Survival analysis and regression models
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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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Abstract Introduction Early cessation of EBF has the short and long term effect for the welfare of infants including the life-long impacts of poor school performance, reduced productivity, and impaired intellectual development. Objective of the study : the main objective of this study was to compare the performance of CPH model and AFT models in analyzing EBF data in Ethiopia, 2016 EDHS. Specifically, the study aimed to identify the major predictor variables of the duration of EBF based on 2016 EDHS data. Methodology: The secondary data is obtained from Ethiopian Demographic and Health Survey (EDHS), 2016. The outcome variable of this study was the duration of EBF in month. To achieve the objective of the study, descriptive survival analysis like the median survival time, Kaplan Meier survival estimate and log-rank test were used to compare the estimated survival probability among different levels of predictor variables at 5 percent significant level. The Cox proportional hazard regression and Accelerated failure time model were fitted and their results were compared using model comparison criterion such as AIC, BIC. Results: of 1092 interviewed mothers, 15.3 % of them were discontinued EBF and 84.7% of them were exclusively breastfed (censored). The estimated median duration of EBF was 3 months. Based on estimated Kaplan Meier survival curve and log-rank test, it was found that there was a statistically significant difference in survivor experience of discontinuing EBF over each duration with respect to place of delivery, maternal education, husband education, mode of delivery and employment status. The fitted CPH and AFT model indicated that mode of delivery, wealth index, and employment status was found as significant predictors of EBF duration. Moreover, comparatively Weibull AFT model performed better in analyzing EBF data. According to the fitted model, mothers who were in poor wealth index category and who gave birth by cesarean shortens the duration of EBF by 16% and 29% respectively. On the other hand, employed mothers were improved the duration of EBF by 26%. Conclusion: Weibull AFT model is performed better in analyzing EBF data. A mother who was unemployed, poor wealth index, and gave birth by cesarean shortens the duration of EBF than their counterparts. Therefore, special emphasis should be given for mothers who are unemployed, who are economically poor, and give birth by cesarean to improve the duration of EBF.
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Accelerated Failure Time (AFT) models can be used for the analysis of time to event data to estimate the effects of covariates on acceleration/deceleration of the survival time. The effect of the covariate is measured through a log-linear model taking logarithm of the survival time as the outcome or dependent variable. Hence, the effect of covariate is multiplicative on time scale, and the results of AFT models may be easier to interpret as the covariate effects are directly expressed in terms of time ratio (TR). Some AFT models are applied to the data on time to death of hospitalized Acute Liver Failure (ALF) patients in All India Institute of Medical Sciences, New Delhi, India to identify the prognostic factors. This type of study is being carried out for the first time in Indian population using retrospective data of ALF patients using AFT models.
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The restricted mean survival time is a clinically easy-to-interpret measure that does not require any assumption of proportional hazards. We focus on two ways to directly model the survival time and adjust the covariates. One is to calculate the pseudo-survival time for each subject using leave-one-out, and then perform a model analysis using all pseudo-values to adjust for covariates. The pseudo-survival time is used to reflect information of censored subjects in the model analysis. The other method adjusts for covariates using subjects for whom the time-to-event was observed while adjusting for the censored subjects using the inverse probability of censoring weighting (IPCW). This paper evaluates the performance of these two methods in terms of the power to detect group differences through a simple example dataset and computer simulations. The simple example illustrates the intuitive behavior of the two methods. With the method using pseudo-survival times, it is difficult to interpret the pseudo-values. We confirm that the pseudo-survival times are different from the actual data obtained in a primary biliary cholangitis clinical trial because of the many censored data. In the simulations, the method using IPCW is found to be more powerful. Even in the case of group differences with respect to the censor incidence rates and covariates, the method using IPCW maintains a nominal significance level for the type-1 error rate. We conclude that the IPCW method should be used to estimate the restricted mean survival time when adjusting the covariates.
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One of the primary problems facing statisticians who work with survival data is the loss of information that occurs with right-censored data. This research considers trying to recover some of this endpoint information through the use of a prognostic covariate which is measured on each individual. We begin by defining a survival estimate which uses time-dependent covariates to more precisely get at the underlying survival curves in the presence of censoring. This estimate has a smaller asymptotic variance than the usual Kaplan-Meier in the presence of censoring and reduces to the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) in situations where the covariate is not prognostic or no censoring occurs. In addition, this estimate remains consistent when the incorporated covariate contains information about the censoring process as well as survival information. Because the Kaplan-Meier estimate is known to be biased in this situation due to informative censoring, we recommend use of our estimate.
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The proportional hazard (PH) model and its extension are used comprehensively to assess the effect of an intervention in the presence of covariates. The assumptions of PH model may not hold where the effect of the intervention is to accelerate the onset of an event. The accelerated failure time (AFT) model is the alternative when the PH assumption does not hold. The aim of this paper is to formulate a model that yields biological plausible and interpretable estimates of the effect of important covariates on survival time. The data consists of 1236 tuberculosis patients admitted in randomized controlled clinical trial. A total of six covariates are considered for modeling. The AFT model gives better prediction than the Cox PH model. Keywords: Accelerated failure time model; proportional hazards model; time dependent covariate, tuberculosis.
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Time-dependent covariates in survival analysis EDWARD D. LUSTBADER EDWARD D. LUSTBADER Institute for Cancer Research, Fox ChasePhiladelphia, Pennsylvania Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 67, Issue 3, 1980, Pages 697–698, https://doi.org/10.1093/biomet/67.3.697 Published: 01 December 1980 Article history Received: 01 September 1979 Revision received: 01 March 1980 Published: 01 December 1980
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In many cases, an endpoint measures time from the point of randomisation to some well-defined event, such as time to death in oncology or time to rash healing in herpes zoster. The data from such an endpoint invariably have a special feature known as censoring . Censoring in clinical trials usually occurs because the patient is still alive at the end of the period of follow-up. This chapter discusses Kaplan-Meier curves, which are used to display the data and to calculate summary statistics. It explains the logrank and Gehan-Wilcoxon tests, which are simple two-group comparisons for censored survival data, and extend these ideas to incorporate baseline covariates and factors. The chapter argues that calculating the mean is not possible in general because of censoring and that survival times/time to event values are not available for all subjects.
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Origins of survival analysis lie in actuarial science of lifetimes and death rates of people. As a discipline, survival analysis has grown from many applications in biomedical fields. This chapter discusses life tables, Kaplan–Meier survival curves, and Cox’s proportional hazards regression model, which take into account censoring of the lifetimes. The Kaplan–Meier estimator gives a nonparametric estimate of the survival distribution. The chapter illustrates standard error calculation for Kaplan–Meier survival curves. Logrank test provides a test of the overall difference between two survival distributions by cumulating the differences between the survival distributions at the observed death times. Cox’s proportional hazards model postulates how the hazard rate depends on the covariates. It has a nonparametric component and a parametric component. Hence, the model is also called a semiparametric model.
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