The bulk of statistical artillery has long consisted of models with additive error structure, arising primarily from concepts employed in controlled experiments. In such models, variability among independent random variables is partitioned into an (often linear) fixed systematic component and an (often Gaussian) additive random component. In less controlled settings, these modeling techniques have been extended to deal with grouped data such as are obtained in studies involving clustered or repeated sampling, longitudinal analysis, and small-area estimation. In these settings it is often necessary to employ models with nonlinear systematic components, leading to nonlinear mixed models.
Recent analytic techniques such as Estimating Functions and Markov Chain Monte Carlo methods have allowed greater consideration of models in which the random component is not additive in nature. If the systematic component is not taken as fixed, additional variability may be incorporated through the use of random data model parameters, resulting in marginal distributions for response random variables that take the form of general mixture distributions. Although such models lend themselves naturally to the incorporation of prior information, analysis may or may not proceed using hierarchical Bayesian methods.
In models with linear systematic components, mixed an mixture approaches to modeling variability lead to the same marginal distribution for response variables. This no longer remains true in the nonlinear setting. The purpose of this session is to provide a comparison of nonlinear mixed and mixture approaches to forming statistical models. The session is targeted at two questions: `what substantive considerations might lead one to prefer one of these approaches over the other in a particular problem?', and `what are some topics in each approach that need further development?'. By addressing these questions, we hope to provide greater understanding of both the similarities and differences between these approaches to modeling variability.
Theme Session: No
Applied Session: No
Session Chair & Affiliation: Alicia Carriquiry, Dept. of Statistics, Iowa State Univ.
Mailing Address: Department of Statistics Snedecor Hall Iowa State University Ames, Iowa 50011-1210 Ph: (515) 294-7782, Fax: (515) 294-4040, email: alicia@iastate.edu
Session Organizer & Affiliation: Mark S. Kaiser, Dept. of Statistics, Iowa State Univ.
Mailing Address: Department of Statistics Snedecor Hall Iowa State University Ames, Iowa 50011-1210 Ph: (515) 294-8871, Fax: (515) 294-4040, email: mskaiser@iastate.edu
Participant No. 1 & Affiliation: James S. Hodges, Division of Biostatistics, Univ. Minnestoa Co-Authors & Affiliations: Dan Sargent, Mayo Clinic
Title of Paper: Possible research strategies for nonlinear hierarchical models
Key Words: Degrees of Freedom, Diagnostics, Cox Models
Abstract: This talk considers two issues in linear hierarchical models -- diagnostics and model-fitting -- as possible starting places for research in nonlinear hierarchical models
Diagnostics: Hodges (1994) gave a way to reformulate a linear hierarchical models as a linear model with heteroskedastic errors. This trick has a variety of uses including constructing diagnostics and putting a rigorous basis under fractional degrees of freedom, and it is easily extended to nonlinear hierarchical models. Whether it is useful in nonlinear hierarchical models, as it is for linear hierarchical models, is speculative at this point.
Model-fitting: Treating the Bayes machinery shorn of ideology and simply as a way to generate procedures, Bayesian procedures should have better frequency properties than competitors that involve maximizing a likelihood or solving an estimating equation, because integrating is a smoother operation than maximizing, and that seems to matter in these problems.
Mailing Address for Participant No. 1: Division of Biostatistics School of Public Health University of Minnesota - Twin Cities 2221 University Ave. SE Ste 200 Minneapolis, Minnesota 55414-3075 Ph: (612) 626-9626, Fax: (612) 626-8892, email: hodges@gopher.ccbr.umn.edu
Participant No. 2 & Affiliation: Mark S. Kaiser, Dept. of Statistics, Iowa State Univ. Co-Authors & Affiliations: Noel Cressie, Dept. of Statistics, Iowa State Univ.
Title of Paper: Statistical issues and scientific connections in the use of nonlinear mixed and mixture models
Key Words: Fixed and Random Regression Coefficients, Dependence Structure
Abstract: This talk centers on the connections between substantive objectives and the formulation of statistical models that contain random elements in the systematic model component. Such models may be developed by using what has become known as `mixed' model structure or by using the concepts of `mixture' distributions. For linear models these approaches give the same structure in marginal distributions of response variables, but this is no longer true in the nonlinear setting. The situation may be characterized in nonlinear settings by the phrase `what you see is not always what you want to get'. This concept is illustrated through introduction of the terms `fixed regression coefficient' models and `random regression coefficient' models. The former correspond to the mixed model approach, while the later may be developed only through the use of general mixture models. Two issues are discussed relative to the use of these modeling approaches, (1) whether or not the scientific phenomenon of interest is expressed directly through the marginal expectations of response variables, and (2) the effect of these model specifications on the variance of the marginal response distribution.
Mailing Address for Participant No. 2: Department of Statistics Snedecor Hall Iowa State University Ames, Iowa 50011-1210 Ph: (515) 294-8871, Fax: (515) 294-4040, email: mskaiser@iastate.edu
Participant No. 3 & Affiliation: Sidhartha Chib, Washington University
Title of Paper: Random Effects Models and Non-Ignorable Missingness: MCMC\ Approaches
Key Words: Longitudinal models, discrete response, missing data
Abstract: Models for longitudinal data have grown in sophistication over the years and, for continuous responses, mixed models can be routinely fit. Interesting problems of modeling, estimation and inference arise when the responses are not continuous and some of the data in each cluster is missing. This talk discusses the fitting of non-ignorable missing data models for both continuous and binary data. Some practical approaches based on Markov Chain Monte methods are developed and illustrated using simulated and real data.
Mailing Address for Participant No. 3: John M. Olin School of Business Washington University - St. Louis Campus Box 1133 1 Brookings Dr. St Louis, Missouri 63130-4899 Ph: (314) 935-4657, Fax: (314) 935-6359, email: chib@simon.wustl.edu