Testing equality restrictions in generalized linear models for multinomial data

Based on f{\phi } -divergences an estimator of the generalized linear models for multinomial data under linear restrictions on the parameters is considered. New test statistics, also based on f{\phi } -divergences are considered as alternatives to the classical ones for testing a hypothesis about linear restrictions on the parameters. The asymptotic distribution of them is obtained under the null hypothesis as well as under contiguous local hypotheses. An application of the estimators and the tests is illustrated in a numerical example and in simulation studies.     

Recursive residuals for linear mixed models

Quality & Quantity - This paper presents and extends the concept of recursive residuals and their estimation to an important class of statistical models, Linear Mixed Models (LMM). Recurrence...     

Simultaneous optimal estimation in linear mixed models

Simultaneous optimal estimation in linear mixed models is considered. A necessary and sufficient condition is presented for the least squares estimator of the fixed effects and the analysis of variance estimator of the variance components to be of uniformly minimum variance simultaneously in a general variance components model. That is, the matrix obtained by orthogonally projecting the covariance matrix onto the orthogonal complement space of the column space of the design matrix is symmetric, each eigenvalue of the matrix is a linear combinations of the variance components and the number of all distinct eigenvalues of the matrix is equal to the the number of the variance components. Under this condition, uniformly optimal unbiased tests and uniformly most accurate unbiased confidence intervals are constructed for the parameters of interest. A necessary and sufficient condition is also given for the equivalence of several common estimators of variance components. Two examples of their application are given.     

Estimation under linear regression models

Summary Royall and Herson considered balanced samples for ensuring robustness of standard ratio estimator under polynomial superpopulation models. Here we formulate a post-sample estimator of Royall type which remains robust (in respect of bias) under a wide class of polynomial regression models.     

A simple approach for the analysis of generalizea linear mixed models

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi-likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi-likelihood, using iteratively re-weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.     

Generalized linear mixed models with informative dropouts and missing covariates

Generalized linear mixed models (GLMM) are useful in many longitudinal data analyses. In the presence of informative dropouts and missing covariates, however, standard complete-data methods may not be applicable. In this article, we consider a likelihood method and an approximate method for GLMM with informative dropouts and missing covariates. The methods are implemented by Monte–Carlo EM algorithms combined with Gibbs sampler. The approximate method may lead to inconsistent estimators but is computationally more efficient than the likelihood method. The two methods are evaluated via a simulation study for longitudinal binary data, and appear to perform reasonably well. A dataset on mental distress is analyzed in details.     

On restricted estimation in linear models

Efficient estimation of models under linear restrictions on parameters has received little attention. We present a unified approach which takes account of both full rank and less than full rank design and constraint matrices. The procedure is numerically fast, accurate, and space efficient. An example is included.     

Two kinds of variance/covariance estimates in linear mixed models

For longitudinal data, the within-subject covariance matrix plays an important role in statistical inference and it is of great interest to investigate this. In the paper, two kinds of estimators are investigated for the random effect covariance matrix D ₁ and the error variance σ ² in linear mixed models. One is to estimate D ₁ first and then to estimate σ ²; the other kind is to estimate σ ² first and then for D ₁. Both kinds of estimators are consistent. The covariance matrices of these covariance estimators and the variances of these two error variance estimators are calculated. In particular, the mean square errors of these estimators are also derived for one dimensional random effects. Besides, a simulation study is conducted to investigate the performances of these estimators.     

On two methods for solving and estimating linear simultaneous equations under rational expectations

A method of Chow (1983) and a method of Dagli and Taylor (1982) for solving and estimating linear simultaneous equations under rational expectations are compared. The latter solution is shown to be a special case of the former in the sense of imposing a set of restrictions on the parameters of the former solution. Statistical methods to test the restrictions implicit in the latter solution are suggested. An illustrated model is provided to demonstrate the two methods, with the Dagli-Taylor method found to give inconsistent estimates when the restrictions are not met.     

On the natural selection rule in general linear models

We consider a problem of selecting the best treatment in a general linear model. We look at the properties of the natural selection rule. It is shown that the natural selection rule is minimax under to “0–1” loss function and it is a Bayes rule under a monotone permutation invariant loss function with respect to a permutation invariant prior for every variance balanced design. Some other condition on the design matrix is given so that a Bayes rule with respect to a normal prior will be of simple structure.     

On the identification of the significant independent variables in linear models

Summary Methods for the identification of the significant independent variables in multiple linear regression and in the multiple regression approach to non-orthogonal analysis of variance and covariance are discussed. “Forward Ranking” and “Backward Ranking” (by order of importance) of the independent variables are defined, and the backward method is shown to avoid the disadvantageous effects of “Compounds” upon the ranking. For non-orthogonal analysis of variance, a unique orthogonal decomposition of the regression sum of squares (due to all ANOVA effects) is shown to be possible when the groups of independent variables (representing the effects) are ranked by the criterion of “Non-Significance” and under “Restricted Admissibility.” A computer program is outlined which incorporates the proposed methods.

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Zusammenfassung Methoden für die Identifizierung der signifikanten unabh?ngigen Ver?nderlichen in der mehrfachen linearen Regressionsrechnung und im Regressionsverfahren für nichtorthogonale Varianz- und Kovarianzanalyse werden besprochen. „Vorw?rtsgerichtetes“ und „rückw?rtsgerichtetes“ Rangordnen (nach Bedeutung) der unabh?ngigen Ver?nderlichen werden definiert, und es wird gezeigt, da? beim rückw?rtsgerichteten Rangordnen die nachteiligen Wirkungen von „Verb?nden“ auf das Ordnen vermieden werden. Für den Fall der nichtorthogonalen Varianzanalyse wird gezeigt, da? eine eindeutige orthogonale Zerlegung der Quadratsumme für die Regression (erkl?rt durch die Gesamtheit der Haupt- und Wechselwirkungen in der Varianzanalyse) erreicht werden kann, wenn die Gruppen der unabh?ngigen Ver?nderlichen, die die Haupt- und Wechselwirkungen repr?sentieren, nach dem Rangordnungskriterium „Nicht-Signifikanz“ und unter „Beschr?nkter Zul?ssigkeit“ geordnet werden. Ein Rechenprogramm wird erl?utert, welches auf den vorgeschlagenen Methoden basiert.

     

On equalities of estimations of parametric functions under a general linear model and its restricted models

Estimations of parametric functions under a general linear model and its restricted models involve some complicated operations of matrices and their generalized inverses. In the past several years, a powerful tool—the matrix rank method was utilized to manipulate various complicated matrix expressions that involve generalized inverses of matrices. In this paper, we use this method to derive necessary and sufficient conditions for six equalities of the ordinary least-squares estimators and the best linear unbiased estimators of parametric functions to equal under a general linear model and its corresponding restricted model.     

Estimation under two stage random permutation models

Summary Estimation of the population mean under assumptions of non-informativeness of labels in a two stage finite population of distinguishable units has been studied. Under the random permutation model, for the two stage set up, sample mean, the natural estimator, is found to be the best.     

Canonical forms and tests of hypotheses Part II: Multivariate mixed linear models

Abstract  In the literature on multivariate analysis of variance, exact test procedures are restricted to linear models with fixed effects only. In this paper tests are presented for multivarite linear hypotheses with respect to mixed models, which constitude a generalization of (univariate) regular models described by R oebruck (1982). Furthermore it is shown, that the matrices, which are used to compute the test statistics, can be derived from the univariate "sums of squares" in the same manner as in the case of fixed models. The applicability of this theory is demonstrated by two examples.     

Retransformation bias: A look at the box-cox transformation to linear balanced mixed ANOVA models

Summary After a Box-Cox transformation to data following a linear balanced mixed ANOVA model, final results may be presented after retransformation to the original scale of measurement. Consequently, estimation of means which may be unbiased in the transformed scale will not be so after retransformation. In this article, the bias introduced together with the corresponding variance is assessed. It is found that whereas bias may not be a serious problem, the variances are inflated for positive transformation parameter the closer it is to zero.     

Optimal designs under two related models for interference

We consider two related models for interference: one having different directional neighbour effects and one with the same neighbour effects. We show that optimal designs for one model can be obtained from optimal designs for the other model. Acknowledgement. We are grateful to H. Kushner for advance notice of his work. SP was supported by the Isfahan University of Technology, Iran.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.