Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

On locally optimal invariant unbiased tests for the variance components ratio in mixed linear models

In the paper the problem of testing of two-sided hypotheses for variance components in mixed linear models is considered. When the uniformly most powerful invariant test does not exist (see e.g. Das and Sinha, in Proceedings of the second international Tampere conference in statistics, 1987; Gnot and Michalski, in Statistics 25:213–223, 1994; Michalski and Zmyślony, in Statistics 27:297–310, 1996) then to conduct the optimal statistical inference on model parameters a construction of a test with locally best properties is desirable, cf. Michalski (in Tatra Mountains Mathematical Publications 26:1–21, 2003). The main goal of this article is the construction of the locally best invariant unbiased test for a single variance component (or for a ratio of variance components). The result has been obtained utilizing Andersson’s and Wijsman’s approach connected with a representation of density function of maximal invariant (Andersson, in Ann Stat 10:955–961, 1982; Wijsman, in Proceedings of fifth Berk Symp Math Statist Prob 1:389–400, 1967; Wijsman, in Sankhyā A 48:1–42, 1986; Khuri et al., in Statistical tests for mixed linear models, 1998) and from generalized Neyman–Pearson Lemma (Dantzig and Wald, in Ann Math Stat 22:87–93, 1951; Rao, in Linear statistical inference and its applications, 1973). One selected real example of an unbalanced mixed linear model is given, for which the power functions of the LBIU test and Wald’s test (the F-test in ANOVA model) are computed, and compared with the attainable upper bound of power obtained by using Neyman–Pearson Lemma.     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Confidence intervals for variance component ratios in unbalanced linear mixed models

Methods for constructing confidence intervals for variance component ratios in general unbalanced mixed models are developed. The methods are based on inverting the distribution of the signed root of the log-likelihood ratio statistic constructed from either the restricted maximum likelihood or the full likelihood. As this distribution is intractable, the inversion is rather based on using a saddlepoint approximation to its distribution. Apart from Wald's exact method, the resulting intervals are unrivalled in terms of achieving accuracy in overall coverage, underage, and overage. Issues related to the proper “reference set” with which to judge the coverage as well as issues connected to variance ratios being nonnegative with lower bound 0 are addressed. Applications include an unbalanced nested design and an unbalanced crossed design.     

On the Properties of the Likelihood Function of Spanos' Conditional t Heteroskedastic Model

This article investigates the properties of the likelihood function of Spanos’ conditional t heteroskedastic model (Spanos, 1994 Spanos , A. ( 1994 ). On modeling heteroskedasticity: The student's t and elliptical linear regression models . Econometric Theor. 10 : 286 – 315 .[Crossref], [Web of Science ®] , [Google Scholar]) On modeling heteroskedasticity: the student's t and elliptical linear regression models. It is shown that estimability of the degrees of freedom of t distribution and the block-diagonality of the information matrix of the joint likelihood function with respect to conditional mean parameters and remaining parameters hold for the model. The joint maximum likelihood estimator and its inference based on the t-statistic and χ²-statistic are examined in finite samples by simulation when the degrees of freedom is known and unknown.     

Quadratic estimation in mixed linear models with two variance components

Explicit expressions for Bayes invariant quadratic estimates, biased and unbiased, are presented and proved to cover the entire class of admissible estimates in the considered classes. An unbalanced genetic model is studied for demonstration.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Influence diagnostics and outlier tests for semiparametric mixed models

Summary. Semiparametric mixed models are useful in biometric and econometric applications, especially for longitudinal data. Maximum penalized likelihood estimators (MPLEs) have been shown to work well by Zhang and co-workers for both linear coefficients and nonparametric functions. This paper considers the role of influence diagnostics in the MPLE by extending the case deletion and subject deletion analysis of linear models to accommodate the inclusion of a nonparametric component. We focus on influence measures for the fixed effects and provide formulae that are analogous to those for simpler models and readily computable with the MPLE algorithm. We also establish an equivalence between the case or subject deletion model and a mean shift outlier model from which we derive tests for outliers. The influence diagnostics proposed are illustrated through a longitudinal hormone study on progesterone and a simulated example.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.     

Estimation of variance components in linear mixed measurement error models

In this paper, we consider a linear mixed model with measurement errors in fixed effects. We find the corrected score function estimators for the variance components. An iterative algorithm is proposed for estimating the parameters. The computations on each iteration of this algorithm are those associated with computing estimates of fixed and random effects for given values of the variance components. We also derive the consistency of the estimators under regularity conditions. The simulation study shows that for relatively small sample size the corrected estimators perform very well. Finally, an example of real data is given for illustration.     

Likelihood ratio tests for multivariate normality

This paper presents some powerful omnibus tests for multivariate normality based on the likelihood ratio and the characterizations of the multivariate normal distribution. The power of the proposed tests is studied against various alternatives via Monte Carlo simulations. Simulation studies show our tests compare well with other powerful tests including multivariate versions of the Shapiro–Wilk test and the Anderson–Darling test.     

Generalized linear mixed models: a review and some extensions

Breslow and Clayton (J Am Stat Assoc 88:9–25,1993) was, and still is, a highly influential paper mobilizing the use of generalized linear mixed models in epidemiology and a wide variety of fields. An important aspect is the feasibility in implementation through the ready availability of related software in SAS (SAS Institute, PROC GLIMMIX, SAS Institute Inc., URL , 2007), S-plus (Insightful Corporation, S-PLUS 8, Insightful Corporation, Seattle, WA, URL , 2007), and R (R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, URL , 2006) for example, facilitating its broad usage. This paper reviews background to generalized linear mixed models and the inferential techniques which have been developed for them. To provide the reader with a flavor of the utility and wide applicability of this fundamental methodology we consider a few extensions including additive models, models for zero-heavy data, and models accommodating latent clusters.     

Generalised partial linear single-index mixed models for repeated measures data

In this paper, we propose generalised partial linear single-index mixed models for analysing repeated measures data. A penalised quasi-likelihood approach using P-spline is used to estimate the nonparametric function, linear parameters, and single-index coefficients. Asymptotic properties of the estimators are developed when the dimension of spline basis grows with increasing sample size. Simulation examples and two applications: the study of health effects of air pollution in North Carolina, and treatment effect of naltrexone on health costs for alcohol-dependent individuals, illustrate the effectiveness of our approach.     

Reference priors for linear models with general covariance structures

We develop a new class of reference priors for linear models with general covariance structures. A general Markov chain Monte Carlo algorithm is also proposed for implementing the computation. We present several examples to demonstrate the results: Bayesian penalized spline smoothing, a Bayesian approach to bivariate smoothing for a spatial model, and prior specification for structural equation models.     

Bayesian inference for the variance components in general mixed linear models

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.     

Invariant methods for estimating variance components in mixed linear models 2

The subject of this contribution is to present a survey on new methods for variance component estimation, which appeared in the literature in recent years. Starting from mixed models treated in analysis of variance research work on this field turned over to a more general approach in which the covariance matrix of the vector of observations is assumed to be a unknown linear combination of known symmetric matrices. Much interest has been shown in developing some kinds op optimal estimators for the unknown parameters and most results were obtained for estimators being invariant with respect to a certain group of translations. Therefore we restrict attention to this class of estimates. We will deal with minimum variance unbiased estimators, least squared errors estimators, maximum likelihood estimators. Bayes quadratic estimators and show some relations to the mimimum norm quadratic unbiased estimation principle (MINQUE) introduced by C. R. Rao [20]. We do not mention the original motivation of MINQUE since the otion of minimum norm depends on a measure that is not accepted by all statisticians. Also we do‘nt deal with other approaches like the BAYEsian and fiducial methods which were successfully applied by S. Portnoy [18], P. Rusolph [22], G. C. Tiao, W. Y. Tan [28], M. J. K. Healy [9] and others, although in very special situations, only. Additionally we add some new results and also new insight in the properties of known estimators. We give a new characterization of MINQUE in the class of all estimators, extend explicite expressions for locally optimal quadratic estimators given by C. R. Rao [22] to a slightly more general situation and prove complete class theorems useful for the computation of BAYES quadratic estimators. We also investigate situations in which BAYES quadratic unbiased estimators do'nt change if the distribution of the error terms differ from the normal distribution.     

Simultaneous inferences for variance ratios of some mixed linear models

Earlier investigations used a one-sided inequality to consltuct confidence regions for the variance ratios or balanced randoiu models. In this study, confidence regions are based on a two-sided generalisation of this inequality and the results are illustrated by estimating the parameters of some elementary random models.     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.