Meningiomas in children and adolescents: a meta-analysis of individual patient data
Rishi S. KotechaElaine M. PascoeElisabeth J. RushingLucy B. Rorke‐AdamsTed ZwerdlingXing GaoXin LiStephanie GreeneAbbas AmirjamshidiSeung-Ki KimMarco Antônio LimaPo-Cheng HungF. LakhdarNirav MehtaYuguang LiuBhagavatula Indira DeviB. J. SudhirMorten Lund‐JohansenF. GjerrisCatherine ColeNicholas G. Gottardo
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This article introduces readers to survival (failure-time) models, with a focus on Kaplan-Meier curves, Cox regression and sample size estimation.An example is used to show readers how to calculate a Kaplan-Meier curve from first principles.What makes survival data unique is censoring. Readers should understand censoring before undertaking an analysis of survival data.The Cox model continues to set the standard for survival models, and will continue well into the future.
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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
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In medical research, analyzing the time it takes for a phenomenon to occur is sometimes crucial. However, various factors can contribute to the length of survival or observation periods, and removing specific data can lead to bias results. In this paper, we discuss the Kaplan-Meier analysis and Cox proportional hazards regression model, which are the most frequently used methods in survival analysis. For the first step, we shall discuss the temporal concepts needed in survival analysis, such as cohort studies and then the basic statistical functions dealt with in survival analysis. After solidifying the concepts, methods of understanding and practical application of the Kaplan-Meier survival analysis is noted. After that, we will discuss the analysis methods for the Cox proportional hazards regression model, which includes multiple covariates. With the interpretation method of Cox proportional hazards regression result, we then discuss methods for checking the assumptions of the Cox proportional hazards regression, such as log minus log plots. Finally, we briefly explain the concept of time-dependent regression analysis. It is our aim that through this paper, readers can obtain an understanding on survival analysis and learn how to perform it.
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Introduction to Survival Analysis.- Kaplan-Meier Survival Curves and the Log-Rank Test.- The Cox Proportional Hazards Model and Its Characteristics.- Evaluating the Proportional Hazards Assumption.- The Stratified Cox Procedure.- Extension of the Cox Proportional Hazards Model for Time-Dependent Variables.- Parametric Survival Models.- Recurrent Events Survival Analysis.- Competing Risks Survival Analysis.
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Background Although most people with relapsing onset multiple sclerosis (R-MS) eventually transition to secondary progressive multiple sclerosis (SPMS), little is known about disability progression in SPMS. Methods All R-MS patients in the Cardiff MS registry were included. Cox proportional hazards regression was used to examine a) hazard of converting to SPMS and b) hazard of attaining EDSS 6.0 and 8.0 in SPMS. Results 1611 R-MS patients were included. Older age at MS onset (hazard ratio [HR] 1.02, 95%CI 1.01–1.03), male sex (HR 1.71, 95%CI 1.41–2.08), and residual disability after onset (HR 1.38, 95%CI 1.11–1.71) were asso- ciated with increased hazard of SPMS. Male sex (EDSS 6.0 HR 1.41 [1.04–1.90], EDSS 8.0 HR 1.75 [1.14–2.69]) and higher EDSS at SPMS onset (EDSS 6.0 HR 1.31 [1.17–1.46]; EDSS 8.0 HR 1.38 [1.19–1.61]) were associated with increased hazard of reaching disability milestones, while older age at SPMS was associated with a lower hazard of progression (EDSS 6.0 HR 0.94 [0.92–0.96]; EDSS 8.0: HR 0.92 [0.90–0.95]). Conclusions Different factors are associated with hazard of SPMS compared to hazard of disability progres- sion after SPMS onset. These data may be used to plan services, and provide a baseline for comparison for future interventional studies and has relevance for new treatments for SPMS RobertsonNP@cardiff.ac.uk
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A common objective in many medical studies is to investigate the survival time of an individual after being diagnosed with a particular disease or health related condition. In most survival analysis studies the analysis is based on modeling the probability of survival. One of the goals in a survival analysis is usually to model the survival function. This chapter presents three different approaches for modeling a survival function. They are the Kaplan–Meier method of modeling a survival function, the Cox proportional hazards model for a survivor function, and the use of logistic regression for modeling a binary survival response variable. It is important to note that the proportional hazards model is based on the actual survival times and the explanatory variables, a proportional hazards model will provide more information about the survival probabilities than will either the Kaplan–Meier model or a logistic regression model.
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Survival analysis of time to an event such as death or sickness. The survivorship function is estimated via the actuarial method and the Kaplan-Meier curve. Both are compared graphically and via the log-rank test. Stratification is needed when confounding is present. The Cox proportional hazard model incorporates covariates into the survival model. Other parametric and non-parametric models are discussed.
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Survival analysis is used to analyze data from patients who are followed for different periods of time and in whom the outcome of interest, a dichotomous event, may or may not have occurred at the time the study is halted; data from all patients are used in the analysis, including data from patients who dropped out, regardless of the duration of follow-up. This article discusses basic concepts in survival analysis, explains technical terms such as censoring, and provides reasons why ordinary methods of analysis cannot be applied to such data. The Kaplan-Meier survival curve is described, as is the Cox proportional hazards regression and the hazard ratio. Supplementary information includes a data file, graphs with explanations, and additional discussions; these are provided to enhance the reader's experience and understanding.
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The hazard ratio and median survival time are the routine indicators in survival analysis. We briefly introduced the relationship between hazard ratio and median survival time and the role of proportional hazard assumption. We compared 110 pairs of hazard ratio and median survival time ratio in 58 articles and demonstrated the reasons for the difference by examples. The results showed that the hazard ratio estimated by the Cox regression model is unreasonable and not equivalent to median survival time ratio when the proportional hazard assumption is not met. Therefore, before performing the Cox regression model, the proportional hazard assumption should be tested first. If proportional hazard assumption is met, Cox regression model can be used; if proportional hazard assumption is not met, restricted mean survival times is suggested.风险比（hazard ratio，HR）和中位生存时间是生存分析时的常规分析和报告指标。本文简要介绍了HR和中位生存时间的关系以及比例风险假定在这两者之间的作用，分析了检索出的58篇文献中的110对风险比和中位生存时间比的差异，并通过实例阐明了产生这种差异的原因。结果表明，在不满足比例风险假定时，Cox回归模型计算得到的风险比是不合理的，且与中位生存时间之比不等价。因此，在使用Cox回归模型前，应先进行比例风险假定的检验，只有符合比例风险假定时才能使用该模型；当不符合比例风险假定时，建议使用限制性平均生存时间。.
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