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     

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.     

Rényi statistics for testing hypotheses in mixed linear regression models

Rényi divergences are used to propose some statistics for testing general hypotheses in mixed linear regression models. The asymptotic distribution of these tests statistics, of the Kullback–Leibler and of the likelihood ratio statistics are provided, assuming that the sample size and the number of levels of the random factors tend to infinity. A simulation study is carried out to analyze and compare the behavior of the proposed tests when the sample size and number of levels are small.     

Considerations on group-wise identical designs for linear mixed models

We consider a general class of mixed models, where the individual parameter vector is composed of a linear function of the population parameter vector plus an individual random effects vector. The linear function can vary for the different individuals. We show that the search for optimal designs for the estimation of the population parameter vector can be restricted to the class of group-wise identical designs, i.e., for each of the groups defined by the different linear functions only one individual elementary design has to be optimized. A way to apply the result to non-linear mixed models is described.     

Interval estimates for variance components

A new approach to inference on variance components is propounded. This approach not only gives a new justification for Fisher's fiducial, distribution for the “between classes” component, but also leads to a new distribution for the “within classes” component. This latter distribution is studied, and has some intuitively very reasonable properties. Numerical results are given.     

Nonnegative unbiased estimability of linear combinations of two variance components

For a general mixed model with two variance components θ₁ and θ₂, a criterion for a function q₁θ₁+q₂θ₂ to admit an unbiased nonnegative definite quadratic estimator is established in a form that allows answering the question of existence of such an estimator more explicitly than with the use of the criteria known hitherto. An application of this result to the case of a random one-way model shows that for many unbalanced models the estimability criterion is expressible directly by the largest of the numbers of observations within levels, thus extending the criterion established by LaMotte (1973) for balanced models.     

Connections of the Poisson weight function to overdispersion and underdispersion

In this paper, we establish several connections of the Poisson weight function to overdispersion and underdispersion. Specifically, we establish that the logconvexity (logconcavity) of the mean weight function is a necessary and sufficient condition for overdispersion (underdispersion) when the Poisson weight function does not depend on the original Poisson parameter. We also discuss some properties of the weighted Poisson distributions (WPD). We then introduce a notion of pointwise duality between two WPDs and discuss some associated properties. Next, we present some illustrative examples and provide a discussion on various Poisson weight functions used in practice. Finally, some concluding remarks are made.     

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.     

The convolution theorem for estimating linear functionals in indirect nonparametric regression models

Nonparametric regression—directly or indirectly observed—is one of the important statistical models. On one hand it contains two infinite dimensional parameters (the regression function and the error density), and on the other it is of rather simple structure. Therefore, it may serve as an interesting paradigm for illustrating or developing abstract statistical theory for non-Euclidean parameters. In this paper estimation of a linear functional of the indirectly observed regression function is considered, when a deterministic design is used. It should be noted that any Fourier coefficient of an expansion of the regression function in an orthonormal basis is such a functional. Because the design is deterministic the observables are independent but not identically distributed. Local asymptotic normality is established and applied to prove Hájek's convolution theorem for this functional. Pertinent references are Beran [1977. Robust location estimates. Ann. Statist. 5, 431–444] and McNeney and Wellner [2000. Application of convolution theorems in semiparametric models with non-i.i.d. data. J. Statist. Plann. Inference 91, 441–480]. For purposes explained above, however, the paper is kept self-contained and full proofs are provided.     

Characterization of minimax linear estimators in linear regression

Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.     

Lower bounds on Bayes risks for estimating a normal variance: With applications

Brown and Gajek (1990) gave useful lower bounds on Bayes risks, which improve on earlier bounds by various authors. Many of these use the information inequality. For estimating a normal variance using the invariant quadratic loss and any arbitrary prior on the reciprocal of the variance that is a mixture of Gamma distributions, we obtain lower bounds on Bayes risks that are different from Borovkov-Sakhanienko bounds. The main tool is convexity of appropriate functionals as opposed to the information inequality. The bounds are then applied to many specific examples, including the multi-Bayesian setup (Zidek and his coauthors). Subsequent use of moment theory and geometry gives a number of new results on efficiency of estimates which are linear in the sufficient statistic. These results complement earlier results of Donoho, Liu and MacGibbon (1990), Johnstone and MacGibbon (1992) and Vidakovic and DasGupta (1994) for the location case.     

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.     

Regression models for functional data by reproducing kernel Hilbert spaces methods

Non-parametric regression models are developed when the predictor is a function-valued random variable X={_Xt}t∈T$X=\{X_t{\}}_{t\in T}$. Based on a representation of the regression function f(X)$f(X)$ in a reproducing kernel Hilbert space such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variable are performed on stock-exchange and medical data. The results of different regression models are compared.     

A resampling approach to estimate variance components of multilevel models

In a multilevel model for complex survey data, the weight‐inflated estimators of variance components can be biased. We propose a resampling method to correct this bias. The performance of the bias corrected estimators is studied through simulations using populations generated from a simple random effects model. The simulations show that, without lowering the precision, the proposed procedure can reduce the bias of the estimators, especially for designs that are both informative and have small cluster sizes. Application of these resampling procedures to data from an artificial workplace survey provides further evidence for the empirical value of this method. The Canadian Journal of Statistics 40: 150–171; 2012 © 2012 Statistical Society of Canada     

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.     

Optimal designs in random coefficient cubic regression models

In this article we investigate the problem of ascertaining A- and D-optimal designs in a cubic regression model with random coefficients. Our interest lies in estimation of all the parameters or in only those except the intercept term. Assuming the variance ratios to be known, we tabulate D-optimal designs for various combinations of the variance ratios. A-optimality does not pose any new problem in the random coefficients situation.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

Testing for generalized linear mixed models with cluster correlated data under linear inequality constraints

Generalized linear mixed models (GLMMs) are often used for analyzing cluster correlated data, including longitudinal data and repeated measurements. Full unrestricted maximum likelihood (ML) approaches for inference on both fixed‐and random‐effects parameters in GLMMs have been extensively studied in the literature. However, parameter orderings or constraints may occur naturally in practice, and in such cases, the efficiency of a statistical method is improved by incorporating the parameter constraints into the ML estimation and hypothesis testing. In this paper, inference for GLMMs under linear inequality constraints is considered. The asymptotic properties of the constrained ML estimators and constrained likelihood ratio tests for GLMMs have been studied. Simulations investigated the empirical properties of the constrained ML estimators, compared to their unrestricted counterparts. An application to a recent survey on Canadian youth smoking patterns is also presented. As these survey data exhibit natural parameter orderings, a constrained GLMM has been considered for data analysis. The Canadian Journal of Statistics 40: 243–258; 2012 © 2012 Crown in the right of Canada