Survival analysis: Part I - analysis of time-to-event

Abstract

Length of time is a variable often encountered during data analysis. Survival analysis provides simple, intuitive results concerning time-to-event for events of interest, which are not confined to death. This review introduces methods of analyzing time-to-event. The Kaplan-Meier survival analysis, log-rank test, and Cox proportional hazards regression modeling method are described with examples of hypothetical data.

Keywords: Censored data; Cox regression; Hazard ratio; Kaplan-Meier method; Log-rank test; Medical statistics; Power analysis; Proportional hazards; Sample size; Survival analysis.

Fig. 1.

Illustration of survival analysis data from an imaginary experiment with a 6-week study period. Cases 3 and 5 experienced the event and are coded as “event cases”, denoted “X”. Their follow-up periods were 6 and 1 weeks, respectively. The others are all “censored cases,” denoted “C.” Patient 1 participated in the study from the start, but the follow-up was terminated due to unexpected death from an irrelevant cause. Patient 2 participated in the study from 4 to 6 weeks, but then refused to participate further. Case 4 was enrolled in the study in the 2nd week, lost contact in the 4th week, and was considered lost to follow-up. Case 6 was excluded in the 4th week because of uncovered random allocation.

Fig. 2.

Data arranged for Kaplan-Meier analysis. Note that for all participants the observations are rearranged to start at zero. Events are denoted “X,” and censored data are denoted “C.”

Fig. 3.

Example of a Kaplan-Meier survival curve. The axes represent time (x-axis) and survival rate (y-axis). At every time point of event occurrence, the survival curve steps down to the next lower level. Black dots represent points of censored data. At the bottom of the graph, the number of risk and censored cases are presented. It is recommended that specify these numbers with adequate time interval to clarify the clinical meanings.

Fig. 4.

Kaplan-Meier curves for groups A and B. The median survival time (postoperative nausea and vomiting-free time) for group A was 13.0 h (Q1, Q3: 8.0, 24.0 h), and for group B 6.0 h (Q1, Q3: 3.0, 17.0 h). The median survival time was significantly longer in group A than in group B (log-rank test, χ² (1) = 6.802, P = 0.009). Black crosses indicate censored data.

Fig. 5.

Cumulative survival curves of groups A and B estimated using the Cox proportional hazards regression model. The survival curve of group B decreases considerably compared to that of group A as time passes. The hazard ratio for antiemetics was estimated as 2.0 (95% CI: 1.2–3.3, P = 0.009). Group B showed a 2-fold greater hazard of postoperative nausea and vomiting (PONV) compared to group A. Intraoperative fentanyl increased the PONV risk about 4.7-fold (95% CI: 2.8–8.1, P < 0.001) compared to non-use.

Fig. 6.

Log-log function curves for the antiemetic comparison experiment. The Y-axis represents the log-log transformed survival function estimated using the Cox proportional hazards regression model. The basic assumption of Cox proportional hazards regression modeling, a constant hazard ratio over the time (the proportional hazards assumption), can be checked using this plot for only one covariate, the antiemetic. The two curves do not cross over in the early period of observation and remain parallel after that. This plot suggests that this model does not violate the proportional hazards assumption (Created with: IBM^® SPSS^® Statistics ver. 23 [IBM Corp., USA]).