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.     

Generalized pairwise comparisons for censored data: An overview

The method of generalized pairwise comparisons (GPC) is an extension of the well-known nonparametric Wilcoxon–Mann–Whitney test for comparing two groups of observations. Multiple generalizations of Wilcoxon–Mann–Whitney test and other GPC methods have been proposed over the years to handle censored data. These methods apply different approaches to handling loss of information due to censoring: ignoring noninformative pairwise comparisons due to censoring (Gehan, Harrell, and Buyse); imputation using estimates of the survival distribution (Efron, Péron, and Latta); or inverse probability of censoring weighting (IPCW, Datta and Dong). Based on the GPC statistic, a measure of treatment effect, the “net benefit,” can be defined. It quantifies the difference between the probabilities that a randomly selected individual from one group is doing better than an individual from the other group. This paper aims at evaluating GPC methods for censored data, both in the context of hypothesis testing and estimation, and providing recommendations related to their choice in various situations. The methods that ignore uninformative pairs have comparable power to more complex and computationally demanding methods in situations of low censoring, and are slightly superior for high proportions (>40%) of censoring. If one is interested in estimation of the net benefit, Harrell's c index is an unbiased estimator if the proportional hazards assumption holds. Otherwise, the imputation (Efron or Peron) or IPCW (Datta, Dong) methods provide unbiased estimators in case of proportions of drop-out censoring up to 60%.     

Supremum weighted log-rank test and sample size for comparing two-stage adaptive treatment strategies

In two-stage adaptive treatment strategies, patients receivean induction treatment followed by a maintenance therapy, giventhat the patient responded to the induction treatment they received.To test for a difference in the effects of different inductionand maintenance treatment combinations, a modified supremumweighted log-rank test is proposed. The test is applied to adataset from a two-stage randomized trial and the results arecompared to those obtained using a standard weighted log-ranktest. A sample-size formula is proposed based on the limitingdistribution of the supremum weighted log-rank statistic. Thesample-size formula reduces to Eng and Kosorok's sample-sizeformula for a two-sample supremum log-rank test when there isno second randomization. Monte Carlo studies show that the proposedtest provides sample sizes that are close to those obtainedby standard weighted log-rank test under a proportional hazardsalternative. However, the proposed test is more powerful thanthe standard weighted log-rank test under non-proportional hazardsalternatives.     

Nonparametric comparison of two survival-time distributions in the presence of dependent censoring

When testing the null hypothesis that treatment arm-specific survival-time distributions are equal, the log-rank test is asymptotically valid when the distribution of time to censoring is conditionally independent of randomized treatment group given survival time. We introduce a test of the null hypothesis for use when the distribution of time to censoring depends on treatment group and survival time. This test does not make any assumptions regarding independence of censoring time and survival time. Asymptotic validity of this test only requires a consistent estimate of the conditional probability that the survival event is observed given both treatment group and that the survival event occurred before the time of analysis. However, by not making unverifiable assumptions about the data-generating mechanism, there exists a set of possible values of corresponding sample-mean estimates of these probabilities that are consistent with the observed data. Over this subset of the unit square, the proposed test can be calculated and a rejection region identified. A decision on the null that considers uncertainty because of censoring that may depend on treatment group and survival time can then be directly made. We also present a generalized log-rank test that enables us to provide conditions under which the ordinary log-rank test is asymptotically valid. This generalized test can also be used for testing the null hypothesis when the distribution of censoring depends on treatment group and survival time. However, use of this test requires semiparametric modeling assumptions. A simulation study and an example using a recent AIDS clinical trial are provided.     

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.     

On sample size and inference for two-stage adaptive designs

Proschan and Hunsberger (1995, Biometrics 51, 1315-1324) proposed a two-stage adaptive design that maintains the Type I error rate. For practical applications, a two-stage adaptive design is also required to achieve a desired statistical power while limiting the maximum overall sample size. In our proposal, a two-stage adaptive design is comprised of a main stage and an extension stage, where the main stage has sufficient power to reject the null under the anticipated effect size and the extension stage allows increasing the sample size in case the true effect size is smaller than anticipated. For statistical inference, methods for obtaining the overall adjusted p-value, point estimate and confidence intervals are developed. An exact two-stage test procedure is also outlined for robust inference.     

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.     

ERCC1 and TS Expression as Prognostic and Predictive Biomarkers in Metastatic Colon Cancer

In patients with metastatic colon cancer, response to first line chemotherapy is a strong predictor of overall survival (OS). Currently, oncologists lack diagnostic tests to determine which chemotherapy regimen offers the greatest chance for response in an individual patient. Here we present the results of gene expression analysis for two genes, ERCC1 and TS, measured with the commercially available ResponseDX: Colon assay (Response Genetics, Los Angeles, CA) in 41 patients with de novo metastatic colon cancer diagnosed between July 2008 and August 2013 at the University of California, San Diego. In addition ERCC1 and TS expression levels as determined by RNAseq and survival data for patients in TCGA were downloaded from the TCGA data portal. We found that patients with low expression of ERCC1 (n = 33) had significantly longer median OS (36.0 vs. 10.1 mo, HR 0.29, 95% CI .095 to .84, log-rank p = 9.0x10^-6) and median time to treatment to failure (TTF) following first line chemotherapy (14.1 vs. 2.4 mo, HR 0.17, 95% CI 0.048 to 0.58, log-rank p = 5.3x10^-4) relative to those with high expression (n = 4). After accounting for the covariates age, sex, tumor grade and ECOG performance status in a Cox proportional hazard model the association of low ERCC1 with longer OS (HR 0.18, 95% CI 0.14 to 0.26, p = 0.0448) and TTF (HR 0.16, 95% CI 0.14 to 0.21, p = 0.0053) remained significant. Patients with low TS expression (n = 29) had significantly longer median OS (36.0 vs. 14.8 mo, HR 0.25, 95% CI 0.074 to 0.82, log-rank p = 0.022) relative to those with high expression (n = 12). The combined low expression of ERCC1/TS was predictive of response in patients treated with FOLFOX (40% vs. 91%, RR 2.3, Fisher’s exact test p = 0.03, n = 27), but not with FOLFIRI (71% vs. 71%, RR 1.0, Fisher’s exact test p = 1, n = 14). Overall, these findings suggest that measurement of ERCC1 and TS expression has potential clinical utility in managing patients with metastatic colorectal cancer.     

Validation of null model tests using Neyman–Pearson hypothesis testing theory

A long-standing question in ecology is whether interspecific competition affects co-occurrence patterns of species. Null model tests of presence–absence data (NMTPAs) constitute an important approach to address the question, but different tests often give conflicting results when applied to the same data. Neyman–Pearson hypothesis testing theory provides a rigorous and well accepted framework for assessing the validity and optimality of statistical tests. Here, I treat NMTPAs within this framework, and measure the robustness and bias of 72 representative tests. My results indicate that, when restrictive assumptions are met, existing NMTPAs are adequate, but for general testing situations, the use of all existing NMTPAs is unjustified — the tests are nonrobust or biased. For many current applications of NMTPAs, restrictive assumptions appear unmet, so these results illustrate an area in which existing NMTPAs can be improved. In addition to highlighting useful improvements to existing NMTPAs, the results here provide a rigorous framework for developing improved methods.     

Sample size and power for the weighted log-rank test and Kaplan-Meier based tests with allowance for nonproportional hazards

Asymptotic distributions under alternative hypotheses and their corresponding sample size and power equations are derived for nonparametric test statistics commonly used to compare two survival curves. Test statistics include the weighted log-rank test and the Wald test for difference in (or ratio of) Kaplan-Meier survival probability, percentile survival, and restricted mean survival time. Accrual, survival, and loss to follow-up are allowed to follow any arbitrary continuous distribution. We show that Schoenfeld's equation—often used by practitioners to calculate the required number of events for the unweighted log-rank test—can be inaccurate even when the proportional hazards (PH) assumption holds. In fact, it can mislead one to believe that 1:1 is the optimal randomization ratio (RR), when actually power can be gained by assigning more patients to the active arm. Meaningful improvements to Schoenfeld's equation are made. The present theory should be useful in designing clinical trials, particularly in immuno-oncology where nonproportional hazards are frequently encountered. We illustrate the application of our theory with an example exploring optimal RR under PH and a second example examining the impact of delayed treatment effect. A companion R package npsurvSS is available for download on CRAN.     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.     

A Comparison of Procedures for Adaptive Choice of Location Tests in Flexible Two‐Stage Designs

Although linear rank statistics for the two‐sample problem are distribution free tests, their power depends on the distribution of the data. In the planning phase of an experiment, researchers are often uncertain about the shape of this distribution and so the choice of test statistic for the analysis and the determination of the required sample size are based on vague information. Adaptive designs with interim analysis can potentially overcome both problems. And in particular, adaptive tests based on a selector statistic are a solution to the first. We investigate whether adaptive tests can be usefully implemented in flexible two‐stage designs to gain power. In a simulation study, we compare several methods for choosing a test statistic for the second stage of an adaptive design based on interim data with the procedure that applies adaptive tests in both stages. We find that the latter is a sensible approach that leads to the best results in most situations considered here. The different methods are illustrated using a clinical trial example.     

Bayesian survival analysis using a MARS model

A Bayesian multivariate adaptive regression spline fitting approach is used to model univariate and multivariate survival data with censoring. The possible models contain the proportional hazards model as a subclass and automatically detect departures from this. A reversible jump Markov chain Monte Carlo algorithm is described to obtain the estimate of the hazard function as well as the survival curve.     

Statistical methods for describing developmental milestones with censored data: effects of birth weight status and sex in neonatal pigtailed macaques

Neurobehavioral tests are used to assess early neonatal behavioral functioning and detect effects of prenatal and perinatal events. However, common measurement and data collection methods create specific data features requiring thoughtful statistical analysis. Assessment response measurements are often ordinal scaled, not interval scaled; the magnitude of the physical response may not directly correlate with the underlying state of developmental maturity; and a subject's assessment record may be censored. Censoring occurs when the milestone is exhibited at the first test (left censoring), when the milestone is not exhibited before the end of the study (right censoring), or when the exact age of attaining the milestone is uncertain due to irregularly spaced test sessions or missing data (interval censoring). Such milestone data is best analyzed using survival analysis methods. Two methods are contrasted: the non-parametric Kaplan-Meier estimator and the fully parametric interval censored regression. The methods represent the spectrum of survival analyses in terms of parametric assumptions, ability to handle simultaneous testing of multiple predictors, and accommodation of different types of censoring. Both methods were used to assess birth weight status and sex effects on 14 separate test items from assessments on 255 healthy pigtailed macaques. The methods gave almost identical results. Compared to the normal birth weight group, the low birth weight group had significantly delayed development on all but one test item. Within the low birth weight group, males had significantly delayed development for some responses relative to females.     

Bias in Diet Determination: Incorporating Traditional Methods in Bayesian Mixing Models

There are not “universal methods” to determine diet composition of predators. Most traditional methods are biased because of their reliance on differential digestibility and the recovery of hard items. By relying on assimilated food, stable isotope and Bayesian mixing models (SIMMs) resolve many biases of traditional methods. SIMMs can incorporate prior information (i.e. proportional diet composition) that may improve the precision in the estimated dietary composition. However few studies have assessed the performance of traditional methods and SIMMs with and without informative priors to study the predators’ diets. Here we compare the diet compositions of the South American fur seal and sea lions obtained by scats analysis and by SIMMs-UP (uninformative priors) and assess whether informative priors (SIMMs-IP) from the scat analysis improved the estimated diet composition compared to SIMMs-UP. According to the SIMM-UP, while pelagic species dominated the fur seal’s diet the sea lion’s did not have a clear dominance of any prey. In contrast, SIMM-IP’s diets compositions were dominated by the same preys as in scat analyses. When prior information influenced SIMMs’ estimates, incorporating informative priors improved the precision in the estimated diet composition at the risk of inducing biases in the estimates. If preys isotopic data allow discriminating preys’ contributions to diets, informative priors should lead to more precise but unbiased estimated diet composition. Just as estimates of diet composition obtained from traditional methods are critically interpreted because of their biases, care must be exercised when interpreting diet composition obtained by SIMMs-IP. The best approach to obtain a near-complete view of predators’ diet composition should involve the simultaneous consideration of different sources of partial evidence (traditional methods, SIMM-UP and SIMM-IP) in the light of natural history of the predator species so as to reliably ascertain and weight the information yielded by each method.     

Prognostic Relevance of Cytochrome c Oxidase in Primary Glioblastoma Multiforme

Patients with primary glioblastoma multiforme (GBM) have one of the lowest overall survival rates among cancer patients, and reliable biomarkers are necessary to predict patient outcome. Cytochrome c oxidase (CcO) promotes the switch from glycolytic to OXPHOS metabolism, and increased CcO activity in tumors has been associated with tumor progression after chemotherapy failure. Thus, we investigated the relationship between tumor CcO activity and the survival of patients diagnosed with primary GBM. A total of 84 patients with grade IV glioma were evaluated in this retrospective cohort study. Cumulative survival was calculated by the Kaplan-Meier method and analyzed by the log-rank test, and univariate and multivariate analyses were performed with the Cox regression model. Mitochondrial CcO activity was determined by spectrophotometrically measuring the oxidation of cytochrome c. High CcO activity was detected in a subset of glioma tumors (∼30%), and was an independent prognostic factor for shorter progression-free survival and overall survival [P = 0.0087 by the log-rank test, hazard ratio = 3.57 for progression-free survival; P<0.001 by the log-rank test, hazard ratio = 10.75 for overall survival]. The median survival time for patients with low tumor CcO activity was 14.3 months, compared with 6.3 months for patients with high tumor CcO activity. High CcO activity occurs in a significant subset of high-grade glioma patients and is an independent predictor of poor outcome. Thus, CcO activity may serve as a useful molecular marker for the categorization and targeted therapy of GBMs.     

Rare Variant Association Testing by Adaptive Combination of P-values

With the development of next-generation sequencing technology, there is a great demand for powerful statistical methods to detect rare variants (minor allele frequencies (MAFs)<1%) associated with diseases. Testing for each variant site individually is known to be underpowered, and therefore many methods have been proposed to test for the association of a group of variants with phenotypes, by pooling signals of the variants in a chromosomal region. However, this pooling strategy inevitably leads to the inclusion of a large proportion of neutral variants, which may compromise the power of association tests. To address this issue, we extend the -MidP method (Cheung et al., 2012, Genet Epidemiol 36: 675–685) and propose an approach (named ‘adaptive combination of P-values for rare variant association testing’, abbreviated as ‘ADA’) that adaptively combines per-site P-values with the weights based on MAFs. Before combining P-values, we first imposed a truncation threshold upon the per-site P-values, to guard against the noise caused by the inclusion of neutral variants. This ADA method is shown to outperform popular burden tests and non-burden tests under many scenarios. ADA is recommended for next-generation sequencing data analysis where many neutral variants may be included in a functional region.     

A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.