The Joint Modeling of Drug and Positive Control Effects to Estimate Sample Size and Power in Crossover “Thorough” QT/QTc Studies

We present analytical results for computing the power and sample size in a thorough QT/QTc study with a four-period crossover design in which the treatments are placebo, positive control, supratherapeutic dose of investigational drug, and therapeutic dose of investigational drug. An assessment of noninferiority of the supratherapeutic dose to placebo is performed by the intersection–union test and assay sensitivity is tested (union–intersection test) at prespecified time points using positive control within the framework of a linear mixed-effects analysis. The power and sample size estimates are obtained using the joint distribution of statistics to test noninferiority of the supratherapeutic dose to placebo and to test assay sensitivity using positive control.     

Statistical procedures to test for linearity and estimate threshold doses for tumor induction with nonlinear dose-response relationships in bioassays for carcinogenicity

Sublinear shapes of the dose-response curve in the low-dose range of toxicity testing are often postulated to be indicative of a no-effect threshold. We present statistical procedures to test sublinear dose responses in bioassays for carcinogenicity against the hypothesis of linearity and estimate a lower confidence limit for the dose at the postulated breakpoint. First, a control tumor incidence of 0 is assumed. Tumor incidence at dose 1 is allowed to range from 0 to 4 tumor-bearing animals (TBAs) in groups of 50 animals, dose 2 is assumed to result in a tumor incidence of 5-25 TBAs. The null hypothesis of a linear dose response is tested by (i) the likelihood ratio (LR) test and (ii) the minimum chi(2) (MC) method. Validation by simulation showed the MC method to be more conservative than the LR test. At the 5% level with MC, the following observed numbers of TBAs for the dose sequence 0-1-2 resulted in rejection of the hypothesis of linearity: 0-0-6, 0-1-10, 0-2-13, 0-3-16, 0-4-18. Second, the analysis was adapted to allow for a control tumor incidence of 0-4 TBAs/50 and a tumor incidence of 0-10 TBAs/50 at dose 1, and the minimum number of TBAs at dose 2 to reject linearity at the 5% level was calculated. Third, a program is made available to analyze data derived from protocols that include nonstandard dose span and group size. Internet access to the respective statistics software and source file is provided. Examples for nasal tumor induction by formaldehyde and for the induction of renal adenocarcinoma by ochratoxin A are shown. The proposed analysis may be useful to test sublinear sections of the dose response for the possibility of a threshold for carcinogens and to define dose levels that could be used as a starting point for setting exposure standards.     

Wilcoxon–Mann–Whitney Test: Stratify or Not?

The Wilcoxon–Mann–Whitney (WMW) test is the most commonly used nonparametric method to compare two treatments when the underlying distribution of the outcome variable is not normally distributed. In the presence of stratum effects, the van Elteren (vE) test, a stratified WMW test, can be used to adjust for the stratum effect. We provide guidance on how to choose between the two tests in the design phase of clinical trials and in the analysis of clinical data. We show by simulations that both tests preserve the type I error rate regardless of the presence of the stratum effects. Therefore, the test with greater power is preferred. In comparing powers, we found that the WMW test is better when the stratum effects are small, whereas the vE test is better when the stratum effects are large. Finally, when the stratum effects are moderate, the decision depends on the shape of the distribution and the ratio of the number of strata and the number of subjects. In this case, results presented in this article or from similar simulations may be used to determine which test is better.     

Poly-k-Trend Tests for Survival Adjusted Analysis of Tumor Rates Formulated as Approximate Multiple Contrast Test

In long-term rodent carcinogenicity studies without cause of death information, poly-k-adjusted tumor rates are commonly used to assess the carcinogenic potential of a compound. Testing trend on proportions, global tests such as the Cochran–Armitage test and Williams-type test have been proposed. Here are introduced simultaneous confidence intervals and adjusted p-values for multiple contrasts on poly-k-adjusted tumor rates, based on approximation with the multivariate normal distribution. Williams-type contrasts and Dunnett-type comparisons to control are special cases of this approach. In simulation studies, the acceptable performance for finite sample sizes is demonstrated. The methods are applied to a real data example and are implemented in a package for R.     

Bayesian penalized log-likelihood ratio approach for dose response clinical trial studies

In literature, there are a few unified approaches to test proof of concept and estimate a target dose, including the multiple comparison procedure using modeling approach, and the permutation approach proposed by Klingenberg. We discuss and compare the operating characteristics of these unified approaches and further develop an alternative approach in a Bayesian framework based on the posterior distribution of a penalized log-likelihood ratio test statistic. Our Bayesian approach is much more flexible to handle linear or nonlinear dose–response relationships and is more efficient than the permutation approach. The operating characteristics of our Bayesian approach are comparable to and sometimes better than both approaches in a wide range of dose–response relationships. It yields credible intervals as well as predictive distribution for the response rate at a specific dose level for the target dose estimation. Our Bayesian approach can be easily extended to continuous, categorical, and time-to-event responses. We illustrate the performance of our proposed method with extensive simulations and Phase II clinical trial data examples.     

Test Statistics and Confidence Intervals to Establish Noninferiority between Treatments with Ordinal Categorical Data

The problem for establishing noninferiority is discussed between a new treatment and a standard (control) treatment with ordinal categorical data. A measure of treatment effect is used and a method of specifying noninferiority margin for the measure is provided. Two Z-type test statistics are proposed where the estimation of variance is constructed under the shifted null hypothesis using U-statistics. Furthermore, the confidence interval and the sample size formula are given based on the proposed test statistics. The proposed procedure is applied to a dataset from a clinical trial. A simulation study is conducted to compare the performance of the proposed test statistics with that of the existing ones, and the results show that the proposed test statistics are better in terms of the deviation from nominal level and the power.     

A Step-Up Test Procedure to Find the Minimum Effective Dose

It is of great interest to find the minimum effective dose (MED) in dose-response studies. A sequence of decreasing null hypotheses to find the MED is formulated under the assumption of nondecreasing dose response means. A step-up multiple test procedure that controls the familywise error rate (FWER) is constructed based on the maximum likelihood estimators for the monotone normal means. When the MED is equal to one, the proposed test is uniformly more powerful than Hsu and Berger’s test (1999). Also, a simulation study shows a substantial power improvement for the proposed test over four competitors. Three R-codes are provided in Supplemental Materials for this article. Go to the publishers online edition of Journal of Biopharmaceutical Statistics to view the files.     

A Unified Framework for Benchmark Dose Estimation Applied to Mixed Models and Model Averaging

This article develops a framework for benchmark dose estimation that allows intrinsically nonlinear dose–response models to be used for continuous data in much the same way as is already possible for quantal data. This means that the same dose–response model equations may be applied to both continuous and quantal data, facilitating benchmark dose estimation in general for a wide range of candidate models commonly used in toxicology. Moreover, the proposed framework provides a convenient means for extending benchmark dose concepts through the use of model averaging and random effects modeling for hierarchical data structures, reflecting increasingly common types of assay data. We illustrate the usefulness of the methodology by means of a cytotoxicology example where the sensitivity of two types of assays are evaluated and compared. By means of a simulation study, we show that the proposed framework provides slightly conservative, yet useful, estimates of benchmark dose lower limit under realistic scenarios.     

Exact Statistical Tests on Comparing Tumor Incidence Trend in Transgenic Mouse Carcinogenicity Studies

Statistical positive trend tests to predict drugs' tumorigenic potential for long-term carcinogenicity studies, for example, Peto's test, are well accepted in practice and by regulatory agencies. Recently, short-term transgenic mouse studies become widely used as an alternative to lifetime mouse carcinogenicity studies. In general, the tumor incidence rates in a short-term study are much lower than those for a traditional long-term study, so exact positive trend test should be used. However, little is discussed about exact statistical trend tests for transgenic mouse studies. In this article, we evaluated three types of exact methods for testing tumor trend via simulation: exact Cochran-Armitage (C-A) test, exact Peto's test, and a proposed conditional exact Poly-k test. We show, via a simulations study, that the Type I error rate and statistical power are very similar for these three exact tests under various mortality rates and tumorigenic patterns. Overall, we recommend using exact C-A test to compare tumor incidence trend for short-term transgenic mouse studies. When survival adjustment is needed, the proposed conditional exact Poly-k test can be used as an alternative.     

Acute testicular toxicity induced by melamine alone or a mixture of melamine and cyanuric acid in mice

Eight-week-old male Kunming mice were administered either melamine (MA, 30, 140, or 700 mg/kg/day), a melamine and cyanuric acid mixture (MC, each at 15, 70, or 350 mg/kg/day), or vehicle (control) for 3 consecutive days. Testicular toxicity was evaluated on days 1 and 5 after the final exposure. The testicular and epididymal weights and serum testosterone level were significantly decreased in the highest MC group (350 mg/kg/day). Histopathologically, both MA and MC caused obvious lesions in the testis and epididymis, with significant increases in sperm abnormalities. By TEM, the blood–testis barrier was damaged dose dependently. TUNEL staining showed that both MA and MC induced increases in germ cell apoptosis. The Sertoli cell vimentin was collapsed in the treated animals as detected by immunohistochemical staining and Western blotting. This study demonstrated that both MA and MC treatments could disrupt the blood–testis barrier and cause a clear testicular toxicity.     

Comparative nephrotoxicitiy induced by melamine, cyanuric acid, or a mixture of both chemicals in either Sprague-Dawley rats or renal cell lines

Acute nephrotoxicities of melamine (MEL), cyanuric acid (CA), and a mixture of both melamine and cyanuric acid (MC) were comparatively investigated in male Sprague-Dawley rats at 5 doses each with 10-fold dose interval as follows: MEL at 0.0315, 0.315, 3.15, 31.5, and 315 mg/kg; CA at 0.025, 0.25, 2.5, 25, and 250 mg/kg, and MC: [1×: (0.0315 + 0.025), 10×: (0.315 + 0.25), 100×: (3.15 + 2.5), 1000×: (31.5 + 25), and (315 + 250) mg/kg]. No marked adverse effects in renal function were observed in animals treated with MEL alone or CA alone, but evidence related to nephrotoxicity was noted in rats administered MC. Renal calculi and increased kidney weights were found in rats 7 d after daily oral administration of MC. Blood urea nitrogen (BUN) and creatinine were significantly elevated in the high dose MC groups at 100× or 1000×. In addition, elevated numbers of white blood cells (WBC), neutrophils, and lymphocytes in vivo and increased levels of prostaglandin E(2) (PGE(2)) in vitro were found in the MC group. Based on these data, the NOAEL (no-observed-adverse-effect level) for nephrotoxicity for MC was estimated to be 3.15 mg/kg body weight (bw)/d (MEL) plus 2.5 mg/kg bw/d (CA). If a safety factor of 1000 or more were applied to NOAEL, tolerable daily intake (TDI) would be 0.00315 and 0.0025 mg/kg/d or less for MEL and CA, respectively, which is far below the TDI of 0.2 mg/kg/d set by World Health Organization (WHO). In addition, in vitro cytotoxicity assays showed that the ACHN human renal adenocarcinoma cell line was more sensitive to MEL, CA, and MC than the MDCK canine kidney epithelial cell line. The 24-h half maximal inhibitory concentration (IC(50)) values for MEL (4792, 2792 μg/ml) were less than those of CA (9890, 6725 μg/ml, respectively) in MDCK and ACHN cell lines, suggesting that MEL may be more cytotoxic than CA. Furthermore, the 24-h IC(50) value for MC was found to be 208 μg/ml in ACHN cells. Data suggest that NOAELs based upon acute nephrotoxic parameters for MC were low, which might require further reassessment of the current TDI.     

Generalized F Test and Generalized Deviance Test in Two-Way ANOVA Models for Randomized Trials

We consider the problem of detecting treatment effects in a randomized trial in the presence of an additional covariate. By reexpressing a two-way analysis of variance (ANOVA) model in a logistic regression framework, we derive generalized F tests and generalized deviance tests, which provide better power in detecting common location-scale changes of treatment outcomes than the classical F test. The null distributions of the test statistics are independent of the nuisance parameters in the models, so the critical values can be easily determined by Monte Carlo methods. We use simulation studies to demonstrate how the proposed tests perform compared with the classical F test. We also use data from a clinical study to illustrate possible savings in sample sizes.     

Power and Sample Size Calculations for Models from Unknown Distributions

Wald tests and F tests are commonly used for analysis, particularly when the regression model is a generalized linear model. When these tests are proposed for analysis it is important to also estimate the power and sample size during the design phase using this same test. Often, though, the information prior to a study is insufficient to assess whether response variable distributions assumed for power or sample size calculations are appropriate. This article demonstrates that such complete assumptions about the response distribution are not necessary to estimate power and sample size for moderate to large studies using quasi-likelihood methods. This method replaces the need to specify the response variable distribution with the weaker specification of only the mean-to-variance relationship. Complex designs, such as designs with interaction terms, are accommodated by this approach. Results are presented for data from one- and two-parameter exponential family distributions, which are among the most common distributions assumed in the medical, epidemiologic, and social sciences literature. Examples from mixture distributions are also presented. Monte Carlo simulation was used to estimate power for comparison. Quasi-likelihood power estimates were within 0.03 of estimates generated via simulation for most examples presented.     

Flexible parametrization of variance functions for quantal response data derived from counts

Although the Poisson model has been widely used to fit count data, a well-known drawback is that the Poisson mean equals its variance. Many alternative models for counts that are overdispersed relative to Poisson have been developed to solve this issue, including the negative binomial model. In this article, the negative binomial model with a four-parameter logistic mean is proposed to handle these types of counts, with variance that flexibly depends on the mean. Various parameterizations for the variance are considered, including extra-Poisson variability modeled as an exponentiated B-spline. Thus, the proposed model ably captures the leveling off of the mean, i.e., the “lazy-S” shape often encountered for overdispersed dose–response counts, simultaneously taking into account both overdispersion and natural mortality. Two real datasets illustrate the merits of the proposed approach: media colony counts after tuberculosis decontamination, and the number of monkeys killed by Ache hunters over several hunting trips in the Paraguayan tropical forest.     

Mode of action and dose–response framework analysis for receptor-mediated toxicity: The aryl hydrocarbon receptor as a case study

Dioxins and dioxin-like compounds are tumor promoters that cause liver cancer in rats and mice. The aryl hydrocarbon receptor (AHR) has been implicated as a key component in this tumor promotion response. Despite extensive knowledge of the toxicology of dioxins, no mode of action (MOA) hypothesis for their tumorigenicity has been formally documented using the Human Relevance MOA framework developed by the International Programme on Chemical Safety (IPCS). To address this information gap, an expert panel was convened as part of a workshop on receptor-mediated liver tumorigenicity. Liver tumors induced by ligands of the AHR were assessed using data for dioxins and related chemicals as a case study. The panel proposed a MOA beginning with sustained AHR activation, eventually leading to liver tumors via a number of other processes, including increased cell proliferation of previously initiated altered hepatic foci, inhibition of intrafocal apoptosis and proliferation of oval cells. These processes have been identified and grouped as three key events within the hepatocarcinogenic MOA: (1) sustained AHR activation, (2) alterations in cellular growth and homeostasis and (3) pre-neoplastic tissue changes. These key events were identified through application of the Bradford-Hill considerations in terms of both their necessity for the apical event/adverse outcome and their human relevance. The panel identified data supporting the identification and dose–response behavior of key events, alteration of the dose–response by numerous modulating factors and data gaps that potentially impact the MOA. The current effort of applying the systematic frameworks for identifying key events and assessing human relevance to the AHR activation in the tumorigenicity of dioxins and related chemicals is novel at this time. The results should help direct future regulatory efforts and research activities aimed at better understanding the potential human cancer risks associated with dioxin exposure.     

Sample Size Analysis for Pharmacogenetic Studies

Pharmacogenetic studies identify the genetic factors that influence the intersubject variation in drug response. This article proposes a general framework to determine sample size in pharmacogenetic studies. Simple closed form solutions for the sample size are derived for continuous and binary outcomes. To extend the application to pharmacogenomic studies, where a large number of gene-treatment interactions are evaluated simultaneously, we advocate the use of false discovery rate (FDR) in controlling false positive proportion. We adapt the method proposed by Shaoand Tseng (2007) to facilitate adjustment for correlation among multiple tests for better control of false positives and power. A real example is given and simulation studies are carried out to demonstrate the performance of the proposed method.     

两生存率检验所需样本容量的模拟

本文以模拟试验展示以前提出的两生存率检验所需样本容量测定方法的行为。取α=0.05(单侧),β=0.1,按不同数值的最小临床承认疗效差量或两生存率和终检率交叉组合得出32种方案结合Weibull生存分布形状参数1/3—3形成160个试验集各重复1000次。在几乎全部指数生存分布试验集,该检验的观测功效符合预定功效。在如此大范围形状参数Weibull分布下,变异甚微。这提示,这种方法对于指数生存分布十分精确,对于不同形态的实际分布,仍然适用。     

Unified Sample Size Calculations Using the Competing Probability

Sample size formulas are functions of two categories of quantities. The first category does not depend on the analysis variables and contains randomization ratio, Type I error rate, and power. The second category depends on the analysis variables and includes means and standard deviations for the two-sample t-test, competing probabilities for the Wilcoxon–Mann–Whitney test, and median survival times for the logrank test. Quantities in the second category depend on subject matter knowledge and are much harder to specify than those in the first category. In addition, these quantities have different interpretations. In this article we propose using the competing probability as the only second category quantity when calculating sample sizes for the aforementioned three commonly used tests in clinical trials. Doing so unifies the interpretation of and sheds new light on the two-sample t-tests and the logrank tests in terms of sample size calculations.     

Discussion of “Executive Summary of the PhRMA Working Group on Adaptive Designs in Clinical Drug Development”

Four multidose animal carcinogenicity assay trend test procedures based on the chi-square, Hoel-Walburg, log-rank, and Peto procedures are compared under conditions of competing risks. A total of 42 different risk populations are simulated in which a simulated animal can contract one or more of five different tumor types. The risk populations contain such different risks as incidence of tumor increasing with dose, time of death increasing or decreasing with dose, tumor decreasing time of death without necessarily causing death (nonrepresentativeness), and combinations of the above. Each population is analyzed using each of the four procedures for linear dose-response effect for each tumor separately and for all animals with tumors.

Results show the Peto procedure to be the most robust of the four, giving rejection rates near the nominal for all competing risk situations and having the best power to reject when rejection is called for.     

Alternative Analysis Methods for Time to Event Endpoints Under Nonproportional Hazards: A Comparative Analysis

Abstract

The log-rank test is most powerful under proportional hazards (PH). In practice, non-PH patterns are often observed in clinical trials, such as in immuno-oncology; therefore, alternative methods are needed to restore the efficiency of statistical testing. Three categories of testing methods were evaluated, including weighted log-rank tests, Kaplan–Meier curve-based tests (including weighted Kaplan–Meier and restricted mean survival time), and combination tests (including Breslow test, Lee’s combo test, and MaxCombo test). Nine scenarios representing the PH and various non-PH patterns were simulated. The power, Type I error, and effect estimate of each method were compared. In general, all tests control Type I error well. There is not a single most powerful test across all scenarios. In the absence of prior knowledge regarding the underlying or non-PH patterns, the MaxCombo test is relatively robust across patterns. Since the treatment effect changes over time under non-PH, the overall profile of the treatment effect may not be represented comprehensively based on a single measure. Thus, multiple measures of the treatment effect should be prespecified as sensitivity analyses to describe the totality of the data. Supplementary materials for this article are available online.