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Link Functions (link + function)
Selected AbstractsRegression modelling of correlated data in ecology: subject-specific and population averaged response patternsJOURNAL OF APPLIED ECOLOGY, Issue 5 2009John Fieberg Summary 1.,Statistical methods that assume independence among observations result in optimistic estimates of uncertainty when applied to correlated data, which are ubiquitous in applied ecological research. Mixed effects models offer a potential solution and rely on the assumption that latent or unobserved characteristics of individuals (i.e. random effects) induce correlation among repeated measurements. However, careful consideration must be given to the interpretation of parameters when using a nonlinear link function (e.g. logit). Mixed model regression parameters reflect the change in the expected response within an individual associated with a change in that individual's covariates [i.e. a subject-specific (SS) interpretation], which may not address a relevant scientific question. In particular, a SS interpretation is not natural for covariates that do not vary within individuals (e.g. gender). 2.,An alternative approach combines the solution to an unbiased estimating equation with robust measures of uncertainty to make inferences regarding predictor,outcome relationships. Regression parameters describe changes in the average response among groups of individuals differing in their covariates [i.e. a population-averaged (PA) interpretation]. 3.,We compare these two approaches [mixed models and generalized estimating equations (GEE)] with illustrative examples from a 3-year study of mallard (Anas platyrhynchos) nest structures. We observe that PA and SS responses differ when modelling binary data, with PA parameters behaving like attenuated versions of SS parameters. Differences between SS and PA parameters increase with the size of among-subject heterogeneity captured by the random effects variance component. Lastly, we illustrate how PA inferences can be derived (post hoc) from fitted generalized and nonlinear-mixed models. 4.,Synthesis and applications. Mixed effects models and GEE offer two viable approaches to modelling correlated data. The preferred method should depend primarily on the research question (i.e. desired parameter interpretation), although operating characteristics of the associated estimation procedures should also be considered. Many applied questions in ecology, wildlife management and conservation biology (including the current illustrative examples) focus on population performance measures (e.g. mean survival or nest success rates) as a function of general landscape features, for which the PA model interpretation, not the more commonly used SS model interpretation may be more natural. [source] Generalized Linear Models in Family StudiesJOURNAL OF MARRIAGE AND FAMILY, Issue 4 2005Zheng WU Generalized linear models (GLMs), as defined by J. A. Nelder and R. W. M. Wedderburn (1972), unify a class of regression models for categorical, discrete, and continuous response variables. As an extension of classical linear models, GLMs provide a common body of theory and methodology for some seemingly unrelated models and procedures, such as the logistic, Poisson, and probit models, that are increasingly used in family studies. This article provides an overview of the principle and the key components of GLMs, such as the exponential family of distributions, the linear predictor, and the link function. To illustrate the application of GLMs, this article uses Canadian national survey data to build an example focusing on the number of close friends among older adults. The article concludes with a discussion of the strengths and weaknesses of GLMs. [source] Model-Checking Techniques Based on Cumulative ResidualsBIOMETRICS, Issue 1 2002D. Y. Lin Summary. Residuals have long been used for graphical and numerical examinations of the adequacy of regression models. Conventional residual analysis based on the plots of raw residuals or their smoothed curves is highly subjective, whereas most numerical goodness-of-fit tests provide little information about the nature of model misspecification. In this paper, we develop objective and informative model-checking techniques by taking the cumulative sums of residuals over certain coordinates (e.g., covariates or fitted values) or by considering some related aggregates of residuals, such as moving sums and moving averages. For a variety of statistical models and data structures, including generalized linear models with independent or dependent observations, the distributions of these stochastic processes under the assumed model can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be easily generated by computer simulation. Each observed process can then be compared, both graphically and numerically, with a number of realizations from the Gaussian process. Such comparisons enable one to assess objectively whether a trend seen in a residual plot reflects model misspecification or natural variation. The proposed techniques are particularly useful in checking the functional form of a covariate and the link function. Illustrations with several medical studies are provided. [source] A comparison between multivariate Slash, Student's t and probit threshold models for analysis of clinical mastitis in first lactation cowsJOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 5 2006Y-M. Chang Summary Robust threshold models with multivariate Student's t or multivariate Slash link functions were employed to infer genetic parameters of clinical mastitis at different stages of lactation, with each cow defining a cluster of records. The robust fits were compared with that from a multivariate probit model via a pseudo-Bayes factor and an analysis of residuals. Clinical mastitis records on 36 178 first-lactation Norwegian Red cows from 5286 herds, daughters of 245 sires, were analysed. The opportunity for infection interval, going from 30 days pre-calving to 300 days postpartum, was divided into four periods: (i) ,30 to 0 days pre-calving; (ii) 1,30 days; (iii) 31,120 days; and (iv) 121,300 days of lactation. Within each period, absence or presence of clinical mastitis was scored as 0 or 1 respectively. Markov chain Monte Carlo methods were used to draw samples from posterior distributions of interest. Pseudo-Bayes factors strongly favoured the multivariate Slash and Student's t models over the probit model. The posterior mean of the degrees of freedom parameter for the Slash model was 2.2, indicating heavy tails of the liability distribution. The posterior mean of the degrees of freedom for the Student's t model was 8.5, also pointing away from a normal liability for clinical mastitis. A residual was the observed phenotype (0 or 1) minus the posterior mean of the probability of mastitis. The Slash and Student's t models tended to have smaller residuals than the probit model in cows that contracted mastitis. Heritability of liability to clinical mastitis was 0.13,0.14 before calving, and ranged from 0.05 to 0.08 after calving in the robust models. Genetic correlations were between 0.50 and 0.73, suggesting that clinical mastitis resistance is not the same trait across periods, corroborating earlier findings with probit models. [source] Some robust design strategies for percentile estimation in binary response modelsTHE CANADIAN JOURNAL OF STATISTICS, Issue 4 2006Stefanie Biedermann Abstract For the problem of percentile estimation of a quantal response curve, the authors determine multiobjective designs which are robust with respect to misspecifications of the model assumptions. They propose a maximin approach based on efficiencies which leads to designs that are simultaneously efficient with respect to various choices of link functions and parameter regions. Furthermore, the authors deal with the problems of designing model and percentile robust experiments. They give various examples of such designs, which are calculated numerically. Quelques plans d'expérience robustes pour I'estirnation des centiles dans les modèles de reponse binaire Préoccupés par l'estimation des centiles d'une courbe de réponse quantale, les auteurs identifient des plans d'expérience multi-objectifs qui s'avèrent robustes m,me si les postulats du modèle ont été mal spécifiés. Ils proposent une approche maximin à base d'efficacités qui conduit à des plans efficaces à la fois pour divers choix de fonctions de lien et d'ensembles de valeurs pour les paramètres. Les auteurs abordent aussi la conception de modèles et de plans d'expérience robustes pour l'estimation des centiles. Ils fournissent plusieurs exemples de tels plans, obtenus numériquement. [source] Univariate and multirater ordinal cumulative link regression with covariate specific cutpointsTHE CANADIAN JOURNAL OF STATISTICS, Issue 4 2000Hemant Ishwaran Abstract The author considers a reparameterized version of the Bayesian ordinal cumulative link regression model as a tool for exploring relationships between covariates and "cutpoint" parameters. The use of this parameterization allows one to fit models using the leapfrog hybrid Monte Carlo method, and to bypass latent variable data augmentation and the slow convergence of the cutpoints which it usually entails. The proposed Gibbs sampler is not model specific and can be easily modified to handle different link functions. The approach is illustrated by considering data from a pediatric radiology study. RÉSUMÉ L'auteur propose une nouvelle paramé'trisation du modèle de régression ordinale bayésien à lien cumu-latif dont il se sert pour explorer la relation entre des covariables et des "points de coupure." Cette reparamétrisation permet d'ajuster les modèles par une méthode de Monte-Carlo à saute-mouton modifiée, évitant ainsi le besoin d'augmentation de données de la variable latente et la lenteur de convergence des points de coupure qui en découle souvent. L'échantillonneur de Gibbs qui est proposé n'est pas spécifique au modèle et peut ,tre adapté facilement à d'autres fonctions de lien. La méthode est illustrée au moyen d'une étude de radiologie pédiatrique [source] A general class of hierarchical ordinal regression models with applications to correlated roc analysisTHE CANADIAN JOURNAL OF STATISTICS, Issue 4 2000Hemant Ishwaran Abstract The authors discuss a general class of hierarchical ordinal regression models that includes both location and scale parameters, allows link functions to be selected adaptively as finite mixtures of normal cumulative distribution functions, and incorporates flexible correlation structures for the latent scale variables. Exploiting the well-known correspondence between ordinal regression models and parametric ROC (Receiver Operating Characteristic) curves makes it possible to use a hierarchical ROC (HROC) analysis to study multilevel clustered data in diagnostic imaging studies. The authors present a Bayesian approach to model fitting using Markov chain Monte Carlo methods and discuss HROC applications to the analysis of data from two diagnostic radiology studies involving multiple interpreters. RÉSUMÉ Les auteurs s'intéressent à une classe assez vaste de modèles de régression ordinale avec paramètres de localisation et d'échelle, laquelle permet la sélection adaptative de fonctions de lien s'exprimant comme mélanges finis de fonctions de répartition normales et fournit des structures de correlation flexibles pour les variables d'échelle latentes. En exploitant la correspondance bien connue entre les modèles de régression ordinale et les courbes d'efficacité paramétriques (CEP) des tests diagnostiques, il est possible d'analyser des données d'imagerie médicate diagnostique regroupées à plusieurs niveaux au moyen d'une CEP hiéiarchique. Les auteurs décrivent une approche bayésienne pour l'ajustement de tels modèles au moyen des méthodes de Monte Carlo à cha,ne de Markov et présentent deux applications concrètes concernant l'interprétation de clichés radiologiques [source] Efficiency measure, modelling and estimation in combined array designsAPPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 4 2003Tak Mak Abstract In off-line quality control, the settings that minimize the variance of a quality characteristic are unknown and must be determined based on an estimated dual response model of mean and variance. The present paper proposes a direct measure of the efficiency of any given design-estimation procedure for variance minimization. This not only facilitates the comparison of different design-estimation procedures, but may also provide a guideline for choosing a better solution when the estimated dual response model suggests multiple solutions. Motivated by the analysis of an industrial experiment on spray painting, the present paper also applies a class of link functions to model process variances in off-line quality control. For model fitting, a parametric distribution is employed in updating the variance estimates used in an iteratively weighted least squares procedure for mean estimation. In analysing combined array experiments, Engel and Huele (Technometrics, 1996; 39:365) used log-link to model process variances and considered an iteratively weighted least squares leading to the pseudo-likelihood estimates of variances as discussed in Carroll and Ruppert (Transformation and Weighting in Regression, Chapman & Hall: New York). Their method is a special case of the approach considered in this paper. It is seen for the spray paint data that the log-link may not be satisfactory and the class of link functions considered here improves substantially the fit to process variances. This conclusion is reached with a suggested method of comparing ,empirical variances' with the ,theoretical variances' based on the assumed model. Copyright © 2003 John Wiley & Sons, Ltd. [source] | ||||||||||