Testing Linear Regression Models in Non Regular Case

We propose a new statistic for testing linear hypotheses in the non parametric regression model in the case of a homoscedastic error structure and fixed design. In contrast to most models suggested in the literature, our procedure is applicable in the non parametric model case without regularity condition, and also under either the null or the alternative hypotheses. We show the asymptotic normality of the test statistic under the null hypothesis and the alternative one. A simulation study is conducted to investigate the finite sample properties of the test with application to regime switching.     

Confidence interval estimation and hypothesis testing for a common correlation coefficient

Several procedures have been proposed for testing the hypothesis that all off-diagonal elements of the correlation matrix of a multivariate normal distribution are equal. If the hypothesis of equal correlation can be accepted, it is then of interest to estimate and perhaps test hypotheses for the common correlation. In this paper, two versions of five different test statistics are compared via simulation in terms of adequacy of the normal approximation, coverage probabilities of confidence intervals, control of Type I error, and power. The results indicate that two test statistics based on the average of the Fisher z-transforms of the sample correlations should be used in most cases. A statistic based on the sample eigenvalues also gives reasonable results for confidence intervals and lower-tailed tests.     

Multiplicity and flexibility in clinical trials

Flexible designs offer a large amount of flexibility in clinical trials with control of the type I error rate. This allows the combination of trials from different clinical phases of a drug development process. Such combinations require designs where hypotheses are selected and/or added at interim analysis without knowing the selection rule in advance so that both flexibility and multiplicity issues arise. The paper reviews the basic principles and some of the common methods for reaching flexibility while controlling the family-wise error rate in the strong sense. Flexible designs have been criticized because they may lead to different weights for the patients from the different stages when reassessing sample sizes. Analyzing the data in a conventional way avoids such unequal weighting but may inflate the multiple type I error rate. In cases where the conditional type I error rates of the new design (and conventional analysis) are below the conditional type I error rates of the initial design the conventional analysis may, however, be done without inflating the type I error rate. Focusing on a parallel group design with two treatments and a common control, we use this principle to investigate when we can select one treatment, reassess sample sizes and test the corresponding null hypotheses by the conventional level alpha z-test without compromising on the multiple type I error rate.     

Bayesian two-stage design for drug screening trials with switching hypothesis tests based on continuous endpoints

Abstract

In this paper, we propose a Bayesian two-stage design with changing hypothesis test by bridging a single-arm study and a double-arm randomized trial in one phase II clinical trial based on continuous endpoints rather than binary endpoints. We have also calibrated with respect to frequentist and Bayesian error rates. The proposed design minimizes the Bayesian expected sample size if the new candidate has low or high efficacy activity subject to the constraint upon error rates in both frequentist and Bayesian perspectives. Tables of designs for various combinations of design parameters are also provided.     

Hypothesis testing for Markov chain Monte Carlo

Testing between hypotheses, when independent sampling is possible, is a well developed subject. In this paper, we propose hypothesis tests that are applicable when the samples are obtained using Markov chain Monte Carlo. These tests are useful when one is interested in deciding whether the expected value of a certain quantity is above or below a given threshold. We show non-asymptotic error bounds and bounds on the expected number of samples for three types of tests, a fixed sample size test, a sequential test with indifference region, and a sequential test without indifference region. Our tests can lead to significant savings in sample size. We illustrate our results on an example of Bayesian parameter inference involving an ODE model of a biochemical pathway.     

BOUNDS FOR THE AVERAGE SAMPLE NUMBER FUNCTION FOR A BINOMIAL SEQUENTIAL TEST

Recently, Billard (1977) developed a partial sequential procedure for comparing a null hypothesis against a two-sided alternative hypothesis when the parameter under test is that of the binomial distribution. In that paper, approximations to the operating characteristic and average sample number function were derived. In this note, bounds to the average sample number function are derived. Using numerical results a comparison of the approximations, bounds and empirical values is made.     

A rank test for stochastically ordered alternatives

A rank test based on the number of ‘near-matches’ among within-block rankings is proposed for stochastically ordered alternatives in a randomized block design with t treatments and b blocks. The asymptotic relative efficiency of this test with respect to the Page test is computed as number of blocks increases to infinity. A sequential analog of the above test procedure is also considered. A repeated significance test procedure is developed and average sample number is computed asymptotically under the null hypothesis as well as under a sequence of contiguous alternatives.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Multivariate nonparametric methods in two-way balanced designs: performances and limitations in small samples

Investigations of multivariate population are pretty common in applied researches, and the two-way crossed factorial design is a common design used at the exploratory phase in industrial applications. When assumptions such as multivariate normality and covariance homogeneity are violated, the conventional wisdom is to resort to nonparametric tests for hypotheses testing. In this paper we compare the performances, and in particular the power, of some nonparametric and semi-parametric methods that have been developed in recent years. Specifically, we examined resampling methods and robust versions of classical multivariate analysis of variance (MANOVA) tests. In a simulation study, we generate data sets with different configurations of factor''s effect, number of replicates, number of response variables under null hypothesis, and number of response variables under alternative hypothesis. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the tradeoff between power and type I error, the strategic impact of increasing number of response variables, and the favourable performance of one test when the alternative is sparse. A real case study from an industrial engineering experiment in thermoformed packaging production is used to compare and illustrate the application of the various methods.     

Evaluations of FWER-controlling methods in multiple hypothesis testing

Simultaneously testing a family of n null hypotheses can arise in many applications. A common problem in multiple hypothesis testing is to control Type-I error. The probability of at least one false rejection referred to as the familywise error rate (FWER) is one of the earliest error rate measures. Many FWER-controlling procedures have been proposed. The ability to control the FWER and achieve higher power is often used to evaluate the performance of a controlling procedure. However, when testing multiple hypotheses, FWER and power are not sufficient for evaluating controlling procedure’s performance. Furthermore, the performance of a controlling procedure is also governed by experimental parameters such as the number of hypotheses, sample size, the number of true null hypotheses and data structure. This paper evaluates, under various experimental settings, the performance of some FWER-controlling procedures in terms of five indices, the FWER, the false discovery rate, the false non-discovery rate, the sensitivity and the specificity. The results can provide guidance on how to select an appropriate FWER-controlling procedure to meet a study’s objective.     

A parametric bootstrap approach for two-way error component regression models

In this article, the two-way error component regression model is considered. For the nonhomogenous linear hypothesis testing of regression coefficients, a parametric bootstrap (PB) approach is proposed. Simulation results indicate that the PB test, regardless of the sample sizes, maintains the Type I error rates very well and outperforms the existing generalized variable test, which may far exceed the intended significance level when the sample sizes are small or moderate. Real data examples illustrate the proposed approach work quite satisfactorily.     

Bayesian (mean) most powerful tests

A fundamental theorem in hypothesis testing is the Neyman‐Pearson (N‐P) lemma, which creates the most powerful test of simple hypotheses. In this article, we establish Bayesian framework of hypothesis testing, and extend the Neyman‐Pearson lemma to create the Bayesian most powerful test of general hypotheses, thus providing optimality theory to determine thresholds of Bayes factors. Unlike conventional Bayes tests, the proposed Bayesian test is able to control the type I error.     

C334. On the neutron and proton masses: a numerological case study

The intercomponent rank test suggested by Thompson (1991a) for the bivariate two sample problem is compared with the intracomponent rank test discussed by Puri and Sen (1971) and Hettmansperger (1984) and with the Hotelling T ² test. Asymptotic relative efficiencies are discussed and the results of a simulation study are presented. Power studies show that for small sample sizes and small Type 1 error rates, say n = 5 and α = .01, the intercomponent rank test of Thompson (1991a) is somewhat liberal and the intracomponent test is quite conservative. For larger sample sizes and larger Type 1 error rates, both rank tests have improved properties under the null hypothesis. In almost all simulated cases, the intercomponent test is more powerful. In light of these studies it is suggested that the intercomponent rank test of Thompson (1991a), which has the added advantage of being easily computed with standard statistical software, is a strong competitor to the intracomponent rank test.     

A relative error-based estimation with an increasing number of parameters

The least product relative error (LPRE) estimator and test statistic to test linear hypotheses of regression parameters in the multiplicative regression model are studied when the number of covariate variables increases with the sample size. Some properties of the LPRE estimator and test statistic are obtained such as consistency, Bahadur presentation, and asymptotic distributions. Furthermore, we extend the LPRE to a more general relative error criterion and provide their statistical properties. Numerical studies including simulations and two real examples show that the proposed estimation performs well.     

Adaptive designs for IVPT data with mixed scaled average bioequivalence

In vitro permeation tests (IVPT) offer accurate and cost-effective development pathways for locally acting drugs, such as topical dermatological products. For assessment of bioequivalence, the FDA draft guidance on generic acyclovir 5% cream introduces a new experimental design, namely the single-dose, multiple-replicate per treatment group design, as IVPT pivotal study design. We examine the statistical properties of its hypothesis testing method—namely the mixed scaled average bioequivalence (MSABE). Meanwhile, some adaptive design features in clinical trials can help researchers make a decision earlier with fewer subjects or boost power, saving resources, while controlling the impact on family-wise error rate. Therefore, we incorporate MSABE in an adaptive design combining the group sequential design and sample size re-estimation. Simulation studies are conducted to study the passing rates of the proposed methods—both within and outside the average bioequivalence limits. We further consider modifications to the adaptive designs applied for IVPT BE trials, such as Bonferroni's adjustment and conditional power function. Finally, a case study with real data demonstrates the advantages of such adaptive methods.     

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

A fixed sequence Bonferroni procedure for testing multiple endpoints

A method for controlling the familywise error rate combining the Bonferroni adjustment and fixed testing sequence procedures is proposed. This procedure allots Type I error like the Bonferroni adjustment, but allows the Type I error to accumulate whenever a null hypothesis is rejected. In this manner, power for hypotheses tested later in a prespecified order will be increased. The order of the hypothesis tests needs to be prespecified as in a fixed sequence testing procedure, but unlike the fixed sequence testing procedure all hypotheses can always be tested, allowing for an a priori method of concluding a difference in the various endpoints. An application will be in clinical trials in which mortality is a concern, but it is expected that power to distinguish a difference in mortality will be low. If the effect on mortality is larger than anticipated, this method allows a test with a prespecified method of controlling the Type I error rate. Copyright © 2003 John Wiley & Sons, Ltd.     

On the poisson index of dispersion

A commonly used statistic for testing hypotheses concerning count data is the index of dispersion or variance test (c.f., Fisher 1950, Rao and Chakravarti 1956). The statistic is essentially the ratio of the sample variance to the sample mean,The hypothesis being tested is commonly referred to as the homogeneity hypothesis, often without further detailed specification. However, a closer study of the literature indicates that there are really two different hypotheses of interest.     

Testing diagonality of high-dimensional covariance matrix under non-normality

Under non-normality, this article is concerned with testing diagonality of high-dimensional covariance matrix, which is more practical than testing sphericity and identity in high-dimensional setting. The existing testing procedure for diagonality is not robust against either the data dimension or the data distribution, producing tests with distorted type I error rates much larger than nominal levels. This is mainly due to bias from estimating some functions of high-dimensional covariance matrix under non-normality. Compared to the sphericity and identity hypotheses, the asymptotic property of the diagonality hypothesis would be more involved and we should be more careful to deal with bias. We develop a correction that makes the existing test statistic robust against both the data dimension and the data distribution. We show that the proposed test statistic is asymptotically normal without the normality assumption and without specifying an explicit relationship between the dimension p and the sample size n. Simulations show that it has good size and power for a wide range of settings.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.