Viewpoint: observations on scaled average bioequivalence

The two one-sided test procedure (TOST) has been used for average bioequivalence testing since 1992 and is required when marketing new formulations of an approved drug. TOST is known to require comparatively large numbers of subjects to demonstrate bioequivalence for highly variable drugs, defined as those drugs having intra-subject coefficients of variation greater than 30%. However, TOST has been shown to protect public health when multiple generic formulations enter the marketplace following patent expiration. Recently, scaled average bioequivalence (SABE) has been proposed as an alternative statistical analysis procedure for such products by multiple regulatory agencies. SABE testing requires that a three-period partial replicate cross-over or full replicate cross-over design be used. Following a brief summary of SABE analysis methods applied to existing data, we will consider three statistical ramifications of the proposed additional decision rules and the potential impact of implementation of scaled average bioequivalence in the marketplace using simulation. It is found that a constraint being applied is biased, that bias may also result from the common problem of missing data and that the SABE methods allow for much greater changes in exposure when generic-generic switching occurs in the marketplace.     

A likelihood-based approach for multivariate one-sided tests with missing data

Inequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.     

A unifying approach to non-inferiority, equivalence and superiority tests via multiple decision processes

Two approaches of multiple decision processes are proposed for unifying the non-inferiority, equivalence and superiority tests in a comparative clinical trial for a new drug against an active control. One is a method of confidence set with confidence coefficient 0.95 improving the conventional 0.95 confidence interval in the producer's risk and also the consumer's risk in some cases. It requires to include 0 within the region as well as to clear the non-inferiority margin so that a trial with somewhat large number of subjects and inappropriately large non-inferiority margin for proving non-inferiority of a drug that is actually inferior should be unsuccessful. The other is the closed testing procedure which combines the one- and two-sided tests by applying the partitioning principle and justifies the switching procedure by unifying the non-inferiority, equivalence and superiority tests. In particular regarding the non-inferiority, the proposed method justifies simultaneously the old Japanese Statistical Guideline (one-sided 0.05 test) and the International Guideline ICH E9 (one-sided 0.025 test). The method is particularly attractive, changing the strength of the evidence of relative efficacy of the test drug against a control at five levels according to the achievement of the clinical trial. The meaning of the non-inferiority test and also the rationale of switching from it to superiority test will be discussed.     

A new unit root test against ESTAR based on a class of modified statistics

This paper proposes a new unit root test against a nonlinear exponential smooth transition autoregressive model. This model receives much attention in international macroeconomics as it has been successfully applied to a variety of financial time series. The new test is build upon the nonstandard testing approach of Abadir and Distaso (J Econom 140:695–718, 2007) who introduce a class of modified statistics for testing joint hypotheses when one of the alternatives is one-sided. The asymptotic properties of the suggested unit root test are derived. In a Monte Carlo study the popular Dickey–Fuller-type test proposed by Kapetanios et al. (J Econom 112:359–379, 2003) is compared to the new test. The results suggest that the new test is generally superior in terms of power. An application to a real effective exchange rate underlines its usefulness.     

Goodness-of-Fit for Longitudinal Count Data with Overdispersion

In this article, we develop a method for checking the estimation equations, which is for joint estimation of the regression parameters and the overdispersion parameters, based on one dimension projected covariate. This method is different from the general testing methods in that our proposed method can be applied to high-dimensional response while the classical testing methods can not be extended to high dimension problem simply to construct a powerful test. Furthermore, the properties of the test statistics are investigated and Nonparametric Monte Carlo Test (NMCT) is suggested to determine the critical values of the test statistics under null hypothesis.     

A SCORE TEST FOR SEASONAL FRACTIONAL INTEGRATION AND COINTEGRATION

This paper develops a time domain score statistic for testing fractional integration at zero and seasonal frequencies in quarterly time series models. Further, it introduces the notion of fractional cointegration at different frequencies between two seasonally integrated, I(1) series. In testing problems involving seasonal fractional cointegration, it is argued that the alternative hypothesis is one-sided for which the usual score test may not be appropriate. Therefore, based on ideas in Silvapulle and Silvapulle (1995), a one-sided score statistic is constructed. A simulation study finds that the score statistic generally has desirable size and power properties in moderately sized samples. The score test is applied to the quarterly Australian consumption function. The income and consumption series are found to be I(1) at zero and seasonal frequencies and these two series are not cointegrated at any frequency.     

Testing Homogeneity of Gini Indices Against Simple-Ordered Alternative

Gini index is one of the most popular indices employed for measuring inequality in income and wealth. In this article, we propose a multi-sample test for testing Gini indices against simple-ordered alternatives. The exact critical points through simulation were computed for the proposed test in case of two-parameter exponential distribution; however, the proposed test can be applied even in case of other distributions like Pareto, Weibull, etc. Construction of simultaneous one-sided confidence intervals (SOCIs) was proposed along with an illustration. Power of the test is computed and some power comparisons are also made.     

A One-Sided Rank Test for Multivariate Censored Data

An asymptotically distribution-free test is proposed for testing the equality of two multivariate failure distributions against a particular one-sided alternative based on censored observations. This test may be interpreted as a multivariate one-sided Gehan test. The consistency of the test is established. An illustrative example is given.     

Change Detection in INAR(p) Processes Against Various Alternative Hypotheses

Change in the coefficients or the mean of the innovation of an INAR(p) process is a sign of disturbance that is important to detect. The proposed methods can test for change in any one of these quantities separately, or in any collection of them. They make both one-sided and two-sided tests possible, furthermore, they can be used to test against the “epidemic” alternative. The tests are based on a CUSUM process using CLS estimators of the parameters. Under the one-sided and two-sided alternatives, consistency of the tests is proved and the properties of the change-point estimator are also explored.     

ONE-SIDED ESTIMATING AND TESTING PROBLEMS FOR LOCATION MODELS FROM GROUPED SAMPLES

This article deals with one-sided problems for location models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjointed intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. This article considers one-sided estimating and testing problems for location models. Some methods for computing the maximum likelihood estimates of the parameters subject to order restrictions are proposed and a numerical example by the method is given.     

Methods for one-sided testing of the difference between proportions and sample size considerations related to non-inferiority clinical trials

Assessment of non-inferiority is often performed using a one-sided statistical test through an analogous one-sided confidence limit. When the focus of attention is the difference in success rates between test and active control proportions, the lower confidence limit is computed, and many methods exist in the literature to address this objective. This paper considers methods which have been shown to be popular in the literature and have surfaced in this research as having good performance with respect to controlling type I error at the specified level. Performance of these methods is assessed with respect to power and type I error through simulations. Sample size considerations are also included to aid in the planning stages of non-inferiority trials focusing on the difference in proportions. Results suggest that the appropriate method to use depends on the sample size allocation of subjects in the test and active control groups.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

The p-values for one-sided hypothesis testing in univariate linear calibration

In this article, we focus on the one-sided hypothesis testing for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved. The observations of the response variable corresponding to known values of the explanatory variable are used to make inferences on a single unknown value of the explanatory variable. We apply the generalized inference to the calibration problem, and take the generalized p-value as the test statistic to develop a new p-value for one-sided hypothesis testing, which we refer to as the one-sided posterior predictive p-value. The behavior of the one-sided posterior predictive p-value is numerically compared with that of the generalized p-value, and simulations show that the proposed p-value is quite satisfactory in the frequentist performance.     

Detecting a change in variances

The likelihood ratio test for testing for a change in a sequence of variances of normal populations is derived. The alternative hypothesis considered is of a one-sided nature. For the test, the conservativeness of the Sidak bound is shown and the asymptotic version of the Sidak bound is also constructed. These bound are compared with the Bonferroni bound and the Worsley bound, using the Monte Carlo method. Finally Hsu's data of stock market returns is reanalysed, using the test.     

The M-Bayesian Credible Limits of the Reliability Derived from Binomial Distribution

This article introduces a new method, M-Bayesian credible limit method, to estimate reliability derived from binomial distribution, in the case of zero-failure data. Firstly, the definition of one-sided and two-sided M-Bayesian credible limits are provided, and moreover, formulas of one-sided and two-sided M-Bayesian credible limits are also provided. secondly, properties of one-sided and two-sided M-Bayesian credible limits are discussed, and we will see that the M-Bayesian credible limit method is superior to the corresponding classical confidence limit method. Finally, the new estimation method is applied to a numerical example. Through the example the efficiency and easiness of operation of this method are commended.     

A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances

This article considers the twin problems of testing for autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH disturbances in the linear regression model. A feature of these testing problems, ignored by the standard Lagrange multiplier test, is that they are onesided in nature. A test that exploits this one-sided aspect is constructed based on the sum of the scores. The small-sample-size and power properties of two versions of this test under both normal and leptokurtic disturbances are investigated via a Monte Carlo experiment. The results indicate that both versions of the new test typically have superior power to two versions of the Lagrange multiplier test and possibly also more accurate asymptotic critical values.     

The sample breakdown points of tests

Several authors have taken the worst case breakdown measures in analyzing the robustness of a test. In general, these kinds of measures give only a rough picture of breakdown robustness of a test. To overcome this limitation, a new kind of breakdown measure of a test is defined as the smallest proportion of arbitrary outliers in the sample that can distort the test decision. It is called as the sample breakdown point of a test in this paper. A distinct advantage of this new measure is that it is directly concerned with the test decision based on the present sample and with the critical region of the test. The sample breakdown points of several commonly used tests of one-sided or two-sided hypotheses are calculated and their asymptotic properties are also established. By Monte Carlo simulations and asymptotic analysis, we show that the acceptance breakdown of the t-test and the Hotelling T²-test is slightly better than that of the sample mean test. Finally, we prove that, for a one-sided hypothesis testing of location, the sign test has the maximum sample breakdown points asymptotically within a class of M-tests and score-tests.     

Exact Tests for 2 × 2 Contingency Tables

Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.     

Multivariate tests comparing binomial probabilities, with application to safety studies for drugs

Summary.  In magazine advertisements for new drugs, it is common to see summary tables that compare the relative frequency of several side-effects for the drug and for a placebo, based on results from placebo-controlled clinical trials. The paper summarizes ways to conduct a global test of equality of the population proportions for the drug and the vector of population proportions for the placebo. For multivariate normal responses, the Hotelling T ²-test is a well-known method for testing equality of a vector of means for two independent samples. The tests in the paper are analogues of this test for vectors of binary responses. The likelihood ratio tests can be computationally intensive or have poor asymptotic performance. Simple quadratic forms comparing the two vectors provide alternative tests. Much better performance results from using a score-type version with a null-estimated covariance matrix than from the sample covariance matrix that applies with an ordinary Wald test. For either type of statistic, asymptotic inference is often inadequate, so we also present alternative, exact permutation tests. Follow-up inferences are also discussed, and our methods are applied to safety data from a phase II clinical trial.