survivalmallit

Survivalmallit, or survival models, are statistical methods for analyzing time-to-event data. The defining focus is the duration until an event of interest occurs (for example death, disease onset, or equipment failure). In many datasets the exact event time is unknown for some observations, a situation called censoring, which these models handle explicitly.

Key concepts include the survival function S(t) = P(T>t), which gives the probability of surviving beyond time t, and the hazard function h(t), describing the instantaneous risk of the event at time t given survival to that time. The cumulative hazard H(t) aggregates the hazard over time. Models relate covariates to these functions in order to explain differences in survival experience among individuals or groups.

Major classes of survival models include: non-parametric methods such as the Kaplan-Meier estimator for S(t) and the log-rank test for group comparisons; semi-parametric models, most notably the Cox proportional hazards model, which links covariates to relative risk without fully specifying the baseline hazard; and parametric models (e.g., exponential, Weibull, log-normal, Gompertz) that assume a distribution for the event time to allow extrapolation. Extensions cover competing risks, where multiple event types are possible, and multi-state models that describe transitions between health states.

Assumptions and diagnostics typically involve independent censoring and, for some models, proportional hazards. Model checks use residuals (such as Schoenfeld residuals), time-varying covariates, and goodness-of-fit measures, with validation often performed on external data or through calibration.

Applications and data considerations: Survival models are widely used in clinical trials, epidemiology, and reliability engineering. Data may be right-censored, left-truncated, or interval-censored, and methods are chosen to reflect the censoring pattern and study design.

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