Survival Analysis and Interpretation of Time-to-Event Data: The Tortoise and the Hare

Abstract

Survival analysis, or more generally, time-to-event analysis, refers to a set of methods for analyzing the length of time until the occurrence of a well-defined end point of interest. A unique feature of survival data is that typically not all patients experience the event (eg, death) by the end of the observation period, so the actual survival times for some patients are unknown. This phenomenon, referred to as censoring, must be accounted for in the analysis to allow for valid inferences. Moreover, survival times are usually skewed, limiting the usefulness of analysis methods that assume a normal data distribution. As part of the ongoing series in Anesthesia & Analgesia, this tutorial reviews statistical methods for the appropriate analysis of time-to-event data, including nonparametric and semiparametric methods-specifically the Kaplan-Meier estimator, log-rank test, and Cox proportional hazards model. These methods are by far the most commonly used techniques for such data in medical literature. Illustrative examples from studies published in Anesthesia & Analgesia demonstrate how these techniques are used in practice. Full parametric models and models to deal with special circumstances, such as recurrent events models, competing risks models, and frailty models, are briefly discussed.

Figure 1.

Survival (survivor) function estimated by the Kaplan-Meier method, including 95% confidence bands. Censoring is indicated by vertical marks (at 5 and 21 d). The number of patients at risk at different time points is displayed on the graph. The point on the x-axis where the horizontal dashed line at a survival probability of .5 intersects the curve represents the estimated median survival time (17 d).

Figure 2.

Kaplan-Meier curves displaying the estimated survival probability for 4 different groups of patients after lung cancer surgery. Patients either did or did not perioperatively receive flurbiprofen axetil (FA) and dexamethasone (DXM) (reprinted with permission from Huang et al). Each vertical step in the curve indicates one or more events (ie, deaths), and right-censored patients are indicated by a vertical mark in the curve at the censoring time. A visual inspection suggests that survival seems to be more favorable for patients who received FA and DXM, compared with patients who received none of these 2 drugs. The log-rank test indicates a significant difference between the survival curves.