Increasing sample sizes do not always increase the power of UMPU-tests for 2×2 tables

We consider the uniformly most powerful unbiased (UMPU) one-sided test for the comparison of two proportions based on sample sizes m and n, i.e., the randomized version of Fisher's exact one-sided test. It will be shown that the power function of the one-sided UMPU-test based on sample sizes m and n can coincide with the power function of the UMPU-test based on sample sizes m+1 and n for certain levels on the entire parameter space. A characterization of all such cases with identical power functions is derived. Finally, this characterization is closely related to number theoretical problems concerning Fermat-like binomial equations. Some consequences for Fisher's original exact test will be discussed, too.     

Bayesian Calibration of p‐Values from Fisher's Exact Test

p‐Values are commonly transformed to lower bounds on Bayes factors, so‐called minimum Bayes factors. For the linear model, a sample‐size adjusted minimum Bayes factor over the class of g‐priors on the regression coefficients has recently been proposed (Held & Ott, The American Statistician 70(4), 335–341, 2016). Here, we extend this methodology to a logistic regression to obtain a sample‐size adjusted minimum Bayes factor for 2 × 2 contingency tables. We then study the relationship between this minimum Bayes factor and two‐sided p‐values from Fisher's exact test, as well as less conservative alternatives, with a novel parametric regression approach. It turns out that for all p‐values considered, the maximal evidence against the point null hypothesis is inversely related to the sample size. The same qualitative relationship is observed for minimum Bayes factors over the more general class of symmetric prior distributions. For the p‐values from Fisher's exact test, the minimum Bayes factors do on average not tend to the large‐sample bound as the sample size becomes large, but for the less conservative alternatives, the large‐sample behaviour is as expected.     

Small‐sample testing inference in symmetric and log‐symmetric linear regression models

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.     

On the convergence of the critical values of uniformly most powerful unbiased tests for two-sided hypotheses

Summary The behavior of the critical values of two-sided uniformly most powerful unbiased tests is studied under the condition of weak convergence of the distribution of the test statistic.     

Multivariate residual‐based finite‐sample tests for serial dependence and ARCH effects with applications to asset pricing models

In this paper, we propose several finite‐sample specification tests for multivariate linear regressions (MLR). We focus on tests for serial dependence and ARCH effects with possibly non‐Gaussian errors. The tests are based on properly standardized multivariate residuals to ensure invariance to error covariances. The procedures proposed provide: (i) exact variants of standard multivariate portmanteau tests for serial correlation as well as ARCH effects, and (ii) exact versions of the diagnostics presented by Shanken ( 1990 ) which are based on combining univariate specification tests. Specifically, we combine tests across equations using a Monte Carlo (MC) test method so that Bonferroni‐type bounds can be avoided. The procedures considered are evaluated in a simulation experiment: the latter shows that standard asymptotic procedures suffer from serious size problems, while the MC tests suggested display excellent size and power properties, even when the sample size is small relative to the number of equations, with normal or Student‐t errors. The tests proposed are applied to the Fama–French three‐factor model. Our findings suggest that the i.i.d. error assumption provides an acceptable working framework once we allow for non‐Gaussian errors within 5‐year sub‐periods, whereas temporal instabilities clearly plague the full‐sample dataset. Copyright © 2009 John Wiley & Sons, Ltd.     

Score tests for zero covariances in recursive linear models for grouped or censored data

The score test statistic for testing whether an error covariance is zero is derived for a normal linear recursive model for fully observed, censored or grouped data. The test, which is obtained by regarding non-zero error covariances as arising from correlated random parameter variation, is shown to be closely related to the Information Matrix test. It turns out that the statistic, which is asymptotically N[0,1] under the null, examines the sample covariance of appropriately defined residuals.     

Evaluating estimation methods for ordinal data in structural equation modeling

This study examined the performance of two alternative estimation approaches in structural equation modeling for ordinal data under different levels of model misspecification, score skewness, sample size, and model size. Both approaches involve analyzing a polychoric correlation matrix as well as adjusting standard error estimates and model chi-squared, but one estimates model parameters with maximum likelihood and the other with robust weighted least-squared. Relative bias in parameter estimates and standard error estimates, Type I error rate, and empirical power of the model test, where appropriate, were evaluated through Monte Carlo simulations. These alternative approaches generally provided unbiased parameter estimates when the model was correctly specified. They also provided unbiased standard error estimates and adequate Type I error control in general unless sample size was small and the measured variables were moderately skewed. Differences between the methods in convergence problems and the evaluation criteria, especially under small sample and skewed variable conditions, were discussed.     

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.     

System‐Equation ADL Test for Threshold Cointegration with an Application to the Term Structure of Interest Rates

This paper proposes a new system‐equation test for threshold cointegration based on a threshold vector autoregressive distributed lag (ADL) model. The new test can be applied when the cointegrating vector is unknown and when weak exogeneity fails. The asymptotic null distribution of the new test is derived, critical values are tabulated and finite‐sample properties are examined. In particular, the new test is shown to have good size, so the bootstrap is not required. The new test is illustrated using the long‐term and short‐term interest rates. We show that the system‐equation model can shed light on both asymmetric adjustment speeds and asymmetric adjustment roles. The latter is unavailable in the single‐equation testing strategy.     

Stratified studies about risk ratios in 2 × 2 tables with structural zero

Incomplete correlated 2 × 2 tables are common in some infectious disease studies and two‐step treatment studies in which one of the comparative measures of interest is the risk ratio (RR). This paper investigates the two‐stage tests of whether K RRs are homogeneous and whether the common RR equals a freewill constant. On the assumption that K RRs are equal, this paper proposes four asymptotic test statistics: the Wald‐type, the logarithmic‐transformation‐based, the score‐type and the likelihood ratio statistics to test whether the common RR equals a prespecified value. Sample size formulae based on hypothesis testing method and confidence interval method are proposed in the second stage of test. Simulation results show that sample sizes based on the score‐type test and the logarithmic‐transformation‐based test are more accurate to achieve the predesigned power than those based on the Wald‐type test. The score‐type test performs best of the four tests in terms of type I error rate. A real example is used to illustrate the proposed methods.     

Comparison of three‐dimensional ROC surfaces for clustered and correlated markers,with a proteomics application

We propose a non‐parametric test to compare two correlated diagnostic tests for a three‐category classification problem. Our development was motivated by a proteomic study where the objectives are to detect glycan biomarkers for liver cancer and to compare the discrimination ability of various markers. Three distinct disease categories need to be identified from this analysis. We therefore chose to use three‐dimensional receiver operating characteristic (ROC) surfaces and volumes under the ROC surfaces to describe the overall accuracy for different biomarkers. Each marker in this study might include a cluster of similar individual markers and thus was considered as a hierarchically structured sample. Our proposed statistical test incorporated the within‐marker correlation as well as the between‐marker correlation. We derived asymptotic distributions for three‐dimensional ROC surfaces and subsequently implemented bootstrap methods to facilitate the inferences. Simulation and real‐data analysis were included to illustrate our methods. Our distribution‐free test may be simplified for paired and independent two‐sample comparisons as well. Previously, only parametric tests were known for clustered and correlated three‐category ROC analyses.     

Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.     

Exact inference for the difference of Laplace location parameters

We consider exact procedures for testing the equality of means (location parameters) of two Laplace populations with equal scale parameters based on corresponding independent random samples. The test statistics are based on either the maximum likelihood estimators or the best linear unbiased estimators of the Laplace parameters. By conditioning on certain quantities we manage to express their exact distributions as mixtures of ratios of linear combinations of standard exponential random variables. This allows us to find their exact quantiles and tabulate them for several sample sizes. The powers of the tests are compared either numerically or by simulation. Exact confidence intervals for the difference of the means corresponding to those tests are also constructed. The exact procedures are illustrated via a real data example.     

Trend‐stationary GNP: evidence from a new exact pointwise most powerful invariant unit root test

There has been a substantial debate whether GNP has a unit root. However, statistical tests have had little success in distinguishing between unit‐root and trend‐reverting specifications because of poor statistical properties. This paper develops a new exact small‐sample, pointwise most powerful unit root test that is invariant to the unknown mean and scale of the time series tested, that generates exact small‐sample critical values, powers and p‐values, that has power which approximates the maximum possible power, and that is highly robust to conditional heteroscedasticity. This test decisively rejects the unit root null hypothesis when applied to annual US real GNP and US real per capita GNP series. This paper also develops a modified version of the test to address whether a time series contains a permanent, unit root process in addition to a temporary, stationary process. It shows that if these GNP series contain a unit root process in addition to the stationary process, then it is most likely very small. Copyright © 2001 John Wiley & Sons, Ltd.     

Relative Efficiency of the Fuzzy p-Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy  p -value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the "complete data" test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed "complete data" relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald-type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy  p -value approach in latent variable or missing data problems and suggest some viable alternatives.     

On the distribution of the sample autocorrelation coefficients

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation-based tests are provided. Actual size properties of the Box–Pierce and Ljung–Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box–Pierce test statistic, we propose an adjusted Box–Pierce test that has a far superior size property than the traditional Box–Pierce and Ljung–Box tests.     

Approximate Bayesian Computation for a Class of Time Series Models

In the following article, we consider approximate Bayesian computation (ABC) for certain classes of time series models. In particular, we focus upon scenarios where the likelihoods of the observations and parameter are intractable, by which we mean that one cannot evaluate the likelihood even up to a non‐negative unbiased estimate. This paper reviews and develops a class of approximation procedures based upon the idea of ABC, but specifically maintains the probabilistic structure of the original statistical model. This latter idea is useful, in that one can adopt or adapt established computational methods for statistical inference. Several existing results in the literature are surveyed, and novel developments with regards to computation are given.     

Precedence-type tests based on record values

Precedence-type tests based on order statistics are simple and efficient nonparametric tests that are very useful in the context of life-testing, and they have been studied quite extensively in the literature; see Balakrishnan and Ng (Precedence-type tests and applications. Wiley, Hoboken, 2006). In this paper, we consider precedence-type tests based on record values and develop specifically record precedence test, record maximal precedence test and record-rank-sum test. We derive their exact null distributions and tabulate some critical values. Then, under the general Lehmann alternative, we derive the exact power functions of these tests and discuss their power under the location-shift alternative. We also establish that the record precedence test is the uniformly most powerful test for testing against the one-parameter family of Lehmann alternatives. Finally, we discuss the situation when we have insufficient number of records to apply the record precedence test and then make some concluding remarks.     

A covariate‐adjusted adaptive design for two‐stage clinical trials with survival data

A randomized two‐stage adaptive design is proposed and studied for allocation of patients to treatments and comparison in a phase III clinical trial with survival time as treatment responses. We consider the possibility of several covariates in the design and analysis. Several exact and limiting properties of the design and the follow‐up inference are studied, both numerically and theoretically. The applicability of the proposed methodology is illustrated by using some real data.     

Panel unit root tests under cross-sectional dependence

In this paper alternative approaches for testing the unit root hypothesis in panel data are considered. First, a robust version of the Dickey-Fuller t -statistic under contemporaneous correlated errors is suggested. Second, the GLS t -statistic is considered, which is based on the t -statistic of the transformed model. The asymptotic power of both tests is compared against a sequence of local alternatives. To adjust for short-run serial correlation of the errors, we propose a pre-whitening procedure that yields a test statistic with a standard normal limiting distribution as N and T tends to infinity. The test procedure is further generalized to accommodate individual specific intercepts or linear time trends. From our Monte Carlo simulations it turns out that the robust OLS t -statistic performs well with respect to size and power, whereas the GLS t -statistic may suffer from severe size distortions in small and moderate sample sizes. The tests are applied to test for a unit root in real exchange rates.