Comparing two crossing hazard rates by Cox proportional hazards modelling

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.

Ball divergence for the equality test of crossing survival curves

It is a very common problem to test survival equality using the right-censored time-to-event data in clinical research. Although the log-rank test is popularly used in various studies, it may become insensitive when the proportional hazards assumption is violated. As follows, there have a variety of statistical methods being proposed to identify the discrepancy between crossing survival curves or hazard functions. The omnibus tests against general alternatives are usually preferred due to their wide applicability to complicated scenarios in real applications. In this paper, we propose two novel statistics to estimate the ball divergence using the right-censored survival data, and then implement them in the equality test on survival time in two independent groups. The simulation analysis demonstrates their efficiency in identifying the survival discrepancy. Compared to the existing methods, our proposed methods present higher power in situations with complex distributions, especially when there is a scale shift between groups. Real examples illustrate its advantage in practical applications.

A win ratio approach for comparing crossing survival curves in clinical trials

Many clinical trials include time-to-event or survival data as an outcome. To compare two survival distributions, the log-rank test is often used to produce a P-value for a statistical test of the null hypothesis that the two survival curves are identical. However, such a P-value does not provide the magnitude of the difference between the curves regarding the treatment effect. As a result, the P-value is often accompanied by an estimate of the hazard ratio from the proportional hazards model or Cox model as a measurement of treatment difference. However, one of the most important assumptions for Cox model is that the hazard functions for the two treatment groups are proportional. When the hazard curves cross, the Cox model could lead to misleading results and the log-rank test could also perform poorly. To address the problem of crossing curves in survival analysis, we propose the use of the win ratio method put forward by Pocock et al. as an estimand for analysing such data. The subjects in the test and control treatment groups are formed into all possible pairs. For each pair, the test treatment subject is labelled a winner or a loser if it is known who had the event of interest such as death. The win ratio is the total number of winners divided by the total number of losers and its standard error can be estimated using Bebu and Lachin method. Using real trial datasets and Monte Carlo simulations, this study investigates the power and type I error and compares the win ratio method with the log-rank test and Cox model under various scenarios of crossing survival curves with different censoring rates and distribution parameters. The results show that the win ratio method has similar power as the log-rank test and Cox model to detect the treatment difference when the assumption of proportional hazards holds true, and that the win ratio method outperforms log-rank test and Cox model in terms of power to detect the treatment difference when the survival curves cross.

Design and analysis of cluster randomized trials with time-to-event outcomes under the additive hazards mixed model

A primary focus of current methods for cluster randomized trials (CRTs) has been for continuous, binary, and count outcomes, with relatively less attention given to right-censored, time-to-event outcomes. In this article, we detail considerations for sample size requirement and statistical inference in CRTs with time-to-event outcomes when the intervention effect parameter is specified through the additive hazards mixed model (AHMM), which includes a frailty term to explicitly account for the dependency between the failure times. First, we discuss improved inference for the treatment effect parameter via bias-corrected sandwich variance estimators and randomization-based test under AHMM, addressing potential small-sample biases in CRTs. Next, we derive a new sample size formula for AHMM analysis of CRTs accommodating both equal and unequal cluster sizes. When the cluster sizes vary, our sample size formula depends on the mean and coefficient of variation of cluster sizes, based on which we articulate the impact of cluster size variation in CRTs with time-to-event outcomes. Furthermore, we obtain the insight that the classical variance inflation factor for CRTs with a non-censored outcome can in fact apply to CRTs with a time-to-event outcome, providing that an appropriate definition of the intraclass correlation coefficient is considered under AHMM. Simulation studies are carried out to illustrate key design and analysis considerations in CRTs with a small to moderate number of clusters. The proposed sample size procedure and analytical methods are further illustrated using the context of the STrategies to Reduce Injuries and Develop Confidence in Elders CRT.

Non-parametric treatment time-lag effect estimation

In general, the change point problem considers inference of a change in distribution for a set of time-ordered observations. This has applications in a large variety of fields, and can also apply to survival data. In survival analysis, most existing methods compare two treatment groups for the entirety of the study period. Some treatments may take a length of time to show effects in subjects. This has been called the time-lag effect in the literature, and in cases where time-lag effect is considerable, such methods may not be appropriate to detect significant differences between two groups. In this paper, we propose a novel non-parametric approach for estimating the point of treatment time-lag effect by using an empirical divergence measure. Theoretical properties of the estimator are studied. The results from the simulated data and the applications to real data examples support our proposed method.

Genetic Modifiers of Age at Onset for Parkinson's Disease in Asians: A Genome-Wide Association Study

BACKGROUND

Age at onset (AAO) is an essential feature of Parkinson's disease (PD) and can help predict disease progression and mortality. Identification of genetic variants influencing AAO of PD could lead to a better understanding of the disease's biological mechanism and provide clinical guidance. However, genetic determinants for AAO of PD remain mostly unknown, especially in the Asian population.

OBJECTIVES

To identify genetic determinants for AAO of PD in the Asian population.

METHODS

We performed a genome-wide association meta-analysis on AAO of PD in 5166 Chinese patients with PD (N_discovery = 3628, N_replication = 1538). We then conducted a further cross-ethnic meta-analysis using our results and summary statistics for the AAO of PD from the European population.

RESULTS

The total heritability of AAO of PD was around 0.10 ~ 0.14, similar to that (~0.11) estimated in populations of European ancestry. One novel significant intergenic locus rs9783733 (NDN; PWRN4) was identified (P = 3.14E-09, beta = 2.30, SE = 0.39). Remarkably, this variant could delay AAO of PD by ~2.43 years, with a more considerable effect on males (~3.18 years) than females (~1.45 years). The variant was suggestively significant in the cross-ethnic meta-analysis and suggested a positive selection in the East Asian population. Additionally, cross-ethnic meta-analysis identified a significant locus rs356203 in SNCA (P = 2.35E-11, beta = -0.71, SE = 0.01).

CONCLUSIONS

These findings improve the current understanding of the genetic etiology of AAO of PD in different ethnic groups, and provide a new target for further research on PD pathogenesis and potential therapeutic options. © 2021 International Parkinson and Movement Disorder Society.

A simulation-free group sequential design with max-combo tests in the presence of non-proportional hazards

Non-proportional hazards (NPH) have been observed in many immuno-oncology clinical trials. Weighted log-rank tests (WLRT) with suitable weights can be used to improve the power of detecting the difference between survival curves in the presence of NPH. However, it is not easy to choose a proper WLRT in practice. A versatile max-combo test was proposed to achieve the balance of robustness and efficiency, and has received increasing attention recently. Survival trials often warrant interim analyses due to their high cost and long durations. The integration and implementation of max-combo tests in interim analyses often require extensive simulation studies. In this report, we propose a simulation-free approach for group sequential designs with the max-combo test in survival trials. The simulation results support that the proposed method can successfully control the type I error rate and offer excellent accuracy and flexibility in estimating sample sizes, with light computation burden. Notably, our method displays strong robustness towards various model misspecifications and has been implemented in an R package.

Effect of Tropisetron on Prevention of Emergence Delirium in Patients After Noncardiac Surgery: A Trial Protocol

IMPORTANCE

Postoperative delirium is a frequent disorder for patients undergoing surgery and is associated with poor outcomes. Delirium may occur in the immediate period after anesthesia administration and surgery. Tropisetron, which is frequently administrated for postoperative nausea and vomiting, is also a partial agonist of α7 nicotinic acetylcholine receptors associated with neuroprotective effects. Tropisetron may be the potential pharmacological treatment to decrease delirium after noncardiac surgery.

OBJECTIVE

To perform a randomized clinical trial to determine the efficacy and safety of tropisetron for prevention of emergence delirium in patients undergoing noncardiac surgery.

DESIGN, SETTING, AND PARTICIPANTS

This single-center, 2-arm randomized, double-blind, placebo-controlled trial will include 1508 patients undergoing noncardiac surgery. The intervention group will receive 5 mg of intravenous tropisetron before anesthesia induction, and patients in the control group will receive a placebo. The primary end point is the incidence of emergence delirium within 1 hour after tracheal tube removal, measured by the Confusion Assessment Method for the Intensive Care Unit score. The main secondary outcome is the incidence of postoperative delirium measured at 3 days of follow-up. An intention-to-treat principle will be used for all analyses.

DISCUSSION

Delirium remains the most common neuropsychiatric complication for patients after surgery. This will be the first randomized clinical study to evaluate whether tropisetron is effective in preventing emergence delirium. Results from this study will provide evidence for alteration of daily practice.

TRIAL REGISTRATION

ClinicalTrials.gov Identifier: NCT04027751.

A resampling-based test for two crossing survival curves

The area between two survival curves is an intuitive test statistic for the classical two-sample testing problem. We propose a bootstrap version of it for assessing the overall homogeneity of these curves. Our approach allows ties in the data as well as independent right censoring, which may differ between the groups. The asymptotic distribution of the test statistic as well as of its bootstrap counterpart are derived under the null hypothesis, and their consistency is proven for general alternatives. We demonstrate the finite sample superiority of the proposed test over some existing methods in a simulation study and illustrate its application by a real-data example.

Anti-GD2 antibody-containing immunotherapy postconsolidation therapy for people with high-risk neuroblastoma treated with autologous haematopoietic stem cell transplantation

BACKGROUND

Neuroblastoma is a rare malignant disease that primarily affects children. The tumours mainly develop in the adrenal medullary tissue, and an abdominal mass is the most common presentation. High-risk disease is characterised by metastasis and other primary tumour characteristics resulting in increased risk for an adverse outcome. The GD2 carbohydrate antigen is expressed on the cell surface of neuroblastoma tumour cells and is thus a promising target for anti-GD2 antibody-containing immunotherapy.

OBJECTIVES

To assess the efficacy of anti-GD2 antibody-containing postconsolidation immunotherapy after high-dose chemotherapy (HDCT) and autologous haematopoietic stem cell transplantation (HSCT) compared to standard therapy after HDCT and autologous HSCT in people with high-risk neuroblastoma. Our primary outcomes were overall survival and treatment-related mortality. Our secondary outcomes were progression-free survival, event-free survival, early toxicity, late non-haematological toxicity, and health-related quality of life.

SEARCH METHODS

We searched the electronic databases CENTRAL (2018, Issue 9), MEDLINE (PubMed), and Embase (Ovid) on 20 September 2018. We searched trial registries and conference proceedings on 28 October 2018. Further searches included reference lists of recent reviews and relevant articles as well as contacting experts in the field. There were no limits on publication year or language.

SELECTION CRITERIA

Randomised controlled trials evaluating anti-GD2 antibody-containing immunotherapy after HDCT and autologous HSCT in people with high-risk neuroblastoma.

DATA COLLECTION AND ANALYSIS

Two review authors independently performed study selection, abstracted data on study and participant characteristics, and assessed risk of bias and GRADE. Any differences were resolved by discussion, with third-party arbitration unnecessary. We performed analyses according to the guidelines of the Cochrane Handbook for Systematic Reviews of Interventions. We used the five GRADE considerations, that is study limitations, consistency of effect, imprecision, indirectness, and publication bias, to judge the quality of the evidence.

MAIN RESULTS

We identified one randomised controlled trial that included 226 people with high-risk neuroblastoma who were pre-treated with autologous HSCT. The study randomised 113 participants to receive immunotherapy including isotretinoin, granulocyte-macrophage colony-stimulating factor (GM-CSF), interleukin-2, and ch14.18, a type of anti-GD2 antibody also known as dinutuximab. The study randomised another 113 participants to receive standard therapy including isotretinoin.The results on overall survival favoured the dinutuximab-containing immunotherapy group (hazard ratio (HR) 0.50, 95% confidence interval (CI) 0.31 to 0.80; P = 0.004). The results on event-free survival also favoured the dinutuximab-containing immunotherapy group (HR 0.61, 95% CI 0.41 to 0.92; P = 0.020). Randomised data on adverse events were not reported separately. The study did not report progression-free survival, late non-haematological toxicity, and health-related quality of life as separate endpoints. We graded the quality of the evidence as moderate.

AUTHORS' CONCLUSIONS

The evidence base favours dinutuximab-containing immunotherapy compared to standard therapy concerning overall survival and event-free survival in people with high-risk neuroblastoma pre-treated with autologous HSCT. Randomised data on adverse events are lacking, therefore more research is needed before definitive conclusions can be made regarding this outcome.

Effect of Human Papillomavirus Vaccine to Interrupt Recurrence of Vulvar and Anal Neoplasia (VIVA): A Trial Protocol

IMPORTANCE

Human papillomavirus (HPV), particularly HPV type 16, causes most anal and vulvar high-grade squamous intraepithelial lesions (HSIL), which are precursors to cancer. After initial treatment of HSIL, more than 30% of patients will have disease recurrence, with even higher recurrence among HIV-positive individuals and men who have sex with men. Recurrences can be debilitating and lead to significant morbidity and medical expense. Observational studies suggest a possible therapeutic benefit of the licensed HPV vaccines in reducing recurrent lesions in previously infected persons.

OBJECTIVE

To test whether the licensed prophylactic HPV vaccine (Gardasil-9) can reduce the risk of HSIL recurrence by 50% in previously unvaccinated individuals recently treated for anal or vulvar HSIL.

DESIGN, SETTING, AND PARTICIPANTS

This is a trial protocol for a randomized, double-blind, placebo-controlled, proof-of-concept clinical trial. Eligible participants are aged 27 to 69 at study start and have not received prior HPV vaccination, have had anal or vulvar HSIL diagnosed on or after January 1, 2014, and have no evidence of HSIL recurrence at screening. Persons infected with HIV are eligible for the study provided they are receiving antiretroviral therapy. Target enrollment is 345 individuals. The primary outcome is time to histopathologically confirmed recurrence of HSIL. Differences in the risk for recurrence of HSIL will be evaluated using Cox proportional hazard models. Additional analyses include (1) frequency of HSIL recurrence; (2) role of HPV antibodies in deterring recurrence; (3) role of HPV persistence in recurrence, as measured by HPV genotype or HPV-16 variant lineage determined using swab samples collected at months 0, 18, and 36; and (4) incidence of adverse events. The study will be conducted at the University of Washington Virology Research Clinic from 2017 through 2022. Participants will be followed up for up to 36 months in the clinic, and up to 42 months by telephone.

DISCUSSION

Management of persistent or rapidly recurring anogenital HSIL remains challenging. Results from this study will provide evidence on whether incorporating the nonavalent HPV vaccine into routine care can decrease recurrence of anal and vulvar HSIL.

TRIAL REGISTRATION

ClinicalTrials.gov identifier: NCT03051516.

Issues possibly associated with misinterpreting survival data: A method study

OBJECTIVE

Proper interpretation of survival data of clinical cancer studies may be difficult and pitfalls related to the nature of Kaplan-Meier analyses might end up in mistaken inferences. The purpose of the present work is to raise awareness of those pitfalls and to prevent errors in future studies.

STUDY DESIGN AND SETTING

While evaluating a randomized controlled trial, we came across some issues possibly associated with misinterpreting survival data. We thoroughly reviewed the reporting of survival analyses, statistical approaches, baseline characteristics, and choice of primary end point. The reported data were derived from people with high-risk neuroblastoma. Thus, the trial focused on survival. We reenacted survival functions by deducing the data of various treatment groups from pictured survival functions to estimate the concerning hazard ratios.

RESULTS

Opposed to the reporting of the trial, we did not identify a significant difference between treatment groups with respect to overall survival. We were not able to appreciate an effective crossing of survival curves. With respect to event-free survival, we focused on comparable treatment groups and we did not identify a significant difference between treatment groups, thereby again opposing the reporting of the trial.

CONCLUSIONS

The present work exemplifies statistical issues that were apparently difficult to detect and that are possibly associated with misinterpreting survival functions. These issues include assumed crossing of survival curves, statistical approach changed in follow-up, different pretreatment between groups, and event-free survival used as primary outcome. Careful handling might prevent similar potential misinterpretation in future studies.

Evaluation of the treatment time-lag effect for survival data

Medical treatments often take a period of time to reveal their impact on subjects, which is the so-called time-lag effect in the literature. In the survival data analysis literature, most existing methods compare two treatments in the entire study period. In cases when there is a substantial time-lag effect, these methods would not be effective in detecting the difference between the two treatments, because the similarity between the treatments during the time-lag period would diminish their effectiveness. In this paper, we develop a novel modeling approach for estimating the time-lag period and for comparing the two treatments properly after the time-lag effect is accommodated. Theoretical arguments and numerical examples show that it is effective in practice.

Partitioned log-rank tests for the overall homogeneity of hazard rate functions

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.

A systematic review of using and reporting survival analyses in acute lymphoblastic leukemia literature

Backgrounds

Survival analysis is commonly used to determine the treatment effect among acute lymphoblastic leukemia (ALL) patients who undergo allogeneic stem cell transplantation (allo-SCT) or other treatments. The aim of this study was to evaluate the use and reporting of survival analyses in these articles.

Methods

We performed a systematic review by searching the MEDLINE, EMBASE and Cochrane library databases from inception to April 2015. Clinical trials of patients with ALL comparing allo-SCT compared to another treatment were included. We included only studies that used survival analysis as a part of the statistical methods.

Results

There were 14 studies included in the review. Sample size estimation was described in 4 (29 %) studies. Only 4 (29 %) studies reported the list of covariates assessed in the Cox regression and 6 (43 %) studies provided a description of censorship. All studies reported survival curves using the Kaplan-Meier method. The comparisons between groups were investigated using the log-rank test and Wilcoxon test. Crossing survival curves were observed in 11(79 %) studies. The Cox regression model was incorporated in 10 (71 %) studies. None of the studies assessed the Cox proportional hazards assumption or goodness-of-fit.

Conclusions

The use and reporting of survival analysis in adult ALL patients undergoing allo-SCT have significant limitations. Notably, the finding of crossing survival curves was common and none of the studies assessed for the proportional hazards assumption. We encourage authors, reviewers and editors to improve the quality of the use and reporting of survival analysis in the hematology literature.

Comparison of multiple hazard rate functions

Many robust tests have been proposed in the literature to compare two hazard rate functions, however, very few of them can be used in cases when there are multiple hazard rate functions to be compared. In this article, we propose an approach for detecting the difference among multiple hazard rate functions. Through a simulation study and a real-data application, we show that the new method is robust and powerful in many situations, compared with some commonly used tests.

Statistical inference methods for two crossing survival curves: a comparison of methods

A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.

Model selection and diagnostics for joint modeling of survival and longitudinal data with crossing hazard rate functions

Comparison of two hazard rate functions is important for evaluating treatment effect in studies concerning times to some important events. In practice, it may happen that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. Also, besides survival data, there could be longitudinal data available regarding some time-dependent covariates. When jointly modeling the survival and longitudinal data in such cases, model selection and model diagnostics are especially important to provide reliable statistical analysis of the data, which are lacking in the literature. In this paper, we discuss several criteria for assessing model fit that have been used for model selection and apply them to the joint modeling of survival and longitudinal data for comparing two crossing hazard rate functions. We also propose hypothesis testing and graphical methods for model diagnostics of the proposed joint modeling approach. Our proposed methods are illustrated by a simulation study and by a real-data example concerning two early breast cancer treatments.

The consequences of proportional hazards based model selection

For testing the efficacy of a treatment in a clinical trial with survival data, the Cox proportional hazards (PH) model is the well-accepted, conventional tool. When using this model, one typically proceeds by confirming that the required PH assumption holds true. If the PH assumption fails to hold, there are many options available, proposed as alternatives to the Cox PH model. An important question which arises is whether the potential bias introduced by this sequential model fitting procedure merits concern and, if so, what are effective mechanisms for correction. We investigate by means of simulation study and draw attention to the considerable drawbacks, with regard to power, of a simple resampling technique, the permutation adjustment, a natural recourse for addressing such challenges. We also consider a recently proposed two-stage testing strategy (2008) for ameliorating these effects.

Confidence intervals for the first crossing point of two hazard functions

The phenomenon of crossing hazard rates is common in clinical trials with time to event endpoints. Many methods have been proposed for testing equality of hazard functions against a crossing hazards alternative. However, there has been relatively few approaches available in the literature for point or interval estimation of the crossing time point. The problem of constructing confidence intervals for the first crossing time point of two hazard functions is considered in this paper. After reviewing a recent procedure based on Cox proportional hazard modeling with Box-Cox transformation of the time to event, a nonparametric procedure using the kernel smoothing estimate of the hazard ratio is proposed. The proposed procedure and the one based on Cox proportional hazard modeling with Box-Cox transformation of the time to event are both evaluated by Monte-Carlo simulations and applied to two clinical trial datasets.

Crossing Hazard Functions in Common Survival Models

Crossing hazard functions have extensive applications in modeling survival data. However, existing studies in the literature mainly focus on comparing crossed hazard functions and estimating the time at which the hazard functions cross, and there is little theoretical work on conditions under which hazard functions from a model will have a crossing. In this paper, we investigate crossing status of hazard functions from the proportional hazards (PH) model, the accelerated hazard (AH) model, and the accelerated failure time (AFT) model. We provide and prove conditions under which the hazard functions from the AH and the AFT models have no crossings or a single crossing. A few examples are also provided to demonstrate how the conditions can be used to determine crossing status of hazard functions from the three models.