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The dependence structure in a $d$-variate continuous random vector $\mathbf{X}$ is characterized by its unique copula. Starting from the fact that many copulas can be extracted from the global $d$-dimensional copula of $\mathbf{X}$, a very general framework is proposed here for testing that a given collection of induced $p$-dimensional copulas from a multivariate distribution are identical. Many hypotheses of interest in copula modeling fall into this category, including bivariate symmetry (diagonal, radial, joint), exchangeability, as well as various types of equality of copulas. Here, a broad class of test statistics is defined around a matrix representation of the null hypothesis and quadratic functionals including Cramer–von Mises and characteristic function mappings. Since the null hypotheses to be tested are composite by nature, the computation of $\mathrm{P}$-values is achieved using multiplier bootstrap versions of the test statistics. The sample properties of the method are investigated when testing for several types of bivariate symmetry, exchangeability, equality of non-overlapping and overlapping copulas and equality of all bivariate copulas. The general conclusion is that the tests are good at keeping their nominal level and are powerful against a wide variety of alternatives, showing the relevance and reliability of the methodology for the modeling of multivariate datasets with the help of copulas.

We consider the one-way ANOVA problem of testing the equality of several normal means when the variances are not assumed to be equal. This is a generalization of the Behrens-Fisher problem, but even in this special case there is no exact test and the actual size of any test depends on the values of the nuisance parameters. Therefore, controlling the actual size of the test is of main concern. In this article, we first consider a test using the concept of generalized p-value. Extensive simulation studies show that the actual size of this test does not exceed the nominal level, for practically all values of the nuisance parameters, but the test is not too conservative either, in the sense that the actual size of the test can be very close to the nominal level for some values of the nuisance parameters. We then use this test to propose a simple F-test, which has similar properties but avoids the computations associated with generalized p-values. Because of its simplicity, both conceptually as well as computa...

Among statistical inferences, one of the main interests is drawing the inferences about the log-normal means since the log-normal distribution is a well-known candidate model for analyzing positive and right-skewed data. In the past, the researchers only focused on one or two log-normal populations or used the large sample theory or quadratic procedure to deal with several log-normal distributions. In this article, we focus on making inferences on several log-normal means based on the modification of the quadratic method, in which the researchers often used the vector of the generalized variables to deal with the means of the symmetric distributions. Simulation studies show that the quadratic method performs well only for symmetric distributions. However, the modified procedure fits both symmetric and skew distribution. The numerical results show that the proposed modified procedure can provide the confidence interval with coverage probabilities close to the nominal level and the hypothesis testing perfor...

Linear mixed-effects models are a powerful tool for the analysis of longitudinal data. The aim of this paper is to study model averaging for linear mixed-effects models. The asymptotic distribution of the frequentist model average estimator is derived, and a confidence interval procedure with an actual coverage probability that tends to the nominal level in large samples is developed. The two confidence intervals based on the model averaging and based on the full model are shown to be asymptotically equivalent. A simulation study shows good finite sample performance of the model average estimators.

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are ...

It is well known that the standard independent, identically distributed (iid) bootstrap of the mean is inconsistent in a location model with infinite variance (α-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the iid bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the iid bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower con...

This article investigates three procedures on the multiple comparisons with a control in the presence of unequal error variances. The advantages of the proposed methods are illustrated through two examples. The performance of the proposed methods and other alternative methods is compared by simulation studies. The results show that the typical methods assuming equal variance will have inflated error rate and may lead to erroneous inference when the equal variance assumption fails. In addition, the simulation study shows that the proposed approaches always control the family-wise error rate at a specified nominal level α, while some established methods are liberal and have inflated error rate in some scenarios.

Publication bias in meta-analysis is a serious issue as it may lead to biased estimates which appear to be precise. A popular method for detecting and adjusting the publication bias is the trim and fill method. This study uses simulated meta-analysis to quantify the effects of publication bias on the overall meta-analysis estimates of continuous data where the absolute mean difference was utilized as the measure of effect. It additionally evaluates the performance of the trim and fills method for adjusting the publication bias in terms of statistical bias, the standard errors and the coverage probability. The results demonstrate that if the publication bias is not adjusted it could lead to up to 40% biased in treatment effect estimates. Utilization of the trim and fill method has reduced the bias in the overall effect estimate by more than half. It is optimum in presence of moderate underlying bias but has minimal effects in presence of low and severe bias. Additionally, the trim and fill method improves the coverage probability by more than half when subjected to the same level of publication bias as those of the unadjusted data. However, the method tends to produce false positive results. A sensitivity analysis suggests that the trim and fill method will incorrectly adjust the data for publication bias between 10-45% of the time (for the 5% nominal level). Although the data was incorrectly adjusted, it was found that the Percentage Relative Bias (PRB) introduced into the estimates due to this adjustment is minimal (min: 0.007%, max: 0.109%) and coverage probability for estimates based on this incorrectly adjusted data is not significantly different from those of which is correctly not adjusted. Therefore the trim and fill method is recommended be routinely used when conducting meta-analysis.

Summary In the classical theory of randomized trials (RTs), the treatment effect is defined as a mean difference of potential outcomes. To achieve nominal coverage probability for confidence intervals (CIs) on treatment effects in RTs, certain assumptions are necessary. Specifically, one must either make assumptions about the joint distribution of potential outcomes or enroll subjects in the trial by random sampling of the target population on which the treatment effect is defined. In practice, no such sampling usually takes place and assumptions about the joint distribution of potential outcomes cannot be verified based on observed data. Furthermore, the most common of these assumptions, such as treatment-unit additivity (TUA) or independence are biologically implausible in most RTs involving human subjects. Hence, it is not usually possible to construct CIs on treatment effects with nominal coverage probability. However, for any joint distribution of potential outcomes, the standard estimator of the variance of the difference of two independent sample means produces CIs with asymptotic coverage at least at the nominal level. This interpretation of CIs as conservative bounds may not always hold in conventional regression models applied to RT data. Resume Dans la theorie classique des experiences randomisees (Randomized Trials ou RTs), l'effet-traitement est defini comme une difference entre reponses potentielles. Pour que les intervalles de confiance sur ces effets-traitements atteignent la probabilite de couverture nominale, des hypotheses sont necessaires. Soit ces hypotheses portent sur la loi jointe des reponses potentielles, soit elles concernent la facon dont sont selectionnes, par echantillonnage de populations-cibles, les sujets participant a l'experience. En pratique, de tels echantillonnages sont fictifs, et les hypotheses sur les lois jointes de reponses potentielles ne peuvent etre testees sur base d'observations. En outre, les plus repandues de ces hypotheses, telle l'hypothese d'additivite traitement-sujet (treatment-unit additivity ou TUA), sont biologiquement peu plausibles dans les experiences faisant intervenir des sujets humains. Il est donc en general impossible de construire pour les effets-traitements des intervalles de confiance dont le taux de couverture egale le niveau de confiance nominal. L'estimateur standard de la variance de la difference de deux moyennes empiriques independantes fournit des intervalles de confiance dont le niveau de couverture est au moins egal au niveau nominal. Cette interpretation conservative des intervalles de confiance n'est toutefois pas toujours correcte pour les modeles de regression conventionnels dans le cadre d'observations provenant d'experiences randomisees.

Historical Simulation (HS) and its variant, the Filtered Historical Simulation (FHS), are the most popular Value-at-Risk forecast methods at commercial banks. These forecast methods are traditionally evaluated by means of the unconditional backtest. This paper formally shows that the unconditional backtest is always inconsistent for backtesting HS and FHS models, with a power function that can be even smaller than the nominal level in large samples. Our findings have fundamental implications in the determination of market risk capital requirements, and also explain Monte Carlo and empirical findings in previous studies. We also propose a data-driven weighted backtest with good power properties to evaluate HS and FHS forecasts. A Monte Carlo study and an empirical application with three US stocks confirm our theoretical findings. The empirical application shows that multiplication factors computed under the current regulatory framework are downward biased, as they inherit the inconsistency of the unconditional backtest. Research funded by the Spanish Plan Nacional de I + D + I, reference number SEJ2007 62908.

The aim of this article is to introduce the general form (so called RT class) of the robust and classical Jarque–Bera (JB) test based on the location functional. We introduce the two-step procedure which is optimal for testing against the individual or contaminated Pareto alternative. As a reference for such a contamination we consider different Pareto distributions. We also give practical guidelines for robust testing for normality against short- and heavy-tailed alternatives. We concentrate mainly on simulation results for moderate and small samples. However, we also prove consistency and asymptotic distribution for introduced tests. We show that as the suitable measure of nominal level of Pareto tail parameter we may take the t-Hill estimator introduced in the article. To guarantee the consistency of the whole procedure, we also prove the consistency of t-Hill estimator. The introduced general class of robust tests of the normality is illustrated at the selected datasets of financial time series.

Statistical procedures for the equality of two and k univariate distributions based on samples of dependent observations are proposed in this work. The test statistics are L"2 distances of standard empirical and characteristic function processes. The p-values of the tests are obtained from a version of the multiplier central limit theorem whose asymptotic validity is established. Simple formulas for the test statistics and their multiplier versions in terms of multiplication of matrices are provided. Simulations under many patterns of dependence characterized by copulas show the good behavior of the tests in small samples, both in terms of their power and of their ability to keep their nominal level under the null hypothesis.

Testing non-inferiority in active-controlled clinical trials examines whether a new procedure is, to a pre-specified amount, no worse than an existing procedure. To assess non-inferiority between two procedures using clustered matched-pair binary data, two new statistical tests are systematically compared to existing tests. The calculation of corresponding confidence interval is also proposed. None of the tests considered requires structural within-cluster correlation or distributional assumptions. The results of an extensive Monte Carlo simulation study illustrate that the performance of the statistics depends on several factors including the number of clusters, cluster size, probability of success in the test procedure, the homogeneity of the probability of success across clusters, and the intra-cluster correlation coefficient (ICC). In evaluating non-inferiority for a clustered matched-pair study, one should consider all of these issues when choosing an appropriate test statistic. The ICC-adjusted test statistic is generally recommended to effectively control the nominal level when there is constant or small variability of cluster sizes. For a greater number of clusters, the other test statistics maintain the nominal level reasonably well and have higher power. Therefore, with the carefully designed clustered matched-pair study, a combination of the statistics investigated may serve best in data analysis. Finally, to illustrate the practical application of the recommendations, a real clustered matched-pair collection of data is used to illustrate testing non-inferiority.

Classical inference considers sampling variability to be the only source of uncertainty, and does not address the issue of bias caused by contamination. Naive robust intervals replace the classical estimates by their robust counterparts without considering the possible bias of the robust point estimates. Consequently, the asymptotic coverage proportion of these intervals of any nominal level will invariably tend to zero for any proportion of contamination.In this study, we attempt to achieve reasonable coverage percentages by constructing globally robust confidence intervals that adjust for the bias of the robust point estimates. We improve these globally robust intervals by considering the direction of the bias of the robust estimates used. We compare the proposed intervals with the existing ones through an extensive simulation study. The proposed methods have reasonable coverage percentage while the existing method show very poor coverage as sample size increases.DOI: http://dx.doi.org/10.3329/dujs.v60i1.10347 Dhaka Univ. J. Sci. 60(1): 109-113 2012 (January)

We introduce two simple methods for the statistical comparison of the temporal pattern of life-cycle events between two populations. The methods are based on a translation of stage-frequency data into individual 'times in stage'. For example, if the stage-k individuals in a set of samples consist of three individuals counted at time t(1) and two counted at time t(2), the observed times in stage k would be (t(1), t(1), t(1), t(2), t(2)). Times in stage then can be compared between two populations by performing stage-specific t-tests or by testing for equality of regression lines of time versus stage between the two populations. Simulations show that our methods perform at close to the nominal level, have good power against a range of alternatives, and have much better operating characteristics than a widely-used phenology model from the literature.

Shift in research and development strategy from developing follow-on or 'me-too' drugs to differentiated medical products with potentially better efficacy than the standard of care (e.g., first-in-class, best-in-class, and bio-betters) highlights the scientific and commercial interests in establishing superiority even when a non-inferiority design, adequately powered for a pre-specified non-inferiority margin, is appropriate for various reasons. In this paper, we propose a group sequential design to test superiority at interim analyses in a non-inferiority trial. We will test superiority at the interim analyses using conventional group sequential methods, and we may stop the study because of better efficacy. If the study fails to establish superior efficacy at the interim and final analyses, we will test the primary non-inferiority hypothesis at the final analysis at the nominal level without alpha adjustment. Whereas superiority/non-inferiority testing no longer has the hierarchical structure in which the rejection region for testing superiority is a subset of that for testing non-inferiority, the impact of repeated superiority tests on the false positive rate and statistical power for the primary non-inferiority test at the final analysis is essentially ignorable. For the commonly used O'Brien-Fleming type alpha-spending function, we show that the impact is extremely small based upon Brownian motion boundary-crossing properties. Numerical evaluation further supports the conclusion for other alpha-spending functions with a substantial amount of alpha being spent on the interim superiority tests. We use a clinical trial example to illustrate the proposed design.

A correlation curve measures the strength of the association between two variables locally at different values of x . The purpose of this study is to obtain point-wise confidence intervals for a correlation curve using wild bootstrap techniques. Empirical coverage probabilities are found to be close to the specified nominal level. Bootstrapping is an attractive alternative to confidence intervals based on asymptotic expressions that have slow rate of convergence.

SummaryObjectives: Methods for change point (also sometimes referred to as threshold or breakpoint) detection in binary sequences are not new and were introduced as early as 1955. Much of the research in this area has focussed on asymptotic and exact conditional methods. Here we develop an exact unconditional test.Methods: An unconditional exact test is developed which assumes the total number of events as random instead of conditioning on the number of observed events. The new test is shown to be uniformly more powerful than Worsley’s exact conditional test and means for its efficient numerical calculations are given. Adaptions of methods by Berger andBoos are made to deal with the issue that the unknown event probability imposes a nuisance parameter. The methods are compared in a Monte Carlo simulation study and applied to a cohort of patients undergoing traumatic orthopaedic surgery involving external fixators where a change in pin site infections is investigated.Results: The unconditional test controls the type I error rate at the nominal level and is uniformly more powerful than (or to be more precise uniformly at least as powerful as) Worsley’s exact conditional test which is very conservative for small sample sizes. In the application a beneficial effect associated with the introduction of a new treatment procedure for pin site care could be revealed.Conclusions: We consider the new test an effective and easy to use exact test which is recommended in small sample size change point problems in binary sequences.

The high energy physics unfolding problem is an important statistical inverse problem in data analysis at the Large Hadron Collider (LHC) at CERN. The goal of unfolding is to make nonparametric inferences about a particle spectrum from measurements smeared by the finite resolution of the particle detectors. Previous unfolding methods use ad hoc discretization and regularization, resulting in confidence intervals that can have significantly lower coverage than their nominal level. Instead of regularizing using a roughness penalty or stopping iterative methods early, we impose physically motivated shape constraints: positivity, monotonicity, and convexity. We quantify the uncertainty by constructing a nonparametric confidence set for the true spectrum, consisting of all those spectra that satisfy the shape constraints and that predict the observations within an appropriately calibrated level of fit. Projecting that set produces simultaneous confidence intervals for all functionals of the spectrum, including averages within bins. The confidence intervals have guaranteed conservative frequentist finite-sample coverage in the important and challenging class of unfolding problems for steeply falling particle spectra. We demonstrate the method using simulations that mimic unfolding the inclusive jet transverse momentum spectrum at the LHC. The shape-constrained intervals provide usefully tight conservative inferences, while the conventional methods suffer from severe undercoverage., 44 pages, 9 figures, to appear in the Annals of Applied Statistics

In most empirical studies, once the best model has been selected according to a certain criterion, subsequent analysis is conducted conditionally on the chosen model. In other words, the uncertainty of model selection is ignored once the best model has been chosen. However, the true data-generating process is in general unknown and may not be consistent with the chosen model. In the analysis of productivity and technical efficiencies in the stochastic frontier settings, if the estimated parameters or the predicted efficiencies differ across competing models, then it is risky to base the prediction on the selected model. Buckland et al. (Biometrics 53:603–618, 1997) have shown that if model selection uncertainty is ignored, the precision of the estimate is likely to be overestimated, the estimated confidence intervals of the parameters are often below the nominal level, and consequently, the prediction may be less accurate than expected. In this paper, we suggest using the model-averaged estimator based on the multimodel inference to estimate stochastic frontier models. The potential advantages of the proposed approach are twofold: incorporating the model selection uncertainty into statistical inference; reducing the model selection bias and variance of the frontier and technical efficiency estimators. The approach is demonstrated empirically via the estimation of an Indian farm data set.

An important question that arises in clinical trials is how many additional observations, if any, are required beyond those originally planned. This has satisfactorily been answered in the case of two-treatment double-blind clinical experiments. However, one may be interested in comparing a new treatment with its competitors, which may be more than one. This problem is addressed in this investigation involving responses from arbitrary distributions, in which the mean and the variance are not functionally related. First, a solution in determining the initial sample size for specified level of significance and power at a specified alternative is obtained. Then it is shown that when the initial sample size is large, the nominal level of significance and the power at the pre-specified alternative are fairly robust for the proposed sample size re-estimation procedure. An application of the results is made to the blood coagulation functionality problem considered by Kropf et al. [Multiple comparisons of treatme...

We consider the coverage rate of model-averaged confidence intervals for the treatment means in a factorial experiment, when we use a normal linear model in the analysis. Model-averaging provides a useful compromise between using the full model (containing all main effects and interactions) and a “best model” obtained by some model-selection process. Use of the full model guarantees perfect coverage, whereas use of a best model is known to lead to narrow intervals with poor coverage. Model-averaging allows us to achieve good coverage using intervals that are also narrower than those from the full model. We compare four information criteria that might be used for model-averaging in this setting: A I C , A I C c , A I C c ∗ and B I C . In this setting, if the full model is “truth”, all the criteria will have perfect coverage rates asymptotically. We use simulation to assess the coverage rates and interval widths likely to be achieved by a confidence interval with a nominal coverage of 95%. Our results suggest that A I C performs best in terms of coverage rate; across a wide range of scenarios and replication levels, it consistently provides coverage rates within 1.5% points of the nominal level, while also leading to reductions in interval-width of up to 30%, compared to the full model. A I C c performed worst overall, with a coverage rate that was up to 5.2% points too low. We recommend that model-averaging become standard practise when summarising the results of a factorial experiment in terms of the treatment means, and that A I C be used to perform the model-averaging.

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L2-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.

When testing geometrically irregular parametric hypotheses, the bootstrap is an intuitively appealing method to circumvent difficult distribution theory. It has been shown, however, that the usual bootstrap is inconsistent in estimating the asymptotic distributions involved in such problems. This paper is concerned with the asymptotic size of likelihood ratio tests when critical values are computed using the inconsistent bootstrap. We clarify how the asymptotic size of such a test can be obtained from the size of the corresponding bootstrap test in the relevant limiting normal experiment. For boundary problems, that is, hypotheses given by convex cones, we show the bootstrap test to always be anticonservative, and we compute the size numerically for different two-dimensional examples. The examples illustrate that the size can be below or above the nominal level, and reveal that the relationship between the size of the test and the geometry of the considered hypotheses is surprisingly subtle. Copyright 2011, Oxford University Press.

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.

In analytical chemistry, the evaluation on performance accuracy of an analytical method is an important issue. When an adjusted or new method (test method) is developed, the linear measurement error model is commonly used to compare it with another reference method. For this routine practice, the measurements on the reference method can be placed on the x-axis, whereas those of the test method on the y-axis, then the slope of this linear relationship indicates the agreement between them and also the performance of the test method. Under the assumption that both variables are subject to heteroscedastic measurement errors, a novel approach based on the concepts of a generalized pivotal quantity (GPQ) is proposed to construct confidence intervals for the slope. Its performance is compared with two maximum likelihood estimation (MLE)-based approaches through simulation studies. It is shown that the proposed GPQ-based approach is capable of maintaining the empirical coverage probabilities close to the nominal level and yielding reasonable expected lengths. The GPQ-based approach can be recommended in practical use because of its easy implementation and better performance than the MLE-based approaches in most simulation scenarios. Two real datasets are given to illustrate the approaches. Copyright © 2011 John Wiley & Sons, Ltd.

This paper investigates homogeneity test of rate ratios in stratified matched-pair studies on the basis of asymptotic and bootstrap-resampling methods. Based on the efficient score approach, we develop a simple and computationally tractable score test statistic. Several other homogeneity test statistics are also proposed on the basis of the weighted least-squares estimate and logarithmic transformation. Sample size formulae are derived to guarantee a pre-specified power for the proposed tests at the pre-given significance level. Empirical results confirm that (i) the modified score statistic based on the bootstrap-resampling method performs better in the sense that its empirical type I error rate is much closer to the pre-specified nominal level than those of other tests and its power is greater than those of other tests, and is hence recommended, whilst the statistics based on the weighted least-squares estimate and logarithmic transformation are slightly conservative under some of the considered settings; (ii) the derived sample size formulae are rather accurate in the sense that their empirical powers obtained from the estimated sample sizes are very close to the pre-specified nominal powers. A real example is used to illustrate the proposed methodologies.

In clinical trials, it is often desirable to test for superiority conditioned on establishment of noninferiority based on the same primary end-point. According to a guidance document issued by the European regulatory agency Committee for Proprietary Medicinal Products in 2001, no type I error rate adjustment is necessary for switching between superiority and non-inferiority because the family-wise type I error rate is controlled at the same nominal level. However, Ng raised the issues of switching between superiority and noninferiority even though there is no inflation of the family-wise type I error rate and showed that the fake discovery rate could be increased.

The problems of constructing tolerance intervals for the binomial and Poisson distributions are considered. Closed-form approximate equal-tailed tolerance intervals (that control percentages in both tails) are proposed for both distributions. Exact coverage probabilities and expected widths are evaluated for the proposed equal-tailed tolerance intervals and the existing intervals. Furthermore, an adjustment to the nominal confidence level is suggested so that an equal-tailed tolerance interval can be used as a tolerance interval which includes a specified proportion of the population, but does not necessarily control percentages in both tails. Comparison of such coverage-adjusted tolerance intervals with respect to coverage probabilities and expected widths indicates that the closed-form approximate tolerance intervals are comparable with others, and less conservative, with minimum coverage probabilities close to the nominal level in most cases. The approximate tolerance intervals are simple and easy to compute using a calculator, and they can be recommended for practical applications. The methods are illustrated using two practical examples.

We derive likelihood ratio (LR) tests for the null hypothesis of equivalence that the normal means fall into a practical indifference zone. The LR test can easily be constructed and applied to k ≥ 2 treatments. Simulation results indicate that the LR test might be slightly anticonservative statistically, but when the sample sizes are large, it always produces the nominal level for mean configurations under the null hypothesis. More powerful than the studentized range test, the LR test is a straightforward application that requires only current existing statistical tables, with no complicated computations.

Adverse impact is often assessed by evaluating whether the success rates for 2 groups on a selection procedure are significantly different. Although various statistical methods have been used to analyze adverse impact data, Fisher's exact test (FET) has been widely adopted, especially when sample sizes are small. In recent years, however, the statistical field has expressed concern regarding the default use of the FET and has proposed several alternative tests. This article reviews Lancaster's mid-P (LMP) test (Lancaster, 1961), an adjustment to the FET that tends to have increased power while maintaining a Type I error rate close to the nominal level. On the basis of Monte Carlo simulation results, the LMP test was found to outperform the FET across a wide range of conditions typical of adverse impact analyses. The LMP test was also found to provide better control over Type I errors than the large-sample Z-test when sample size was very small, but it tended to have slightly lower power than the Z-test under some conditions.

This paper tries to formulate conclusions about the indexation of old-age pensions. Pensions can be adjusted and indexed taking into consideration a wage increase. The point of indexation with regard to wages lies in the increment of pensions on to a new, higher nominal level of consumption opportunities (the pension increases), but leaving it at the same relative or potential level of consumption opportuni­ties (the pensions-to-earnings ratio remains constant). Pensions can also be adjusted and indexed according to an increase in the price level. The adjustment of pensions with respect to the price level maintains the real level of consumption (a person is always able to buy the same set of goods). The aim of this study is to identify the conditions of efficient indexation; to summarize the methods of indexation; to draw con­clusions as to which of these methods maximizes the wealth of taxpayers.

When competing risks data arise, information on the actual cause of failure for some subjects might be missing. Therefore, a cause-specific proportional hazards model together with multiple imputation (MI) methods have been used to analyze such data. Modelling the cumulative incidence function is also of interest, and thus we investigate the proportional subdistribution hazards model (Fine and Gray model) together with MI methods as a modelling approach for competing risks data with missing cause of failure. Possible strategies for analyzing such data include the complete case analysis as well as an analysis where the missing causes are classified as an additional failure type. These approaches, however, may produce misleading results in clinical settings. In the present work we investigate the bias of the parameter estimates when fitting the Fine and Gray model in the above modelling approaches. We also apply the MI method and evaluate its comparative performance under various missing data scenarios. Results from simulation experiments showed that there is substantial bias in the estimates when fitting the Fine and Gray model with naive techniques for missing data, under missing at random cause of failure. Compared to those techniques the MI-based method gave estimates with much smaller biases and coverage probabilities of 95 per cent confidence intervals closer to the nominal level. All three methods were also applied on real data modelling time to AIDS or non-AIDS cause of death in HIV-1 infected individuals.

The mid-p is defined as the sum of the probabilities of all outcomes more extreme than an observed value, plus half of the probabilities of all outcomes exactly as extreme. On the one hand, it offers greater power than the standard p-value, but on the other, tests based on the mid-p statistic may have greater Type I error than their nominal level. This article investigates the mid p-value's properties under the estimated truth paradigm, which views p-values as estimators of the truth. The mid-p is shown to minimize the maximum risk for one-sided and two-sided tests.

A recent comparative analysis of alternative interval estimation approaches and procedures has shown that confidence intervals (CIs) for true raw scores determined with the Score method—which uses the normal approximation to the binomial distribution—have actual coverage probabilities that are closest to their nominal level. It has also recently been shown that the skew-normal distribution yields a better approximation to the binomial distribution than the normal distribution. This article thus evaluates the benefits of using the skew-normal approximation for interval estimation of true scores. Three different CIs for true scores based on the skew-normal approximation are considered, and a simulation study is conducted whose results reveal that none of these skew-normal CIs is more accurate than the conventional Score CI in terms of probability coverage. Thus, skew-normal CIs, which incur a substantial computational cost, do not seem to be an advisable alternative for interval estimation of true scores.

New tests based on the ratio of generalized variances are presented to compare covariance matrices from dependent normal populations. Monte Carlo simulation concluded that the tests considered controlled the Type I error, providing empirical probabilities that were consistent with the nominal level stipulated.

We propose a sample size calculation approach for the estimation of sensitivity and specificity of diagnostic tests with multiple observations per subjects. Many diagnostic tests such as diagnostic imaging or periodontal tests are characterized by the presence of multiple observations for each subject. The number of observations frequently varies among subjects in diagnostic imaging experiments or periodontal studies. Nonparametric statistical methods for the analysis of clustered binary data have been recently developed by various authors. In this paper, we derive a sample size formula for sensitivity and specificity of diagnostic tests using the sign test while accounting for multiple observations per subjects. Application of the sample size formula for the design of a diagnostic test is discussed. Since the sample size formula is based on large sample theory, simulation studies are conducted to evaluate the finite sample performance of the proposed method. We compare the performance of the proposed sample size formula with that of the parametric sample size formula that assigns equal weight to each observation. Simulation studies show that the proposed sample size formula generally yields empirical powers closer to the nominal level than the parametric method. Simulation studies also show that the number of subjects required increases as the variability in the number of observations per subject increases and the intracluster correlation increases.

The ANOVA method and permutation tests, two heritages of Fisher, have been extensively studied. Several permutation strategies have been proposed by others to obtain a distribution-free test for factors in a fixed effect ANOVA (i.e., single error term ANOVA). The resulting tests are either approximate or exact. However, there exists no universal exact permutation test which can be applied to an arbitrary design to test a desired factor. An exact permutation strategy applicable to fixed effect analysis of variance is presented. The proposed method can be used to test any factor, even in the presence of higher-order interactions. In addition, the method has the advantage of being applicable in unbalanced designs (all-cell-filled), which is a very common situation in practice, and it is the first method with this capability. Simulation studies show that the proposed method has an actual level which stays remarkably close to the nominal level, and its power is always competitive. This is the case even with very small datasets, strongly unbalanced designs and non-Gaussian errors. No other competitor show such an enviable behavior.

While intent-to-treat (ITT) analysis is widely accepted for superiority trials, there remains debate about its role in non-inferiority trials. It has often been said that ITT analysis tends to be anti-conservative in demonstrating non-inferiority, suggesting that per-protocol (PP) analysis may be preferable for non-inferiority trials, despite the inherent bias of such analyses. We propose using randomization-based g-estimation analyses that more effectively preserve the integrity of randomization than do the more widely used PP analyses. Simulation studies were conducted to investigate the impacts of different types of treatment changes on the conservatism or anti-conservatism of analyses using the ITT, PP, and g-estimation methods in a time-to-event outcome. The ITT results were anti-conservative for all simulations. Anti-conservativeness increased with the percentage of treatment change and was more pronounced for outcome-dependent treatment changes. PP analysis, in which treatment-switching cases were censored at the time of treatment change, maintained type I error near the nominal level for independent treatment changes, whereas for outcome-dependent cases, PP analysis was either conservative or anti-conservative depending on the mechanism underlying the percentage of treatment changes. G-estimation analysis maintained type I error near the nominal level even for outcome-dependent treatment changes, although information on unmeasured covariates is not used in the analysis. Thus, randomization-based g-estimation analyses should be used to supplement the more conventional ITT and PP analyses, especially for non-inferiority trials.

The size properties of a two-stage test in a panel data model are investigated where in the …rst stage a Hausman (1978) speci…cation test is used as a pretest of the random eects speci…cation and in the second stage, a simple hypothesis about a component of the parameter vector is tested, using a t-statistic that is based on either the random eects or the …xed eects estimator depending on the outcome of the Hausman pretest. It is shown that the asymptotic size of the two-stage test depends on the degree of time variation in the regressors and on the variance of the error term relative to the variance of the individual speci…c eect and equals 1 for empirically relevant speci…cations of the parameter space. Monte carlo simulations document that the size distortion is well re‡ected in …nite samples. The size distortion is caused mainly by the poor power properties of the pretest that lead to frequent unjusti…ed inference based on the random eects estimator in the second stage. However, it is also shown that the conditional size of the test, conditional on the Hausman pretest rejecting the pretest null hypothesis, exceeds the nominal level of the test. Given the results in the paper, the recommendation then is to use a t-statistic based on the …xed eects estimator instead of using the two-stage procedure.

This paper focuses on the model averaging method for semiparametric models.We aim to extend the framework of the parameter models given by Hjort and Claeskens in 2003.Although Claeskens and Carroll considered exactly the same 问题 in 2007, the two methods are not identical.We derive the asymptotic distribution of the model averaging estimator,and also develop a confidence interval with an actual coverage probability that tends toward the nominal level in large samples.Both simulation study and real data analysis show that the proposed model averaging method performs well.

Introduction A common quality indicator for monitoring and comparing hospitals is based on death within 30 days of admission. An important use is to determine whether a hospital has higher or lower mortality than other hospitals. Thus, the ability to identify such outliers correctly is essential. Two approaches for detection are: 1) calculating the ratio of observed to expected number of deaths (OE) per hospital and 2) including all hospitals in a logistic regression (LR) comparing each hospital to a form of average over all hospitals. The aim of this study was to compare OE and LR with respect to correctly identifying 30-day mortality outliers. Modifications of the methods, i.e., variance corrected approach of OE (OE-Faris), bias corrected LR (LR-Firth), and trimmed mean variants of LR and LR-Firth were also studied. Materials and methods To study the properties of OE and LR and their variants, we performed a simulation study by generating patient data from hospitals with known outlier status (low mortality, high mortality, non-outlier). Data from simulated scenarios with varying number of hospitals, hospital volume, and mortality outlier status, were analysed by the different methods and compared by level of significance (ability to falsely claim an outlier) and power (ability to reveal an outlier). Moreover, administrative data for patients with acute myocardial infarction (AMI), stroke, and hip fracture from Norwegian hospitals for 2012–2014 were analysed. Results None of the methods achieved the nominal (test) level of significance for both low and high mortality outliers. For low mortality outliers, the levels of significance were increased four- to fivefold for OE and OE-Faris. For high mortality outliers, OE and OE-Faris, LR 25% trimmed and LR-Firth 10% and 25% trimmed maintained approximately the nominal level. The methods agreed with respect to outlier status for 94.1% of the AMI hospitals, 98.0% of the stroke, and 97.8% of the hip fracture hospitals. Conclusion We recommend, on the balance, LR-Firth 10% or 25% trimmed for detection of both low and high mortality outliers.

We consider a clinical trial with a primary and a secondary endpoint where the secondary endpoint is tested only if the primary endpoint is significant. The trial uses a group sequential procedure with two stages. The familywise error rate (FWER) of falsely concluding significance on either endpoint is to be controlled at a nominal level α. The Type I error rate for the primary endpoint is controlled by choosing any α-level stopping boundary, e.g., the standard O’Brien-Fleming or the Pocock boundary. Given any particular α-level boundary for the primary endpoint, we study the problem of determining the boundary for the secondary endpoint to control the FWER. We study this FWER analytically and numerically and find that it is maximized when the correlation coefficient ρ between the two endpoints equals 1. For the four combinations consisting of O’Brien-Fleming and Pocock boundaries for the primary and secondary endpoints, the critical constants required to control the FWER are computed for different values of ρ. An ad-hoc boundary is proposed for the secondary endpoint to address a practical concern that may be at issue in some applications. Numerical studies indicate that the O’Brien-Fleming boundary for the primary endpoint and the Pocock boundary for the secondary endpoint generally gives the best primary as well as secondary power performance. The Pocock boundary may be replaced by the ad-hoc boundary for the secondary endpoint with a very little loss of secondary power if the practical concern is at issue. A clinical trial example is given to illustrate the methods.

Summary The purpose of toxicological studies is a safety assessment of compounds (e.g. pesticides, pharmaceuticals, industrial chemicals and food additives) at various dose levels. Because a mistaken declaration that a really non-equivalent dose is equivalent could have dangerous consequences, it is important to adopt reliable statistical methods that can properly control the family-wise error rate. We propose a new stepwise confidence interval procedure for toxicological evaluation based on an asymmetric loss function. The new procedure is shown to be reliable in the sense that the corresponding family-wise error rate is well controlled at or below the pre-specified nominal level. Our simulation results show that the new procedure is to be preferred over the classical confidence interval procedure and the stepwise procedure based on Welch's approximation in terms of practical equivalence/safety. The implementation and significance of the new procedure are illustrated with two real data sets: one from a reproductive toxicological study on Nitrofurazone in Swiss CD-1 mice, and the other from a toxicological study on Aconiazide.

PurposeThe purpose of this paper is to improve the measurement of nominal level and actual strength of China's intellectual property protection (IPP), and examine whether the increase of actual protection strength (APS) is positive or negative impact on China's provincial economic growth.Design/methodology/approachA modified approach, based on Ginarte‐Park's (GP's) and HL's approaches, is used to measure nominal level and APS of China's intellectual property rights (IPR) from 1995 to 2007. The pooled EGLS method (cross‐section fixed effect) is used to estimate the effect of China's IPP and other variables on provincial economic growth.FindingsThe paper proves that China's APS appears an increase with a phase. China's IPP level by GP approach is on the high side, whereas China's IPP level by HL approach is slightly on the low side. Nominal level of China's IPP is largely influenced by the legislation level, whereas APS mostly embodies the effect of implementing law level. The increase of China's APS has significant positive impact on provincial economic growth. However, at the outset of building an independent innovation country, too strong IPP is bad for the development of innovation capability, and bad for provincial economic growth.Research limitations/implicationsBecause the APS is unknown, it is impossible to use APS as the dependent variable to estimate the weights of the main influencing factors. The method that the paper assumes three main factors the same weights is second best choice. Thus, several different weights are supplemented to measure the distribution values of China's APS.Practical implicationsChina's APS quantified by a modified approach and strong evidences can be used to estimate the effect on economic growth. Policy effectiveness could be maximized at seeking the endogenous benefit balance between strengthening IPP and promoting economic development.Originality/valueThe paper proposes a modified approach to measure the APS of China's IPR, and proves that the reinforcement of China's APS is beneficial to promoting provincial economic growth. However, at the outset, too strong IPP is harmful.

This article describes an adaptive test which can be used to analyze repeated measures data that have some missing values. Several tests for parallelism and for group effect are compared. The multivariate adaptive test for parallelism is compared to the likelihood ratio test and to a test based on the mixed model approach. The simulation results show that all of these tests had significance levels that were at or below their nominal level of significance, but they also demonstrate that there are large differences in power between these tests. Unless the researcher knew that the errors were multivariate normal the adaptive test for parallelism is recommended, even if there are values that are missing at random or missing completely at random. A univariate adaptive test for group effect is recommended unless the random effects were known to be approximately normally distributed.

We assume X and Y be two independent random variables and define θ=P (X < Y ). The inference for θ can be found in various fields. This paper not only compares several methods for constructing the confidence interval for θ in a small sample but also proposes some new methods. The intervals derived by these new methods show good performance in a small sample, and their actual coverage probability is close to the nominal level. In addition, one of the biggest advantages of our methods is that it does not require complicated calculations.

Several copula goodness-of-fit approaches are examined, three of which are proposed in this paper. Results are presented from an extensive Monte Carlo study, where we examine the effect of dimension, sample size and strength of dependence on the nominal level and power of the different approaches. While no approach is always the best, some stand out and conclusions and recommendations are made. A novel study of p-value variation due to permutation order, for approaches based on Rosenblatt's transformation is also carried out. Results show significant variation due to permutation order for some of the approaches based on this transform. However, when approaching rejection regions, the additional variation is negligible.

The score test statistics for testing zero inflation and covariance parameter are proposed in the bivariate zero-inflated Poisson (BZIP) regression model. The Monte Carlo studies show that the score test and likelihood ratio test for testing zero inflation underestimate the nominal significance level, while the score test for covariance parameter keeps the significance level close to the nominal one. To overcome this nominal level underestimation, we propose a bootstrap method of the score test for the testing problem of zero inflation. An empirical example with covariates is provided to illustrate the results. In addition, score test for zero inflation is also proposed in the BZIP model, which allows a flexible dependence structure using copula.

This paper considers inference based on a test statistic that has a limit distribution that is discontinuous in a parameter. The paper shows that subsampling and m out of n bootstrap tests based on such a test statistic often have asymptotic size—defined as the limit of exact size—that is greater than the nominal level of the tests. This is due to a lack of uniformity in the pointwise asymptotics. We determine precisely the asymptotic size of such tests under a general set of high-level conditions that are relatively easy to verify. The results show that the asymptotic size of subsampling and m out of n bootstrap tests is distorted in some examples but not in others.