Showing papers on "Random effects model published in 1987"

Multilevel Statistical Models

Abstract: Contents Dedication Preface Acknowledgements Notation A general classification notation and diagram Glossary Chapter 1 An introduction to multilevel models 1.1 Hierarchically structured data 1.2 School effectiveness 1.3 Sample survey methods 1.4 Repeated measures data 1.5 Event history and survival models 1.6 Discrete response data 1.7 Multivariate models 1.8 Nonlinear models 1.9 Measurement errors 1.10 Cross classifications and multiple membership structures. 1.11 Factor analysis and structural equation models 1.12 Levels of aggregation and ecological fallacies 1.13 Causality 1.14 The latent normal transformation and missing data 1.15 Other texts 1.16 A caveat Chapter 2 The 2-level model 2.1 Introduction 2.2 The 2-level model 2.3 Parameter estimation 2.4 Maximum likelihood estimation using Iterative Generalised Least Squares (IGLS) 2.5 Marginal models and Generalized Estimating Equations (GEE) 2.6 Residuals 2.7 The adequacy of Ordinary Least Squares estimates. 2.8 A 2-level example using longitudinal educational achievement data 2.9 General model diagnostics 2.10 Higher level explanatory variables and compositional effects 2.11 Transforming to normality 2.12 Hypothesis testing and confidence intervals 2.13 Bayesian estimation using Markov Chain Monte Carlo (MCMC) 2.14 Data augmentation Appendix 2.1 The general structure and maximum likelihood estimation for a multilevel model Appendix 2.2 Multilevel residuals estimation Appendix 2.3 Estimation using profile and extended likelihood Appendix 2.4 The EM algorithm Appendix 2.5 MCMC sampling Chapter 3. Three level models and more complex hierarchical structures. 3.1 Complex variance structures 3.2 A 3-level complex variation model example. 3.3 Parameter Constraints 3.4 Weighting units 3.5 Robust (Sandwich) Estimators and Jacknifing 3.6 The bootstrap 3.7 Aggregate level analyses 3.8 Meta analysis 3.9 Design issues Chapter 4. Multilevel Models for discrete response data 4.1 Generalised linear models 4.2 Proportions as responses 4.3 Examples 4.4 Models for multiple response categories 4.5 Models for counts 4.6 Mixed discrete - continuous response models 4.7 A latent normal model for binary responses 4.8 Partitioning variation in discrete response models Appendix 4.1. Generalised linear model estimation Appendix 4.2 Maximum likelihood estimation for generalised linear models Appendix 4.3 MCMC estimation for generalised linear models Appendix 4.4. Bootstrap estimation for generalised linear models Chapter 5. Models for repeated measures data 5.1 Repeated measures data 5.2 A 2-level repeated measures model 5.3 A polynomial model example for adolescent growth and the prediction of adult height 5.4 Modelling an autocorrelation structure at level 1. 5.5 A growth model with autocorrelated residuals 5.6 Multivariate repeated measures models 5.7 Scaling across time 5.8 Cross-over designs 5.9 Missing data 5.10 Longitudinal discrete response data Chapter 6. Multivariate multilevel data 6.1 Introduction 6.2 The basic 2-level multivariate model 6.3 Rotation Designs 6.4 A rotation design example using Science test scores 6.5 Informative response selection: subject choice in examinations 6.6 Multivariate structures at higher levels and future predictions 6.7 Multivariate responses at several levels 6.8 Principal Components analysis Appendix 6.1 MCMC algorithm for a multivariate normal response model with constraints Chapter 7. Latent normal models for multivariate data 7.1 The normal multilevel multivariate model 7.2 Sampling binary responses 7.3 Sampling ordered categorical responses 7.4 Sampling unordered categorical responses 7.5 Sampling count data 7.6 Sampling continuous non-normal data 7.7 Sampling the level 1 and level 2 covariance matrices 7.8 Model fit 7.9 Partially ordered data 7.10 Hybrid normal/ordered variables 7.11 Discussion Chapter 8. Multilevel factor analysis, structural equation and mixture models 8.1 A 2-stage 2-level factor model 8.2 A general multilevel factor model 8.3 MCMC estimation for the factor model 8.4 Structural equation models 8.5 Discrete response multilevel structural equation models 8.6 More complex hierarchical latent variable models 8.7 Multilevel mixture models Chapter 9. Nonlinear multilevel models 9.1 Introduction 9.2 Nonlinear functions of linear components 9.3 Estimating population means 9.4 Nonlinear functions for variances and covariances 9.5 Examples of nonlinear growth and nonlinear level 1 variance Appendix 9.1 Nonlinear model estimation Chapter 10. Multilevel modelling in sample surveys 10.1 Sample survey structures 10.2 Population structures 10.3 Small area estimation Chapter 11 Multilevel event history and survival models 11.1 Introduction 11.2 Censoring 11.3 Hazard and survival funtions 11.4 Parametric proportional hazard models 11.5 The semiparametric Cox model 11.6 Tied observations 11.7 Repeated events proportional hazard models 11.8 Example using birth interval data 11.9 Log duration models 11.10 Examples with birth interval data and children s activity episodes 11.11 The grouped discrete time hazards model 11.12 Discrete time latent normal event history models Chapter 12. Cross classified data structures 12.1 Random cross classifications 12.2 A basic cross classified model 12.3 Examination results for a cross classification of schools 12.4 Interactions in cross classifications 12.5 Cross classifications with one unit per cell 12.6 Multivariate cross classified models 12.7 A general notation for cross classifications 12.8 MCMC estimation in cross classified models Appendix 12.1 IGLS Estimation for cross classified data. Chapter 13 Multiple membership models 13.1 Multiple membership structures 13.2 Notation and classifications for multiple membership structures 13.3 An example of salmonella infection 13.4 A repeated measures multiple membership model 13.5 Individuals as higher level units 13.5.1 Example of research grant awards 13.6 Spatial models 13.7 Missing identification models Appendix 13.1 MCMC estimation for multiple membership models. Chapter 14 Measurement errors in multilevel models 14.1 A basic measurement error model 14.2 Moment based estimators 14.3 A 2-level example with measurement error at both levels. 14.4 Multivariate responses 14.5 Nonlinear models 14.6 Measurement errors for discrete explanatory variables 14.7 MCMC estimation for measurement error models Appendix 14.1 Measurement error estimation 14.2 MCMC estimation for measurement error models Chapter 15. Smoothing models for multilevel data. 15.1 Introduction 15.2. Smoothing estimators 15.3 Smoothing splines 15.4 Semi parametric smoothing models 15.5 Multilevel smoothing models 15.6 General multilevel semi-parametric smoothing models 15.7 Generalised linear models 15.8 An example Fixed Random 15.9 Conclusions Chapter 16. Missing data, partially observed data and multiple imputation 16.1 Creating a completed data set 16.2 Joint modelling for missing data 16.3 A two level model with responses of different types at both levels. 16.4 Multiple imputation 16.5 A simulation example of multiple imputation for missing data 16.6 Longitudinal data with attrition 16.7 Partially known data values 16.8 Conclusions Chapter 17 Multilevel models with correlated random effects 17.1 Non-independence of level 2 residuals 17.2 MCMC estimation for non-independent level 2 residuals 17.3 Adaptive proposal distributions in MCMC estimation 17.4 MCMC estimation for non-independent level 1 residuals 17.5 Modelling the level 1 variance as a function of explanatory variables with random effects 17.6 Discrete responses with correlated random effects 17.7 Calculating the DIC statistic 17.8 A growth data set 17.9 Conclusions Chapter 18. Software for multilevel modelling References Author index Subject index

Estimation and Comparison of Changes in the Presence of Information Right Censoring by Modeling the Censoring Process.

Abstract: : In estimating and comparing the rates of change of a continuous variable between two groups, the unweighted averages of individual simple least squares estimates from each group are often used. Under a linear random effects model, when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change. However, with censored of missing data, these estimates are no longer efficient when compared to generalized least squares estimates. When, in addition, the right censoring process is dependent upon the individual rates of change (i.e., informative right censoring), the generalized least squares estimates will be biased. Likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process. In realistic situations, we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored. (Author)

Limited Dependent Variable Models Using Panel Data

Abstract: This paper presents a survey of the methods used in the estimation of limited dependent variable models with panel data. It first reviews some issues in the analysis of panel data when the dependent variables are continuous. The problems of fixed effects vs. random effects and serious correlation vs. state dependence are discussed with reference to continuous data. The paper then discusses these problems with reference to the panel logit, panel probit, and panel tobit models. The paper presents a comparative assessment of these models.

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

Regression Methods for Poisson Process Data

Abstract: This article is directed toward situations where individuals can experience repeated events, and data on an individual consist of the number and occurrence times of events, along with concomitant variables. Methods of regression analysis are presented, based on Poisson process and proportional intensity assumptions. These include parametric and semi-parametric approaches to model fitting, model assessment, and the treatment of random effects. In addition, insight is gained as to the central role of Poisson and mixed Poisson regression analysis of counts in these methods, and as to the effects of unobserved heterogeneity on semi-parametric analyses. The methods in the article are based on the proportional intensity Poisson process model, for which an individual with given fixed covariate vector x has repeated events occur according to a nonhomogeneous Poisson process with intensity function λx(t) = λ0(t)exp(x′β). Estimation of β and the baseline intensity λ0(t) are considered when λ0(t) is specifi...

Semiparametric analysis of random effects linear models from binary panel data

Abstract: Andersen (1970) considered the problem of inference on random effects linear models from binary response panel data. He showed that inference is possible if the disturbances for each panel member are known to be white noise with the logistic distribution and if the observed explanatory variables vary over time. A conditional maximum likelihood estimator consistently estimates the model parameters up to scale. The present paper shows that inference remains possible if the disturbances for each panel member are known only to be time-stationary with unbounded support and if the explanatory variables vary enough over time. A conditional version of the maximum score estimator (Manski, 1975, 1985) consistently estimates the model parameters up to scale.

Diagnostics for Group Effects in Regression Analysis

Abstract: The diagnostic tools examined in this article are applicable to regressions estimated with panel data or cross-sectional data drawn from a population with grouped structure. The diagnostic tools considered include (a) tests for the existence of group effects under both fixed and random effects models, (b) checks for outlying groups, and (c) specification tests for comparing the fixed and random effects models. A group-specific counterpart to the studentized residual is introduced. The methods are illustrated using a hedonic housing price regression.

Trend in correlated proportions

Abstract: A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.

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).

What is an Analysis of Variance

Abstract: The analysis of variance is usually regarded as being concerned with sums of squares of numbers and independent quadratic forms of random variables. In this paper, an alternative interpretation is discussed. For certain classes of dispersion models for finite or infinite arrays of random variables, a form of generalized spectral analysis is described and its intuitive meaning explained. The analysis gives a spectral decomposition of each dispersion in the class, incorporating an analysis of the common variance, and an associated orthogonal decomposition of each of the random variables. One by-product of this approach is a clear understanding of the similarity between the spectral decomposition for second-order stationary processes and the familiar linear models with random effects.

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.

Some considerations in the analysis of rates of change in longitudinal studies.

Abstract: This paper discusses and compares several estimators of mean rate of change in unbalanced longitudinal data based on a model with randomly distributed regression coefficients across individuals. The estimators are unweighted and weighted means of these coefficients. The paper also evaluates commonly used variance estimates corresponding to the estimators. Results show that in situations of very slight imbalance, the choice of method is not critical. When imbalance is substantial, however, one should weight the regression coefficients by their estimated precision. An example using data from a nutritional study on premature neonates illustrates some issues encountered in the analysis of longitudinal clinical data sets.

Econometric Analysis of Discrete Choice: With Applications on the Demand for Housing in the U.S. and West-Germany

Abstract: 1: Introduction.- 1.1 Substantive Issues.- 1.2 Methodological Issues.- 1.3 Organization of the Book.- One: Econometric Foundations Discrete Choice Analysis.- 2: The Random utility Maximization Hypothesis.- 2.1 Continuous Versus Discrete Choice.- 2.2 Starting from the Utility Function.- 2.3 Starting from Choice Probabilities.- 3: Functional Specification of Discrete Choice Models.- 3.1 The Probit Model.- 3.2 The Logit Model.- 3.3 Extensions of the Logit Approach.- 3.4 The Linear Probability Model.- 3.5 Nonparametric Specifications.- 4: The Nested Multinominal Logit Model.- 4.1 Hierarchical Choice.- 4.2 Relation to the Random Utility Maximization Hypothesis.- 4.3 Estimation by Maximum Likelihood.- 4.4 Specification Tests.- 4.5 Summary.- 5: Panel Data.- 5.1 Discrete Choice for Pooled Cross-Sectional Data.- 5.2 Fixed Effects.- 5.3 Random Effects.- 5.4 Fixed Effects versus Random Effects Specifications.- 6: Economical Sampling and Estimation Techniques.- 6.1 Random Sampling of Alternatives.- 6.2 Choice Based Sampling.- 6.3 Fitting Aggregate Probability Shares.- 6.4 Estimation with Grouped Data.- 6.5 Goodness-of-Fit Measures.- Two: Applications: The Demand for Housing in the United States and West Germany.- 7: Housing Choices.- 7.1 Discrete Choice Description of Housing Demand.- 7.2 Explanatory Variables.- 7.3 Data Sources.- 8: Housing Preferences in the United States and West Germany.- 8.1 Introduction.- 8.2 Specification of Price Variables and Hedonic Regression.- 8.3 Specification of Income Variables and Permanent Income Estimation.- 8.4 Specification of Demographic Variables.- 8.5 Specification of Nesting Structures.- 8.6 Empirical Results.- 8.7 Conclusions.- 9: The Household Formation Decision.- 9.1 Introduction.- 9.2 Background and Nucleus Decomposition.- 9.3 Specification of the Demand Equations.- 9.4 Empirical Results.- 9.5 Conclusions.- 10: Tracing Housing Choices Over Time.- 10.1 Introduction.- 10.2 Data and Specification of the Demand Equations.- 10.3 Comparison: Panel Data and Cross-Sectional Analysis.- 10.4 Conclusions.

for Comparing Heuristics: An Example from Capacity Assignment Strategies in Computer Network Design

Abstract: An analysis of variance (ANOVA) model is developed for determining the existence of significant differences among strategies employing heuristics. Use of the model is illustrated in an application involving capacity assignment for networks utilizing the dynamic hierarchy architecture, in which the apex node is reassigned in response to changing environments. The importance of the model lies in the structure provided to the evaluation of heuristics, a major need in the assessment of benefits of artificial-intelligence applications. A nested three-factor design with fixed and random effects provides a numerical example of the model.

A statistical technique for comparing heuristics: an example from capacity assignment strategies in computer network design

Abstract: An analysis of variance (ANOVA) model is developed for determining the existence of significant differences among strategies employing heuristics. Use of the model is illustrated in an application involving capacity assignment for networks utilizing the dynamic hierarchy architecture, in which the apex node is reassigned in response to changing environments. The importance of the model lies in the structure provided to the evaluation of heuristics, a major need in the assessment of benefits of artificial-intelligence applications. A nested three-factor design with fixed and random effects provides a numerical example of the model.

A Random-Effects Logit Model for Panel Data

Abstract: A random-effects panel logit model is proposed, in which the unmeasured attributes of an individual are represented by a discrete-valued random variable, the distribution of which is binomial with a known number of support points. The maximum-likelihood estimator of the unknown parameters of the model are derived, and the performance of the ML estimators is investigated in a series of Monte-Carlo experiments. Several further extensions of the framework are also suggested, including application to discrete event-history data.

Allocating Observations in the Random Effects Balanced One-Way ANOVA

Abstract: To design a random effects one-way analysis of variance experiment one needs to determine both a, the number of treatments that are to be included in the experiment, as well as n, the number of times that each treatment is to be replicated needs to be d..

Fisher scoring algorithm for variance component analysis with hierarchically nested random effects

Abstract: We discuss several issues arising in applications of variance component analysis for data with multilevel structure, with specific reference to the Fisher scoring algorithm of Longford (1987). An adaptation of the algorithm for the exponential family, with particular relevance to discrete data, is presented.

Variance components estimation for the balanced random effects model under mixed prior distributions

Abstract: For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.

Estimation in one–way classification with heteroscedastic error

Abstract: For an unbalanced one way calssification under the random effect model the problem of estimation of the fixed effect parameter and the variance is considered. Tje error variance which are funtionally related to the above set of parameters are assumed to fall into k classes with constant error varaince for a class. The asymptotic properties of the proposed estimate is established for increasing number of classes k, assuming the number of observations in the classes form a fixed sequence

A Note on Heritability of a Dichotomous Trait

Abstract: In a random effects model for a dichotomous trait, we prove that the heritability of the dichotomous trait given in Dempster & Lerner (1950) is strictly smaller than the intraclass correlation. A numerical comparison is given.