Bayesian Functional Data Analysis Using WinBUGS

We provide user friendly software for Bayesian analysis of functional data models using \pkg{WinBUGS}~1.4. The excellent properties of Bayesian analysis in this context are due to: (1) dimensionality reduction, which leads to low dimensional projection bases; (2) mixed model representation of functional models, which provides a modular approach to model extension; and (3) orthogonality of the principal component bases, which contributes to excellent chain convergence and mixing properties. Our paper provides one more, essential, reason for using Bayesian analysis for functional models: the existence of software.

Bayesian Analysis for Penalized Spline Regression Using WinBUGS

Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. Good mixing properties of the MCMC chains are obtained by using low-rank thin-plate splines, while simulation times per iteration are reduced employing WinBUGS specific computational tricks.

MULTILEVEL SPARSE FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS

The basic observational unit in this paper is a function. Data are assumed to have a natural hierarchy of basic units. A simple example is when functions are recorded at multiple visits for the same subject. Di et al. (2009) proposed Multilevel Functional Principal Component Analysis (MFPCA) for this type of data structure when functions are densely sampled. Here we consider the case when functions are sparsely sampled and may contain as few as 2 or 3 observations per function. As with MFPCA, we exploit the multilevel structure of covariance operators and data reduction induced by the use of principal component bases. However, we address inherent methodological differences in the sparse sampling context to: 1) estimate the covariance operators; 2) estimate the functional scores and predict the underlying curves. We show that in the sparse context 1) is harder and propose an algorithm to circumvent the problem. Surprisingly, we show that 2) is easier via new BLUP calculations. Using simulations and real data analysis we show that the ability of our method to reconstruct underlying curves with few observations is stunning. This approach is illustrated by an application to the Sleep Heart Health Study, which contains two electroencephalographic (EEG) series at two visits for each subject

Bayesian Functional Data Analysis Using WinBUGS

We provide user friendly software for Bayesian analysis of functional data models using pkg{WinBUGS}~1.4. The excellent properties of Bayesian analysis in this context are due to: (1) dimensionality reduction, which leads to low dimensional projection bases; (2) mixed model representation of functional models, which provides a modular approach to model extension; and (3) orthogonality of the principal component bases, which contributes to excellent chain convergence and mixing properties. Our paper provides one more, essential, reason for using Bayesian analysis for functional models: the existence of software

Multilevel functional principal component analysis

The Sleep Heart Health Study (SHHS) is a comprehensive landmark study of sleep and its impacts on health outcomes. A primary metric of the SHHS is the in-home polysomnogram, which includes two electroencephalographic (EEG) channels for each subject, at two visits. The volume and importance of this data presents enormous challenges for analysis. To address these challenges, we introduce multilevel functional principal component analysis (MFPCA), a novel statistical methodology designed to extract core intra- and inter-subject geometric components of multilevel functional data. Though motivated by the SHHS, the proposed methodology is generally applicable, with potential relevance to many modern scientific studies of hierarchical or longitudinal functional outcomes. Notably, using MFPCA, we identify and quantify associations between EEG activity during sleep and adverse cardiovascular outcomes.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS206 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Analysis for Penalized Spline Regression Using Win BUGS

Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. MCMC mixing is substantially improved from the previous versions by using low{rank thin{plate splines instead of truncated polynomial basis. Simulation time per iteration is reduced 5 to 10 times using a computational trick

COX MODELS WITH NONLINEAR EFFECT OF COVARIATES MEASURED WITH ERROR: A CASE STUDY OF CHRONIC KIDNEY DISEASE INCIDENCE

We propose, develop and implement the simulation extrapolation (SIMEX) methodology for Cox regression models when the log hazard function is linear in the model parameters but nonlinear in the variables measured with error (LPNE). The class of LPNE functions contains but is not limited to strata indicators, splines, quadratic and interaction terms. The first order bias correction method proposed here has the advantage that it remains computationally feasible even when the number of observations is very large and multiple models need to be explored. Theoretical and simulation results show that the SIMEX method outperforms the naive method even with small amounts of measurement error. Our methodology was motivated by and applied to the study of time to chronic kidney disease (CKD) progression as a function of baseline kidney function and applied to the Atherosclerosis Risk in Communities (ARIC), a large epidemiological cohort stud

CORRECTED CONFIDENCE BANDS FOR FUNCTIONAL DATA USING PRINCIPAL COMPONENTS

Functional principal components (FPC) analysis is widely used to decompose and express functional observations. Curve estimates implicitly condition on basis functions and other quantities derived from FPC decompositions; however these objects are unknown in practice. In this paper, we propose a method for obtaining correct curve estimates by accounting for uncertainty in FPC decompositions. Additionally, pointwise and simultaneous confidence intervals that account for both model- based and decomposition-based variability are constructed. Standard mixed-model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition. A bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities. Iterated expectation and variance formulas combine both sources of uncertainty by combining model-based conditional estimates across the distribution of decompositions. Our method compares favorably to competing approaches in simulation studies that include both densely- and sparsely-observed functions. We apply our method to sparse observations of CD4 cell counts and to dense white-matter tract profiles. Code for the analyses and simulations is publicly available, and our method is implemented as the IVfpca() function in the R package refund on CRAN

Spatially Adaptive Bayesian P-Splines with Heteroscedastic Errors

An increasingly popular tool for nonparametric smoothing are penalized splines (P-splines) which use low-rank spline bases to make computations tractable while maintaining accuracy as good as smoothing splines. This paper extends penalized spline methodology by both modeling the variance function nonparametrically and using a spatially adaptive smoothing parameter. These extensions have been studied before, but never together and never in the multivariate case. This combination is needed for satisfactory inference and can be implemented effectively by Bayesian \mbox{MCMC}. The variance process controlling the spatially-adaptive shrinkage of the mean and the variance of the heteroscedastic error process are modeled as log-penalized splines. We discuss the choice of priors and extensions of the methodology,in particular, to multivariate smoothing using low-rank thin plate splines. A fully Bayesian approach provides the joint posterior distribution of all parameters, in particular, of the error standard deviation and penalty functions. In the multivariate case we produce maps of the standard deviation and penalty functions. Our methodology can be implemented using the Bayesian software WinBUGS