Using weighted Kaplan-Meier statistics in nonparametric comparisons of paired censored survival outcomes

Asymptotic convergence in distribution of the area bounded by prevalence-weighted Kaplan-Meier curves using empirical process modelling

The Kaplan-Meier product-limit estimator is a simple and powerful tool in time to event analysis. An extension exists for populations stratified into cohorts where a population survival curve is generated by weighted averaging of cohort-level survival curves. For making population-level comparisons using this statistic, we analyse the statistics of the area between two such weighted survival curves. We derive the large sample behaviour of this statistic based on an empirical process of product-limit estimators. This estimator was used by an interdisciplinary National Institutes of Health-Social Security Administration team in the identification of medical conditions to prioritize for adjudication in disability benefits processing.

Efficacy and safety of four doses of glycopyrrolate/formoterol fumarate delivered via a metered dose inhaler compared with the monocomponents in patients with moderate-to-severe COPD

Purpose

To determine the efficacy and safety of glycopyrrolate/formoterol fumarate metered dose inhaler (GFF MDI 36/9.6, 36/7.2, 18/9.6, 9/9.6 µg) using innovative co-suspension delivery technology, compared with glycopyrrolate (GP) MDI 36 µg and formoterol fumarate (FF) MDI 9.6 µg, in patients with moderate-to-severe COPD.

Methods

In this Phase IIb, randomized, double-blind, balanced incomplete-block, two-period, cross-over study (NCT01349816), patients received treatment twice-daily for 7 days. The primary efficacy endpoint was forced expiratory volume in 1 second (FEV₁) area under the curve from 0 to 12 hours (AUC_0–12) on Day 7. Secondary efficacy endpoints were peak change from baseline in FEV₁ through 2 hours; time to onset of action (≥10% improvement in mean FEV₁); proportion of patients achieving ≥12% improvement in FEV₁ on Day 1; peak change from baseline in inspiratory capacity (IC) on Days 1 and 7; change from baseline in morning pre-dose FEV₁; peak change from baseline in FEV₁ through 6 hours; and change from baseline in mean evening 12-hour post-dose trough FEV₁ on Day 7. Safety was assessed.

Results

All 185 randomized patients received treatment. All doses of GFF MDI significantly improved the primary endpoint compared with GP MDI 36 µg (all P≤0.0137). For peak change in FEV₁ and IC and time to onset of action secondary endpoints, ≥2 doses of GFF MDI demonstrated superiority to GP MDI 36 µg. No significant differences were observed between GFF MDI and FF MDI 9.6 µg for primary and secondary endpoints. The incidence of adverse events was similar between treatments.

Conclusion

While all doses of GFF MDI were superior to GP MDI 36 µg for the primary end-point, in this study neither superiority of GFF MDI to FF MDI 9.6 µg nor a clear dose-response was observed. All treatments were well tolerated with no unexpected safety findings.

A phase IIb randomized, chronic-dosing, incomplete block, cross-over study of glycopyrronium, delivered via metered dose inhaler, compared with a placebo and an active control in patients with moderate-to-severe COPD

Background

Long-acting muscarinic antagonist (LAMA) and long-acting β₂-agonist (LABA) bronchodilators are key to the pharmacologic treatment of chronic obstructive pulmonary disease (COPD). This Phase IIb study investigated the safety and efficacy of four doses of the LAMA glycopyrronium (GP) delivered using co-suspension delivery technology via metered dose inhaler (MDI). The study was part of a wider clinical trial program performed to determine the optimal dose of GP MDI, the LABA formoterol fumarate dihydrate (FF) MDI, and glycopyrronium/formoterol fumarate dihydrate (GFF) MDI fixed-dose combination to be taken forward into Phase III studies.

Methods

In this randomized, double-blind, 7-day chronic-dosing, three-period incomplete block, cross-over study, patients with moderate-to-severe COPD received two of the four doses of GP MDI (28.8 μg, 14.4 μg, 7.2 μg, and 3.6 μg) twice daily (BID), and either placebo MDI BID or open-label ipratropium MDI 34 μg four times daily. The primary efficacy endpoint was forced expiratory volume in 1 s (FEV₁) area under the curve from 0 to 12 h (AUC_0–12) relative to baseline on Day 7. Secondary and exploratory efficacy endpoints were assessed on Days 1 and 7. Safety and tolerability were evaluated throughout the study.

Results

All GP MDI treatments were superior to placebo MDI for the primary efficacy endpoint (all p < 0.0001). However, only GP MDI 28.8 μg and 14.4 μg demonstrated statistical superiority to placebo MDI for all secondary efficacy endpoints analyzed in this study, with the exception of GP MDI 14.4 μg versus placebo MDI for the proportion of patients achieving ≥12% improvement in FEV₁. No nominally significant differences were observed between GP MDI 28.8 μg and GP MDI 14.4 μg for any of the endpoints. All doses of GP MDI were well tolerated, with no unexpected safety findings.

Conclusions

This study indicated that there was no advantage of GP MDI 28.8 μg compared with GP MDI 14.4 μg. It therefore added to the evidence from the Phase I/II clinical trial program, which identified GP MDI 14.4 μg as the most appropriate dose for use in the Phase III clinical studies.

Trial registration

ClinicalTrials.gov (NCT01350128). Registered May 09, 2011.

Data maturity and follow-up in time-to-event analyses

We propose methods to determine the minimum number of subjects remaining at risk after which Kaplan-Meier survival plots for time-to-event outcomes should be curtailed, as, once the number remaining at risk drops below this minimum, the survival estimates are no longer meaningful in the context of the investigation. The size of the decrease of the Kaplan-Meier survival estimate S(t) at time t if one extra event should occur is considered in two ways. In the first approach, the investigator sets a maximum acceptable absolute decrease in S(t) should one extra event occur. In the second, a minimum acceptable number of subjects still at risk is calculated by comparing the size of the decrease in S(t) if an extra event should occur with the variability of the survival estimate had all subjects been followed to that time (confidence interval approach). We recommend calculating both limits for the number still at risk and then making an informed choice in the context of the particular investigation. We explore further how the amount of information actually available can assist in considering issues of data maturity for studies whose outcome of interest is a survival percentage at a particular time point. We illustrate the approaches with a number of published studies having differing sample sizes and censoring issues. In particular, one study was the subject of some controversy regarding how far in time the Kaplan-Meier plot should be extended. The proposed methods allow for limits to be calculated simply using the output provided by most statistical packages.

Comparing conditional survival functions with missing population marks in a competing risks model

In studies involving nonparametric testing of the equality of two or more survival distributions, the survival curves can exhibit a wide variety of behaviors such as proportional hazards, early/late differences, and crossing hazards. As alternatives to the classical logrank test, the weighted Kaplan-Meier (WKM) type statistic and their variations were developed to handle these situations. However, their applicability is limited to cases where the population membership is available for all observations, including the right censored ones. Quite often, failure time data are confronted with missing population marks for the censored observations. To alleviate this, a new WKM-type test is introduced based on imputed population marks for the censored observations leading to fractional at-risk sets that estimate the underlying risk for the process. The asymptotic normality of the proposed test under the null hypothesis is established, and the finite sample properties in terms of empirical size and power are studied through a simulation study. Finally, the new test is applied on a study of subjects undergoing bone marrow transplantation.

Sample size determination for paired right-censored data based on the difference of Kaplan-Meier estimates

Sample size determination is essential to planning clinical trials. Jung (2008) established a sample size calculation formula for paired right-censored data based on the logrank test, which has been well-studied for comparing independent survival outcomes. An alternative to rank-based methods for independent right-censored data, advocated by Pepe and Fleming (1989), tests for differences between integrated weighted Kaplan-Meier estimates and is more sensitive to the magnitude of difference in survival times between groups. In this paper, we employ the concept of the Pepe-Fleming method to determine an adequate sample size by calculating differences between Kaplan-Meier estimators considering pair-wise correlation. We specify a positive stable frailty model for the joint distribution of paired survival times. We evaluate the performance of the proposed method by simulation studies and investigate the impacts of the accrual times, follow-up times, loss to follow-up rate, and sensitivity of power under misspecification of the model. The results show that ignoring the pair-wise correlation results in overestimating the required sample size. Furthermore, the proposed method is applied to two real-world studies, and the R code for sample size calculation is made available to users.

Utilizing the integrated difference of two survival functions to quantify the treatment contrast for designing, monitoring, and analyzing a comparative clinical study

BACKGROUND

Consider a comparative, randomized clinical study with a specific event time as the primary end point. In the presence of censoring, standard methods of summarizing the treatment difference are based on Kaplan-Meier curves, the logrank test, and the point and interval estimates via Cox's procedure. Moreover, for designing and monitoring the study, one usually utilizes an event-driven scheme to determine the sample sizes and interim analysis time points.

PURPOSE

When the proportional hazards (PHs) assumption is violated, the logrank test may not have sufficient power to detect the difference between two event time distributions. The resulting hazard ratio estimate is difficult, if not impossible, to interpret as a treatment contrast. When the event rates are low, the corresponding interval estimate for the 'hazard ratio' can be quite large due to the fact that the interval length depends on the observed numbers of events. This may indicate that there is not enough information for making inferences about the treatment comparison even when there is no difference between two groups. This situation is quite common for a postmarketing safety study. We need an alternative way to quantify the group difference.

METHODS

Instead of quantifying the treatment group difference using the hazard ratio, we consider an easily interpretable and model-free parameter, the integrated survival rate difference over a prespecified time interval, as an alternative. We present the inference procedures for such a treatment contrast. This approach is purely nonparametric and does not need any model assumption such as the PHs. Moreover, when we deal with equivalence or noninferiority studies and the event rates are low, our procedure would provide more information about the treatment difference. We used a cardiovascular trial data set to illustrate our approach.

RESULTS

The results using the integrated event rate differences have a heuristic interpretation for the treatment difference even when the PHs assumption is not valid. When the event rates are low, for example, for the cardiovascular study discussed in this article, the procedure for the integrated event rate difference provides tight interval estimates in contrast to those based on the event-driven inference method.

LIMITATIONS

The design of a trial with the integrated event rate difference may be more complicated than that using the event-driven procedure. One may use simulation to determine the sample size and the estimated duration of the study.

CONCLUSIONS

The procedure discussed in this article can be a useful alternative to the standard PHs method in the survival analysis.

Analyzing survival curves at a fixed point in time for paired and clustered right-censored data

In clinical trials, information about certain time points may be of interest in making decisions about treatment effectiveness. Rather than comparing entire survival curves, researchers can focus on the comparison at fixed time points that may have a clinical utility for patients. For two independent samples of right-censored data, Klein et al. (2007) compared survival probabilities at a fixed time point by studying a number of tests based on some transformations of the Kaplan-Meier estimators of the survival function. However, to compare the survival probabilities at a fixed time point for paired right-censored data or clustered right-censored data, their approach would need to be modified. In this paper, we extend the statistics to accommodate the possible within-paired correlation and within-clustered correlation, respectively. We use simulation studies to present comparative results. Finally, we illustrate the implementation of these methods using two real data sets.

The simultaneous use of weighted logrank and weighted Kaplan-Meier statistics with clustered right-censored data

Clustered right-censored data often arise from tumorigenicity experiments and clinical trials. For testing the equality of two survival functions, Jung and Jeong extended weighted logrank (WLR) tests to two independent samples of clustered right-censored data, while the weighted Kaplan-Meier (WKM) test can be derived from the work of O'Gorman and Akritas. The weight functions in both classes of tests (WLR and WKM) can be selected to be more sensitive to detect a certain alternative; however, since the exact alternative is unknown, it is difficult to specify the selected weights in advance. Since WLR is rank-based, it is not sensitive to the magnitude of the difference in survival times. Although WKM is constructed to be more sensitive to the magnitude of the difference in survival times, it is not sensitive to late hazard differences. Therefore, in order to combine the advantages of these two classes of tests, this paper developed a class of versatile tests based on simultaneously using WLR and WKM for two independent samples of clustered right-censored data. The comparative results from a simulation study are presented and the implementation of the versatile tests to two real data sets is illustrated.

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.

Simultaneous group sequential analysis of rank-based and weighted Kaplan-Meier tests for paired censored survival data

This research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log-rank (PWLR) and standardized paired weighted Kaplan-Meier (PWKM) tests. During a trial, these tests may alternately assume the role of the more extreme statistic. By monitoring PEMAX, the maximum between the absolute values of the standardized PWLR and PWKM, one combines advantages of rank-based (RB) and non-RB paired testing paradigms. Simulations show that monitoring treatment differences using PEMAX maintains type I error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards (PH) as well as non-PH situations. Hence, PEMAX preserves power more robustly than individually monitored PWLR and PWKM, while maintaining a reasonably simple approach to design and analysis of results. An example from the Early Treatment Diabetic Retinopathy Study (ETDRS) is given.

Risk-adjusted manipulation of melphalan dose before stem cell transplantation in patients with amyloidosis is associated with a lower response rate

High-dose chemotherapy and autologous stem cell transplantation are used increasingly to treat patients with light-chain-related amyloidosis (AL). Treatment-related mortality is approximately 15%. To enable more patients to undergo stem cell transplantation, a risk-adapted strategy has been developed to treat with lower chemotherapy doses those patients who are at excessive risk. It is unclear whether reducing the chemotherapy dose in patients at excessive risk of treatment toxicity reduces the overall response. We retrospectively reviewed 171 AL patients who underwent conditioning chemotherapy with stem cell transplantation. The patients comprised two groups: those receiving standard high-dose melphalan and those receiving intermediate-dose melphalan. Responses were categorized as hematologic response, which used criteria for myeloma response. The two groups showed statistically significant differences; the overall response rates were 75% in the high-dose group and 53% in the intermediate-dose group although treatment-related mortality was the same in both groups. Reducing the melphalan dose appeared to render more AL patients eligible for stem cell transplantation but sacrificed an element of response. Methods are needed to reduce treatment-related toxicity so that more patients can receive full-dose conditioning chemotherapy.

Protection of cultured Cyprinus carpio against a lethal viral disease by an attenuated virus vaccine

Massive mortality of koi and common carp--Cyprinus carpio species--has been observed since 1998 in many countries worldwide, resulting in severe economic losses. The cause of the disease is an as yet unclassified large DNA virus, designated carp nephritis gill necrosis virus (CNGV) or koi herpes virus (KHV). Previously, we demonstrated that the wild type CNGV lost its pathogenecity following serial transfer in cell culture, and that clones isolated from the attenuated population can be used as a prophylactic vaccine. Here, we describe the basic conditions required for proper fish immunization so that a protection protocol may be devised. We demonstrated that carps are very sensitive to the pathogenic and the attenuated viruses, and short immersion of fish in water containing the viruses is sufficient for infection. The infection of fish with the pathogenic and the attenuated viruses is temperature-restricted; fish held at the non-permissive temperature, immediately following infection, were not affected by the pathogenic virus, and were not rendered resistant to the disease. Thus, propagation of the virus in the fingerlings is a pre-requisite for immunization. In order to increase the number of random mutations in the genome of the attenuated virus, and thus, reduce the possibility of the attenuated virus reverting to pathogenic, we irradiated it and selected additional clones appropriate for vaccination. The results of our study suggest that a safe and efficient prophylactic vaccine can be developed by selecting an appropriate attenuated virus.

Non-parametric paired two-sample tests for censored survival data incorporating longitudinal covariate information

In this manuscript, we present non-parametric two-sample tests for paired censored survival data incorporating longitudinal covariate information. These tests take advantage of information collected at baseline and post-baseline to provide efficiency gains when censoring is uninformative. Additionally, these methods adjust for potential bias from informative censoring that is captured by the baseline and longitudinal covariates. Finite sample properties are investigated with simulation, and we illustrate methodology with an example from the Early Treatment Diabetic Retinopathy Study.

Dependency issues in survival analyses of 55,782 primary hip replacements from 47,355 patients

Artificial hip joints are used in only one hip for about 85 per cent of the patients and in both hips (bilateral) for about 15 per cent of the patients. The occurrence of bilateral prostheses and the influence they have in survival analyses of joint arthroplasties are seldom considered. In this study we therefore focus on issues related to bilateral primary hip prostheses, time to revision surgery, and some commonly used statistical methods. We used information from 47,355 patients with 55,782 primary hip prostheses reported to the Norwegian Arthroplasty Register between 1987 and 2000. Due to the large number of diagnoses, fixation techniques for the prostheses, and combination of prostheses brands, we furthermore considered a 'homogeneous' subset of 8703 prostheses from 7930 patients with primary osteoarthritis, and Charnley prosthesis fixed with antibiotic-containing Palacos cement. Kaplan-Meier curves for all prostheses, ignoring that some patients have bilateral prostheses, were compared with Kaplan-Meier curves using only the first inserted prostheses, and with survival curves modified for patients with bilateral prostheses. Cox regression analyses were used to assess explanatory variables and to adjust for confounding factors. The results from the ordinary Cox regression analyses were compared with results from a marginal model, a shared gamma frailty model, and a model using a time dependent covariate to condition on failures in the opposite hip. We found no practical difference between the three calculated survival curves for the hip replacement data. The ordinary Cox-model and the marginal model gave equivalent results. In the shared gamma frailty model estimates for the risk factors were comparable with the former two approaches. The estimated frailty variance was higher when all data were used, even after adjustment for confounding factors. For the 'homogeneous' data the estimated frailty variance was negligible. Using a time dependent covariate to condition on previous revisions in the opposite hip, we found a higher risk of revision for the remaining primary hip prosthesis if the opposite hip had been revised (RR = 3.49, p < 0.0001). There was no difference in risk for revision between right and left hip prostheses. If the time interval between the two primary operations was more than two years, for the full data, the first hip prosthesis had an increased risk of revision compared to prostheses in patients with only one prosthesis (RR = 1.25, p = 0.01). For the 'homogeneous' data no statistically significant difference was found between unilateral and bilateral prostheses. A revision in one hip, for patients with bilateral prostheses, is a risk factor for revision of the other hip. Thus, in analyses of prostheses survival, dependencies between two hip prostheses from one patient should be considered. However, ignoring possible dependencies does not necessarily have an impact on the results on standard risk factors.

Group sequential monitoring of years of life saved with paired censored survival data

This research develops non-parametric methodology for sequential monitoring of paired time-to-event data when comparing years of life saved, or years where any unpleasant outcome is delayed, is of interest. The recommended family of test statistics uses integrated differences in survival estimates that are available during the study period, where adjustments are made for dependence in the survival and censoring outcomes under comparison. In the context of paired censored survival data, the joint asymptotic closed form distribution of these sequentially monitored test statistics is developed and shown to have a dependent increments structure. Simulations verifying nice operating characteristics of the proposed monitoring methods also reveal consequences of ignoring an underlying paired data structure in terms of size and power properties. A motivating example is also presented via the Early Treatment Diabetic Retinopathy Study, which did not have methods available for sequentially monitoring paired censored survival data at the time.