Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ²-distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ²-approximation for the proposed tests substantially.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

Rank-based empirical likelihood inference on medians of k populations

We propose a nonparametric method, called rank-based empirical likelihood (REL), for making inferences on medians and cumulative distribution functions (CDFs) of k populations. The standard distribution-free approach to testing the equality of k medians requires that the k population distributions have the same shape. Our REL-ratio (RELR) test for this problem requires fewer assumptions and can effectively use the symmetry information when the distributions are symmetric. Furthermore, our RELR statistic does not require estimation of variance, and achieves asymptotic pivotalness implicitly. When the k populations have equal medians we show that the REL method produces valid inferences for the common median and CDFs of k populations. Simulation results show that the REL approach works remarkably well in finite samples. A real data example is used to illustrate the proposed REL method.     

A two sample ordered alternative test for means and variances

A two sample test of likelihood ratio type is proposed, assuming normal distribution theory, for testing the hypothesis that two samples come from identical normal populations versus the alternative that the populations are normal but vary in mean value and variance with one population having a smaller mean and smaller variance than the other. The small sample and large sample distribution of the proposed statistic are derived assuming normality. Some computations are presented which show the speed of convergence of small sample critical values to their asymptotic counterparts. Comparisons of local power of the proposed test are made with several potential competing tests. Asymptotics for the test statistic are derived when underlying distributions are not necessarily normal.     

Hypothesis Testing in Functional Comparative Calibration Models

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

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.     

A note on a parametric extension of rank tests to the analysis of multifactor experiments within blocks

A general rank test procedure based on an underlying multinomial distribution is suggested for randomized block experiments with multifactor treatment combinations within each block. The Wald statistic for the multinomial is used to test hypotheses about the within–block rankings. This statistic is shown to be related to the one–sample Hotellingt's T² statistic, suggesting a method for computing the test statistic using the standard statistical computer packages.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.     

Computing Wald criteria for nested hypotheses

We present a numerically convenient procedure for computing Wald criteria for nested hypotheses. Similar to Szroeter’s (1983) generalized Wald test, the suggested procedure does not require explicit derivation of the restrictions implied by the null hypothesis and hence its use might eliminate an intricate step in testing linear and nonlinear hypotheses. We show that the traditional Wald test, Szroeter’s (1983) generalized Wald test and our procedure are asymptotically equivalent under H₀. A class of nonlinear transformations of the restrictions for which the Wald statistic is asymptotically invariant is discussed. Finally, we illustrate the use of our procedure for testing the common factor restrictions in a dynamic regression model.     

Small-sample likelihood inference in extreme-value regression models

We deal with a general class of extreme-value regression models introduced by Barreto-Souza and Vasconcellos [Bias and skewness in a general extreme-value regression model, Comput. Statist. Data Anal. 55 (2011), pp. 1379–1393]. Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as χ² with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite-sample performance of the three classical tests (likelihood ratio, Wald, and score), the gradient test that has been recently proposed by Terrell [The gradient statistic, Comput. Sci. Stat. 34 (2002), pp. 206–215], and the adjusted likelihood ratio test obtained in this article. Our simulations favour the latter. Applications of our results are presented.     

Sample Size Determination for Logistic Regression: A Simulation Study

This article considers the different methods for determining sample sizes for Wald, likelihood ratio, and score tests for logistic regression. We review some recent methods, report the results of a simulation study comparing each of the methods for each of the three types of test, and provide Mathematica code for calculating sample size. We consider a variety of covariate distributions, and find that a calculation method based on a first order expansion of the likelihood ratio test statistic performs consistently well in achieving a target level of power for each of the three types of test.     

Gini index based goodness-of-fit test for the logistic distribution

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.     

The indirect method for stochastic logistic growth models

We developed the indirect method for stochastic logistic growth models involving both birth and death rates in the drift and diffusion coefficients, and not only propose two indirect estimators, but also construct a likelihood ratio-type indirect statistic for testing hypotheses concerning parameters. Simulations show that the proposed two indirect estimators can correct the discretization bias, and the proposed indirect test possesses very good estimated power and size.     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

Calculating power for the comparison of dependent κ-coefficients

Summary. In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ -statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ -statistics. The Wald test statistic has a non-central χ ²-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.     

Comparison of treatment regimes with adjustment for auxiliary variables

Treatment regimes are algorithms for assigning treatments to patients with complex diseases, where treatment consists of more than one episode of therapy, potentially with different dosages of the same agent or different agents. Sequentially randomized clinical trials are usually designed to evaluate and compare the effect of different treatment regimes. In such designs, eligible patients are first randomly assigned to receive one of the initial treatments. Patients meeting some criteria (e.g. no progressive disease) are then randomized to receive one of the maintenance treatments. Usually, the procedure continues until all treatment options are exhausted. Such multistage treatment assignment results in treatment regimes consisting of initial treatments, intermediate responses and second-stage treatments. However, methods for efficient analysis of sequentially randomized trials have only been developed very recently. As a result, earlier clinical trials reported results based only on the comparison of stage-specific treatments. In this article, we propose a model that applies to comparisons of any combination of any number of treatment regimes regardless of the number of stages of treatment adjusted for auxiliary variables. Contrasts of treatment regimes are tested using the Wald chi-square method. Both the model and Wald chi-square tests of contrasts are illustrated through a simulation study and an application to a high-risk neuroblastoma study to complement the earlier results reported on this study.     

Multiple comparison procedures for finding non-maximum normal means

Abstract

In this study, we discuss multiple comparison procedures for finding normal means which are not maximum among several normal means. Specifically, we propose the single step procedure, the sequentially rejective step down procedure and the step up procedure. For the single step procedure we determine the critical value for a specified significance level. For the sequentially rejective step down procedure and the step up procedure we determine the critical value at each step of the test for a specified significance level. For three procedures we formulate the power of the test under a specified alternative hypothesis. We give some numerical examples regarding critical values and power of the test intended to compare three procedures.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Wald statistic for iiomogeneity of multiple parameters

The paper proposes Wald statistic for testing homogeneity of parameters of k populations and gives the asymptotic expansion of the distribution function under a null hypothesis and of power function of it under contiguous alternatives. It shows that Wald statistic is the same as Nagao's statistic (1973) for testing homogeneity of covariance matrices of multivariate normal populations.