The analytical ultracentrifuge has had wide application in the characterization of the structure, interaction and function of macromolecules in solution. This includes the determination of molecular weight, the characterization of shape, the determination of subunit stoichiometry, the quantification of ligand binding, the quantification of ligand-binding-promoted conformational changes and the characterization of macromolecular assembly processes (Schachman, 1992). In recent years, however, other technologies have been developed which can make some of the relevent measurements with equal or greater precision, often using less material. For instance, molecular weight can now be routinely determined by gene or protein sequencing and by mass spectroscopic methods, especially electrospray (ES) and matrix-assisted laser-desorption (MALD) MS (Carr, et al, 1991). Ligand binding can be quantified by classical spectroscopic and radiochemical methods as well as by microcalorimetry (Freire, et al, 1990). Ligand binding promoted conformational changes can be characterized by time- resolved fluorescence anisotropy (Lakowicz, 1983; Beechem, et al, 1986; Waxman, et al, 1994) although, the centrifuge still has a major impact in this area (Kirschner and Schachman, 1973a; Kirschner and Schachman, 1973b; Howlett and Schachman, 1977; Eisenstein, et al, 1990) The thermodynamic characterization of macromolecular assembly processes, however, is an area where the analytical ultracentrifuge has few rivals (Adams and Lewis, 1968; Roark and Yphantis, 1969; Hensley, et al, 1975b; Blackburn and Noltman, 1981; Minton and Lewis, 1981; Wilf and Minton, 1981; Correia, et al, 1985; Duong, et al., 1986; Hensley, et al, 1986; Lewis and Youle, 1986; Chatelier and Minton, 1987; Ross, et al, 1991; Rivas, et al, 1992).

This chapter focuses on parameter estimates from nonlinear models. The methods require some extra computing after the model has been fitted to a data set but the computing is efficient and easily accomplished. Fitting nonlinear models to data relies heavily on procedures used to fit linear models. The chapter presents a review of fitting linear models, including how to assess the quality of parameter estimates for such fits and discusses fitting nonlinear models and application of linear model methods for assessing the quality of parameter estimates for nonlinear models. The chapter also discusses a more accurate and valid procedure for characterizing the behavior of estimates of parameters in nonlinear models, illustrating the approach and the insights it gives using a model for frontal elution affinity chromatography and a compartment model. Profile plots can be extremely useful in nonlinear model building because they remove the gross dangers involved when using linear approximation standard errors and confidence regions.

Abstract. Autoregressive intergrated moving average (ARIMA) times series models are nonlinear in the parameters and so summarizing the inferential results for such models can be difficult. A common approach is to present parameter joint and marginal inference regions based on the linear approximation, and although such approximate regions are easy to calculate, it is not known generally whether they approximate the true regions adequately. In this paper we present exact approaches for summarizing the inferential results for time series model parameters, called profile t and profile trace plots, which are based on the work of Bates and Watts. Calculations for the profile plots are simple and can be used to determine exact regions, and so can be used to assess the accuracy of linear approximation regions. In addition to developing the profile plots for time series models, the main finding of the paper is that, for ARIMA model parameters, linear approximation regions are very satisfactory except when a parameter estimate is within about two standard errors of the stationarity or invertibility region boundary.

Free Access Appendix 4. Data Sets Used in Problems Douglas M. Bates, Douglas M. Bates Department of Statistics, University of Wisconsin, Madison, WisconsinSearch for more papers by this authorDonald G. Watts, Donald G. Watts Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, CanadaSearch for more papers by this author Book Author(s):Douglas M. Bates, Douglas M. Bates Department of Statistics, University of Wisconsin, Madison, WisconsinSearch for more papers by this authorDonald G. Watts, Donald G. Watts Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, CanadaSearch for more papers by this author First published: 26 August 1988 https://doi.org/10.1002/9780470316757.app4Book Series:Wiley Series in Probability and Statistics AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Nonlinear Regression Analysis and Its Applications RelatedInformation

Free Access Appendix 2. QR Decompositions Using Householder Transformations Douglas M. Bates, Douglas M. Bates Department of Statistics, University of Wisconsin, Madison, WisconsinSearch for more papers by this authorDonald G. Watts, Donald G. Watts Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, CanadaSearch for more papers by this author Book Author(s):Douglas M. Bates, Douglas M. Bates Department of Statistics, University of Wisconsin, Madison, WisconsinSearch for more papers by this authorDonald G. Watts, Donald G. Watts Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, CanadaSearch for more papers by this author First published: 26 August 1988 https://doi.org/10.1002/9780470316757.app2Book Series:Wiley Series in Probability and Statistics AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat Nonlinear Regression Analysis and Its Applications RelatedInformation

A statistical design criterion is applied to the problem of estimating dielectric constants in the model proposed by Havriliak and Negami. Under this criterion, the volume of the joint inference region of the parameter estimates is minimized. Because a computer program must be used to determine the optimal design points, simple approximating equations are presented for more practical use.

Wiley-Interscience Paperback Series Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. The authors have put together an extraordinary presentation of concepts and methods concerning the use and analysis of nonlinear regression models ...highly recommend[ed] ...for anyone needing to use and/or understand issues concerning the analysis of nonlinear regression -Technometrics [This book] provides a good balance of relevant theory and application with many examples ...[and it] provides the most balanced approach to theory and application appropriate for a first course in nonlinear regression modeling for graduate statistics students. -Mathematical Reviews [This book] joins a distinguished list of publications with a reputation for balancing technical rigor with readability, and theory with application. [It] upholds tradition ...[and is] a worthwhile reference for the marketing researcher with a serious interest in linear models. -Journal of Marketing Research This book offers a balanced presentation of the theoretical, practical, and computational aspects of nonlinear regression and provides background material on linear regression, including the geometrical development for linear and nonlinear least squares. authors employ real data sets throughout, and their extensive use of geometric constructs and continuing examples makes the progression of ideas appear very natural. book also includes pseudocode for computing algorithms.