Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double‐blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae. Copyright © 2013 John Wiley & Sons, Ltd.     

协整参数的自举推断

 针对完全修正最小二乘(full-modified ordinary least square,简称FMOLS）估计方法,给出一种协整参数的自举推断程序,证明零假设下自举统计量与检验统计量具有相同的渐近分布。关于检验功效的研究表明,虽然有约束自举的实际检验水平表现良好,但如果零假设不成立,自举统计量的分布是不确定的,因而其经验分布不能作为检验统计量精确分布的有效估计。实际应用中建议使用无约束自举,因为无论观测数据是否满足零假设,其自举统计量与零假设下检验统计量都具有相同的渐近分布。最后,利用蒙特卡洛模拟对自举推断和渐近推断的有限样本表现进行比较研究。     

Exact sample-size determination in testing non-inferiority under a simple crossover trial

For testing the non-inferiority (or equivalence) of an experimental treatment to a standard treatment, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of an exact test procedure proposed elsewhere for a simple crossover design, we develop an exact sample-size calculation procedure with respect to the OR of patient response rates for a desired power of detecting non-inferiority at a given nominal type I error. We note that the sample size calculated for a desired power based on an asymptotic test procedure can be much smaller than that based on the exact test procedure under a given situation. We further discuss the advantage and disadvantage of sample-size calculation using the exact test and the asymptotic test procedures. We employ an example by studying two inhalation devices for asthmatics to illustrate the use of sample-size calculation procedure developed here.     

An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

On the accuracy of approximate studentization

With a parametric model, a measure of departure for an interest parameter is often easily constructed but frequently depends in distribution on nuisance parameters; the elimination of such nuisance parameter effects is a central problem of statistical inference. Fraser & Wong (1993) proposed a nuisance-averaging or approximate Studentization method for eliminating the nuisance parameter effects. They showed that, for many standard problems where an exact answer is available, the averaging method reproduces the exact answer. Also they showed that, if the exact answer is unavailable, as say in the gamma-mean problem, the averaging method provides a simple approximation which is very close to that obtained from third order asymptotic theory. The general asymptotic accuracy, however, of the method has not been examined. In this paper, we show in a general asymptotic context that the averaging method is asymptotically a second order procedure for eliminating the effects of nuisance parameters.     

Quantile based tests for exponentiality against DMRQ and NBUE alternatives

Quantile-based reliability analysis has received much attention recently. We propose new quantile-based tests for exponentiality against decreasing mean residual quantile function (DMRQ) and new better than used in expectation (NBUE) classes of alternatives. The exact null distribution of the test statistic is derived when the alternative class is DMRQ. The asymptotic properties of both the test statistics are studied. The performance of the proposed tests with other existing tests in the literature is evaluated through simulation study. Finally, we illustrate our test procedure using real data sets.     

An efficient numerical algorithm for exact inference in meta analysis

The performance of commonly used asymptotic inference procedures for the random-effects model used in meta analysis relies on the number of studies. When the number of studies is moderate or small, the exact inference procedure is more reliable than the asymptotic counterparts. However, the related numerical computation may be demanding and an obstacle of routine use of the exact method. In this paper, we proposed a novel numerical algorithm for constructing the exact 95% confidence interval of the location parameter in the random-effects model. The algorithm is much faster than the naive method and may greatly facilitate the use of the more appropriate exact inference procedure in meta analysis. Numerical studies and real data examples are used to illustrate the advantage of the proposed method.     

The Multi-Sample Block-Scalar Sphericity Test: Exact and Near-Exact Distributions for Its Likelihood Ratio Test Statistic

In this article the authors show how by adequately decomposing the null hypothesis of the multi-sample block-scalar sphericity test it is possible to obtain the likelihood ratio test statistic as well as a different look over its exact distribution. This enables the construction of well-performing near-exact approximations for the distribution of the test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior for increasing sample sizes as well as for increasing number of variables and/or populations involved.     

A test for zero point triserial correlation coefficient and its relationship with tests for homogeneity of means with ordered alternatives

The point triserial correlation coefficient is defined and, under appropriate order restrictions, an exact test that this correlation coefficient equals zero is developed. The power function of that test is derived and partially tabulated. The general problem of testing for homogeneity of means under ordered alternatives is discussed. The available procedures for performing such tests are considered, are seen to provide alternative approaches to the test developed herein, and are compared with that test. An exact test for the equality of dependent point triserial correlation coefficients is described through application of a procedure suggested by Wolfe ‘1976’     

Likelihood ratio test for simultaneous testing of the mean and the variance of a normal distribution

In this paper, we consider the simultaneous testing of the mean and the variance of a normal distribution. The exact distribution of the likelihood ratio test statistic is obtained, which is not available in the literature. The critical points of the exact test are reported. We also consider some of the other exact and asymptotic tests. The powers of these tests are compared using the Monte Carlo simulations.     

Gini index based goodness-of-fit test for the logistic distribution

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.     

Designing historical control studies with survival endpoints using exact statistical inference

Historical control trials compare an experimental treatment with a previously conducted control treatment. By assigning all recruited samples to the experimental arm, historical control trials can better identify promising treatments in early phase trials compared with randomized control trials. Existing designs of historical control trials with survival endpoints are based on asymptotic normal distribution. However, it remains unclear whether the asymptotic distribution of the test statistic is close enough to the true distribution given relatively small sample sizes in early phase trials. In this article, we address this question by introducing an exact design approach for exponentially distributed survival endpoints, and compare it with an asymptotic design in both real examples and simulation examples. Simulation results show that the asymptotic test could lead to bias in the sample size estimation. We conclude the proposed exact design should be used in the design of historical control trials.     

On the Gini Mean Difference Test for Circular Data

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributions under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency.     

A Cauchy estimator test for autocorrelation

This article presents a new test for serial correlation in an observed stationary time series. Rather than using the traditional portmanteau tests based on the sample autocorrelation function, we propose a test based on the Cauchy estimator of correlation. A goodness-of-fit statistic for fitted autoregressive moving average models is also derived and the asymptotic distribution of this statistic is quantified. The test can be employed using either this asymptotic distribution or by using Monte-Carlo quantiles. The small sample behaviour is studied via simulation and the Monte-Carlo-based test seems to be more precise. The method is demonstrated on monthly asset returns for Facebook, Incorporated.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

Randomized allocation procedure for testing a normal mean with known variance

This paper provides an alternative test procedure for the problem of testing normal mean against two-sided alternative with known variance for costly trials. The goal is to carry out the test procedure with a smaller sample size if the alternative is true. The sample size is determined in an adaptive fashion which takes all the previous observations into account for adaptation. Some exact and asymptotic results related to the test and design are studied.     

Separation and Lack of Separation of Subpopulation in the Mixed Distributions

This is a comparative study between the estimates of parameters of mixed distributions in the case of the possibility of separating the units of subpopulation or the absence of that possibility under the progressive type I censored test data. An iterative procedure is developed and tested numerically to obtain new estimators and their variance–covariance matrix. Finally, we will use the exact distribution of the maximum likelihood estimators as well as its asymptotic distribution and the parametric bootstrap method; then, we will discuss the construction of confidence intervals for the mean parameter and their performance is assessed through Monte Carlo simulations.     

Sequential Monitoring Variance Changes in Location Model

In this paper, a moving monitoring procedure is proposed to detect potential variance change of the location model with dependent errors. The procedure is motivated by the problem that the existing square CUSUM test is insensitive to a late variance change of the location model. The asymptotic distribution of the statistics under the null hypothesis and the consistency under the alternative hypothesis are derived. Simulations show that our monitoring procedure compared to the square CUSUM test offers better power and can more quickly detect change. Moreover, the effectiveness of our procedure is illustrated by applying it to two data sets.     

Asymmetry models for square contingency tables: exact tests via algebraic statistics

Square contingency tables with the same row and column classification occur frequently in a wide range of statistical applications, e.g. whenever the members of a matched pair are classified on the same scale, which is usually ordinal. Such tables are analysed by choosing an appropriate loglinear model. We focus on the models of symmetry, triangular, diagonal and ordinal quasi symmetry. The fit of a specific model is tested by the chi-squared test or the likelihood-ratio test, where p-values are calculated from the asymptotic chi-square distribution of the test statistic or, if this seems unjustified, from the exact conditional distribution. Since the calculation of exact p-values is often not feasible, we propose alternatives based on algebraic statistics combined with MCMC methods.