Simple Stepwise Tests of Hypotheses and Multiple Comparisons

Holm's (1979) step-down and Hochberg's (1988) step-up procedures for tests of multiple hypotheses are simple to apply and are widely used. Holm's procedure controls the familywise error rate (FWE), while Hochberg's is more powerful. This paper investigates a step-down procedure (labelled CS) of Seneta & Chen (1997) which is a sharpening of Holm's, takes into account the degree of association between test statistics, and also controls the FWE. Computation for the CS procedure may be minimized by using the procedure as an adjustment to Holm's. The computational steps are detailed, and the adjustment is then illustrated by an application to a text-book example of multiple comparisons, in which step-wise procedures are shown to perform better than the usual Tukey T -comparison. Simulation investigations in a standard comparison with a control setting show that the CS step–down procedure is more powerful than Hochberg's step-up procedure and the procedure of Simes (1986), especially in regard to error rate, and not much less powerful than an optimal, but very specific, step-up procedure of Dunnett & Tamhane (1992).     

Generalizing weak instrument robust IV statistics towards multiple parameters,unrestricted covariance matrices and identification statistics

We generalize the weak instrument robust score or Lagrange multiplier and likelihood ratio instrumental variables (IV) statistics towards multiple parameters and a general covariance matrix so they can be used in the generalized method of moments (GMM). The GMM extension of Moreira's [2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio statistic towards GMM preserves its expression except that it becomes conditional on a statistic that tests the rank of a matrix. We analyze the spurious power decline of Kleibergen's [2002. Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803, 2005. Testing parameters in GMM without assuming that they are identified. Econometrica 73, 1103–1124] score statistic and show that an independent misspecification pre-test overcomes it. We construct identification statistics that reflect if the confidence sets of the parameters are bounded. A power study and the possible shapes of confidence sets illustrate the analysis.     

Unconditional tests for association in 2 × 2 contingency tables in the total sum fixed design

The asymptotic approach and Fisher's exact approach have often been used for testing the association between two dichotomous variables. The asymptotic approach may be appropriate to use in large samples but is often criticized for being associated with unacceptable high actual type I error rates for small to medium sample sizes. Fisher's exact approach suffers from conservative type I error rates and low power. For these reasons, a number of exact unconditional approaches have been proposed, which have been seen to be generally more powerful than exact conditional counterparts. We consider the traditional unconditional approach based on maximization and compare it to our presented approach, which is based on estimation and maximization. We extend the unconditional approach based on estimation and maximization to designs with the total sum fixed. The procedures based on the Pearson chi‐square, Yates's corrected, and likelihood ratio test statistics are evaluated with regard to actual type I error rates and powers. A real example is used to illustrate the various testing procedures. The unconditional approach based on estimation and maximization performs well, having an actual level much closer to the nominal level. The Pearson chi‐square and likelihood ratio test statistics work well with this efficient unconditional approach. This approach is generally more powerful than the other p‐value calculation methods in the scenarios considered.     

Inverse-probability-weighted logrank test for stratified survival data with missing measurements

The stratified logrank test can be used to compare survival distributions of several groups of patients, while adjusting for the effect of some discrete variable that may be predictive of the survival outcome. In practice, it can happen that this discrete variable is missing for some patients. An inverse-probability-weighted version of the stratified logrank statistic is introduced to tackle this issue. Its asymptotic distribution is derived under the null hypothesis of equality of the survival distributions. A simulation study is conducted to assess behavior of the proposed test statistic in finite samples. An analysis of a medical dataset illustrates the methodology.     

A directional test of exponentiality based on maximum correlations

In Fortiana and Grané (J Stat Plann Infer 108:85–97), we study a scale-free statistic, based on Hoeffding’s maximum correlation, for testing exponentiality. This statistic admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence can be written as a quotient of L-statistics. In this paper, we propose a scale-free adaptive statistic for testing exponentiality with optimal power against a specific alternative and obtain its exact distribution. An empirical power study shows that the test based on this new statistic has the same level of performance than the best tests in the statistical literature.     

基于Sieve Bootstrap方法的季节时间序列模型检验问题研究

研究目标：完善季节时间序列模型建模理论,解决建模过程烦琐、各类检验方法的结论差异大以及模型误设定问题。研究方法：基于对各季节时间序列模型的数理分析及比较,提出合理的模型检验程序;再运用Sieve Bootstrap方法,给出季节性单位根检验及确定性季节过程检验的统计量的临界值,并比较基于Sieve Bootstrap的检验方法与HEGY检验、BT检验的异同。研究发现：本文提出的检验程序能有效识别模型,检验统计量有限样本性质优良;实证分析表明,本文提出的检验程序及方法能更有效地识别中国宏观经济数据中的季节性。研究创新：将Sieve Bootstrap方法应用于季节时间序列的平稳性检验及趋势性检验中。研究价值：提出季节时间序列模型检验程序及检验方法,促进其在季节性经济数据中的应用。     

Tests based on φ-divergences for bivariate symmetry

In this paper we introduce a family of test statistics for testing symmetry based on φ-divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistic as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to see that some new test statistics offer an attractive alternative to the classical Pearson test statistic for the problem of symmetry. Received: May 2000     

On specification testing of ordered discrete choice models

We discuss how to test the specification of an ordered discrete choice model against a general alternative. Two main approaches can be followed: tests based on moment conditions and tests based on comparisons between parametric and nonparametric estimations. Following these approaches, various statistics are proposed and their asymptotic properties are discussed. The performance of the statistics is compared by means of simulations. An easy-to-compute variant of the standard moment-based statistic yields the best results in models with a single explanatory variable. In models with various explanatory variables the results are less conclusive, since the relative performance of the statistics depends on both the fit of the model and the type of misspecification that is considered.     

基于参数空间的ST-ECM模型的协整检验问题研究

在非线性平滑转移误差修正模型（ST-ECM）的协整检验中,由于存在未识别参数而使协整检验统计量构造困难,同时由于目前文献普遍使用的泰勒展开近似法并不能精确替代原始非线性模型,从而导致协整检验统计量功效较低。本文首先在遍历未识别参数的参数空间的基础上构造了ST-ECM模型协整检验的supF统计量,推导了supF统计量的极限分布并说明了其收敛性质。接着,蒙特卡洛仿真模拟结果显示,supF统计量在ST-ECM模型协整检验中具有良好的检验水平和功效,且supF统计量的功效明显优于EG统计量、F*NEC统计量和inft统计量。最后,本文对亚洲六个国家的利率期限结构预期假说进行了验证,结果表明中国、新加坡和泰国三个国家的利率期限结构预期假说成立且存在非线性调整效应,supF统计量较其他统计量具有更高的检验功效。     

A statistical procedure for testing social reciprocity at group,dyadic and individual levels

This paper examines statistical analysis of social reciprocity, that is, the balance between addressing and receiving behaviour in social interactions. Specifically, it focuses on the measurement of social reciprocity by means of directionality and skew-symmetry statistics at different levels. Two statistics have been used as overall measures of social reciprocity at group level: the directional consistency and the skew-symmetry statistics. Furthermore, the skew-symmetry statistic allows social researchers to obtain complementary information at dyadic and individual levels. However, having computed these measures, social researchers may be interested in testing statistical hypotheses regarding social reciprocity. For this reason, it has been developed a statistical procedure, based on Monte Carlo sampling, in order to allow social researchers to describe groups and make statistical decisions.     

Testing for structural change in regression quantiles

Most studies in the structural change literature focus solely on the conditional mean, while under various circumstances, structural change in the conditional distribution or in conditional quantiles is of key importance. This paper proposes several tests for structural change in regression quantiles. Two types of statistics are considered, namely, a fluctuation type statistic based on the subgradient and a Wald type statistic, based on comparing parameter estimates obtained from different subsamples. The former requires estimating the model under the null hypothesis, and the latter involves estimation under the alternative hypothesis. The tests proposed can be used to test for structural change occurring in a pre-specified quantile, or across quantiles, which can be viewed as testing for change in the conditional distribution with a linear specification of the conditional quantile function. Both single and multiple structural changes are considered. We derive the limiting distributions under the null hypothesis, and show they are nuisance parameter free and can be easily simulated. A simulation study is conducted to assess the size and power in finite samples.     

A cautionary note on methods of comparing programmatic efficiency between two or more groups of DMUs in data envelopment analysis

In some applications of data envelopment analysis (DEA) there may be doubt as to whether all the DMUs form a single group with a common efficiency distribution. The Mann–Whitney rank statistic has been used to evaluate if two groups of DMUs come from a common efficiency distribution under the assumption of them sharing a common frontier and to test if the two groups have a common frontier. These procedures have subsequently been extended using the Kruskal–Wallis rank statistic to consider more than two groups. This technical note identifies problems with the second of these applications of both the Mann–Whitney and Kruskal–Wallis rank statistics. It also considers possible alternative methods of testing if groups have a common frontier, and the difficulties of disaggregating managerial and programmatic efficiency within a non-parametric framework.      

Small sample behaviour of some nonparametric multi-sample location tests in the presence of dispersion differences

This paper describes the small and moderate sample behaviour of three studentized non-parametric statistics by Sen and a modified Wilcoxon statistic by Potthoff for testing the equality of location parameters in the presence of dispersion differences. Based on the agreement between the asymptotic and small sample distributions of the statistics as well as a comparison of their power under heteroscedasticity, the studentized Wilcoxon and the studentized Brown and Mood tests arc preferred in the two-sample and c-sample situations, respectively.     

Small‐scale Inference: Empirical Bayes and Confidence Methods for as Few as a Single Comparison

Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision‐theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect‐size interval estimates are not only effective multiple comparison procedures but also they might replace p‐values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.     

THE ASYMPTOTIC BEHAVIOR OF SIMPLE LINEAR RANK STATISTICS

The concept of a simple linear rank statistic is essential in nonparametric theory. These statistics may be used for testing the null-hypothesis that all observations are independent and identically distributed against classes of alternatives indicated by the choice of regression constants and scores. Special cases are the two-sample Linear rank statistics and Spearman's rank correlation coefficient rho under the null-hypothesis of independence.
In this paper we shall give a survey of the asympotic distribution theory of simple linear rank statistics.     

Building logistic models by means of a non parametric goodness of fit test: a case study

A sequence of logistic models is fitted to data from a Dutch follow-up study on preterm infants (POPS). To examine the adequacy of the model, a recently developed non parametric method to check goodness of fit is applied (le Cessie and Van Houwelingen (1991)). This method uses a test statistic based upon kernel regression methods.
In this paper the problem of choosing a "best" bandwidth, corresponding to the greatest power of the test statistic, is avoided by computing the test statistic for a range of different bandwidths. Testing is then based upon the asymptotic distribution of the maximum of the test statistics.
The testing method is used as a goodness of fit criterion, and the contribution of each individual observation to the test statistic is used as a diagnostic tool to localize deviations of the model, and to determine directions in which the model can be improved.     

Pitfalls of testing non-nested hypotheses by the lagrange multiplier method

In applied research in econometrics a general model determined from the current knowledge of economic theory often establishes a ‘natural’ method of embedding a number of otherwise non-nested hypotheses. Under these circumstances, significant tests of various hypotheses can be carried out within the classical framework, and tests of non-nested or separate families of hypotheses do not require development of new statistical methods. The application of some suitable variant of likelihood ratio testing procedure will be quite appropriate.There are, however, many ocassions in applied econometrics where the hypotheses under consideration are intended to provide genuine rival explanations of the same given phenomenon and the state of economic theory is not such as to furnish us with a general model that contains both of the rival hypotheses in a ‘natural’ and theoretically consistent manner. A number of investigators have advocated that even when a ‘natural’ comprehensive model containing both of the hypotheses under consideration cannot be obtained from theoretical considerations, it is still appropriate to base significant tests of non-nested hypotheses upon a combined model ‘artificially’ constructed from the rival alternatives. Moreover, in a recent paper on the application of Lagrange Multiplier (LM) tests to model specification, T.S. Breusch and A.R. Pagan (1980) have claimed that Cox's test statistic is connected to an LM or ‘score’ statistic derived from the application of the LM method to an exponentially combined model earlier employed by A.C. Atkinson (1970).Although the use of ‘artificially’ constructed comprehensive models fortesting separate families of hypotheses is analytically tempting, nevertheless it is subject to two major difficulties. Firstly, in many cases of interest in econometrics, the structural parameters under the combined hypothesis are not identified. Secondly, the log likelihood function of the artificially constructed model has singularities under both the null and alternative hypotheses.The paper firstly examines the derivation of LM statistics in the case of non-nested hypotheses and shows that Atkinson's general test statistic, or Breusch and Pagan's result, can be regarded as an LM test if the parameters of the alternative hypothesis are known. The paper also shows that unless all the parameters of the combined models are identified, no meaningful test of the separate families of the hypotheses by the artificial embedding procedure is possible, and in the identified case an expression for the LM statistic which avoids the problem of the singularity of the information matrix under the null and the alternative hypotheses is obtained.The paper concludes that none of the artificially embedding procedures are satisfactory for testing non-nested models and should be abandoned. It, however, emphasizes that despite these difficulties associated with the use of artificial embedding procedures, Cox's original statistic (which is not derived as an LM statistic and does not depend on any arbitrary synthetic combination of hypotheses) can still be employed as a useful procedure for testing the rival hypotheses often encountered in applied econometrics.     

Marginal likelihood and unit roots

We develop new tests for the hypothesis of unit roots that are based on the marginal likelihood of the general linear model. The marginal likelihood allows the incorporation of invariance arguments in the likelihood function. It turns out that marginal likelihood tests for unit roots appear to be more powerful than other unit root tests. For some basic models power functions almost coincide with the power envelopes, even in small samples. General correlation structures can be incorporated, either by standard likelihood procedures or by adjustments of the test statistics on the basis of asymptotic distributions.     

Multiple aspect testing for case-control designs

When testing for the equality of two distributions in a case-control design with treatment effects presumed to act possibly on more than one aspect, different tests may be properly considered for testing for different features of a null hypothesis, leading to the multiple aspect testing issue. Two different aspects are therefore of interest: the location-aspect, based on the comparison of location indexes, and the distributional-aspect, based on the comparison of the empirical distribution functions. A simulation study shows that the combined testing procedure exhibits a good robust overall performance, and an application in biomedical research is also presented.     

A model of fractional cointegration, and tests for cointegration using the bootstrap

The paper proposes a framework for modelling cointegration in fractionally integrated processes, and considers methods for testing the existence of cointegrating relationships using the parametric bootstrap. In these procedures, ARFIMA models are fitted to the data, and the estimates used to simulate the null hypothesis of non-cointegration in a vector autoregressive modelling framework. The simulations are used to estimate p-values for alternative regression-based test statistics, including the F goodness-of-fit statistic, the Durbin–Watson statistic and estimates of the residual d. The bootstrap distributions are economical to compute, being conditioned on the actual sample values of all but the dependent variable in the regression. The procedures are easily adapted to test stronger null hypotheses, such as statistical independence. The tests are not in general asymptotically pivotal, but implemented by the bootstrap, are shown to be consistent against alternatives with both stationary and nonstationary cointegrating residuals. As an example, the tests are applied to the series for UK consumption and disposable income. The power properties of the tests are studied by simulations of artificial cointegrating relationships based on the sample data. The F test performs better in these experiments than the residual-based tests, although the Durbin–Watson in turn dominates the test based on the residual d.