7.3 Survival analysis techniques

Survival analysis techniques are crucial tools in demographic research, allowing us to study time-to-event data like mortality rates. These methods help us estimate survival probabilities, compare patterns between groups, and assess how different factors impact survival times.

Key concepts include survival and hazard functions, censoring, and non-parametric estimators like Kaplan-Meier and Nelson-Aalen. We'll also explore how to interpret survival curves and use log-rank tests to compare survival between groups.

Principles of Survival Analysis

Key Concepts and Objectives

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• Survival analysis focuses on analyzing time-to-event data where the outcome variable is the time until an event of interest occurs (death, disease occurrence, machine failure)
• Survival time or failure time refers to the time to the event of interest
• Primary objectives include estimating survival probabilities, comparing survival patterns between groups, and assessing the impact of covariates on survival times
• Key concepts include the survival function (probability of surviving beyond a given time) and the hazard function (instantaneous risk of the event occurring at a given time)

Censoring in Survival Data

• Survival data is often characterized by censoring where the exact survival time is unknown for some individuals due to loss to follow-up, study termination, or competing events
• Right censoring occurs when the event of interest has not been observed by the end of the study period or the individual is lost to follow-up
• Left censoring occurs when the event of interest has already occurred before the start of the observation period
• Interval censoring occurs when the event of interest is known to have occurred within a certain time interval, but the exact time is unknown

Kaplan-Meier and Nelson-Aalen Estimators

Kaplan-Meier (KM) Estimator

• Non-parametric method for estimating the survival function from censored survival data
• Calculates the probability of surviving beyond a given time by multiplying the conditional probabilities of surviving each event time, given survival up to that point
• Step function that changes value only at event times and accounts for censoring by adjusting the risk set at each event time
• Widely used due to its simplicity, robustness, and ability to handle censored data without making parametric assumptions about the distribution of survival times

Nelson-Aalen (NA) Estimator

• Non-parametric method for estimating the cumulative hazard function (integral of the hazard function over time)
• Calculated by summing the increments in the cumulative hazard at each event time, where the increment is the number of events divided by the number at risk
• Can be used to derive an estimate of the survival function through the relationship between the survival function and the cumulative hazard function
• Widely used due to its simplicity, robustness, and ability to handle censored data without making parametric assumptions about the distribution of survival times

Survival Curve Interpretation

Graphical Representation of Survival Probabilities

• Survival curves, typically generated using the Kaplan-Meier estimator, graphically represent the estimated survival probabilities over time
• Y-axis represents the estimated survival probability, and x-axis represents time
• Vertical drops in the survival curve indicate event times, while horizontal lines represent periods where no events occur
• Censored observations are often marked on the survival curve using tick marks or other symbols (crosses, circles)

Comparing Survival Curves Between Groups

• When comparing survival curves, consider the overall shape of the curves, the median survival time (time at which the survival probability is 0.5), and any notable differences in survival probabilities at specific time points
• Crossing survival curves may indicate a violation of the proportional hazards assumption, which assumes that the hazard ratio between groups remains constant over time
• Area under the survival curve (AUC) can be used as a summary measure to compare overall survival experiences between groups, with larger AUCs indicating better survival
• Confidence intervals for the survival curves can be used to assess the uncertainty in the estimated survival probabilities and to determine if differences between groups are statistically significant

Log-Rank Tests for Survival Comparisons

Non-Parametric Hypothesis Test

• Log-rank test, also known as the Mantel-Cox test, is a non-parametric hypothesis test used to compare the survival curves of two or more groups
• Null hypothesis: no difference in the survival curves between the groups being compared
• Alternative hypothesis: there is a difference in the survival curves between the groups

Test Statistic Calculation

• Log-rank test calculates the expected number of events in each group at each event time, assuming that the null hypothesis is true, and compares it to the observed number of events
• Test statistic is calculated by summing the differences between the observed and expected number of events across all event times and squaring the result
• Test statistic follows a chi-square distribution with degrees of freedom equal to the number of groups minus one

Interpretation and Multiple Comparisons

• A small p-value (typically < 0.05) from the log-rank test indicates that there is sufficient evidence to reject the null hypothesis and conclude that there is a significant difference in survival between the groups
• When comparing more than two groups, pairwise log-rank tests can be performed to determine which specific groups differ significantly from each other, with appropriate adjustments for multiple comparisons (Bonferroni correction, Tukey's HSD)
• Log-rank test is sensitive to differences in the overall survival curves and may not detect differences that occur only at specific time points or differences in the shape of the curves