Sample size calculation for the combination test under nonproportional hazards

For sample size calculation in clinical trials with survival endpoints, the logrank test, which is the optimal method under the proportional hazard (PH) assumption, is predominantly used. In reality, the PH assumption may not hold. For example, in immuno-oncology trials, delayed treatment effects are often expected. The sample size without considering the potential violation of the PH assumption may lead to an underpowered study. In recent years, combination tests such as the maximum weighted logrank test have received great attention because of their robust performance in various hazards scenarios. In this paper, we propose a flexible simulation-free procedure to calculate the sample size using combination tests. The procedure extends the Lakatos' Markov model and allows for complex situations encountered in a clinical trial, like staggered entry, dropouts, etc. We evaluate the procedure using two maximum weighted logrank tests, one projection-type test, and three other commonly used tests under various hazards scenarios. The simulation studies show that the proposed method can achieve the target power for all compared tests in most scenarios. The combination tests exhibit robust performance under correct specification and misspecification scenarios and are highly recommended when the hazard-changing patterns are unknown beforehand. Finally, we demonstrate our method using two clinical trial examples and provide suggestions about the sample size calculations under nonproportional hazards.     

A note on P-values under group sequential testing and nonproportional hazards

Summary .   Group sequential designs are often used for periodically assessing treatment efficacy during the course of a clinical trial. Following a group sequential test, P -values computed under the assumption that the data were gathered according to a fixed sample design are no longer uniformly distributed under the null hypothesis of no treatment effect. Various sample space orderings have been proposed for computing proper P -values following a group sequential test. Although many of the proposed orderings have been compared in the setting of time-invariant treatment effects, little attention has been given to their performance when the effect of treatment within an individual varies over time. Our interest here is to compare two of the most commonly used methods for computing proper P -values following a group sequential test, based upon the analysis time (AT) and Z -statistic orderings, with respect to resulting power functions when treatment effects on survival are delayed. Power under the AT ordering is shown to be heavily influenced by the presence of a delayed treatment effect, while power functions corresponding to the Z -statistic ordering remain robust under time-varying treatment effects.     

Estimating cumulative treatment effects in the presence of nonproportional hazards

Summary .   Often in medical studies of time to an event, the treatment effect is not constant over time. In the context of Cox regression modeling, the most frequent solution is to apply a model that assumes the treatment effect is either piecewise constant or varies smoothly over time, i.e., the Cox nonproportional hazards model. This approach has at least two major limitations. First, it is generally difficult to assess whether the parametric form chosen for the treatment effect is correct. Second, in the presence of nonproportional hazards, investigators are usually more interested in the cumulative than the instantaneous treatment effect (e.g., determining if and when the survival functions cross). Therefore, we propose an estimator for the aggregate treatment effect in the presence of nonproportional hazards. Our estimator is based on the treatment-specific baseline cumulative hazards estimated under a stratified Cox model. No functional form for the nonproportionality need be assumed. Asymptotic properties of the proposed estimators are derived, and the finite-sample properties are assessed in simulation studies. Pointwise and simultaneous confidence bands of the estimator can be computed. The proposed method is applied to data from a national organ failure registry.     

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.     

On the inadmissibility of Watterson's estimator

We consider the estimation of the scaled mutation parameter θ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.     

On correcting the overestimation of the permutation-based false discovery rate estimator

MOTIVATION: Recent attempts to account for multiple testing in the analysis of microarray data have focused on controlling the false discovery rate (FDR), which is defined as the expected percentage of the number of false positive genes among the claimed significant genes. As a consequence, the accuracy of the FDR estimators will be important for correctly controlling FDR. Xie et al. found that the standard permutation method of estimating FDR is biased and proposed to delete the predicted differentially expressed (DE) genes in the estimation of FDR for one-sample comparison. However, we notice that the formula of the FDR used in their paper is incorrect. This makes the comparison results reported in their paper unconvincing. Other problems with their method include the biased estimation of FDR caused by over- or under-deletion of DE genes in the estimation of FDR and by the implicit use of an unreasonable estimator of the true proportion of equivalently expressed (EE) genes. Due to the great importance of accurate FDR estimation in microarray data analysis, it is necessary to point out such problems and propose improved methods. RESULTS: Our results confirm that the standard permutation method overestimates the FDR. With the correct FDR formula, we show the method of Xie et al. always gives biased estimation of FDR: it overestimates when the number of claimed significant genes is small, and underestimates when the number of claimed significant genes is large. To overcome these problems, we propose two modifications. The simulation results show that our estimator gives more accurate estimation.     

A causal proportional hazards estimator for the effect of treatment actually received in a randomized trial with all-or-nothing compliance

Survival data from randomized trials are most often analyzed in a proportional hazards (PH) framework that follows the intention-to-treat (ITT) principle. When not all the patients on the experimental arm actually receive the assigned treatment, the ITT-estimator mixes its effect on treatment compliers with its absence of effect on noncompliers. The structural accelerated failure time (SAFT) models of Robins and Tsiatis are designed to consistently estimate causal effects on the treated, without direct assumptions about the compliance selection mechanism. The traditional PH-model, however, has not yet led to such causal interpretation. In this article, we examine a PH-model of treatment effect on the treated subgroup. While potential treatment compliance is unobserved in the control arm, we derive an estimating equation for the Compliers PROPortional Hazards Effect of Treatment (C-PROPHET). The jackknife is used for bias correction and variance estimation. The method is applied to data from a recently finished clinical trial in cancer patients with liver metastases.     

On the thermal noise limit of cellular membranes

Comparison of thermal noise limits and the effects of low frequency electromagnetic fields (LFEMF) on the cellular membrane have important implications for the study of bioelectro-magnetism in this regime. Over a decade ago, Weaver and Astumian developed a model to show that thermal noise can limit the efficacy of LFEMF. A recent report by Kaune [Kaune (2002) Bioelectromagnetics 23:622-628], however, contradicted their findings. Kaune assumes that the conductance noise current of cell membrane can be decomposed into two components, where one of them is identical regarding all segments (coherent), while the other is different (incoherent). Besides, this decomposition is not unequivocal and contradicts to the statistical independence of the segment noise currents, and therefore to the second law of thermodynamics as well. We suggest the procedure based on the method of symmetrical components, by the means of which we can re-interpret the result of Kaune in a correct way.     

On the existence of limit cycles in motion field

In this paper we show the existence of limit cycles in the vector field generated by the perspective projection on the image plane of the velocity field of a moving surface. The existence of limit cycles is proved with the Poincaré-Bendixon theorem, in the case of a rotating smooth non-planar surface and illustrated with computer graphics. The structural stability of the limit cycles is also discussed.     

Physiological performance of silver-fir (Abies alba Mill.) populations under contrasting climates near the south-western distribution limit of the species

Silver-fir (Abies alba Mill.) populations located at the south side of the main Pyrenean axis and Pre-Pyrenees constitute the south-western distribution limit of the species and, therefore, may be responding more noticeably to climate change than those in the centre of the range. The increasing aridity detected in the Pyrenees during the 20th century should affect more negatively the physiological performance of the southernmost silver-fir stand growing under more xeric conditions in comparison with stands growing within the main distribution area under mesic conditions. To evaluate the climatic influence on the performance of silver fir near its distribution limit, we studied several physiological and growth variables in shoots and needles from two silver fir populations located in nearby but climatically contrasting sites: Paco Ezpela (site E) and Gamueta (site G). Site E showed a stronger Mediterranean influence than site G, i.e. the former site was characterized by higher temperatures and leaf-to-air water vapour pressure difference and lower precipitation in summer than the latter site. Silver firs from site E showed lower values of primary and secondary growth, needle length, stomatal conductance, net photosynthesis and photosystem II (PSII) efficiency than individuals from site G. The reduction in net photosynthesis could be ascribed to a low CO₂ availability and to a lower PSII efficiency. We conclude that the physiological differences found between both sites were caused by the more xeric conditions of site E as compared with the more mesic environment in site G. The predicted increase of severe droughts in the southern Pyrenees might cause a decrease in photosynthesis and growth in those silver-fir stands located near the ecological limit of the species.     

Chemical speciation of trace metals in mammalian cell culture media: looking under the hood to boost cellular performance and product quality

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Sufficiency of the number of segregating sites in the limit under finite-sites mutation

We show that the number of segregating sites is a sufficient statistic for the scaled mutation parameter (θ) in the limit as the number of sites tends to infinity and there is free recombination between sites. We assume that the mutation parameter at each site tends to zero such than the total mutation parameter (θ) is constant in the limit. Our results show that Watterson’s estimator is the maximum likelihood estimator in this case, but that it estimates a composite parameter which is different for different mutation models. Some of our results hold when recombination is limited, because Watterson’s estimator is an unbiased, method-of-moments estimator regardless of the recombination rate. The quantity it estimates depends on the details of how mutations occur at each site.     

Leakage and slow allostery limit performance of single drug-sensing aptazyme molecules based on the hammerhead ribozyme

Engineered “aptazymes” fuse in vitro selected aptamers with ribozymes to create allosteric enzymes as biosensing components and artificial gene regulatory switches through ligand-induced conformational rearrangement and activation. By contrast, activating ligand is employed as an enzymatic cofactor in the only known natural aptazyme, the glmS ribozyme, which is devoid of any detectable conformational rearrangements. To better understand this difference in biosensing strategy, we monitored by single molecule fluorescence resonance energy transfer (FRET) and 2-aminopurine (AP) fluorescence the global conformational dynamics and local base (un)stacking, respectively, of a prototypical drug-sensing aptazyme, built from a theophylline aptamer and the hammerhead ribozyme. Single molecule FRET reveals that a catalytically active state with distal Stems I and III of the hammerhead ribozyme is accessed both in the theophylline-bound and, if less frequently, in the ligand-free state. The resultant residual activity (leakage) in the absence of theophylline contributes to a limited dynamic range of the aptazyme. In addition, site-specific AP labeling shows that rapid local theophylline binding to the aptamer domain leads to only slow allosteric signal transduction into the ribozyme core. Our findings allow us to rationalize the suboptimal biosensing performance of the engineered compared to the natural aptazyme and to suggest improvement strategies. Our single molecule FRET approach also monitors in real time the previously elusive equilibrium docking dynamics of the hammerhead ribozyme between several inactive conformations and the active, long-lived, Y-shaped conformer.