Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation

Summary .   In some clinical studies comparing treatments in terms of their survival curves, researchers may anticipate that the survival curves will cross at some point, leading to interest in a long-term survival comparison. However, simple comparison of the survival curves at a fixed point may be inefficient, and use of a weighted log-rank test may be overly sensitive to early differences in survival. We formulate the problem as one of testing for differences in survival curves after a prespecified time point, and propose a variety of techniques for testing this hypothesis. We study these methods using simulation and illustrate them on a study comparing survival for autologous and allogeneic bone marrow transplants.     

Improved Logrank-Type Tests for Survival Data Using Adaptive Weights

Summary .  For testing for treatment effects with time-to-event data, the logrank test is the most popular choice and has some optimality properties under proportional hazards alternatives. It may also be combined with other tests when a range of nonproportional alternatives are entertained. We introduce some versatile tests that use adaptively weighted logrank statistics. The adaptive weights utilize the hazard ratio obtained by fitting the model of Yang and Prentice (2005,  Biometrika   92 , 1–17). Extensive numerical studies have been performed under proportional and nonproportional alternatives, with a wide range of hazard ratios patterns. These studies show that these new tests typically improve the tests they are designed to modify. In particular, the adaptively weighted logrank test maintains optimality at the proportional alternatives, while improving the power over a wide range of nonproportional alternatives. The new tests are illustrated in several real data examples.     

A linear rank test for use when the main interest is in differences in cure rates

For diseases with a positive probability of being cured, a family of alternatives to the null hypothesis of equality of survival distributions is introduced, which is designed to focus power against alternatives with differences in cure rates. The optimal linear rank test for this alternative is derived, and found to be substantially more efficient than the log-rank test for this alternative when cure rates are less than 50%, while there is little difference between the tests if the cure rates are 50% or greater. The simple test based on the difference of Kaplan-Meier estimates of the proportion cured is also examined, and found to be fully efficient for this alternative with no censoring, while its efficiency rapidly drops as censoring is increased. The new test is not a pure test of equality of cure rates when the data are censored, but rather is a test of equality of survival distributions that focuses power against late differences in the survival curves.     

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.     

Nonparametric Tests for Interval‐Censored Failure Time Data

This paper discusses two sample nonparametric comparison of survival functions when only interval‐censored failure time data are available. The problem considered often occurs in, for example, biological and medical studies such as medical follow‐up studies and clinical trials. For the problem, we present and study several nonparametric test procedures that include methods based on both absolute and squared survival differences as well as simple survival differences. The presented tests provide alternatives to existing methods, most of which are rank‐based tests and not sensitive to nonproportional or nonmonotone alternatives. Simulation studies are performed to evaluate and compare the proposed methods with existing methods and suggest that the proposed tests work well for nonmonotone alternatives as well as monotone alternatives. An illustrative example is presented.     

Stratified modestly weighted log-rank tests in settings with an anticipated delayed separation of survival curves

Delayed separation of survival curves is a common occurrence in confirmatory studies in immuno-oncology. Many novel statistical methods that aim to efficiently capture potential long-term survival improvements have been proposed in recent years. However, the vast majority do not consider stratification, which is a major limitation considering that most large confirmatory studies currently employ a stratified primary analysis. In this article, we combine recently proposed weighted log-rank tests that have been designed to work well under a delayed separation of survival curves, with stratification by a baseline variable. The aim is to increase the efficiency of the test when the stratifying variable is highly prognostic for survival. As there are many potential ways to combine the two techniques, we compare several possibilities in an extensive simulation study. We also apply the techniques retrospectively to two recent randomized clinical trials.     

A class of permutation tests for stratified survival data

We propose a class of permutation tests for stratified survival data. The tests are derived using the framework of Fay and Shih (1998, Journal of the American Statistical Association 93, 387-396), which creates tests by permuting scores based on a functional of estimated distribution functions. Here the estimated distribution function for each possibly right-, left-, or interval-censored observation is based on a shrinkage estimator similar to the nonparametric empirical estimator of Ghosh, Lahiri, and Tiwari (1989, Communications in Statistics--Theory and Methods 18, 121-146), and permutation is carried out within strata. The proposed test with a weighted Mann-Whitney functional is similar to the permutation form of the stratified log-rank test when there is a large strata effect or the sample size in each stratum is large and is similar to the permutation form of the ordinary log-rank test when there is little strata effect. Thus, the proposed test unifies the advantages of both the stratified and ordinary log-rank tests. By changing the functional, we may obtain a stratified Prentice-Wilcoxon test or a difference in means test with similar unifying properties. We show through simulations the advantage of the proposed test over existing tests for uncensored and right-censored data.     

Generalized log-rank tests for partly interval-censored failure time data

In this paper, we consider incomplete survival data: partly interval-censored failure time data where observed data include both exact and interval-censored observations on the survival time of interest. We present a class of generalized log-rank tests for this type of survival data and establish their asymptotic properties. The method is evaluated using simulation studies and illustrated by a set of real data from a diabetes study.     

The efficiency of a weighed log-rank test under a percent error misspecification model for the log hazard ratio.

For comparison of two survival distributions, it is natural to use a weighted log-rank test with weight function given by the log hazard ratio function that is anticipated a priori. This paper investigates the efficiency of this test when the a priori estimate of the log hazard ratio is subject to a specified percentage error. The test is shown to be the maximum efficiency robust test over the class of alternatives in question and a simple expression for the maximum efficiency is established.     

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.     

The influence of competing-risks setting on the choice of hypothesis test for treatment effect

There is considerable debate regarding the choice of test for treatment difference in a randomized clinical trial in the presence of competing risks. This question arose in the study of standard and new antiepileptic drugs (SANAD) trial comparing new and standard antiepileptic drugs. This paper provides simulation results for the log-rank test comparing cause-specific hazard rates and Gray's test comparing cause-specific cumulative incidence curves. To inform the analysis of the SANAD trial, competing-risks settings were considered where both events are of interest, events may be negatively correlated, and the degree of correlation may differ in the 2 treatment groups. In settings where there are effects in opposite directions for the 2 event types, a likely situation for the SANAD trial, Gray's test has greater power to detect treatment differences than log-rank analysis. For the epilepsy application, conclusions were qualitatively similar for both log-rank and Gray's tests.     

Radiation-induced rectal complications are not influenced by age: a dose fractionation study in the rat

Radiation-induced complications of the rectum are an important dose-limiting factor in radiotherapy of pelvic malignancies. In general, animal studies demonstrated no differences in acute and late normal tissue toxicity with age, but little is known about rectal complications in relation to age. For this purpose, an extensive histological and dose fractionation study was carried out on the rectum of young (12 weeks) and older (77-80 weeks) rats. In this paper, the results of dose fractionation are presented in relation to age at the time of irradiation. Young and older animals were irradiated with single and fractionated doses. After irradiation, rectal complications could lead to occlusion and stenosis, eventually resulting in the clinical symptoms of a megacolon and a possible fistula. For each dose group, cumulative survival rates were obtained with Kaplan-Meier analysis, from which dose-effect curves and the associated LD(50) values for a megacolon/fistula were calculated. The majority of responders died between 8 and 24 weeks after irradiation, irrespective of age. For both age groups, only the fractionation data showed a reduction in the mean latency with increasing dose. In the older age group, 39% of the responders developed a fistula compared to 26% for the younger animals. The LD(50) values increased from around 30 Gy after single doses to nearly 65 Gy after 10 fractions. The increases in LD(50) values with the number of fractions were independent of the age of the rats. For each of the dose fractionation schedules, log-rank testing indicated no significant differences in cumulative survival rates between younger and older animals (P > 0.10). The high alpha/beta ratios obtained for both the young and older animals strongly suggested that the late rectal complications were a consequence of early epithelial injury. Associated histological findings indicated that blood vessel damage, which was already evident at a high incidence at 4 weeks after irradiation, could also play a significant role in the occurrence of consequential late injuries. In conclusion, data obtained for the latent period of rectal occlusion, for the dose-effect curves, for the log-rank testing of cumulative survival rates, and for the alpha/beta ratios strongly support the hypothesis that the incidence of radiation-induced rectal complications is independent of age. Late rectal complications could be a consequence of radiation-induced acute injury.     

The use of frailty hazard models for unrecognized heterogeneity that interacts with treatment: considerations of efficiency and power

Increasingly, genetic studies of tumors of the same histologic diagnosis are elucidating subtypes that are distinct with respect to clinical endpoints such as response to treatment and survival. This raises concerns about the efficiency of using the simple log-rank test for analysis of treatment effect on survival in studies of possibly heterogeneous tumors. Furthermore, such studies, designed under the assumption of homogeneity, may be severely underpowered. We derive analytic approximations for the asymptotic relative efficiency of the simple log-rank test relative to the optimally weighted log-rank test and for the power of the simple log-rank test when applied to subjects with unobserved heterogeneity, as reflected in a continuous frailty, that may interact with treatment. Numerical studies demonstrate that the simple log-rank test may be quite inefficient if the frailty interacts with treatment. Further, there may be a substantial loss of power in the presence of the frailty with or without an interaction with treatment.     

Sample size/power calculation for case-cohort studies

In epidemiologic studies and disease prevention trials, interest often involves estimation of the relationship between some disease endpoints and individual exposure. In some studies, due to the rarity of the disease and the cost in collecting the exposure information for the entire cohort, a case-cohort design, which consists of a small random sample of the whole cohort and all the diseased subjects, is often used. Previous work has focused on analyzing data from the case-cohort design and few have discussed the sample size issues. In this article, we describe two tests for the case-cohort design, which can be treated as a natural generalization of log-rank test in the full cohort design. We derive an explicit form for power/sample size calculation based on these two tests. A number of simulation studies have been used to illustrate the efficiency of the tests for the case-cohort design. An example is provided on how to use the formula.     

Sample size formulae for two-stage randomized trials with survival outcomes

Two-stage randomized trials are growing in importance in developing adaptive treatment strategies, i.e. treatment policies or dynamic treatment regimes. Usually, the first stage involves randomization to one of the several initial treatments. The second stage of treatment begins when an early nonresponse criterion or response criterion is met. In the second-stage, nonresponding subjects are re-randomized among second-stage treatments. Sample size calculations for planning these two-stage randomized trials with failure time outcomes are challenging because the variances of common test statistics depend in a complex manner on the joint distribution of time to the early nonresponse criterion or response criterion and the primary failure time outcome. We produce simple, albeit conservative, sample size formulae by using upper bounds on the variances. The resulting formulae only require the working assumptions needed to size a standard single-stage randomized trial and, in common settings, are only mildly conservative. These sample size formulae are based on either a weighted Kaplan-Meier estimator of survival probabilities at a fixed time-point or a weighted version of the log-rank test.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

Nonparametric rank-based methods for group sequential monitoring of paired censored survival data

This research gives methods for nonparametric sequential monitoring of paired censored survival data in the two-sample problem using paired weighted log-rank statistics with adjustments for dependence in survival and censoring outcomes. The joint asymptotic closed-form distribution of these sequentially monitored statistics has a dependent increments structure. Simulations validating operating characteristics of the proposed methods highlight power and size consequences of ignoring even mildly correlated data. A motivating example is presented via the Early Treatment Diabetic Retinopathy Study.     

Iranian homograft heart valves: assessment of durability and late outcome

Durability and the rate of complications of homograft heart valves, adjusted for patient-related contributors and surgical techniques, rely mainly on the quality of allografts which in turn are mirrored in the donor characteristics and most importantly recovery and processing procedures. Aimed to assess the quality, a study was conducted to figure out the durability and late outcome following homograft replacement with valved conduits procured by the Iranian Tissue Bank. Retrospectively, the pre-implantation, perioperative and follow-up data of 400 non-consecutive recipients of cryopreserved heart valves (222 pulmonary and 178 aortic) from 2006 to 2015 were collected and analyzed in terms of variables reflecting late outcome including adverse events and durability. In the context of durability, the event of interest was defined as the need for homograft replacement and homograft-related death. The mean follow-up time (SD) of study entrants (male/female ratio, 1.4) was 49.8 (36.3) months. Median age at the time of implantation was 11 years. Total 10-years mortality was 21 % (84/400), including 66.7 % early (30-days mortality: 56/84) and 33.3 % late (28/84). Overall late complication rate was 2 %. Median survival time was 120 months (95 % CI 83.3–156.6). The pulmonary valves appeared to be more durable (P value <0.001) and survival probabilities in small sized grafts were lower (P value 0.008). One-, five-, and ten-year graft survival was 82, 76 and 73 %, respectively. The evidences suggest that the homografts function satisfactory with low rate of late complications; nevertheless, more emphasis should be given to make long-term durability comparable.     

Modeling a categorical variable allowing arbitrarily many category choices

This article discusses the modeling of a categorical variable for which subjects can select any number of categories. For c categories, the response variable consists of a cross-classification of c binary components, one pertaining to each category. Using data from a survey (Loughin, T. M. and Scherer, P. N., 1998, Biometrics, 54, 630 637) in which Kansas farmers indicated their primary sources of veterinary information, we discuss simultaneous logit modeling of the binary components of the multivariate response. The use of maximum likelihood or quasi-likelihood fitting provides chi-squared tests with degrees of freedom df = c(r - 1) for testing the independence between each of the c response components and an explanatory variable with r categories. These tests are alternatives to the weighted chi-squared test and the bootstrap test proposed by Loughin and Scherer for this hypothesis.     

Weighted Logrank Testing Procedures for the Simple Tree Alternatives

Weighted logrank testing procedures for comparing r treatments with a control when some of the data are randomly censored are discussed. Four kinds of test statistics for the simple tree alternatives are considered. The weighted logrank statistics based on pairwise ranking scheme is proposed and the covariances of the test statistics are explicitly obtained. This class of test statistics can be viewed as the general statistics of constructing the test procedures for various order restricted alternatives by modifying weights. Four kinds of weighted logrank tests are illustrated with an example. Simulation studies are performed to compare the sizes and the powers of the considered tests with the other.