for the Kaplan-Meier estimator:
  • Let $t_1 < t_2 < ... < t_k$ be the distinct observed event times
  • Let $d_i$ be the number of events at time $t_i$ and $n_i$ be the number of individuals at risk just prior to time $t_i$
  • The Kaplan-Meier estimator is given by $\hat{S}(t) = \prod_{i: t_i \leq t} (1 - \frac{d_i}{n_i})$
• Provides a visual representation of the survival experience of the study population
• Allows for the comparison of survival curves between different groups using the log-rank test
• Limitations include the inability to incorporate covariates directly and the assumption that censoring is non-informative

Cox Proportional Hazards Model

• Semi-parametric regression model that relates the hazard function to a set of covariates
• Assumes that the hazard ratios between groups remain constant over time (proportional hazards assumption)
• The model is defined as $h(t|X) = h_0(t) \exp(\beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p)$, where:
  • $h(t|X)$ is the hazard function for an individual with covariate values $X = (X_1, X_2, ..., X_p)$
  • $h_0(t)$ is the baseline hazard function, which is left unspecified
  • $\beta_1, \beta_2, ..., \beta_p$ are the regression coefficients that quantify the effect of each covariate on the hazard
• Coefficients are estimated using partial likelihood, which accounts for the ordering of the event times without specifying the baseline hazard function
• Hazard ratios (exp($\beta_i$)) represent the multiplicative effect of a one-unit increase in the corresponding covariate on the hazard, assuming all other covariates remain constant
• Allows for the inclusion of both continuous and categorical covariates
• Can be extended to incorporate time-dependent covariates and stratification factors
• Model assumptions (proportional hazards, linearity, and non-informative censoring) should be assessed before interpreting the results

Interpreting Survival Analysis Results

• Kaplan-Meier curves provide a visual representation of the survival experience over time
  • Steep drops indicate time points with a high number of events
  • Wide gaps between curves suggest a substantial difference in survival between groups
  • Crossing curves may indicate non-proportional hazards
• Log-rank test assesses the statistical significance of the difference in survival curves between groups
  • A small p-value (typically < 0.05) suggests a significant difference in survival
• Cox proportional hazards model results are typically presented as hazard ratios (HR) with 95% confidence intervals (CI)
  • An HR > 1 indicates an increased risk of the event for the corresponding covariate, while an HR < 1 indicates a decreased risk
  • The 95% CI provides a range of plausible values for the true HR; if the CI does not include 1, the covariate is considered statistically significant
• Proportional hazards assumption can be assessed using graphical methods (e.g., log-log survival plots or Schoenfeld residuals) or statistical tests (e.g., time-dependent covariates or Grambsch-Therneau test)
• Model fit can be evaluated using measures such as the likelihood ratio test, Wald test, or score test
• Results should be interpreted in the context of the study design, population, and research question, considering potential confounding factors and limitations

Real-World Applications in Biostatistics

• Clinical trials: Evaluating the efficacy and safety of new treatments or interventions
  • Comparing survival outcomes between treatment and control groups
  • Identifying subgroups of patients who may benefit more from a specific treatment
• Epidemiological studies: Investigating risk factors for disease onset or progression
  • Assessing the impact of lifestyle factors (smoking, diet, physical activity) on disease-free survival
  • Examining the role of genetic or environmental factors in the development of chronic diseases
• Prognostic studies: Developing models to predict patient outcomes based on clinical, demographic, or molecular characteristics
  • Identifying prognostic biomarkers that can stratify patients into risk groups
  • Developing risk scores or nomograms to aid in treatment decision-making
• Public health research: Analyzing population-level trends in disease incidence, prevalence, and mortality
  • Evaluating the effectiveness of screening programs or public health interventions
  • Investigating disparities in health outcomes across different socioeconomic or racial/ethnic groups
• Reliability analysis: Assessing the durability or failure rates of medical devices or equipment
  • Comparing the performance of different device designs or materials
  • Identifying factors that contribute to device failure or malfunction