Uniformly most powerful unbiased test in univariate linear calibration

In this paper, we deal with the hypothesis testing problems for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved, and the observations of the response variable corresponding to known values of the explanatory variable are used for making inferences concerning a single unknown value of the explanatory variable. The uniformly most powerful unbiased tests for both one-sided and two-sided hypotheses are constructed and verified. The power behaviour of the proposed tests is numerically compared with that of the existing method, and simulations show that the proposed tests make the powers improved.     

Comparing the overlapping of two independent confidence intervals with a single confidence interval for two normal population parameters

Two overlapping confidence intervals have been used in the past to conduct statistical inferences about two population means and proportions. Several authors have examined the shortcomings of Overlap procedure and have determined that such a method distorts the significance level of testing the null hypothesis of two population means and reduces the statistical power of the test. Nearly all results for small samples in Overlap literature have been obtained either by simulation or by formulas that may need refinement for small sample sizes, but accurate large sample information exists. Nevertheless, there are aspects of Overlap that have not been presented and compared against the standard statistical procedure. This article will present exact formulas for the maximum % overlap of two independent confidence intervals below which the null hypothesis of equality of two normal population means or variances must still be rejected for any sample sizes. Further, the impact of Overlap on the power of testing the null hypothesis of equality of two normal variances will be assessed. Finally, the noncentral t-distribution is used to assess the Overlap impact on type II error probability when testing equality of means for sample sizes larger than 1.     

On equivalence of two variances of a bivariate normal vector

A test for assessing the equivalence of two variances of a bivariate normal vector is constructed. It is uniformly more powerful than the two one-sided tests procedure and the power improvement is substantial. Numerical studies show that it has a type I error close to the test level at most boundary points of the null hypothesis space. One can apply this test to paired difference experiments or 2×2 crossover designs to compare the variances of two populations with two correlated samples. The application of this test on bioequivalence in variability is presented. We point out that bioequivalence in intra-variability implies bioequivalence in variability, however, the latter has a larger power.     

Graphical comparison of means with a standard in the one-way layout

Some simple test procedures are considered for comparing several group means with a standard value when the data are in a one-way layout. The underlying distributions are assumed to be normal with possibly unequal variances. The tests are based on a union-intersection formulation and can be applied in a form similar to a Shewhart control chart. Both two-sided and one-sided alternatives are considered. The power of the tests can be obtained from tables of a non-central t distribution. Implementation of the tests is illustrated with a numerical example. The tests help identify any group means different from the standard and might lead to a decision about rejecting the null hypothesis before all the group means are observed. The resulting savings in time and resources might be valuable in applications where the number of groups is large and the cost of acquiring data is high. For situations where the normality assumption is untenable, a non-parametric procedure, based on one-sample sign tests is considered.     

A Test of Non Null Hypothesis for Linear Trends in Proportions

Usual tests for trends stand under null hypothesis. This article presents a test of non null hypothesis for linear trends in proportions. A weighted least squares method is used to estimate the regression coefficient of proportions. A non null hypothesis is defined as its expectation equal to a prescribed regression coefficient margin. Its variance is used to construct an equation of basic relationship for linear trends in proportions along the asymptotic normal method. Then follow derivations for the sample size formula, the power function, and the test statistic. The expected power is obtained from the power function and the observed power is exhibited by Monte Carlo method. It reduces to the classical test for linear trends in proportions on setting the margin equal to zero. The agreement between the expected and the observed power is excellent. It is the non null hypothesis test matched with the classical test and can be applied to assess the clinical significance of trends among several proportions. By contrast, the classical test is restricted in testing the statistical significance. A set of data from a website is used to illustrate the methodology.     

Union–intersection permutation solution for two-sample equivalence testing

One of the well-known problems with testing for sharp null hypotheses against two-sided alternatives is that, when sample sizes diverge, every consistent test rejects the null with a probability converging to one, even when it is true. This kind of problem emerges in practically all applications of traditional two-sided tests. The main purpose of the present paper is to overcome this very intriguing impasse by considering a general solution to the problem of testing for an equivalence null interval against a two one-sided alternative. Our goal is to go beyond the limitations of likelihood-based methods by working in a nonparametric permutation framework. This solution requires the nonparameteric Combination of dependent permutation tests, which is the methodological tool that achieves Roy’s Union–intersection principle. To obtain practical solutions, the related algorithm is presented. To appreciate its effectiveness for practical purposes, a simple example and some simulation results are also presented. In addition, for every pair of consistent partial test statistics it is proved that, if sample sizes diverge, when the effect lies in the open equivalence interval, the Rejection probability (RP) converges to zero. Analogously, if the effect lies outside that interval, the RP converges to one.     

A Nonparametric Test Procedure Based on a Group of Quantile Tests

In this article, we consider nonparametric test procedures based on a group of quantile test statistics. We consider the quadratic form for the two-sided test and the maximal and summing types of statistics for the one-sided alternatives. Then we derive the null limiting distributions of the proposed test statistics using the large sample approximation theory. Also, we consider applying the permutation principle to obtain the null distribution. In this vein, we may consider the supremum type, which should use the permutation principle for obtaining the null distribution. Then we illustrate our procedure with an example and compare the proposed tests with other existing tests including the individual quantile tests by obtaining empirical powers through simulation study. Also, we comment on the related discussions to this testing procedure as concluding remarks. Finally we prove the lemmas and theorems in the appendices.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

Testing for Persistence in the Error Component Model: A One-Sided Approach

This article proposes new simple testing procedures for the joint null hypothesis of absence of persistent effects, in the form of random effects and first-order serial correlation in the error component model. The fact that the presence of random effects is clearly of a one-sided nature, together with the fact that in many empirical applications researchers worry about positive serial correlation leaves room for a power gain that arises from restricting the parameter space under the alternative hypothesis, compared to existing procedures that allow for two-sided alternatives. A Monte Carlo experiment shows that the proposed statistics have good size and power performance in very small samples like those typically used in applied work in panel data. An empirical example illustrates the usefulness of the proposed statistics.     

Simultaneous inferences for ordered exponential location parameters under unbalanced data and heteroscedasticity of scale parameters

A test procedure for testing homogeneity of location parameters against simple ordered alternative is proposed for k(k ≥ 2) members of two parameter exponential distribution under unbalanced data and heteroscedasticity of the scale parameters. The relevant one-sided and two-sided simultaneous confidence intervals (SCIs) for all k(k ? 1)/2 ordered pairwise differences of location parameters are also proposed. Simulation-based study revealed that the proposed procedure is better than the recently proposed procedure in terms of power, coverage probability, and average volume of SCIs. The implementation of proposed procedure is demonstrated through real life data.     

A Bayesian analysis for the multivariate point null testing problem

A Bayesian test for the point null testing problem in the multivariate case is developed. A procedure to get the mixed distribution using the prior density is suggested. For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test. With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence.     

On group sequential tests for data in unequally sized groups and with unknown variance

We present a method to generalise the scope of application of group sequential tests designed for equally sized groups of normal observations with known variance. Preserving the significance levels against which standardised statistics are compared leads to tests for unequally grouped data which maintain Type I error probabilities to a high degree of accuracy. The same approach can be followed when observations have unknown variance by setting critical values for Studentised statistics at percentiles of the appropriate t-distributions. This significance level approach is equally applicable to group sequential one-sided tests and two-sided tests, possibly with early stopping permitted to accept the null hypothesis. In applications to equivalence testing, tests are required to maintain a specified power, rather than Type I error rate: such tests can be constructed by defining the standardised test statistics used in the significance level approach with respect to appropriately chosen hypotheses.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.     

A revision of the tukey multiple comparisons procedure to control the probability of committing at most one type i error

Halperin et al. (1988) suggested an approach which allows for k Type I errors while using Scheffe's method of multiple comparisons for linear combinations of p means. In this paper we apply the same type of error control to Tukey's method of multiple pairwise comparisons. In fact, the variant of the Tukey (1953) approach discussed here defines the error control objective as assuring with a specified probability that at most one out of the p(p-l)/2 comparisons between all pairs of the treatment means is significant in two-sided tests when an overall null hypothesis (all p means are equal) is true or, from a confidence interval point of view, that at most one of a set of simultaneous confidence intervals for all of the pairwise differences of the treatment means is incorrect. The formulae which yield the critical values needed to carry out this new procedure are derived and the critical values are tabulated. A Monte Carlo study was conducted and several tables are presented to demonstrate the experimentwise Type I error rates and the gains in power furnished by the proposed procedure     

The Equivalence of Two Proportions Revisited

The classic conditional test for checking that the difference between two independent proportions is not null may not be appropriate in many circumstances. Dunnett & Gent (1977) showed that in clinical trials, in studies of drugs, etc, the aim is to prove the practical equality (equivalence) of both proportions. On other occasions the aim may be the opposite: i.e. to prove that the two proportions are substantially different (biologically significant). Both cases are usually solved by two one-sided tests (TOST test). In this article, this procedure is shown to be conservative and two true two-sided tests for each case are proposed.     

Comparison of testing procedures utilizing P-values and Bayes factors in some common situations

Bayesian alternatives to classical tests for several testing problems are considered. One-sided and two-sided sets of hypotheses are tested concerning an exponential parameter, a Binomial proportion, and a normal mean. Hierarchical Bayes and noninformative Bayes procedures are compared with the appropriate classical procedure, either the uniformly most powerful test or the likelihood ratio test, in the different situations. The hierarchical prior employed is the conjugate prior at the first stage with the mean being the test parameter and a noninformative prior at the second stage for the hyper parameter(s) of the first stage prior. Fair comparisons are attempted in which fair means the likelihood of making a type I error is approximately the same for the different testing procedures; once this condition is satisfied, the power of the different tests are compared, the larger the power, the better the test. This comparison is difficult in the two-sided case due to the unsurprising discrepancy between Bayesian and classical measures of evidence that have been discussed for years. The hierarchical Bayes tests appear to compete well with the typical classical test in the one-sided cases.     

Generalized Likelihood Ratio Tests and Uniformly Most Powerful Tests

For testing a one-sided hypothesis in a one-parameter family of distributions, it is shown that the generalized likelihood ratio (GLR) test coincides with the uniformly most powerful (UMP) test, assuming certain monotonicity properties for the likelihood function. In particular, the equivalence of GLR tests and UMP tests holds for one-parameter exponential families. In addition, the relationship between GLR and UMPU (UMP unbiased) tests is considered when testing two-sided hypotheses.     

Bayesian inference method for model validation and confidence extrapolation

This paper presents a Bayesian-hypothesis-testing-based methodology for model validation and confidence extrapolation under uncertainty, using limited test data. An explicit expression of the Bayes factor is derived for the interval hypothesis testing. The interval method is compared with the Bayesian point null hypothesis testing approach. The Bayesian network with Markov Chain Monte Carlo simulation and Gibbs sampling is explored for extrapolating the inference from the validated domain at the component level to the untested domain at the system level. The effect of the number of experiments on the confidence in the model validation decision is investigated. The probabilities of Type I and Type II errors in decision-making during the model validation and confidence extrapolation are quantified. The proposed methodologies are applied to a structural mechanics problem. Numerical results demonstrate that the Bayesian methodology provides a quantitative approach to facilitate rational decisions in model validation and confidence extrapolation under uncertainty.     

Classical testing in functional linear models

We extend four tests common in classical regression – Wald, score, likelihood ratio and F tests – to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.