Abstract: Regression analysis is usually carried out under the hypothesis that one of the variables is normally distributed with constant variance, its mean being a function of the other variables. This assumption is not always satisfied, and in most cases difficult to ascertain.

Abstract: The authors present general descriptive models for spatiotemporal MEG (magnetoencephalogram) data and show the separability of the linear moment parameters and nonlinear location parameters in the MEG problem. A forward model with current dipoles in a spherically symmetric conductor is used as an example: however, other more advanced MEG models, as well as many EEG (electroencephalogram) models, can also be formulated in a similar linear algebra framework. A subspace methodology and computational approach to solving the conventional least-squares problem is presented. A new scanning approach, equivalent to the statistical MUSIC method, is also developed. This subspace method scans three-dimensional space with a one-dipole model, making it computationally feasible to scan the complete head volume. >

Abstract: Recent computational models of color vision demonstrate that it is possible to achieve exact color constancy over a limited range of lights and surfaces described by linear models. The success of these computational models hinges on whether any sizable range of surface spectral reflectances can be described by a linear model with about three parameters. In the first part of this paper, I analyze two large sets of empirical surface spectral reflectances and examine three conjectures concerning constraints on surface reflectance: that empirical surface reflectances fall within a linear model with a small number of parameters, that empirical surface reflectances fall within a linear model composed of band-limited functions with a small number of parameters, and that the shape of the spectral-sensitivity curves of human vision enhance the fit between empirical surface reflectances and a linear model. I conclude that the first and second conjectures hold for the two sets of spectral reflectances analyzed but that the number of parameters required to model the spectral reflectances is five to seven, not three. A reanalysis of the empirical data that takes human visual sensitivity into account gives more promising results. The linear models derived provide excellent fits to the data with as few as three or four parameters, confirming the third conjecture. The results suggest that constraints on possible surface-reflectance functions and the "filtering" properties of the shapes of the spectral-sensitivity curves of photoreceptors can both contribute to color constancy. In the last part of the paper I derive the relation between the number of photoreceptor classes present in vision and the "filtering" properties of each class. The results of this analysis reverse a conclusion reached by Barlow: the "filtering" properties of human photoreceptors are consistent with a trichromatic visual system that is color constant.

Abstract: A solution to multivariate state-space modeling, forecasting, and smoothing is discussed. We allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation, the observational equation, or both. An adaptive Monte Carlo integration technique known as the Gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework. The methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space. Missing data problems (including the k-step ahead prediction problem) also are easily incorporated into this framework. We illustrate the broad applicability of our approach with two examples: a problem involving nonnormal error distributions in a linear model setting and a one-step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters.

Abstract: A previous paper showed that a simple prospective model of voting and party identification subsumed much of the social-psychological and retrospective voting literatures, in the sense that it rigorously implied their key findings and added many new ones as well. This paper extends the argument by showing that the same prospective voting model has drastic implications for conventional statistical specifications in voting research. First, linear models should be discarded in favor of a particular nonlinear specification. Second, demographics should be dropped from the list of independent variables.

Abstract: : A fully Bayesian analysis of linear and nonlinear population models has previously been unavailable, as a consequence of the seeming impossibility of performing the necessary numerical Integrations in the complex multi- parameter structures typically arising in such models It is demonstrated that, for a variety of linear and nonlinear population models, a fully Bayesian analysis can be implemented in a straightforward manner using the Gibbs sampler The approach is illustrated with examples involving challenging problems of outliers and mean-variance relationships in population modelling

Abstract: This paper describes the problem of informative censoring in longitudinal studies where the primary outcome is rate of change in a continuous variable. Standard approaches based on the linear random effects model are valid only when the data are missing in a non-ignorable fashion. Informative censoring, which is a special type of non-ignorably missing data, occurs when the probability of early termination is related to an individual subject's true rate of change. When present, informative censoring causes bias in standard likelihood-based analyses, as well as in weighted averages of individual least-squares slopes. This paper reviews several methods proposed by others for analysis of informatively censored longitudinal data, and outlines a new approach based on a log-normal survival model. Maximum likelihood estimates may be obtained via the EM algorithm. Advantages of this approach are that it allows general unbalanced data caused by staggered entry and unequally-timed visits, it utilizes all available data, including data from patients with only a single measurement, and it provides a unified method for estimating all model parameters. Issues related to study design when informative censoring may occur are also discussed.

Abstract: Methods based on periodic regression have been designed for the detection of periodic components in short, noisy, and nonequidistant time series (as they are usually present in medicine and biology). The procedure consists of fitting a set of (cosine) curves to the data, with the analyst choosing the domain of trial periods to be analyzed and the distance between consecutive trial periods. We here describe an interactive program for least-squares rhythmometry written in C language for the Macintosh computer. For any given number of time series to be analyzed at once, the program is able to perform two different kinds of analyses: (a) linear in time, for the sequential fit of trial periods; and (b) linear in frequency, for the sequential fit of harmonic components from an initial fundamental period. For each series and for each trial period fitted to the data, the program gives the following information: fitted period; percent rhythm; p value from testing the assumption of zero amplitude; rhythm-adjusted mean or mesor, amplitude, and acrophase, each with corresponding standard errors and 95% confidence intervals when the component is statistically significant; and (when required by the analyst) p values from tests of sinusoidality, normality of residuals, and homogeneity of variance. Additionally, the program provides a summary report for each time series analyzed, including descriptive statistics such as the number of data analyzed for that series, minimum, maximum, arithmetic mean, standard deviation, standard error, 90% range, and 50% range. The analyst is also able to transform the data before doing any rhythmometric analysis. Transformations already integrated in the program include square root, logarithm, inverse, data as percentage of mean, data as percentage of mesor, and elimination of values outside +/- 3 SD from the mean. When several periods are suspected to be statistically significant, a multiple-component analysis can be also used by the concomitant least-squares fit of several harmonics. The program allows the simultaneous analysis of several periods in several variables from several individuals, with limitations depending solely on internal memory availability and speed requirements from the user. When series from different subjects or different variables in the same subject are available for analysis, a parameter test also included in the program can be used for comparison of rhythm characteristics at any given period. All information required in a single analysis is given by the analyst in the form of self-explanatory commands grouped in different "menus."(ABSTRACT TRUNCATED AT 400 WORDS)

Abstract: Recently developed methods for power analysis expand the options available for study design. We demonstrate how easily the methods can be applied by (1) reviewing their formulation and (2) describing their application in the preparation of a particular grant proposal. The focus is a complex but ubiquitous setting: repeated measures in a longitudinal study. Describing the development of the research proposal allows demonstrating the steps needed to conduct an effective power analysis. Discussion of the example also highlights issues that typically must be considered in designing a study. First, we discuss the motivation for using detailed power calculations, focusing on multivariate methods in particular. Second, we survey available methods for the general linear multivariate model (GLMM) with Gaussian errors and recommend those based on F approximations. The treatment includes coverage of the multivariate and univariate approaches to repeated measures, MANOVA, ANOVA, multivariate regression, and ...

Abstract: Structural equation modelling (SEM) is a modern statistical method that allows one to evaluate causal hypotheses on a set of intercorrelated nonexperimental data The sample variances and covariances, and possibly the means, are compared to those predicted by a theory-based hypothetical model after optimal estimation of the parameters of the model The goodness-of-fit of the empirical data to the hypothesized model is evaluated statistically This review describes the underlying statistical theory and rationale of SEM Both confirmatory factor analysis and latent variable path models are discussed The applicability of SEM to assessment of reliability and validity is noted A detailed example is provided, and several examples from the medical literature are briefly reviewed Cautions regarding the possible misuse or misinterpretation of the technique are also mentioned Possible future directions for the use of SEM in medical research are suggested Two appendices provide more technical details

Abstract: The problem considered is that of predicting a linear combination of the fixed and random effects of a mixed-effects linear model. More generally, the problem considered is that of predicting an unobservable random variable from a set of observable random variables. The best linear-unbiased predictor depends on parameters which generally are unknown. Various exact or approximate expressions are given for the mean squared error (MSE) of the predictor obtained by replacing the unknown parameters with estimates. Several estimators of the MSE are investigated.

Abstract: For reliable information on operating plants it is essential to design measuring points well by selecting directly measured quantities from the set of all measurable quantities. This article deals with a new method for optimizing measurement design. It is based on multiple Gauss-Jordan elimination of the system of linear mathematical model equations and solves the problem of instrumentation design in new plants as well as the problem of optimizing existing measuring systems. Optimization methods for linear objective functions and for objective functions of general type are proposed. The method also offers a complex classification of quantities (observability and redundancy). After the optimization, the problem is presolved and is ready for an optimal processing of measured data. The mathematical model is reduced to the minimum set of equations and quantities relevant to the solution of a given problem. From a numerical standpoint, the solution is efficient.

Abstract: A nonlinear regression model is proposed as an alternative to the Box-Cox regression model for nonnegative variables. The functional form contains linear, exponential, and reciprocal models as special cases. Unlike Box-Cox type approaches, the proposed estimators of the conditional mean function are robust to conditional variance and other distributional misspecifications. Computationally simple, robust Lagrange multiplier statistics for restricted versions of the model are derived. Scale invariant t-statistics are proposed and the Lagrange multiplier statistic for exclusion restrictions is shown to be scale invariant. Copyright 1992 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Abstract: A simple estimator for β is proposed for the model y=x'β+g(1)+error, g smooth but unknown. The approach is to approximate the estimating equation obtained from a semiparametric likelihood and in the simplest case reduces to minimizing the distance between the pseudoresiduals y-x'β and a local linear cross-validated estimate of them. When the errors are independent with finite variance, the bias and variance of the estimate are computed and compared against the least squares estimate with g known

Abstract: Random coefficient regression models are important in representing linear models with heteroscedastic errors and in unifying the study of classical fixed effects and random effects linear models. For prediction intervals and for bootstrapping in random coefficient regressions, it is necessary to estimate the distributions of the random coefficients consistently. We show that this is often possible and provide practical representative estimators of these distributions.

Abstract: Linear models are the by far most commonly used approach for describing physical signals and systems. As a result, the theory of linear models is quite extensive in areas like control theory and si ...

Abstract: SUMMARY We propose a method of obtaining an M-estimator in a linear model when the responses are subject to right censoring. The central limit theorem for the estimator using squared error loss, i.e. least squares, is derived using counting process martingale techniques. The estimation method is applied to the Stanford heart transplant data for illustration.

Abstract: The goal of atmospheric data assimilation is to determine the most accurate representation of the signal from the available observations. The optimality of a data assimilation scheme measures how much information has been extracted from the observations. It is possible to quantify the optimality of the scheme using on-line performance diagnostics. Such a diagnostic is the proposed lagged innovation covariance procedure. This diagnostic has been developed from Kalman filter theory. Its characteristics are examined using a simple scalar model, a univariate one-dimensional linear advection model, and a linear quasigeostrophic model. The model results are compared with actual lagged innovation covariances derived from the innovation sequences of an operational data assimilation system.

Abstract: SUMMARY General aspects of nonlinearity in the context of component of variance models are discussed, and two special topics are examined in detail. Firstly, simple procedures, both formal and informal, are proposed for describing departures from normal-theory linear models. Transformation models are shown to be a special case of a more general formulation, and data on blood pressure are analyzed in illustration. Secondly, an approximate likelihood is proposed and its accurate performance is examined numerically using examples of exponential regression and the analysis of several related 2 x 2 tables. In the latter example, the approximate score test has improved power over the MantelHaenszel test.

Abstract: Stationary waves generated over orography in a linear model and a general circulation model (GCM) are compared to examine how the atmosphere's response is established for small mountains and how linear theory breaks down over large orographic features. Both models have nine vertical levels and are low-resolution (R15) spectral models. The linear model solves the stationary linear primitive equations. The GCM's control integration uses zonally uniform and hemispherically symmetric boundary conditions, with a global swamp surface. Five experiments are performed by perturbing the GCM with Gaussian mountains of various heights introduced in midlatitudes. The stationary wave model is linearized about zonal mean fields from the GCM climatology. The linear model's response to a Gaussian mountain at 45°N latitude is dominated by a single wave train radiating toward the southeast. For mountain heights between 0.7 and 2 km, the GCM's stationary waves are similar to the linear model response to orography, a...

Abstract: Time series analyses of the agenda‐setting function of the mass media suggest that the strength of media effects might vary not only between, but also within issues. This leads to the question of whether the relationship between media and public agenda can be described best with linear or nonlinear models. Two data sets were used to compare linear and nonlinear models: A content analysis of the main German television news shows and 53 weekly surveys on 16 issues. The results indicate that nonlinear models are in some cases superior to the linear model in terms of explained variance. This leads to some theoretical considerations about the nature and the process of agenda setting.