Showing papers on "Mixed model published in 1987"

Maximum likelihood computations with repeated measures: application of the EM algorithm

Abstract: The purpose of this article is to consider the use of the EM algorithm (Dempster, Laird, and Rubin 1977) for both maximum likelihood (ML) and restricted maximum likelihood (REML) estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure (Anderson 1973). Several models and methods of analysis have been proposed in recent years for repeated measures data; Ware (1985) presented an overview. Because the EM algorithm is a general-purpose, iterative method for computing ML estimates with incomplete data, it has often been used in this particular setting (Dempster et al. 1977; Andrade and Helms 1984; Jennrich and Schluchter 1985). There are two apparently different approaches to using the EM algorithm in this setting. In one application, each experimental unit is observed under a standard protocol specifying measurements at each of n occasions (or under n conditions), and incompleteness implies that the number of measurements actua...

Plane answers to complex questions: the theory of linear models

Abstract: Introduction.- Estimation.- Testing.- One-Way ANOVA.- Multiple Comparison Techniques.- Regression Analysis.- Multifactor Analysis of Variance.- Experimental Design Models.- Analysis of Covariance.- General Gauss-Markov Models.- Split Plot Models.- Mixed Models and Variance Components.- Model Diagnostics.- Variable Selection.- Collinearity and Alternative Estimates.-

A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects

Abstract: On decrit un algorithme qui utilise des formules explicites pour l'inverse et le determinant de la matrice de covariance donnee par La Motte (1972) et evite l'inversion des grandes matrices

The Analysis of Repeated Measures Designs Involving Multiple Dependent Variables

Abstract: Experimental designs involving repeated measures may be analyzed using traditional ANOVA methods (if all assumptions are met), or, given sufficient sample size, with a MANOVA-type procedure (if only some assumptions are met). Recent statistical advances have made possible the extension of these procedures to repeated measures experimental designs involving multiple dependent variables. This tutorial presents the concepts, complete examples (including computer control commands), and interpretations for four methods of testing differences among means in a mixed model repeated measures design. The four methods of analysis are: The traditional ANOVA and the MANOVA method for the single dependent variable case, and a Multivariate Mixed Model analysis and a Doubly Multivariate analysis for the multiple dependent variable case.

Prediction of Future Observations in Growth Curve Models

Abstract: The problem of predicting a future measurement on an individual given the past measurements is discussed under nonparametric and parametric growth models. The efficiencies of different methods of prediction are assessed by cross-validation or leave-one-out technique in each of three data sets and the results are compared. Under nonparametric models, direct and inverse regression methods of prediction are described and their relative advantages and disadvantages are discussed. Under parametric models polynomial and factor analytic type growth curves are considered. Bayesian and empirical Bayesian methods are used to deal with unknown parameters. A general finding is that much of the information for forecasting is contained in the immediate past few observations or a few summary statistics based on past data. A number of data reduction methods are suggested and analyses based on them are described. The usefulness of the leave-one-out technique in model selection is demonstrated. A new method of calibration is introduced to improve prediction.

A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects

Abstract: SUMMARY A fast Fisher scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects is described. The algorithm uses explicit formulae for the inverse and the determinant of the covariance matrix, given by LaMotte (1972), and avoids inversion of large matrices. Description of the algorithm concentrates on computational aspects for large sets of data. Computational methods for maximum likelihood estimation in unbalanced variance component models were developed by Hemmerle & Hartley (1973) using the W-transfor- mation, and by Patterson & Thompson (1971); see also Thompson (1980). These methods were reviewed by Harville (1977) who also discussed a variety of applications for the variance component models. Computational problems may arise when the number of clusters or random coefficients is large because inversion of very large matrices is required, and so there are severe limitations on the size of practical problems that can be handled. Goldstein (1986) and Aitkin & Longford (1986) present arguments for routine use of variance component models in educational context, but their arguments are applicable for a much wider range of problems including social surveys, longitudinal data, repeated measurements or experiments and multivariate analysis. The formulation of the general EM algorithm by Dempster, Laird & Rubin (1977) has led to development of alternative computational algorithms for variance component analysis by Dempster, Rubin & Tsutakawa (1981), Mason, Wong & Entwisle (1984) and others. These algorithms avoid inversion of large matrices, but may be very slow on complex problems, a common feature of EM algorithms. Convergence is especially slow when the variance components are small. The present paper gives details of a Fisher scoring algorithm for the unbalanced nested random effects model which converges rapidly and does not require the inversion of large matrices. The algorithm exploits the formulae for the inverse and the determinant of the irregularly patterned covariance matrix of the observations given by LaMotte (1972). The analysis presented by Aitkin & Longford (1986) uses software based on this algorithm. For another example see Longford (1985).

A locally most powerful test for homogeneity with many strata

Abstract: SUMMARY A locally most powerful test is proposed for homogeneity of nuisance parameters from many strata in the mixed model setting. The test is based on the difference between the square of the score and the observed information about the nuisance parameters, all evaluated at the null hypothesis. Several examples including constant normal mean with possibly different variances and matched case-control studies are presented in detail. The relative efficiencies of the unstratified estimator of parameter of interest versus the stratified one are calculated. Two data sets on matched case-control studies are used for illustration.

A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions

Abstract: Large-sample covariance matrices for the analysis of variance (ANOVA), minimum norm quadratic unbiased estimator (MINQUE), restricted maximum likelihood (REML), and maximum likelihood (ML) estimates of variance components are presented for the unbalanced one-way model when the underlying distributions are not necessarily normal. The limiting variances depend on the design sequence, on the actual values of the variance components, and on the kurtosis parameters of the underlying distributions. (The skewness parameters and other moments do not affect the limiting distributions.) Because all estimates are consistent and asymptotically normal, it is reasonable to compare the estimates using their asymptotic variances. Thus the efficiency of one estimate relative to another is defined as the ratio of their asymptotic variances, and these efficiencies are evaluated numerically and analytically for a variety of nonnormal situations. Various authors, including Hocking and Kutner (1975), Corbeil and Searl...

Mixed-model analysis of a censored normal distribution with reference to animal breeding.

Abstract: A mixed-model procedure for analysis of censored data assuming a multivariate normal distribution is described. A Bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left-censored. The procedure can be extended to right- and two-tailed censoring. The model employed is a generalized linear model, and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models. Estimates of variance components are obtained using expressions similar to those employed in the EM algorithm for restricted maximum likelihood (REML) estimation under normality.

Computing for Incomplete Repeated Measures

Abstract: SUMMARY Repeated-measures experiments involve two or more intended measurements per subject. If the within-subjects design is the same for each subject and no data are missing, then the analysis is relatively simple and there are readily available programs that do the analysis automatically. However, if the data are incomplete, and do not have the same arrangement for each subject, then the analysis becomes much more difficult. Beginning with procedures that are not optimal but are comparatively simple, we discuss unbalanced linear model analysis and then normal maximum likelihood (ML) procedures. Included are ML and REML (restricted maximum likelihood) estimators for the mixed model and also estimators for a model that allows arbitrary within-subject covariance matrices. The objective is to give procedures that can be implemented with available software.

Restricted maximum likelihood to estimate variance components for mixed models with two random factors

Abstract: A Restricted Maximum Likelihood procedure is described to estimate variance components for a univariate mixed model with two random factors. An EM-type algorithm is presented with a reparameterisation to speed up the rate of convergence. Computing strategies are outlined for models common to the analysis of animal breeding data, allowing for both a nested and a crossclassified design of the 2 random factors. Two special cases are considered : firstly, the total number of levels of fixed effects is small compared to the number of levels of both random factors ; secondly, one fixed effect with a large number of levels is to be fitted in addition to other

On the E-Optimality of Certain Asymmetrical Designs under Mixed Effects Model.

Abstract: TheE-optimality of the following designs within the class of all proper and connected designs with givenb, k andv under mixed effects model are established. All these designs are known to satisfy the same optimality property under fixed effects model whenk v. From the results proved here, theE-optimality of designs (ii, (iii), (iv) and (v) under fixed effects model in the situation whenk >v also follows.

On the optimalizes of the mbgdd's under mixed effects model

Abstract: In the present paper, under the assumption of a mixed effects model with random block effects, the type 1 optimality of the most balanced group divisible designs of type 1 has been established within the general class of all proper and connected block designs with k

Computable MINQUE-Type Estimates of Variance Components

Abstract: The minimum norm quadratic unbiased estimator type (MINQUE type) of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity. The resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties: assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates (MIVQUE's) approaches unity when the true variance ratios are identical and tend to infinity. Assuming nonnormal effect distributions in the model with two variance components, the estimates are asymptotically efficient: in a sequence of designs where the number of classes and the number of observations on each class approach infinity, it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadrati...

Bayesian estimation of mixed weibull exponential in life testing

Abstract: This paper deals with estimating the parameters of a mixed Weibull exponential model using a Bayesian method for type I censored samples. A numerical comparison between maximum likelihood and Bayes results has been carried out, using a numerical example and computer facilities for different prior information. Bayesian results in the cases of mixed exponential (complete and censored), single exponential and single Weibull may be consider as special cases of the results of this paper. The problem can be extended to the case of more than two causes of failure.