The effect of number of clusters and cluster size on statistical power and Type I error rates when testing random effects variance components in multilevel linear and logistic regression models

When using multilevel regression models that incorporate cluster-specific random effects, the Wald and the likelihood ratio (LR) tests are used for testing the null hypothesis that the variance of the random effects distribution is equal to zero. We conducted a series of Monte Carlo simulations to examine the effect of the number of clusters and the number of subjects per cluster on the statistical power to detect a non-null random effects variance and to compare the empirical type I error rates of the Wald and LR tests. Statistical power increased with increasing number of clusters and number of subjects per cluster. Statistical power was greater for the LR test than for the Wald test. These results applied to both the linear and logistic regressions, but were more pronounced for the latter. The use of the LR test is preferable to the use of the Wald test.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Score Tests for Testing Independence in the Zero-Truncated Bivariate Poisson Models

In this study, score test statistics for testing independence in the zero-truncated bivariate Poisson distributions are proposed. The Monte Carlo study shows that the score tests proposed in this article keep the significance level close to the nominal one, but the LR and Wald tests over-reject the null hypothesis when it is true. The score tests for testing independence in the zero-truncated bivariate Poisson regression models are also derived in this study.     

Jackknife empirical likelihood tests for distribution functions

It has been a long history for testing whether the underlying distribution belongs to a particular family. In this paper, we propose some jackknife empirical likelihood tests via estimating equations. The proposed new tests allow one to add more relevant constraints so as to improve the powers. A simulation study shows the effectiveness of the new tests.     

Simulated classical tests in multinomial probit models

This paper compares the application of different versions of the simulated counterparts of the Wald test, the score test, and the likelihood ratio test in one- and multiperiod multinomial probit models. Monte Carlo experiments show that the use of the simple form of the simulated likelihood ratio test delivers relatively robust results regarding the testing of several multinomial probit model specifications. In contrast, the inclusion of the Hessian matrix of the simulated loglikelihood function into the simulated score test and (in the multiperiod multinomial probit model) particularly the inclusion of the quasi-maximum likelihood theory into the simulated likelihood ratio test leads to substantial computational problems. The combined application of the quasi-maximum likelihood theory with the simulated Wald test or the simulated score test is not systematically superior to the application of the other versions of these two simulated classical tests either. Neither an increase in the number of observations nor in the number of random draws in the incorporated Geweke-Hajivassiliou-Keane simulator systematically lead to more precise conformities between the frequencies of type I errors and the basic significance levels. An increase in the number of observations only decreases the frequencies of type II errors, particularly regarding the simulated classical testing of multiperiod multinomial probit model specifications.     

A comment on “testing the lucas critique: A review ”

We derive general distribution tests based on the method of maximum entropy (ME) density. The proposed tests are derived from maximizing the differential entropy subject to given moment constraints. By exploiting the equivalence between the ME and maximum likelihood (ML) estimates for the general exponential family, we can use the conventional likelihood ratio (LR), Wald, and Lagrange multiplier (LM) testing principles in the maximum entropy framework. In particular, we use the LM approach to derive tests for normality. Monte Carlo evidence suggests that the proposed tests are compatible with and sometimes outperform some commonly used normality tests. We show that the proposed tests can be extended to tests based on regression residuals and non-i.i.d. data in a straightforward manner. An empirical example on production function estimation is presented.     

Local power and size properties of the LR,Wald, score and gradient tests in dispersion models

We derive asymptotic expansions for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

面板数据单位根似然比检验研究

本文采用似然比类检验统计量进行面板单位根检验（简称为LR检验）研究,在局部备择假设成立的条件下,推导了其在无确定项、仅含截距项以及存在线性时间趋势项三种模型下所对应的渐近分布与局部渐近势函数。Monte Carlo模拟结果显示,当面板数据中含确定项（截距项或时间趋势项）时,LR检验水平比LLC和IPS检验水平更接近于给定的显著性检验水平;此外,当面板数据中包含发散个体时,经水平修正后的LR检验势要远远高于经水平修正后的LLC与IPS检验势,其中,经水平修正后的LLC与IPS检验势接近于零。     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Rényi statistics for testing composite hypotheses in general exponential models

We introduce a family of Rényi statistics of orders r?∈?R for testing composite hypotheses in general exponential models, as alternatives to the previously considered generalized likelihood ratio (GLR) statistic and generalized Wald statistic. If appropriately normalized exponential models converge in a specific sense when the sample size (observation window) tends to infinity, and if the hypothesis is regular, then these statistics are shown to be χ²-distributed under the hypothesis. The corresponding Rényi tests are shown to be consistent. The exact sizes and powers of asymptotically α-size Rényi, GLR and generalized Wald tests are evaluated for a concrete hypothesis about a bivariate Lévy process and moderate observation windows. In this concrete situation the exact sizes of the Rényi test of the order r?=?2 practically coincide with those of the GLR and generalized Wald tests but the exact powers of the Rényi test are on average somewhat better.     

Multi-sample tests for axial data from Watson distributions

The Watson distribution is one of the most used distributions for modeling axial data. In some situations, it is important to investigate if several Watson populations differ significantly. In this paper, we develop likelihood ratio tests and the ANOVA for testing the hypothesis of the equality of the directional parameters of several Watson distributions with different concentrations. We also determine the empirical power of the ANOVA and LR tests for some dimensions of the sphere.     

A monte carlo study of the wald,lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.     

A Monte Carlo comparison of Jarque–Bera type tests and Henze–Zirkler test of multivariate normality

In the paper, tests for multivariate normality (MVN) of Jarque-Bera type, based on skewness and kurtosis, have been considered. Tests proposed by Mardia and Srivastava, and the combined tests based on skewness and kurtosis defined by Jarque and Bera have been taken into account. In the Monte Carlo simulations, for each combination of p = 2, 3, 4, 5 number of traits and n = 10(5)50(10)100 sample sizes 10,000 runs have been done to calculate empirical Type I errors of tests under consideration, and empirical power against different alternative distributions. Simulation results have been compared to the Henze–Zirkler’s test. It should be stressed that no test yet proposed is uniformly better than all the others in every combination of conditions examined.     

Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood

The mixed linear model is a popular method for analysing unbalanced repeated measurement data. The classical statistical tests for parameters in this model are based on asymptotic theory that is unreliable in the small samples that are often encountered in practice. For testing a given fixed effect parameter with a small sample, we develop and investigate refined likelihood ratio (LR) tests. The refinements considered are the Bartlett correction and use of the Cox–Reid adjusted likelihood; these are examined separately and in combination. We illustrate the various LR tests on an actual data set and compare them in two simulation studies. The conventional LR test yields type I error rates that are higher than nominal. The adjusted LR test yields rates that are lower than nominal, with absolute accuracy similar to that of the conventional LR test in the first simulation study and better in the second. The Bartlett correction substantially improves the accuracy of the type I error rates with either the conventional or the adjusted LR test. In many cases, error rates that are very close to nominal are achieved with the refined methods.     

Empirical study of some non-parametric tests for dispersion of correlated data

Results of a computer simulation study of power and robustness of three competitor tests for comparing scales, for use with correlated data: Rothstein, Richardson and Bell (RRB), Arvesen, and Pitman, are presented. It is found that unless one could ímprove the approximate null distributions for Arvesen's and Pitman's test, RRB's procedure is best, having simulated probabilities of Type I error closest to the test's nominal α and being reasonably robust and powerful, for all distributions considered.     

Modified regression coefficient analysis for repeated binary measurements

Myers & Broyles (2000a, 2000b) illustrate that regression coefficient analysis (RCA) is a viable alternative to a generalized estimating equation (GEE) in the analysis of correlated binomial data. Since the regression coefficients (b i ' s ) may have different precisions, we modify RCA by weighting b i ' s by the inverses of their variances for statistical optimality. We perform the simulation study to evaluate the performance of RCA, modified RCA and GEE in terms of empirical type I errors and empirical powers of the regression coefficients in repeated binary measurement designs with and without dropouts. Two thousand data sets are generated using autoregressive (AR(1)) and compound symmetry (CS) correlation structures. We compare the type I errors and powers of RCA, modified RCA and GEE for the analysis of repeated binary measurement data as affected by different dropout mechanisms such as random dropouts and treatment dependent dropouts.     

Empirical likelihood test via estimating equations

It has been a long history for testing whether the underlying distribution belongs to a particular one or a parametric class of distributions. In this paper, we propose some empirical likelihood ratio tests via estimating equations. The proposed new tests allow one to add more relevant constraints so as to improve the powers. A simulation study shows the effectiveness of the new tests. The new method is then used to test employer size and market value distributions of US firms.     

Comparison of powers and exact bahadur slopes of some tests for dimensionality in manova models

In this paper we consider test of dimensionality in MANOVA model. For this testing problem, the likelihood ratio (=LR) test, Lawley-Hotelling (=LH) type test and Bartlett-Nanda-Pillai (=BNP) type test are often used. We obtain the asymptotic expansions of powers of these tests under the local alternatives. Also Bahadur exact slopes of these tests are obtained. Based on these results, we obtain a unified opinion concerning comparison of LR test, LH type test and BNP type test.     

Some tests for a poisson process and their powers with respect to trend in the intensity

Recently many authors have worked on Wei bull process in the area of modelling and analysis. Much less work is done in the area of testing of hypothesis. In this article, some tests for testing the Poisson process against a class of Wei bull process based on the conditional distribution of observations given the sufficient statistic, are proposed. The percentage points of the distributions of the proposed test statistics are simulated. The powers of the tests under alternatives are computed by Monte Carlo method. It is seen that the suggested tests perform well for decreasing intensities.     

Goodness-of-fit tests for lifetime distributions based on Type II censored data

In this article, the general test statistic introduced by Alizadeh Noughabi and Balakrishnan [Goodness of fit using a new estimate of Kullback-Leibler information based on Type II censored data. IEEE Trans Reliab. 2015;64:627–635.] is applied for testing goodness of fit of lifetime distributions based on Type II censored data. The test statistic is constructed based on an estimate of Kullback–Leibler (KL) information. We investigate the properties of the proposed test statistic such as the test statistic is nonnegative, just like KL information. We apply this test statistic to following distributions: Exponential, Weibull, Log-normal and Pareto. The critical values and Type I error of the proposed tests are obtained. It is shown that the proposed tests have an excellent Type I error and hence can be used confidently in practice. Then, by Monte Carlo simulations, the power values of the proposed tests are computed against several alternatives and compared with those of the existing tests. Finally, some real-world reliability data are used for illustrative purpose.