A new generalized p-value approach for testing equality of coefficients of variation in k normal populations

A new generalized p-value method is proposed for testing the equality of coefficients of variation in k normal populations. Simulation studies show that the type I error probabilities are close to the nominal level. The proposed test is also compared with likelihood ratio test, modified Bennett's test and score test through Monte Carlo simulation, the results demonstrate that the generalized p-value method has satisfactory performance in terms of sizes and powers.     

Testing the equality of quantiles for several normal populations

Many procedures exist for testing equality of means or medians to compare several independent distributions. However, the mean or median do not determine the entire distribution. In this article, we propose a new small-sample modification of the likelihood ratio test for testing the equality of the quantiles of several normal distributions. The merits of the proposed test are numerically compared with the existing tests—a generalized p-value method and likelihood ratio test—with respect to their sizes and powers. The simulation results demonstrate that proposed method is satisfactory; its actual size is very close to the nominal level. We illustrate these approaches using two real examples.     

Testing equality of generalized variances of k multivariate normal populations

Generalized variance is a measure of dispersion of multivariate data. Comparison of dispersion of multivariate data is one of the favorite issues for multivariate quality control, generalized homogeneity of multidimensional scatter, etc. In this article, the problem of testing equality of generalized variances of k multivariate normal populations by using the Bartlett's modified likelihood ratio test (BMLRT) is proposed. Simulations to compare the Type I error rate and power of the BMLRT and the likelihood ratio test (LRT) methods are performed. These simulations show that the BMLRT method has a better chi-square approximation under the null hypothesis. Finally, a practical example is given.     

Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

Inferences on the Coefficients of Variation in a Multivariate Normal Population

In this article, the problem of testing the equality of coefficients of variation in a multivariate normal population is considered, and an asymptotic approach and a generalized p-value approach based on the concepts of generalized test variable are proposed. Monte Carlo simulation studies show that the proposed generalized p-value test has good empirical sizes, and it is better than the asymptotic approach. In addition, the problem of hypothesis testing and confidence interval for the common coefficient variation of a multivariate normal population are considered, and a generalized p-value and a generalized confidence interval are proposed. Using Monte Carlo simulation, we find that the coverage probabilities and expected lengths of this generalized confidence interval are satisfactory, and the empirical sizes of the generalized p-value are close to nominal level. We illustrate our approaches using a real data.     

Generalizations of Mathisen's median test for comparing several treatments with a control

A distribution-free test for comparing several treatments with a control is proposed for one-way classified data. It is advantageous to use the test in life-testing experiments where testing time is expensive. The proposed test has a shorter expected duration than a previously proposed test by Slivka(1970). The optimal allocation of the experimental units to the treatments for two situations are given. In a simulation study the power of the test is compared with the power of Slivka's test. An extension of the test for two-way classified data is given     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.     

The limit test statistic distribution of the maximum value test for right-censored data

ABSTRACT

In this paper, the maximum value test is proposed and considered for two-sample problem solving with lifetime data. This test is a distribution-free test under non-censoring and is a not distribution-free test under censoring. The formula of the limit distribution of the proposed maximal value test is represented in the general case. The distribution of the test statistic has been studied experimentally. Also, we propose the estimate of a p-value calculation of the maximum value test instead of the Monte-Carlo simulation. This test is useful and applicable in case of choosing among the logrank test, the Cox–Mantel test, the Q test and Generalized Wilcoxon tests, for instance, the Gehan's Generalized Wilcoxon test and the Peto and Peto's Generalized Wilcoxon test.     

A generalized test variable approach for grain yield comparisons of rice

Traditionally, an assessment for grain yield of rice is to split it into the yield components, including the number of panicles per plant, the number of spikelets per panicle, the 1000-grain weight and the filled-spikelet percentage, such that the yield performance can be individually evaluated through each component, and the products of yield components are employed for grain yield comparisons. However, when using the standard statistical methods, such as the two-sample t-test and analysis of variance, the assumptions of normality and variance homogeneity cannot be fully justified for comparing the grain yields, leading to that the empirical sizes cannot be adequately controlled. In this study, based on the concepts of generalized test variables and generalized p-values, a novel statistical testing procedure is developed for grain yield comparisons of rice. The proposed method is assessed by a series of numerical simulations. According to the simulation results, the proposed method performs reasonably well in Type I error control and empirical power. In addition, a real-life field experiment is analyzed by the proposed method, some productive rice varieties are screened out and suggested for a follow-up investigation.     

A test for detecting etiologic heterogeneity in epidemiological studies

Current statistical methods for analyzing epidemiological data with disease subtype information allow us to acquire knowledge not only for risk factor-disease subtype association but also, on a more profound account, heterogeneity in these associations by multiple disease characteristics (so-called etiologic heterogeneity of the disease). Current interest, particularly in cancer epidemiology, lies in obtaining a valid p-value for testing the hypothesis whether a particular cancer is etiologically heterogeneous. We consider the two-stage logistic regression model along with pseudo-conditional likelihood estimation method and design a testing strategy based on Rao's score test. An extensive Monte Carlo simulation study is carried out, false discovery rate and statistical power of the suggested test are investigated. Simulation results indicate that applying the proposed testing strategy, even a small degree of true etiologic heterogeneity can be recovered with a large statistical power from the sampled data. The strategy is then applied on a breast cancer data set to illustrate its use in practice where there are multiple risk factors and multiple disease characteristics of simultaneous concern.     

Locally optimal test of normality against skew-normality

Efficient Score testing procedures for testing for the skewness parameter of the one-parameter and three-parameter skew-normal distributions are investigated. Approximate percentage points for the test statistics are obtained via the Monte Carlo method. Salvan's locally optimal test is extended and simulation is used to show that the power of the extended test is higher than that of the skewness test.     

Statistical analysis of non-inferiority via non-zero risk difference in stratified matched-pair studies

A stratified study is often designed for adjusting several independent trials in modern medical research. We consider the problem of non-inferiority tests and sample size determinations for a nonzero risk difference in stratified matched-pair studies, and develop the likelihood ratio and Wald-type weighted statistics for testing a null hypothesis of non-zero risk difference for each stratum in stratified matched-pair studies on the basis of (1) the sample-based method and (2) the constrained maximum likelihood estimation (CMLE) method. Sample size formulae for the above proposed statistics are derived, and several choices of weights for Wald-type weighted statistics are considered. We evaluate the performance of the proposed tests according to type I error rates and empirical powers via simulation studies. Empirical results show that (1) the likelihood ratio and the Wald-type CMLE test based on harmonic means of the stratum-specific sample size (SSIZE) weight (the Cochran's test) behave satisfactorily in the sense that their significance levels are much closer to the prespecified nominal level; (2) the likelihood ratio test is better than Nam's [2006. Non-inferiority of new procedure to standard procedure in stratified matched-pair design. Biometrical J. 48, 966–977] score test; (3) the sample sizes obtained by using SSIZE weight are smaller than other weighted statistics in general; (4) the Cochran's test statistic is generally much better than other weighted statistics with CMLE method. A real example from a clinical laboratory study is used to illustrate the proposed methodologies.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

Invariant tests based on M-estimators,estimating functions,and the generalized method of moments

We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)?type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)?type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.     

On Confidence Intervals for Process Capability Indices in a One-Way Random Model

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C _pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.     

Testing Periodicity in Short Series and Application to Gene Expression Data

Testing for periodicity in microarray time series encounters the challenges of short series length, missing values and presence of non-Fourier frequencies. In this article, a test method for such series has been proposed. The method is completely simulation based and finds p-values for test of periodicity through fitting Pearson Type VI distribution. The simulation results compare and reveal the excellence of this method over Fisher's g test for varying series length, frequencies, and error variance. This approach is applied to Caulobacter crescentus cell cycle data in order to demonstrate the practical performance.     

A more powerful unconditional exact test of homogeneity for 2 × c contingency table analysis

The classical unconditional exact p-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact p-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate the 5th percentiles of the distributions of the p-values of the exact test over a range of scenarios and implemented a regression model to predict the values for two-sample multinomial settings. Our results show that the new test is uniformly more powerful than Fisher's, Barnard's, and Boschloo's tests with gains in power as large as several hundred percent in certain scenarios. Lastly, we provide a real-life data example where the unadjusted unconditional exact test wrongly fails to reject the null hypothesis and the corrected unconditional exact test rejects the null appropriately.     

New critical values for the generalized t and generalized rank-sum procedures

Student's t test as well as Wilcoxon's rank-sum test may be inefficient in situations where treatments bring about changes in both location and scale. In order to rectify this situation, O'Brien (1988, Journal of the American Statistical Association 83, 52-61) has proposed two new statistics, the generalized t and generalized rank-sum procedures, which may be much more powerful than their traditional counterparts in such situations. Recently, however, Blair and Morel (1991, Statistics in Medicine in press) have shown that referencing these new statistics to standard F tables as recommended by O'Brien results in inflations of Type I errors. This paper provides tables of critical values which do not produce such inflations. Use of these new critical values results in Type I error rates near nominal levels for the generalized t statistic and slightly conservative rates for the generalized rank-sum test. In addition to the critical values, some new power results are given for the generalized tests.     

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Rao's score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test's statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.