Marginalized Multilevel Models and Likelihood Inference
TL;DR: In this article, an alternative parameterization for the multilevel model in which the marginal mean, rather than the conditional mean given random effects, is regressed on covariates is presented.
Abstract: Hierarchical or ‘‘multilevel’’ regression models typically parameterize the mean response conditional on unobserved latent variables or ‘‘random’’ effects and then make simple assumptions regarding their distribution. The interpretation of a regression parameter in such a model is the change in possibly transformed mean response per unit change in a particular predictor having controlled for all conditioning variables including the random effects. An often overlooked limitation of the conditional formulation for nonlinear models is that the interpretation of regression coefficients and their estimates can be highly sensitive to difficult-to-verify assumptions about the distribution of random effects, particularly the dependence of the latent variable distribution on covariates. In this article, we present an alternative parameterization for the multilevel model in which the marginal mean, rather than the conditional mean given random effects, is regressed on covariates. The impact of random effects model violations on the marginal and more traditional conditional parameters is compared through calculation of asymptotic relative biases. A simple two-level example from a study of teratogenicity is presented where the binomial overdispersion depends on the binary treatment assignment and greatly influences likelihood-based estimates of the treatment effect in the conditional model. A second example considers a three-level structure where attitudes toward abortion over time are correlated with person and district level covariates. We observe that regression parameters in conditionally specified models are more sensitive to random effects assumptions than their counterparts in the marginal formulation.
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TL;DR: The core features of the R package geepack are described, which implements the generalized estimating equations (GEE) approach for fitting marginal generalized linear models to clustered data, through an example of clustered binary data.
Abstract: This paper describes the core features of the R package geepack, which implements the generalized estimating equations (GEE) approach for fitting marginal generalized linear models to clustered data. Clustered data arise in many applications such as longitudinal data and repeated measures. The GEE approach focuses on models for the mean of the correlated observations within clusters without fully specifying the joint distribution of the observations. It has been widely used in statistical practice. This paper illustrates the application of the GEE approach with geepack through an example of clustered binary
data.
1,785 citations
TL;DR: It is argued in general that mixed models involve unverifiable assumptions on the data-generating distribution, which lead to potentially misleading estimates and biased inference and is concluded that the estimation-equation approach of population average models provides a more useful approximation of the truth.
Abstract: Two modeling approaches are commonly used to estimate the associations between neighborhood characteristics and individual-level health outcomes in multilevel studies (subjects within neighborhoods). Random effects models (or mixed models) use maximum likelihood estimation. Population average models typically use a generalized estimating equation (GEE) approach. These methods are used in place of basic regression approaches because the health of residents in the same neighborhood may be correlated, thus violating independence assumptions made by traditional regression procedures. This violation is particularly relevant to estimates of the variability of estimates. Though the literature appears to favor the mixed-model approach, little theoretical guidance has been offered to justify this choice. In this paper, we review the assumptions behind the estimates and inference provided by these 2 approaches. We propose a perspective that treats regression models for what they are in most circumstances: reasonable approximations of some true underlying relationship. We argue in general that mixed models involve unverifiable assumptions on the data-generating distribution, which lead to potentially misleading estimates and biased inference. We conclude that the estimation-equation approach of population average models provides a more useful approximation of the truth.
906 citations
TL;DR: The average annual salary in America in inflation-adjusted 1998 dollars increased from $32,522 in 1970 to $35,864 in 1999, that is, a modest 10 percent increase over three decades, but recent trends in wealth inequality have been equally noteworthy.
Abstract: Many developed countries have experienced a sharp rise in income inequality during the past three decades, and the United States is no exception (1). For example, the average annual salary in America in inflation-adjusted 1998 dollars increased from $32,522 in 1970 to $35,864 in 1999, that is, a modest 10 percent increase over three decades. By contrast over the same period, the average annual compensation of the top 100 chief executive officers rose from $1.3 million (or 39 times the pay of an average worker) to $37.5 million (or more than 1,000 times the pay of an average worker) (2). Recent trends in wealth inequality have been equally noteworthy. The net worth of families in the top decile rose by 69 percent, to $833,600 in 2001, from $493,400 in 1998. By contrast over the same period, the net worth of families in the lowest fifth of income earners rose 24 percent, to $7,900. The median accumulated wealth of families in the top 10 percent of the income distribution was 12 times that of lower-middle-income families through much of the 1990s, but in 2001, the median net worth of the top earners was about 22 times as great (3).
844 citations
TL;DR: The emphasis of this work is on how to define the concept of importance in an unambiguous way and how to assess it in the simultaneous occurrence of correlated input factors and non-additive models.
Abstract: This article deals with global quantitative sensitivity analysis of the Level E model, a computer code used in safety assessment for nuclear waste disposal. The Level E code has been the subject of two international benchmarks of risk assessment codes and Monte Carlo methods and is well known in the literature. We discuss the Level E model with reference to two different settings. In the first setting, the objective is to find the input factor that drives most of the output variance. In the second setting, we strive to achieve a preestablished reduction in the variance of the model output by fixing the smallest number of factors. The emphasis of this work is on how to define the concept of importance in an unambiguous way and how to assess it in the simultaneous occurrence of correlated input factors and non-additive models.
401 citations
TL;DR: This article found that structural characteristics differ in their effects based on the level of aggregation employed, and that more dispersed networks are important for perceived crime and disorder, while economic resources only show a localized effect when aggregating to the block-level and differ based on their outcome; higher average income reduces disorder but increases crime, most likely by increasing the number of attractive targets.
Abstract: This article highlights the importance of considering the proper level of aggregation when estimating neighborhood effects. Using a unique nonrural subsample from a large national survey (the American Housing Survey) at three time points that allows placing respondents in blocks and census tracts, this study tests the appropriate level of aggregation of the structural characteristics hypothesized to affect block-level perceptions of crime and disorder. I find that structural characteristics differ in their effects based on the level of aggregation employed. While the effects of racial/ethnic heterogeneity are fairly robust to the geographical level of aggregation, the stronger effects, when measured at the level of the surrounding census tract, suggest more dispersed networks are important for perceived crime and disorder. In contrast, economic resources only show a localized effect when aggregating to the block-level and differ based on the outcome; higher average income reduces disorder but increases crime, most likely by increasing the number of attractive targets. Additionally, the presence of broken households has a localized effect for social disorder but a more diffuse effect for perceived crime. These findings suggest the need for neighborhood studies of crime rates, as well as the broader neighborhood effects literature, to consider the mechanisms involved when aggregating various structural characteristics.
391 citations
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Book•
01 Jan 1983TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).
23,215 citations
TL;DR: In this article, an extension of generalized linear models to the analysis of longitudinal data is proposed, which gives consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence.
Abstract: SUMMARY This paper proposes an extension of generalized linear models to the analysis of longitudinal data. We introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence. The estimating equations are derived without specifying the joint distribution of a subject's observations yet they reduce to the score equations for multivariate Gaussian outcomes. Asymptotic theory is presented for the general class of estimators. Specific cases in which we assume independence, m-dependence and exchangeable correlation structures from each subject are discussed. Efficiency of the proposed estimators in two simple situations is considered. The approach is closely related to quasi-likelih ood. Some key ironh: Estimating equation; Generalized linear model; Longitudinal data; Quasi-likelihood; Repeated measures.
17,111 citations
TL;DR: The Markov Chain Monte Carlo Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC for NONLINEAR HIERARCHICAL MODELS.
Abstract: INTRODUCING MARKOV CHAIN MONTE CARLO Introduction The Problem Markov Chain Monte Carlo Implementation Discussion HEPATITIS B: A CASE STUDY IN MCMC METHODS Introduction Hepatitis B Immunization Modelling Fitting a Model Using Gibbs Sampling Model Elaboration Conclusion MARKOV CHAIN CONCEPTS RELATED TO SAMPLING ALGORITHMS Markov Chains Rates of Convergence Estimation The Gibbs Sampler and Metropolis-Hastings Algorithm INTRODUCTION TO GENERAL STATE-SPACE MARKOV CHAIN THEORY Introduction Notation and Definitions Irreducibility, Recurrence, and Convergence Harris Recurrence Mixing Rates and Central Limit Theorems Regeneration Discussion FULL CONDITIONAL DISTRIBUTIONS Introduction Deriving Full Conditional Distributions Sampling from Full Conditional Distributions Discussion STRATEGIES FOR IMPROVING MCMC Introduction Reparameterization Random and Adaptive Direction Sampling Modifying the Stationary Distribution Methods Based on Continuous-Time Processes Discussion IMPLEMENTING MCMC Introduction Determining the Number of Iterations Software and Implementation Output Analysis Generic Metropolis Algorithms Discussion INFERENCE AND MONITORING CONVERGENCE Difficulties in Inference from Markov Chain Simulation The Risk of Undiagnosed Slow Convergence Multiple Sequences and Overdispersed Starting Points Monitoring Convergence Using Simulation Output Output Analysis for Inference Output Analysis for Improving Efficiency MODEL DETERMINATION USING SAMPLING-BASED METHODS Introduction Classical Approaches The Bayesian Perspective and the Bayes Factor Alternative Predictive Distributions How to Use Predictive Distributions Computational Issues An Example Discussion HYPOTHESIS TESTING AND MODEL SELECTION Introduction Uses of Bayes Factors Marginal Likelihood Estimation by Importance Sampling Marginal Likelihood Estimation Using Maximum Likelihood Application: How Many Components in a Mixture? Discussion Appendix: S-PLUS Code for the Laplace-Metropolis Estimator MODEL CHECKING AND MODEL IMPROVEMENT Introduction Model Checking Using Posterior Predictive Simulation Model Improvement via Expansion Example: Hierarchical Mixture Modelling of Reaction Times STOCHASTIC SEARCH VARIABLE SELECTION Introduction A Hierarchical Bayesian Model for Variable Selection Searching the Posterior by Gibbs Sampling Extensions Constructing Stock Portfolios With SSVS Discussion BAYESIAN MODEL COMPARISON VIA JUMP DIFFUSIONS Introduction Model Choice Jump-Diffusion Sampling Mixture Deconvolution Object Recognition Variable Selection Change-Point Identification Conclusions ESTIMATION AND OPTIMIZATION OF FUNCTIONS Non-Bayesian Applications of MCMC Monte Carlo Optimization Monte Carlo Likelihood Analysis Normalizing-Constant Families Missing Data Decision Theory Which Sampling Distribution? Importance Sampling Discussion STOCHASTIC EM: METHOD AND APPLICATION Introduction The EM Algorithm The Stochastic EM Algorithm Examples GENERALIZED LINEAR MIXED MODELS Introduction Generalized Linear Models (GLMs) Bayesian Estimation of GLMs Gibbs Sampling for GLMs Generalized Linear Mixed Models (GLMMs) Specification of Random-Effect Distributions Hyperpriors and the Estimation of Hyperparameters Some Examples Discussion HIERARCHICAL LONGITUDINAL MODELLING Introduction Clinical Background Model Detail and MCMC Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC FOR NONLINEAR HIERARCHICAL MODELS Introduction Implementing MCMC Comparison of Strategies A Case Study from Pharmacokinetics-Pharmacodynamics Extensions and Discussion BAYESIAN MAPPING OF DISEASE Introduction Hypotheses and Notation Maximum Likelihood Estimation of Relative Risks Hierarchical Bayesian Model of Relative Risks Empirical Bayes Estimation of Relative Risks Fully Bayesian Estimation of Relative Risks Discussion MCMC IN IMAGE ANALYSIS Introduction The Relevance of MCMC to Image Analysis Image Models at Different Levels Methodological Innovations in MCMC Stimulated by Imaging Discussion MEASUREMENT ERROR Introduction Conditional-Independence Modelling Illustrative examples Discussion GIBBS SAMPLING METHODS IN GENETICS Introduction Standard Methods in Genetics Gibbs Sampling Approaches MCMC Maximum Likelihood Application to a Family Study of Breast Cancer Conclusions MIXTURES OF DISTRIBUTIONS: INFERENCE AND ESTIMATION Introduction The Missing Data Structure Gibbs Sampling Implementation Convergence of the Algorithm Testing for Mixtures Infinite Mixtures and Other Extensions AN ARCHAEOLOGICAL EXAMPLE: RADIOCARBON DATING Introduction Background to Radiocarbon Dating Archaeological Problems and Questions Illustrative Examples Discussion Index
7,399 citations
Book•
01 Jan 1994
TL;DR: In this paper, a generalized linear model for longitudinal data and transition models for categorical data are presented. But the model is not suitable for categric data and time dependent covariates are not considered.
Abstract: 1. Introduction 2. Design considerations 3. Exploring longitudinal data 4. General linear models 5. Parametric models for covariance structure 6. Analysis of variance methods 7. Generalized linear models for longitudinal data 8. Marginal models 9. Random effects models 10. Transition models 11. Likelihood-based methods for categorical data 12. Time-dependent covariates 13. Missing values in longitudinal data 14. Additional topics Appendix Bibliography Index
7,156 citations