Kaplan-Meier Curves, Log-Rank Tests, and Cox Regression for Time-to-Event Data

Related Article, see p 971KEY POINT: Kaplan-Meier curves, log-rank-test, and Cox proportional hazards regression are common examples of “survival analysis” techniques, which are used to analyze the time until an event of interest occurs.In this issue of Anesthesia & Analgesia, Song et al1 report results of a randomized trial in which they studied the onset of labor analgesia with 3 different epidural puncture and maintenance techniques. These authors compared the techniques on the primary outcome of time until adequate analgesia was reached—defined as a visual analog scale (VAS) score of ≤30 mm—with Kaplan-Meier curves, log-rank tests, and Cox proportional hazards regression. In studies addressing the time until an event of interest occurs, some but not all patients will typically have experienced the event at the end of the follow-up period. Patients in whom the even has not occurred—or who are lost to follow-up during the observation period—are said to be “censored.” It is unknown when and, depending on the event, if the event will occur.2 Simply excluding censored patients from the analysis would bias the analysis results. Specific statistical methods are thus needed that can appropriately account for such censored patient observations. Since the event of interest is often death, these analyses are traditionally termed “survival analyses,” and the time until the event occurs is referred to as the “survival time.” However, as done by Song et al,1 these techniques can also be used for the analysis of the time to any other well-defined event. Among the many available survival analysis methods, Kaplan-Meier curves, log-rank tests to compare these curves, and Cox proportional hazards regression are most commonly used. The Kaplan-Meier method estimates the survival function, which is the probability of “surviving” (ie, the probability that the event has not yet occurred) beyond a certain time point. The corresponding Kaplan-Meier curve is a plot of probability (y-axis) against time (x-axis) (Figure). This curve is a step function in which the estimated survival probability drops vertically whenever one or more outcome events occurred with a horizontal time interval between events. Plotting several Kaplan-Meier curves in 1 figure allows for a visual comparison of estimated survival probabilities between treatment or exposure groups; the curves can formally be compared with a log-rank test. The null hypothesis tested by the log-rank test is that the survival curves are identical over time; it thus compares the entire curves rather than the survival probability at a specific time point.Figure.: Kaplan-Meier plot of the percentage of patients without adequate analgesia, redrawn from Figure 2 in Song et al.1 Note that the original figure plotted the probability of adequate analgesia, as this is easily interpretable for readers in the context of the study research aim. In contrast, we present the figure as conventionally done in a Kaplan-Meier curve or plot, with the estimated probability (here expressed as percentage) of “survival” plotted on the y-axis. Vertical drops in the plot indicate that one or more patients reached the end point of experiencing adequate analgesia at the respective time point. CEI indicates continuous epidural infusion; DPE, dural puncture epidural; EP, conventional epidural; PIEB, programmed intermittent epidural bolus.The log-rank test assesses statistical significance but does not estimate an effect size. Moreover, while there is a stratified log-rank test that can adjust the analysis for a few categorical variables, the log-rank test is essentially not useful to simultaneously analyze the relationships of multiple variables on the survival time. Thus, when researchers either desire (a) to estimate an effect size3 (ie, the magnitude of the difference between groups)—as done in the study by Song et al1—or (b) to test or control for effects of several independent variables on survival time (eg, to adjust for confounding in observational research),4 a Cox proportional hazards model is typically used. The Cox proportional hazards regression5 technique does not actually model the survival time or probability but the so-called hazard function. This function can be thought of as the instantaneous risk of experiencing the event of interest at a certain time point (ie, the probability of experiencing the event during an infinitesimally small time period). The event risk is inversely related to the survival function; thus, “survival” rapidly declines when the hazard rate is high and vice versa. The exponentiated regression coefficients in Cox proportional hazards regression can conveniently be interpreted in terms of a hazard ratio (HR) for a 1-unit increase in the independent variable, for continuous independent variables, or versus a reference category, for categorical independent variables. While the HR is not the same as a relative risk, it can for all practical purposes be interpreted as such by researchers who are not familiar with the intricacies of survival analysis techniques. For those wishing to delve deeper into the details and learn more about survival analysis—including but not limited to the topics that we briefly touch on here—we refer to our tutorial on this topic previously published in Anesthesia & Analgesia.2 Importantly, even though the techniques discussed here do not make assumptions on the distribution of the survival times or survival probabilities, these analysis methods have other important assumptions that must be met for valid inferences, as also discussed in more detail in the previous tutorial.2