Topic

Survival analysis and extrapolation

Survival analysis describes the time until an event, such as death or disease progression, while allowing for patients whose follow-up ends early. Models often need survival beyond the end of a trial, so parametric models such as the Weibull and Gompertz, flexible spline models and cure models are fitted and extrapolated.

Concepts in this topic

• Bathtub HazardA bathtub hazard is a U-shaped hazard that is high soon after an event such as surgery, falls to a lower stable level, then rises again as patients age.
• Cause-Specific SurvivalThe probability of surviving without dying from a specific cause of interest, treating deaths from other causes as censoring rather than a competing risk.
• CensoringCensoring occurs in time-to-event data when an individual's event time is not observed exactly but is known only to lie beyond, before, or within a recorded time boundary.
• Constant HazardA constant hazard is an assumption that the instantaneous rate of a specified event among those still at risk remains unchanged over a defined period.
• Cox Proportional HazardsThe Cox proportional hazards model is a semiparametric survival model that relates covariates to the instantaneous event rate while leaving the baseline hazard unspecified.
• Cox-Snell ResidualA Cox-Snell residual is a survival model's fitted cumulative hazard at a patient's observed time, used to check overall fit against a unit exponential.
• Cumulative HazardCumulative hazard, H(t), is the hazard rate summed over time in survival models. Survival equals exp(-H(t)), and unlike a probability H(t) can exceed 1.
• Cure FractionA cure fraction is the model-estimated proportion of a population whose long-term risk from a specified disease-related event becomes negligible under an explicitly defined statistical cure assumption.
• Cure Fraction ModelA survival model separating a population into a cured subgroup facing only background mortality and an uncured subgroup remaining at risk of the disease event.
• Cure ModelA cure model is a survival model that represents a subgroup whose disease-related excess event risk eventually disappears while accounting for the remaining risk specified by the model.
• Cure Rate ModelA cure rate model is a survival model that estimates the cure fraction and survival of uncured patients, used in HTA to extrapolate long-term survival.
• Curve FittingCurve fitting is the estimation of parameters in a chosen mathematical function so that its predictions approximate observed data under a specified fitting criterion.
• Deviance ResidualA deviance residual is a signed measure of how poorly a generalised linear or survival model fits each patient; in a GLM its squares sum to the deviance.
• Excess HazardThe additional death risk attributable specifically to the disease studied, the difference between total observed hazard and the background hazard in a comparable population.
• Exponential DistributionA probability distribution characterised by a constant hazard rate over time, used to model event timing when risk is assumed not to change.
• Exponential ModelA survival model assuming time-to-event data follow an exponential distribution, implying a constant hazard rate throughout the modelled time horizon.
• ExtrapolationExtrapolation projects outcomes beyond the range or follow-up directly observed in the available evidence.
• Fleming-Harrington EstimatorA statistical technique for estimating the survival function from censored data, more flexible than the Kaplan-Meier estimator in weighting events over time.
• Fleming-Harrington TestA hypothesis test comparing survival distributions between groups, generalising the log-rank test by allowing different weights for events at different follow-up times.
• Flexible Parametric ModelA flexible parametric model is a fully specified regression model that represents a transformed survival function with smooth functions of time, commonly restricted cubic splines, so complex hazards and time-varying effects can be estimated and extrapolated.
• Fractional PolynomialA flexible modelling technique representing a non-linear relationship using a small number of power transformations of a continuous variable.
• Frailty ModelA survival model incorporating an unobserved random effect, called frailty, to represent risk heterogeneity not captured by observed covariates.
• Gamma DistributionThe gamma distribution is a continuous probability distribution on positive values, defined here by shape and scale parameters that control its mean, variance, and skew.
• Gamma ModelA survival model assuming time-to-event data follow a gamma distribution, allowing hazard rates that increase, decrease, or stay constant over time.
• Generalized F ModelA highly flexible parametric survival model encompassing several other distributions, including the exponential, Weibull, and log-normal, as special cases.
• Generalized GammaA flexible probability distribution including the gamma, Weibull, and log-normal distributions as special cases, capable of representing many hazard shapes.
• Generalized Gamma ModelA survival model based on the generalised gamma distribution, nesting simpler distributions, such as the Weibull and log-normal, within one framework.
• Gompertz DistributionA probability distribution characterised by a hazard rate rising exponentially with time, originally developed to describe human mortality increasing with age.
• Gompertz ModelA survival model based on the Gompertz distribution, assuming the hazard rate rises exponentially over time, a pattern common in age-related mortality.
• Hazard RateIn survival analysis, the instantaneous rate of an event at a given time among people still free of it; a rate, not a probability, so it can exceed 1.
• Interval CensoringA form of censoring in which an event's exact time is unknown but known to fall within a specific interval, such as between visits.
• Joint ModelA statistical approach analysing a longitudinal outcome, such as a repeated biomarker, alongside a time-to-event outcome, accounting for the link between them.
• Kaplan-Meier CurveA graphical display of the Kaplan-Meier survival estimate, shown as a step function that decreases at each observed event over time.
• Kaplan-Meier EstimatorThe Kaplan–Meier estimator calculates a non-parametric estimate of the survival function when follow-up can be right-censored.
• Kaplan-Meier MethodThe overall approach of estimating and comparing survival functions using the Kaplan-Meier estimator, accounting for censored observations in time-to-event data.
• Landmark AnalysisA survival technique restricting analysis to patients event-free up to a predefined time point, then examining survival from that point forward.
• Left CensoringA form of censoring in which an event is known to have occurred before an observed time, but its exact prior timing is unknown.
• Life Table MethodThe life table (actuarial) method estimates survival from deaths and withdrawals in fixed intervals, treating withdrawals as at risk for half an interval.
• Log-Logistic DistributionA two-parameter continuous distribution for positive event times whose logarithm is logistic; its survival function has a heavy tail, and its hazard may rise and then fall.
• Log-Logistic ModelA survival model based on the log-logistic distribution, representing a hazard function that rises to a peak and then declines.
• Log-Normal DistributionA probability distribution in which the logarithm of the variable follows a normal distribution, often used for hazards that rise then fall.
• Log-Normal ModelA survival model based on the log-normal distribution, capable of representing a hazard that rises then falls, similar in shape to log-logistic.
• Log-Rank TestThe log-rank test is a nonparametric hypothesis test that compares the event-time distributions of two or more groups by accumulating observed-minus-expected events across distinct event times while accounting for right censoring.
• Martingale ResidualA residual used to diagnose the fit of a Cox model, checking the functional form of continuous covariates and identifying influential observations.
• Mixture Cure ModelA cure model treating the population as a mixture of a cured fraction facing only background mortality and an uncured fraction remaining at risk.
• Mixture ModelA statistical model representing a population as a combination of two or more distinct subgroups, each following its own underlying distribution.
• Nelson-Aalen EstimatorA non-parametric method for estimating the cumulative hazard function from censored data, closely related to the Kaplan-Meier estimator.
• Net Survival ModelA survival model estimating the survival a population would experience if the disease studied were the only possible cause of death.
• Non-Mixture Cure ModelAn alternative cure model structure formulating survival directly in terms of the cured fraction without explicitly splitting the population into two subgroups.
• Non-Parametric EstimationA statistical approach to estimating a survival function without assuming the data follow any specific mathematical distribution.
• Non-Parametric SurvivalSurvival data or estimates obtained without assuming a specific distribution, such as a Kaplan-Meier curve calculated directly from observed trial data.
• Parametric EstimationA statistical approach estimating a survival function by assuming the data follow a specific distribution, such as Weibull or log-normal, with estimated parameters.
• Parametric SurvivalSurvival estimates derived from a fitted mathematical distribution rather than directly from observed data alone, allowing extrapolation beyond the data.
• Parametric Survival ModelA parametric survival model specifies a probability distribution for time to a defined event, enabling estimation of survival and hazard functions and projection beyond observed follow-up under stated assumptions.
• Permutation TestA non-parametric hypothesis test determining significance by repeatedly rearranging observed data labels to build an empirical distribution under the null hypothesis.
• Peto TestA Peto–Peto weighted test for comparing time-to-event distributions, using observed-versus-expected events with weights based on pooled survival that emphasize earlier event times.
• Piecewise ModelA survival model dividing follow-up into distinct segments, each modelled with a separate, often simpler function, letting the hazard shape vary in steps.
• Proportional Hazards AssumptionThe Cox model assumption that the hazard ratio between two groups stays constant over follow-up, even as the absolute hazard rate changes.
• Relative SurvivalA survival measure comparing observed survival in a population to expected survival in a comparable general population, without needing cause-of-death data.
• Relative Survival ModelA statistical model estimating disease-specific survival by comparing observed survival against expected general population survival, without individual cause-of-death data.
• Restricted Cubic SplineA restricted cubic spline is a smooth piecewise cubic function of a continuous predictor constrained to be linear beyond its outer knots, allowing regression models to represent nonlinear associations without prescribing one global polynomial shape.
• Restricted Mean Survival TimeA survival summary measure calculated as the area under the survival curve up to a specified time point, an alternative to the hazard ratio.
• Right CensoringThe most common form of censoring, occurring when follow-up ends or a patient is lost before the event of interest has occurred.
• Royston-Parmar ModelA class of flexible parametric survival models using restricted cubic splines to represent a transformed survival or hazard function while remaining fully parametric.
• Scaled SchoenfeldA diagnostic technique testing the proportional hazards assumption by scaling Schoenfeld residuals by their variance to improve test power.
• Schoenfeld ResidualA residual specific to the Cox model, calculated at each event time, used to test whether the proportional hazards assumption holds.
• Semi-Parametric SurvivalA category of survival models, notably the Cox model, that assumes covariate effects parametrically while leaving the baseline hazard unspecified.
• Shared FrailtyAn extension of the frailty model in which individuals within the same group are assumed to share a common unobserved risk effect.
• Spline ModelA model using piecewise polynomial functions joined smoothly at knots to represent a flexible, non-linear relationship such as time and hazard.
• Stratified Log-RankA variant of the log-rank test accounting for stratifying variables, comparing survival within each stratum before combining into an overall statistic.
• Survival AnalysisSurvival analysis examines the time from a defined origin to an event such as death, disease progression, hospitalisation, recovery or treatment discontinuation.
• Survival FunctionA function giving the probability that a person survives without the event of interest beyond a specified point in time.
• Tarone-Ware TestA test comparing survival distributions using a weighting scheme between the log-rank and Peto tests, a compromise depending on when treatment differences emerge.
• Time-Fixed CovariateA patient characteristic in a survival model measured once and assumed constant throughout follow-up, such as baseline age or sex.
• Time-Varying CovariateA patient characteristic in a survival model allowed to change value over follow-up, such as a repeatedly measured biomarker.
• Time-Varying HazardA hazard rate that changes in magnitude over follow-up, unlike a constant hazard that stays the same at every point.
• Unimodal HazardA hazard function that rises to a single peak and then declines over follow-up, rather than a monotonic or bathtub pattern.
• Weibull DistributionA flexible probability distribution capable of a monotonically increasing or decreasing hazard rate depending on its shape parameter.
• Weibull ModelA survival model based on the Weibull distribution, representing an increasing or decreasing hazard over time, though restricted to a monotonic pattern.
• Weighted Log-RankA survival-curve comparison test that weights each event-time observed-minus-expected count to emphasize a prespecified pattern of differences over follow-up.
• Wilcoxon TestA test comparing survival distributions that gives greater weight to earlier event times than later ones, unlike the standard log-rank test.