Interval-specific censoring set adjusted Kaplan-Meier estimator.

We propose a non-parametric approach to reduce the overestimation of the Kaplan-Meier (KM) estimator when the event and censoring times are independent. We adjust the KM estimator based on the interval-specific censoring set, a collection of intervals where censored data are observed between two adjacent event times. The proposed interval-specific censoring set adjusted KM estimator reduces to the KM estimator if there are no censored observations or the sample size tends to infinity and the proposed estimator is consistent, as is the case for the KM estimator. We prove theoretically that the proposed estimator reduces the overestimation compared to the KM estimator and provide a mathematical formula to estimate the variance of the proposed estimator based on Greenwood's approach. We also provide a modified log-rank test based on the proposed estimator. We perform four simulation studies to compare the proposed estimator with the KM estimator when the failure rate is constant, decreasing, increasing, and based on the flexible hazard method. The bias reduction in median survival time and survival rate using the proposed estimator is considerably large, especially when the censoring rate is high. The standard deviations are comparable between the two estimators. We implement the proposed and KM estimator for the Nonalcoholic Fatty Liver Disease patients from a population study. The results show the proposed estimator substantially reduce the overestimation in the presence of high observed censoring rate.

Multi-arm multi-stage survival trial design with arm-specific stopping rule.

Traditional two-arm randomized trial designs have played a pivotal role in establishing the efficacy of medical interventions. However, their efficiency is often compromised when confronted with multiple experimental treatments or limited resources. In response to these challenges, the multi-arm multi-stage designs have emerged, enabling the simultaneous evaluation of multiple treatments within a single trial. In such an approach, if an arm meets efficacy success criteria at an interim stage, the whole trial stops and the arm is selected for further study. However when multiple treatment arms are active, stopping the trial at the moment one arm achieves success diminishes the probability of selecting the best arm. To address this issue, we have developed a group sequential multi-arm multi-stage survival trial design with an arm-specific stopping rule. The proposed method controls the familywise type I error in a strong sense and selects the best promising treatment arm with a high probability.

Analysing lapsing rates among first-time blood donors at a blood centre in Zimbabwe using survival analysis.

Introduction: blood centres are often faced with the problem of donor lapsing resulting in loss of donors from the already strained donor pool. In Zimbabwe, 70% of the donated blood comes from younger donors aged 40 years and below, who at the same time, have high attrition rates. This study seeks to apply the concept of survival analysis in analysing blood donor lapsing rates. Methods: in analysing the donor lapsing and retention rates, data on 450 first-time blood donors at the National Blood Service Zimbabwe, in Harare´s blood bank for the period 2014 to 2017 was extracted from the donors´ database. The Cox proportional hazards (Cox PH) and Kaplan-Meier methods were applied in the analysis. Donor demographic characteristics suspected of having effect on donor lapsing and retention were identified and analysed. Results: the study findings show that 56.9% of the donors had lapsed by the end of the four-year study period. Results from the multiple Cox PH model indicate that donor age had a significant effect on blood donor retention time (p = 0.000918 < 0.05). The hazard ratio (HR) = 0.615 with 95% CI: (0.461; 0.820) shows that the relatively older donors had a lower hazard (38.5% lower) of lapsing compared to the hazard for younger donors. The effect of gender, blood donor group and donation time interval on donor retention and attrition were not statistically significant. Male donors had HR = 1.03; 95% CI (0.537; 1.99) with (p = 0.922 > 0.05) and donors with a 4-month interval between donations had HR = 1.31; 95% CI (0.667; 2.59) with (p = 0.430 > 0.05). Conclusion: the study confirmed the problem of donor attrition faced by blood centres. The age of the donor had a significant effect on the retention time of blood donors before lapsing. The older the blood donor, the lower the risk of lapsing. The Zimbabwe National Blood Service (NBSZ) Blood Centre authorities should have a critical mass of individuals above 40 years as potential blood donors because of their reliability in blood donation according to the study findings.

Sample size calculation for mixture cure model with restricted mean survival time as a primary endpoint.

It is not uncommon for a substantial proportion of patients to be cured (or survive long-term) in clinical trials with time-to-event endpoints, such as the endometrial cancer trial. When designing a clinical trial, a mixture cure model should be used to fully consider the cure fraction. Previously, mixture cure model sample size calculations were based on the proportional hazards assumption of latency distribution between groups, and the log-rank test was used for deriving sample size formulas. In real studies, the latency distributions of the two groups often do not satisfy the proportional hazards assumptions. This article has derived a sample size calculation formula for a mixture cure model with restricted mean survival time as the primary endpoint, and did simulation and example studies. The restricted mean survival time test is not subject to proportional hazards assumptions, and the difference in treatment effect obtained can be quantified as the number of years (or months) increased or decreased in survival time, making it very convenient for clinical patient-physician communication. The simulation results showed that the sample sizes estimated by the restricted mean survival time test for the mixture cure model were accurate regardless of whether the proportional hazards assumptions were satisfied and were smaller than the sample sizes estimated by the log-rank test in most cases for the scenarios in which the proportional hazards assumptions were violated.

Sample Size Calculation Under Nonproportional Hazards Using Average Hazard Ratios.

Many clinical trials assess time-to-event endpoints. To describe the difference between groups in terms of time to event, we often employ hazard ratios. However, the hazard ratio is only informative in the case of proportional hazards (PHs) over time. There exist many other effect measures that do not require PHs. One of them is the average hazard ratio (AHR). Its core idea is to utilize a time-dependent weighting function that accounts for time variation. Though propagated in methodological research papers, the AHR is rarely used in practice. To facilitate its application, we unfold approaches for sample size calculation of an AHR test. We assess the reliability of the sample size calculation by extensive simulation studies covering various survival and censoring distributions with proportional as well as nonproportional hazards (N-PHs). The findings suggest that a simulation-based sample size calculation approach can be useful for designing clinical trials with N-PHs. Using the AHR can result in increased statistical power to detect differences between groups with more efficient sample sizes.

Comparison between asymptotic and re-randomisation tests under non-proportional hazards in a randomised controlled trial using the minimisation method.

BACKGROUND: Pocock-Simon's minimisation method has been widely used to balance treatment assignments across prognostic factors in randomised controlled trials (RCTs). Previous studies focusing on the survival outcomes have demonstrated that the conservativeness of asymptotic tests without adjusting for stratification factors, as well as the inflated type I error rate of adjusted asymptotic tests conducted in a small sample of patients, can be relaxed using re-randomisation tests. Although several RCTs using minimisation have suggested the presence of non-proportional hazards (non-PH) effects, the application of re-randomisation tests has been limited to the log-rank test and Cox PH models, which may result in diminished statistical power when confronted with non-PH scenarios. To address this issue, we proposed two re-randomisation tests based on a maximum combination of weighted log-rank tests (MaxCombo test) and the difference in restricted mean survival time (dRMST) up to a fixed time point τ , both of which can be extended to adjust for randomisation stratification factors. METHODS: We compared the performance of asymptotic and re-randomisation tests using the MaxCombo test, dRMST, log-rank test, and Cox PH models, assuming various non-PH situations for RCTs using minimisation, with total sample sizes of 50, 100, and 500 at a 1:1 allocation ratio. We mainly considered null, and alternative scenarios featuring delayed, crossing, and diminishing treatment effects. RESULTS: Across all examined null scenarios, re-randomisation tests maintained the type I error rates at the nominal level. Conversely, unadjusted asymptotic tests indicated excessive conservatism, while adjusted asymptotic tests in both the Cox PH models and dRMST indicated inflated type I error rates for total sample sizes of 50. The stratified MaxCombo-based re-randomisation test consistently exhibited robust power across all examined scenarios. CONCLUSIONS: The re-randomisation test is a useful alternative in non-PH situations for RCTs with minimisation using the stratified MaxCombo test, suggesting its robust power in various scenarios.

Comparing two hazard curves when there is a treatment time-lag effect.

In cancer and other medical studies, time-to-event (eg, death) data are common. One major task to analyze time-to-event (or survival) data is usually to compare two medical interventions (eg, a treatment and a control) regarding their effect on patients' hazard to have the event in concern. In such cases, we need to compare two hazard curves of the two related patient groups. In practice, a medical treatment often has a time-lag effect, that is, the treatment effect can only be observed after a time period since the treatment is applied. In such cases, the two hazard curves would be similar in an initial time period, and the traditional testing procedures, such as the log-rank test, would be ineffective in detecting the treatment effect because the similarity between the two hazard curves in the initial time period would attenuate the difference between the two hazard curves that is reflected in the related testing statistics. In this paper, we suggest a new method for comparing two hazard curves when there is a potential treatment time-lag effect based on a weighted log-rank test with a flexible weighting scheme. The new method is shown to be more effective than some representative existing methods in various cases when a treatment time-lag effect is present.

Incidence of recurrent ischemic stroke and its associated factors in a tertiary care center in Thailand: a retrospective cohort study.

BACKGROUND: Ischemic stroke (IS) is one of the leading causes of death among non-communicable diseases in Thailand. Patients who have survived an IS are at an increased risk of developing recurrent IS, which can result in worse outcomes and post-stroke complications. OBJECTIVES: The study aimed to investigate the incidence of recurrent IS among patients with first-ever IS during a one-year follow-up period and to determine its associated risk factors. METHODS: Adult patients (aged ≥ 18 years) who were hospitalized at the Stroke Center, King Chulalongkorn Memorial Hospital (KCMH) in Bangkok, Thailand, due to first-ever IS between January and December 2019 and had at least one follow-up visit during the one-year follow-up period were included in this retrospective cohort study. IS diagnosis was confirmed by neurologists and imaging. The log-rank test was used to determine the event-free survival probabilities of recurrent IS in each risk factor. RESULTS: Of 418 patients hospitalized due to first-ever IS in 2019, 366 (87.6%) were included in the analysis. During a total of 327.2 person-years of follow-up, 25 (6.8%) patients developed recurrent IS, accounting for an incidence rate of 7.7 per 100 person-year (95% confidence interval [CI] 5.2-11.3). The median (interquartile range) time of recurrence was 35 (16-73) days. None of the 47 patients with atrial fibrillation developed recurrent IS. The highest incidence rate of recurrent IS occurred within 1 month after the first episode (34 per 100 person-years) compared to other follow-up periods. Patients with small vessel occlusion and large-artery atherosclerosis (LAA) constituted the majority of patients in the recurrent IS episode (48% and 40%, respectively), with LAA exhibiting a higher recurrence rate (13.5%). Additionally, smoking status was found to be associated with an increased risk of recurrence. CONCLUSION: The incidence rate of the recurrence was moderate in our tertiary care setting, with a decreasing trend over time after the first episode. The various subtypes of IS and smoking status can lead to differences in event-free survival probabilities.

Two-stage stratified designs with survival outcomes and adjustment for misclassification in predictive biomarkers.

Biomarker stratified clinical trial designs are versatile tools to assess biomarker clinical utility and address its relationship with clinical endpoints. Due to imperfect assays and/or classification rules, biomarker status is prone to errors. To account for biomarker misclassification, we consider a two-stage stratified design for survival outcomes with an adjustment for misclassification in predictive biomarkers. Compared to continuous and/or binary outcomes, the test statistics for survival outcomes with an adjustment for biomarker misclassification is much more complicated and needs to take special care. We propose to use the information from the observed biomarker status strata to construct adjusted log-rank statistics for true biomarker status strata. These adjusted log-rank statistics are then used to develop sequential tests for the global (composite) hypothesis and component-wise hypothesis. We discuss the power analysis with the control of the type-I error rate by using the correlations between the adjusted log-rank statistics within and between the design stages. Our method is illustrated with examples of the recent successful development of immunotherapy in nonsmall-cell lung cancer.

Methods for non-proportional hazards in clinical trials: A systematic review.

For the analysis of time-to-event data, frequently used methods such as the log-rank test or the Cox proportional hazards model are based on the proportional hazards assumption, which is often debatable. Although a wide range of parametric and non-parametric methods for non-proportional hazards has been proposed, there is no consensus on the best approaches. To close this gap, we conducted a systematic literature search to identify statistical methods and software appropriate under non-proportional hazard. Our literature search identified 907 abstracts, out of which we included 211 articles, mostly methodological ones. Review articles and applications were less frequently identified. The articles discuss effect measures, effect estimation and regression approaches, hypothesis tests, and sample size calculation approaches, which are often tailored to specific non-proportional hazard situations. Using a unified notation, we provide an overview of methods available. Furthermore, we derive some guidance from the identified articles.

A propensity score-integrated approach for leveraging external data in a randomized controlled trial with time-to-event endpoints.

In a randomized controlled trial with time-to-event endpoint, some commonly used statistical tests to test for various aspects of survival differences, such as survival probability at a fixed time point, survival function up to a specific time point, and restricted mean survival time, may not be directly applicable when external data are leveraged to augment an arm (or both arms) of an RCT. In this paper, we propose a propensity score-integrated approach to extend such tests when external data are leveraged. Simulation studies are conducted to evaluate the operating characteristics of three propensity score-integrated statistical tests, and an illustrative example is given to demonstrate how these proposed procedures can be implemented.

The Impact of Mutational Hotspots on Cancer Survival.

BACKGROUND: Cofactors, biomarkers, and the mutational status of genes such as TP53, EGFR, IDH1/2, or PIK3CA have been used for patient stratification. However, many genes exhibit recurrent mutational positions known as hotspots, specifically linked to varying degrees of survival outcomes. Nevertheless, few hotspots have been analyzed (e.g., TP53 and EGFR). Thus, many other genes and hotspots remain unexplored. METHODS: We systematically screened over 1400 hotspots across 33 TCGA cancer types. We compared the patients carrying a hotspot against (i) all cases, (ii) gene-mutated cases, (iii) other mutated hotspots, or (iv) specific hotspots. Due to the limited number of samples in hotspots and the inherent group imbalance, besides Cox models and the log-rank test, we employed VALORATE to estimate their association with survival precisely. RESULTS: We screened 1469 hotspots in 6451 comparisons, where 314 were associated with survival. Many are discussed and linked to the current literature. Our findings demonstrate associations between known hotspots and survival while also revealing more potential hotspots. To enhance accessibility and promote further investigation, all the Kaplan-Meier curves, the log-rank tests, Cox statistics, and VALORATE-estimated null distributions are accessible on our website. CONCLUSIONS: Our analysis revealed both known and putatively novel hotspots associated with survival, which can be used as biomarkers. Our web resource is a valuable tool for cancer research.

Post-surgery survival and associated factors for cardiac patients in Ethiopia: applications of machine learning, semi-parametric and parametric modelling.

INTRODUCTION: Living in poverty, especially in low-income countries, are more affected by cardiovascular disease. Unlike the developed countries, it remains a significant cause of preventable heart disease in the Sub-Saharan region, including Ethiopia. According to the Ethiopian Ministry of Health statement, around 40,000 cardiac patients have been waiting for surgery in Ethiopia since September 2020. There is insufficient information about long-term cardiac patients' post-survival after cardiac surgery in Ethiopia. Therefore, the main objective of the current study was to determine the long-term post-cardiac surgery patients' survival status in Ethiopia. METHODS: All patients attended from 2012 to 2023 throughout the country were included in the current study. The total number of participants was 1520 heart disease patients. The data collection procedure was conducted from February 2022- January 2023. Machine learning algorithms were applied. Gompertz regression was used also for the multivariable analysis report. RESULTS: From possible machine learning models, random survival forest were preferred. It emphasizes, the most important variable for clinical prediction was SPO2, Age, time to surgery waiting time, and creatinine value and it accounts, 42.55%, 25.17%,11.82%, and 12.19% respectively. From the Gompertz regression, lower saturated oxygen, higher age, lower ejection fraction, short period of cardiac center stays after surgery, prolonged waiting time to surgery, and creating value were statistically significant predictors of death outcome for post-cardiac surgery patients' survival in Ethiopia. CONCLUSION: Some of the risk factors for the death of post-cardiac surgery patients are identified in the current investigation. Particular attention should be given to patients with prolonged waiting times and aged patients. Since there were only two fully active cardiac centers in Ethiopia it is far from an adequate number of centers for more than 120 million population, therefore, the study highly recommended to increase the number of cardiac centers that serve as cardiac surgery in Ethiopia.

An Overview of Introductory and Advanced Survival Analysis Methods in Clinical Applications: Where Have we Come so far?

The time-to-event relationship for survival modeling is considered when designing a study in clinical trials. However, because time-to-event data are mostly not normally distributed, survival analysis uses non-parametric data processing and analysis methods, mainly Kaplan-Meier (KM) estimation models and Cox proportional hazards (CPH) regression models. At the same time, the log-rank test can be applied to compare curves from different groups. However, resorting to conventional survival analysis when fundamental assumptions, such as the Cox PH assumption, are not met can seriously affect the results, rendering them flawed. Consequently, it is necessary to examine and report more sophisticated statistical methods related to the processing of survival data, but at the same time, able to adequately respond to the contemporary real problems of clinical applications. On the other hand, the frequent misinterpretation of survival analysis methodology, combined with the fact that it is a complex statistical tool for clinicians, necessitates a better understanding of the basic principles underlying this analysis to effectively interpret medical studies in making treatment decisions. In this review, we first consider the basic models and mechanisms behind survival analysis. Then, due to common errors arising from the inappropriate application of conventional models, we revise more demanding statistical extensions of survival models related to data manipulation to avoid wrong results. By providing a structured review of the most representative statistical methods and tests covering contemporary survival analysis, we hope this review will assist in solving problems that arise in clinical applications.

Model-based hypothesis tests for the causal mediation of semi-competing risks.

Analyzing the causal mediation of semi-competing risks has become important in medical research. Semi-competing risks refers to a scenario wherein an intermediate event may be censored by a primary event but not vice versa. Causal mediation analyses decompose the effect of an exposure on the primary outcome into an indirect (mediation) effect: an effect mediated through a mediator, and a direct effect: an effect not through the mediator. Here we proposed a model-based testing procedure to examine the indirect effect of the exposure on the primary event through the intermediate event. Under the counterfactual outcome framework, we defined a causal mediation effect using counting process. To assess statistical evidence for the mediation effect, we proposed two tests: an intersection-union test (IUT) and a weighted log-rank test (WLR). The test statistic was developed from a semi-parametric estimator of the mediation effect using a Cox proportional hazards model for the primary event and a series of logistic regression models for the intermediate event. We built a connection between the IUT and WLR. Asymptotic properties of the two tests were derived, and the IUT was determined to be a size [Formula: see text] test and statistically more powerful than the WLR. In numerical simulations, both the model-based IUT and WLR can properly adjust for confounding covariates, and the Type I error rates of the proposed methods are well protected, with the IUT being more powerful than the WLR. Our methods demonstrate the strongly significant effects of hepatitis B or C on the risk of liver cancer mediated through liver cirrhosis incidence in a prospective cohort study. The proposed method is also applicable to surrogate endpoint analyses in clinical trials.

Group sequential designs for cancer immunotherapy trial with delayed treatment effect.

Cancer immunotherapy trials are frequently characterized by delayed treatment effects such that the proportional hazards assumption is violated and the log-rank test suffers a substantial loss of statistical power. To increase the efficacy of the trial design, a variety of weighted log-rank tests have been proposed for fixed sample and group sequential trial designs. However, in such a group sequential design, it is often not recommended for futility interim monitoring due to possible delayed treatment effect which could result a high false-negative rate. To resolve this problem, we propose a group sequential design using a piecewise weighted log-rank test which provides an event-driven approach based on number of events after the delayed time. That is, the interim looks will not be conducted until the planned number of events observed after the delay time. Thus, it avoids the possibility of false-negative rate due to the delayed treatment effect. Furthermore, with an event-driven approach, the proposed group sequential design is robust against the underlying survival, accrual and censoring distributions. The group sequential designs using Fleming-Harrington-(ρ,γ) weighted log-rank test and a new weighted log-rank test are also discussed.

Sample size calculation for mixture model based on geometric average hazard ratio and its applications to nonproportional hazard.

With the advent of cancer immunotherapy, some special features including delayed treatment effect, cure rate, diminishing treatment effect and crossing survival are often observed in survival analysis. They violate the proportional hazard model assumption and pose a unique challenge for the conventional trial design and analysis strategies. Many methods like cure rate model have been developed based on mixture model to incorporate some of these features. In this work, we extend the mixture model to deal with multiple non-proportional patterns and develop its geometric average hazard ratio (gAHR) to quantify the treatment effect. We further derive a sample size and power formula based on the non-centrality parameter of the log-rank test and conduct a thorough analysis of the impact of each parameter on performance. Simulation studies showed a clear advantage of our new method over the proportional hazard based calculation across different non-proportional hazard scenarios. Moreover, the mixture modeling of two real trials demonstrates how to use the prior information on the survival distribution among patients with different biomarker and early efficacy results in practice. By comparison with a simulation-based design, the new method provided a more efficient way to compute the power and sample size with high accuracy of estimation. Overall, both theoretical derivation and empirical studies demonstrate the promise of the proposed method in powering future innovative trial designs.

Comparison of methods to testing for differential treatment effect under non-proportional hazards data.

Many tests for comparing survival curves have been proposed over the last decades. There are two branches, one based on weighted log-rank statistics and other based on weighted Kaplan-Meier statistics. If we carefully choose the weight function, a substantial increase in power of tests against non-proportional alternatives can be obtained. However, it is difficult to specify in advance the types of survival differences that may actually exist between two groups. Therefore, a combination test can simultaneously detect equally weighted, early, late or middle departures from the null hypothesis and can robustly handle several non-proportional hazard types with no a priori knowledge of the hazard functions. In this paper, we focus on the most used and the most powerful test statistics related to these two branches which have been studied separately but not compared between them. Through a simulation study, we compare the size and power of thirteen test statistics under proportional hazards and different types of non-proportional hazards patterns. We illustrate the procedures using data from a clinical trial of bone marrow transplant patients with leukemia.

Sequential monitoring of cancer immunotherapy trial with random delayed treatment effect.

Cancer immunotherapy trials are frequently characterized by a delayed treatment effect that violates the proportional hazards assumption. The log-rank test (LRT) suffers a substantial loss of statistical power under the nonproportional hazards model. Various group sequential designs using weighted LRTs (WLRTs) have been proposed under the fixed delayed treatment effect model. However, patients enrolled in immunotherapy trials are often heterogeneous, and the duration of the delayed treatment effect is a random variable. Therefore, we propose group sequential designs under the random delayed effect model using the random delayed distribution WLRT. The proposed group sequential designs are developed for monitoring the efficacy of the trial using the method of Lan-DeMets alpha-spending function with O'Brien-Fleming stopping boundaries or a gamma family alpha-spending function. The maximum sample size for the group sequential design is obtained by multiplying an inflation factor with the sample size for the fixed sample design. Simulations are conducted to study the operating characteristics of the proposed group sequential designs. The robustness of the proposed group sequential designs for misspecifying random delay time distribution and domain is studied via simulations.

The Inverse Log-Rank Test: A Versatile Procedure for Late Separating Survival Curves.

Often in the planning phase of a clinical trial, a researcher will need to choose between a standard versus weighted log-rank test (LRT) for investigating right-censored survival data. While a standard LRT is optimal for analyzing evenly distributed but distinct survival events (proportional hazards), an appropriately weighted LRT test may be better suited for handling non-proportional, delayed treatment effects. The "a priori" misspecification of this alternative may result in a substantial loss of power when determining the effectiveness of an experimental drug. In this paper, the standard unweighted and inverse log-rank tests (iLRTs) are compared with the multiple weight, default Max-Combo procedure for analyzing differential late survival outcomes. Unlike combination LRTs that depend on the arbitrary selection of weights, the iLRT by definition is a single weight test and does not require implicit multiplicity correction. Empirically, both weighted methods have reasonable flexibility for assessing continuous survival curve differences from the onset of a study. However, the iLRT may be preferable for accommodating delayed separating survival curves, especially when one arm finishes first. Using standard large-sample methods, the power and sample size for the iLRT are easily estimated without resorting to complex and timely simulations.