Simultaneous Confidence Bands for Penalized Spline Estimators

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractIn this paper, we propose a unified Bayesian approach for multivariate structured additive distributional regression analysis where inference is applicable to a huge class of multivariate response distributions, comprising continuous, discrete and latent models, and where each parameter of these potentially complex distributions is modelled by a structured additive predictor. The latter is an additive composition of different types of covariate effects e.g. nonlinear effects of continuous variables, random effects, spatial variations, or interaction effects. Inference is realised by a generic, efficient Markov chain Monte Carlo algorithm based on iteratively weighted least squares approximations and with multivariate Gaussian priors to enforce specific properties of functional effects. Examples will be given by illustrations on analysing the joint model of risk factors for chronic and acute childhood malnutrition in India and on ecological regression for German election results.

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“…For non‐parametric effects we choose the 95% simultaneous credible band constructed following Krivobokova et al . ().…”

Section: Bayesian Multivariate Distributional Regressionmentioning

confidence: 94%

Bayesian Structured Additive Distributional Regression for Multivariate Responses

Klein

Kneib

Klasen

et al. 2014

Journal of the Royal Statistical Society Series C: Applied Statistics

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“…Two key issues to maintain the nominal confidence level are to correct for bias in the nonparametric estimation and account for extra variability introduced by selecting the smoothing parameter. For single-level normal functional data, Krivobokova et al (2010) proposed simultaneous confidence band based on Bayesian framework, frequentist framework, and an intermediate mixed model representation framework. For correlated functional data, Craineceanu et al (2012) proposed bootstrap-based approaches.…”

Section: Discussionmentioning

confidence: 99%

A Marginal Approach to Reduced-Rank Penalized Spline Smoothing With Application to Multilevel Functional Data

Chen

Wang

Paik

et al. 2013

Journal of the American Statistical Association

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Multilevel functional data is collected in many biomedical studies. For example, in a study of the effect of Nimodipine on patients with subarachnoid hemorrhage (SAH), patients underwent multiple 4-hour treatment cycles. Within each treatment cycle, subjects’ vital signs were reported every 10 minutes. This data has a natural multilevel structure with treatment cycles nested within subjects and measurements nested within cycles. Most literature on nonparametric analysis of such multilevel functional data focus on conditional approaches using functional mixed effects models. However, parameters obtained from the conditional models do not have direct interpretations as population average effects. When population effects are of interest, we may employ marginal regression models. In this work, we propose marginal approaches to fit multilevel functional data through penalized spline generalized estimating equation (penalized spline GEE). The procedure is effective for modeling multilevel correlated generalized outcomes as well as continuous outcomes without suffering from numerical difficulties. We provide a variance estimator robust to misspecification of correlation structure. We investigate the large sample properties of the penalized spline GEE estimator with multilevel continuous data and show that the asymptotics falls into two categories. In the small knots scenario, the estimated mean function is asymptotically efficient when the true correlation function is used and the asymptotic bias does not depend on the working correlation matrix. In the large knots scenario, both the asymptotic bias and variance depend on the working correlation. We propose a new method to select the smoothing parameter for penalized spline GEE based on an estimate of the asymptotic mean squared error (MSE). We conduct extensive simulation studies to examine property of the proposed estimator under different correlation structures and sensitivity of the variance estimation to the choice of smoothing parameter. Finally, we apply the methods to the SAH study to evaluate a recent debate on discontinuing the use of Nimodipine in the clinical community.

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“…(1995), in which pointwise intervals were derived based on order statistics and scaled using a common credible level. In Krivobokova et al (2010), intervals based on sampled quantiles were scaled by a constant factor to obtain credible bands for functions of penalized spline models. Credible…”

Section: Introductionmentioning

confidence: 99%

“…To assess the performance of the proposed method, we perform a simulation study in which the given method is compared with the approaches in Krivobokova et al. (2010).…”

Section: Introductionmentioning

confidence: 99%

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Simultaneous Credible Bands for Latent Gaussian Models

Sørbye

Rue

2011

Scandinavian J Statistics

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Deterministic Bayesian inference for latent Gaussian models has recently become available using integrated nested Laplace approximations (INLA). Applying the INLAmethodology, marginal estimates for elements of the latent field can be computed efficiently, providing relevant summary statistics like posterior means, variances and pointwise credible intervals. In this paper, we extend the use of INLA to joint inference and present an algorithm to derive analytical simultaneous credible bands for subsets of the latent field. The algorithm is based on approximating the joint distribution of the subsets by multivariate Gaussian mixtures. Additionally, we present a saddlepoint approximation to compute Bayesian contour probabilities, representing the posterior support of fixed parameter vectors of interest. The given methods are applied to various examples from the literature.

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Simultaneous Confidence Bands for Penalized Spline Estimators

Cited by 96 publications

References 34 publications

Bayesian Structured Additive Distributional Regression for Multivariate Responses

Bayesian Structured Additive Distributional Regression for Multivariate Responses

A Marginal Approach to Reduced-Rank Penalized Spline Smoothing With Application to Multilevel Functional Data

Simultaneous Credible Bands for Latent Gaussian Models

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