Introduction to Survival Analysis in Practice

The Cox model-which remains the first choice for analyzing time-to-event data, even for large data sets-relies on the proportional hazards (PH) assumption. When survival data arrive sequentially in chunks, a fast and minimally storage intensive approach to test the PH assumption is desirable. We propose an online updating approach that updates the standard test statistic as each new block of data becomes available and greatly lightens the computational burden. Under the null hypothesis of PH, the proposed statistic is shown to have the same asymptotic distribution as the standard version computed on an entire data stream with the data blocks pooled into one data set. In simulation studies, the test and its variant based on most recent data blocks maintain their sizes when the PH assumption holds and have substantial power to detect different violations of the PH assumption. We also show in simulation that our approach can be used successfully with "big data" that exceed a single computer's computational resources. The approach is illustrated with the survival analysis of patients with lymphoma cancer from the Surveillance, Epidemiology, and End Results Program. The proposed test promptly identified deviation from the PH assumption, which was not captured by the test based on the entire data.

BackgroundSurvival analysis and effect of covariates on survival time is a central research interest. Cox proportional hazards regression remains as a gold standard in the survival analysis. The Cox model relies on the assumption of proportional hazards (PH) across different covariates. PH assumptions should be assessed and handled if violated. Our aim was to investigate the reporting of the Cox regression model details and testing of the PH assumption in survival analysis in total joint arthroplasty (TJA) studies.MethodsWe conducted a review in the PubMed database on 28th August 2019. A total of 1154 studies were identified. The abstracts of these studies were screened for words “cox and “hazard*” and if either was found the abstract was read. The abstract had to fulfill the following criteria to be included in the full-text phase: topic was knee or hip TJA surgery; survival analysis was used, and hazard ratio reported. If all the presented criteria were met, the full-text version of the article was then read. The full-text was included if Cox method was used to analyze TJA survival. After accessing the full-texts 318 articles were included in final analysis.ResultsThe PH assumption was mentioned in 114 of the included studies (36%). KM analysis was used in 281 (88%) studies and the KM curves were presented graphically in 243 of these (87%). In 110 (45%) studies, the KM survival curves crossed in at least one of the presented figures. The most common way to test the PH assumption was to inspect the log-minus-log plots (n = 59). The time-axis division method was the most used corrected model (n = 30) in cox analysis. Of the 318 included studies only 63 (20%) met the following criteria: PH assumption mentioned, PH assumption tested, testing method of the PH assumption named, the result of the testing mentioned, and the Cox regression model corrected, if required.ConclusionsReporting and testing of the PH assumption and dealing with non-proportionality in hip and knee TJA studies was limited. More awareness and education regarding the assumptions behind the used statistical models among researchers, reviewers and editors are needed to improve the quality of TJA research. This could be achieved by better collaboration with methodologists and statisticians and introducing more specific reporting guidelines for TJA studies. Neglecting obvious non-proportionality undermines the overall research efforts since causes of non-proportionality, such as possible underlying pathomechanisms, are not considered and discussed.

Many randomized clinical trials include right censored time to event data, comparing an experimental treatment with a standard treatment or placebo control. In this comparison, one tests whether the two treatments have the same survival function or equivalently the same hazard function over a given time period in order to evaluate effect of treatment. The methodological development of survival analysis for randomized clinical trials with right-censored data that have had the most profound impact are the log-rank test for comparing the equality of two or more survival distributions, and the Cox proportional hazards model for examining the covariate(s) effects on the hazard function. However, when comparing treatments in terms of their time to event distribution, there may be reason to believe that the hazard curves will cross, and in such cases standard comparison techniques could lead to misleading results [16]. Hence, in this study, the performance of new methods for testing treatment effect on randomized clinical trials when the proportional hazards assumption is not satisfied was evaluated based on simulation studies and on two real datasets. New proposed methods are based on combination of early/late treatment effects obtained from stopped/left truncated Cox or equivalently from extended Cox and the overall treatment effect from Cox proportional hazards model. These methods were compared with Cox proportional hazards model [8], pseudo values regression approach based on mean restricted survival time [1, 13] and extended Cox for the time dependent treatment effect [20]. Type I error rate and power of the proposed tests were illustrated based on simulated data under five possible treatment effect. The results of simulations and real data examples on cancer clinical trials showed that the new proposed methods performed reasonably well in case of crossing survival curves compared to Cox proportional hazards model and pseudo values regression approach based on restricted mean survival time. However, they performed about the same compared to extended Cox model. Furthermore, they performed about the same compared to Cox proportional hazards model and extended Cox under the late treatment effect. Using the proposed methods under proportional hazards alternative did not generally yield dramatic decrease in power compared to the Cox model and they allow adjusting for covariate(s).

Background: The Cox proportional hazard (Cox PH) model is the most common and widely used method for analyzing time-to-event data. This approach, however, has been criticized for predicting patient outcomes since it assumes that the outcome is a linear combination of covariates and a constant hazard ratio over time, which is invalid in many cases and may not always be fulfilled, and thus Random Survival Forest (RSF) for time-to-event data has been proposed as an alternative to Cox PH. By considering non-linearity and interrelated covariates, the RSF is recognized as a way to correct bias in Cox proportional hazard models. Methods: The Cox PH model was compared to the Random Survival Forest by a relaxed proportional hazard (PH) assumption in this retrospective cohort study, which included 189 patients who underwent a new algorithmic treatment of an advanced human immunodeficiency virus clinical trial proposed by the World Health Organization from January to December 2018 at Kibong'oto infectious diseases hospital. Results: The study results show that the Random Survival Forest outperforms the Cox PH model in time-to-event data analysis by integrating covariates that do not meet proportional hazard assumptions based on predictive error curves, concordance indices, Brier scores, and integrated Brier scores as predictive indicators. Despite the fact that the PH assumption was relaxed for all covariates in the dataset, the Cox PH model selected only a few, and all deselected covariates were highly significant in the RSF, implying that the Cox PH model was inbuilt with the proportional hazard assumption. Conclusions: As a result, only covariates that satisfied the PH inbuilt assumption were chosen, regardless of whether the assumption was tested or not, rendering it ineffective when compared to a random survival forest. Therefore, having a dataset free from non-linearity and interrelated variables is impossible in the real world; hence, the RSF model outperforms the Cox PH model in predictive accuracy and covariate selection performance.

Re-evaluating Treatments for Advanced Urothelial Carcinoma Using Restricted Mean Survival Time: A Systematic Review and Network Meta-analysis

In most randomized clinical trials (RCTs) with a right-censored time-to-event outcome, the hazard ratio is taken as an appropriate measure of the effectiveness of a new treatment compared with a standard-of-care or control treatment. However, it has long been known that the hazard ratio is valid only under the proportional hazards (PH) assumption. This assumption is formally checked only rarely. Some recent trials, particularly the IPASS trial in lung cancer and the ICON7 trial in ovarian cancer, have alerted researchers to the possibility of gross non-PH, raising the critical question of how such data should be analyzed. Here, we propose the use of the restricted mean survival time at a prespecified, fixed time point as a useful general measure to report the difference between two survival curves. We describe different methods of estimating it and we illustrate its application to three RCTs in cancer. The examples are graded from a trial in kidney cancer in which there is no evidence of non-PH, to IPASS, where the opposite is clearly the case. We propose a simple, general scheme for the analysis of data from such RCTs. Key elements of our approach are Andersen's method of 'pseudo-observations,' which is based on the Kaplan-Meier estimate of the survival function, and Royston and Parmar's class of flexible parametric survival models, which may be used for analyzing data in the presence or in the absence of PH of the treatment effect.

Proportional hazard (PH) models can be formulated with or without assuming a probability distribution for survival times. The former assumption leads to parametric models, whereas the latter leads to the semi-parametric Cox model which is by far the most popular in survival analysis. However, a parametric model may lead to more efficient estimates than the Cox model under certain conditions. Only a few parametric models are closed under the PH assumption, the most common of which is the Weibull that accommodates only monotone hazard functions. We propose a generalization of the log-logistic distribution that belongs to the PH family. It has properties similar to those of log-logistic, and approaches the Weibull in the limit. These features enable it to handle both monotone and nonmonotone hazard functions. Application to four data sets and a simulation study revealed that the model could potentially be very useful in adequately describing different types of time-to-event data.

A Comparison of Alternative Network Meta-Analysis Methods in the Presence of Nonproportional Hazards: A Case Study in First-Line Advanced or Metastatic Renal Cell Carcinoma

PNS92 A Targeted Review of Indirect Treatment Comparison Synthesis Methods That Do Not Rely on the Proportional Hazard Assumption

For sample size calculation in clinical trials with survival endpoints, the logrank test, which is the optimal method under the proportional hazard (PH) assumption, is predominantly used. In reality, the PH assumption may not hold. For example, in immuno-oncology trials, delayed treatment effects are often expected. The sample size without considering the potential violation of the PH assumption may lead to an underpowered study. In recent years, combination tests such as the maximum weighted logrank test have received great attention because of their robust performance in various hazards scenarios. In this paper, we propose a flexible simulation-free procedure to calculate the sample size using combination tests. The procedure extends the Lakatos' Markov model and allows for complex situations encountered in a clinical trial, like staggered entry, dropouts, etc. We evaluate the procedure using two maximum weighted logrank tests, one projection-type test, and three other commonly used tests under various hazards scenarios. The simulation studies show that the proposed method can achieve the target power for all compared tests in most scenarios. The combination tests exhibit robust performance under correct specification and misspecification scenarios and are highly recommended when the hazard-changing patterns are unknown beforehand. Finally, we demonstrate our method using two clinical trial examples and provide suggestions about the sample size calculations under nonproportionalhazards.

Survival analysis is the primary statistical method of analysing time to event data. The most popular method for estimating the survivor function is the Cox-Proportional Hazard model. It assumes that the effect on the hazard function of a particular factor of interest remains unchanged throughout the observation. This is known as Proportional Hazards. Tsiatis assumed that the underlying hazard function is constant over distinct intervals. In the current study, no shape assumption is imposed other than that the hazard function is a smooth function with an arbitrary choice of a smoother. Such an approach involves the implementation of kernel-smoothing of the initial hazard estimate which have proved in studies to provide a trade-off between bias and variance. The cross-validation and plug-in bandwidth selectors are considered to determine the optimal bandwidth, h to be used as a smoothing parameter. Consequently, the survivorship function is estimated using the Cox-Proportional Hazards model. Proper application of the smoothing procedure is seen to improve the statistical performance of the resulting hazard rate estimator. No constraints are implored on the form of the underlying hazard proving to be less bias than Tsiatis’ method. This implies that the kernel smoothed survivorship function is more appropriate than the common standard techniques in survival analysis as it provides piecewise smooth estimates. Coverage probabilities of the estimate are then obtained which are found to be more accurate and closer to the nominal level compared to those estimated by Tsiatis.

The Fundamentals of Survival and Event History Analysis Introduction: What Is Survival and Event History Analysis? Key Concepts and Terminology Censoring and Truncation Mathematical Expression and Relation of Basic Statistical Functions How Do the Survivor, Density and Hazard Function Relate? Why Use Survival and Event History Analysis? Overview of Survival and Event History Models Exercises Using R and Other Computer Programs for Survival and Event History Analysis Introduction: Computer Programs for Survival and Event History Analysis Conducting Serious Data Analysis: Life Lessons Why Use R? Downloading R on Your Personal Computer Add-On Packages Running R Determining and Setting your Working Directory Help and Documentation Importing Data Into R Working With Data: Opening and Accessing Variables from a Data Frame Saving Output as File, Workspace and History and Quitting R Exercises Your First Session: Using the Survival Package and Exploring Data Via Descriptive Statistics and Graphs Your First Session Using the 'Survival' Package In F Loading and Examining the Survival Package and Rcmdrplugin.Survival Plug-In Opening and Examining Data The Surv Object: Packaging the 'Survival Variable' Basic Descriptive Statistics Descriptive Data Exploration with Graphs Exercises Data and Data Reconstruction Introduction: Why Discuss Data and Data Preparation? Sources of Event History Data Single-Episode Data for Single Transition Analyses Multi-Episode Data for Recurrent Event and Frailty Analyses Subject-(Person)-Period Data for Discrete-Time Hazard Models The Counting Process and Episode Splitting A Note on Dates Exercises Non-Parametric Methods: Estimating and Comparing Survival Curves Using the Kaplan-Meier Estimator Introduction The Kaplan-Meier Estimator Producing Kaplan-Meier Estimates Plotting the Kaplan-Meier Survival Curve Testing Differences Between Two Groups Using Survdiff Stratifying the Analysis by a Covariate Exercises The Cox Proportional-Hazards Regression Introduction: Why is The Cox Model So Popular? The Cox Regression Model Estimating and Interpreting The Cox Model with Fixed Covariates The Cox Regression Model with Time-Varying Covariates Exercises Parametric Models Introduction: What are Parametric Models and Why Use Them? Proportional Hazards (Ph) Versus Accelerated Failure Time (Aft) Models The Path to Choosing a Model Estimating and Interpreting Parametric Survival Models Exponential and Piecewise Constant Exponential Model Weibull Model Log-Logistic and Log-Normal Models Additional Parametric Models Finding the Best Fitting Parametric Model Exercises Model Building and Diagnostics Introduction Model Building and Selection of Covariates Assessing the Overall Goodness of Fit of Your Model What is Residual Analysis? Testing Overall Model Adequacy: Cox-Snell Residuals Testing the Proportional Hazards Assumption: Schoenfeld Residuals Checking For Influential Observations: Score Residuals (Dfbeta Statistics) Assessing Nonlinearity: Martingale Residual and Component-Plus-Residual Plots Exercises Correlated and Discrete-Time Survival Data: Frailty, Recurrent Events and Discrete-Time Models Introduction Shared Frailty: Modeling Recurrent Events and Clustering In Groups Other Frailty Models: Unshared, Nested, Joint and Additive Models Estimating Frailty Models in R Example of Frailty Model Estimation and Interpretation Discrete-Time and Count Models Exercises Multiple Events and Entire Histories: Competing Risk, Multistate Models and Sequence Analysis Introduction Competing Risk Models Multistate Models Sequence Analysis: Modeling Entire Histories Exercises Appendix : Datasets Used in this Book

This manuscript focuses on the development of a bootstrap test for validating the proportional hazard (PH) assumption in longevity data, avoiding parametric assumptions on baseline survival and hazard patterns, and subjective interpretations of previously developed graphical tests. Monte Carlo simulations are used to generate new data sets from the estimated Kaplan-Meier survival function, and inferences are then made on the coefficient of variation (CV) of the estimated hazard over time. One-tailed bootstrap intervals can be established, given that the CV could theoretically range between 0 (perfect PH) and +∞ (absolute loss of proportionality between hazard functions). This procedure was tested by simulation, and the obtained results suggested it as a useful statistical tool when Kaplan-Meier assumptions are satisfied. If not, this bootstrap test was robust for medium to large data sets, whereas it could suffer from statistical biases when testing small populations.

In survival analysis, it is routine to test equality of two survival curves, which is often conducted by using the log-rank test. Although it is optimal under the proportional hazards assumption, the log-rank test is known to have little power when the survival or hazard functions cross. To test the overall homogeneity of hazard rate functions, we propose a group of partitioned log-rank tests. By partitioning the time axis and taking the supremum of the sum of two partitioned log-rank statistics over different partitioning points, the proposed test gains enormous power for cases with crossing hazards. On the other hand, when the hazards are indeed proportional, our test still maintains high power close to that of the optimal log-rank test. Extensive simulation studies are conducted to compare the proposed test with existing methods, and three real data examples are used to illustrate the commonality of crossing hazards and the advantages of the partitioned log-rank tests.

Cancer immunotherapy trials often present unique challenges for time-to-event data modeling and analysis due to heterogeneous treatment effects across time or subgroups. A few known scenarios are delayed treatment effects and cure rates, where survival and hazard functions often take complex shapes. Sophisticated survival models like piecewise model are commonly used to capture the shapes of the functions. However, the time of hazard transition for individual patients varies, making it necessary to model it as a latent random variable. In this study, we propose a smoothed piecewise model to account for the random hazard transition time, based on the linear relationship between hazard function and the distribution function of the individual transition time within the smoothing window. We then develop the weighted geometric average hazard ratio (wgAHR) to estimate sample size and power based on the non-centrality parameter of the weighted log-rank statistics under non-proportional hazard (non-PH). We demonstrate that the wgAHR is not only directly linked to the sample size formula, but can also be interpreted as a measure of treatment effect, even when the proportional hazard assumption is violated. Additionally, we provide maxCombo, a combination of weighted log-rank statistics, for robust testing across different non-proportional hazard scenarios. Simulation studies and real trial examples illustrate the performance and robustness of the wgAHR based method in power and sample size estimation in cancer immunotherapy trials.