Gauss-Seidel Estimation of Generalized Linear Mixed Models with Application to Poisson Modeling of Spatially Varying Disease Rates

Generalized linear mixed models (GLMMs) provide an elegant framework for the analysis of correlated data. Due to the non-closed form of the likelihood, GLMMs are often fit by computational procedures like penalized quasi-likelihood (PQL). Special cases of these models are generalized linear models (GLMs), which are often fit using algorithms like iterative weighted least squares (IWLS). High computational costs and memory space constraints often make it difficult to apply these iterative procedures to data sets with very large number of cases. This paper proposes a computationally efficient strategy based on the Gauss-Seidel algorithm that iteratively fits sub-models of the GLMM to subsetted versions of the data. Additional gains in efficiency are achieved for Poisson models, commonly used in disease mapping problems, because of their special collapsibility property which allows data reduction through summaries. Convergence of the proposed iterative procedure is guaranteed for canonical link functions. The strategy is applied to investigate the relationship between ischemic heart disease, socioeconomic status and age/gender category in New South Wales, Australia, based on outcome data consisting of approximately 33 million records. A simulation study demonstrates the algorithm's reliability in analyzing a data set with 12 million records for a (non-collapsible) logistic regression model.

LIKELIHOOD ESTIMATION OF CONJUGACY RELATIONSHIPS IN LINEAR MODELS WITH APPLICATIONS TO HIGH-THROUGHPUT GENOMICS

In the simultaneous estimation of a large number of related quantities, multilevel models provide a formal mechanism for efficiently making use of the ensemble of information for deriving individual estimates. In this article we investigate the ability of the likelihood to identify the relationship between signal and noise in multilevel linear mixed models. Specifically, we consider the ability of the likelihood to diagnose conjugacy or independence between the signals and noises. Our work was motivated by the analysis of data from high-throughput experiments in genomics. The proposed model leads to a more flexible family. However, we further demonstrate that adequately capitalizing on the benefits of a well fitting fully-specified likelihood in the terms of gene ranking is difficult.

A HIDDEN MARKOV MODEL FOR JOINT ESTIMATION OF GENOTYPE AND COPY NUMBER IN HIGH-THROUGHPUT SNP CHIPS

Amplifications and deletions of chromosomal DNA, as well as copy-neutral loss of heterozygosity have been associated with diseases processes. High-throughput single nucleotide polymorphism (SNP) arrays are useful for making genome-wide estimates of copy number and genotype calls. Because neighboring SNPs in high throughput SNP arrays are likely to have dependent copy number and genotype due to the underlying haplotype structure and linkage disequilibrium, hidden Markov models (HMM) may be useful for improving genotype calls and copy number estimates that do not incorporate information from nearby SNPs. We improve previous approaches that utilize a HMM framework for inference in high throughput SNP arrays by integrating copy number, genotype calls, and the corresponding confidence scores when available. Using simulated data, we demonstrate how confidence scores control smoothing in a probabilistic framework. Software for fitting HMMs to SNP array data is available in the R package ICE.

Kernel Estimation of Rate Function for Recurrent Event Data

Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth will produce curve with nicks occurring at the censoring times, whereas there is no such problem with the least squares method. Furthermore, the asymptotic variance of the least squares estimator is shown to be smaller under regularity conditions. However, in the implementation of the bootstrap procedures, the moment method is computationally more efficient than the least squares method because the former approach uses condensed bootstrap data. The performance of the proposed procedures is studied through Monte Carlo simulations and an epidemiological example on intravenous drug users.

Statistical evaluation of time-dependent metabolite concentrations: estimation of post-mortem intervals based on in situ 1H-MRS of the brain

Knowledge of the time interval from death (post-mortem interval, PMI) has an enormous legal, criminological and psychological impact. Aiming to find an objective method for the determination of PMIs in forensic medicine, 1H-MR spectroscopy (1H-MRS) was used in a sheep head model to follow changes in brain metabolite concentrations after death. Following the characterization of newly observed metabolites (Ith et al., Magn. Reson. Med. 2002; 5: 915-920), the full set of acquired spectra was analyzed statistically to provide a quantitative estimation of PMIs with their respective confidence limits. In a first step, analytical mathematical functions are proposed to describe the time courses of 10 metabolites in the decomposing brain up to 3 weeks post-mortem. Subsequently, the inverted functions are used to predict PMIs based on the measured metabolite concentrations. Individual PMIs calculated from five different metabolites are then pooled, being weighted by their inverse variances. The predicted PMIs from all individual examinations in the sheep model are compared with known true times. In addition, four human cases with forensically estimated PMIs are compared with predictions based on single in situ MRS measurements. Interpretation of the individual sheep examinations gave a good correlation up to 250 h post-mortem, demonstrating that the predicted PMIs are consistent with the data used to generate the model. Comparison of the estimated PMIs with the forensically determined PMIs in the four human cases shows an adequate correlation. Current PMI estimations based on forensic methods typically suffer from uncertainties in the order of days to weeks without mathematically defined confidence information. In turn, a single 1H-MRS measurement of brain tissue in situ results in PMIs with defined and favorable confidence intervals in the range of hours, thus offering a quantitative and objective method for the determination of PMIs.

Precise estimation of postoperative cup alignment from single standard X-ray radiograph with gonadal shielding

This paper addresses the problem of estimating postoperative cup alignment from single standard X-ray radiograph with gonadal shielding. The widely used procedure of evaluation of cup orientation following total hip arthroplasty using single standard anteroposterior radiograph is known inaccurate, largely due to the wide variability in individual pelvic position relative to X-ray plate. 2D-3D image registration methods have been introduced to estimate the rigid transformation between a preoperative CT volume and postoperative radiograph(s) for an accurate estimation of the postoperative cup alignment relative to an anatomical reference extracted from the CT data. However, these methods require either multiple radiographs or a radiograph-specific calibration, both of which are not avaiable for most retrospective studies. Furthermore, these methods were only evaluated on X-ray radiograph(s) without gonadal shielding. In this paper, we propose to use a hybrid 2D-3D registration scheme combining an iterative landmark-to-ray registration with a 2D-3D intensity-based registration to estimate the rigid transfromation for a precise estimation of cup alignment. Quantitative and qualitative results evaluated on clinical and cadaveric datasets are given which indicate the validity of our approach.

Anteroposterior pelvic radiographs to assess acetabular retroversion: high validity of the "cross-over-sign"

Acetabular retroversion has been proposed to contribute to the development of osteoarthritis of the hip. For the diagnosis of this condition, conventional AP pelvic radiographs may represent a reliable, easily available diagnostic modality as they can be obtained with a reproducible technique allowing the anterior and posterior acetabular rims to be visible for assessment. This study was designed to: (i) determine cranial, central, and caudal anatomic acetabular version (AV) from cadaveric specimens; (ii) establish the validity and reliability of the radiographic measurements of central acetabular anteversion; and (iii) determine the validity and reliability of the radiographic "cross-over-sign" to detect acetabular retroversion. Using 43 desiccated pelvises (86 acetabuli) the anatomic AVs were measured at three different transverse planes (cranially, centrally, and caudally). From these pelvises, standardized AP pelvic radiographs were obtained. To directly measure central AV, a modified radiographic method is introduced for the use of AP pelvic radiographs. The validity and reliability of this radiographic method and of the radiographic cross-over-sign to detect cranial acetabular retroversion were determined. The mean central and caudal anatomic AVs were approximately 20 degrees , and the mean cranial AV was 8 degrees . Cranial retroversion (AV < 0 degrees ) was present in 19 of 86 hips (22%). A linear correlation was found between the central and cranial AV. Below 10 degrees of central AV, all acetabuli were cranially retroverted. Between 10 degrees and 20 degrees , 30% of the acetabuli were cranially retroverted, and above 20 degrees , only 1 of 45 acetabuli was cranially retroverted. The radiographic measurement of the central AV (20.3 +/- 6.5 degrees ) correlated strongly with the anatomic AV (20.1 +/- 6.4 degrees ). The sensitivity of the cross-over-sign to detect a cranial acetabular anteversion of less than 4 degrees was 96%, its specificity 95%, and the positive predictive and negative predictive values 90% and 98%, respectively. Both the modified radiographic anteversion measurements and the cross-over-sign demonstrated substantial inter- and intraobserver reliability. Retroversion is almost exclusively a problem of the cranial acetabulum. The cranial AV is on average 12 degrees lower than the central AV, with the latter directly measurable from AP pelvic radiographs. A central AV of less than 10 degrees was associated with cranial retroversion. The presence of a positive cross-over-sign is a highly reliable indicator of cranial AV of <4 degrees.

Estimation of sex and age of "virtual skeletons"--a feasibility study

This article presents a feasibility study with the objective of investigating the potential of multi-detector computed tomography (MDCT) to estimate the bone age and sex of deceased persons. To obtain virtual skeletons, the bodies of 22 deceased persons with known age at death were scanned by MDCT using a special protocol that consisted of high-resolution imaging of the skull, shoulder girdle (including the upper half of the humeri), the symphysis pubis and the upper halves of the femora. Bone and soft-tissue reconstructions were performed in two and three dimensions. The resulting data were investigated by three anthropologists with different professional experience. Sex was determined by investigating three-dimensional models of the skull and pelvis. As a basic orientation for the age estimation, the complex method according to Nemeskéri and co-workers was applied. The final estimation was effected using additional parameters like the state of dentition, degeneration of the spine, etc., which where chosen individually by the three observers according to their experience. The results of the study show that the estimation of sex and age is possible by the use of MDCT. Virtual skeletons present an ideal collection for anthropological studies, because they are obtained in a non-invasive way and can be investigated ad infinitum.

Estimation of the flammability zone boundaries with thermodynamic and empirical equations

The flammability zone boundaries are very important properties to prevent explosions in the process industries. Within the boundaries, a flame or explosion can occur so it is important to understand these boundaries to prevent fires and explosions. Very little work has been reported in the literature to model the flammability zone boundaries. Two boundaries are defined and studied: the upper flammability zone boundary and the lower flammability zone boundary. Three methods are presented to predict the upper and lower flammability zone boundaries: The linear model The extended linear model, and An empirical model The linear model is a thermodynamic model that uses the upper flammability limit (UFL) and lower flammability limit (LFL) to calculate two adiabatic flame temperatures. When the proper assumptions are applied, the linear model can be reduced to the well-known equation yLOC = zyLFL for estimation of the limiting oxygen concentration. The extended linear model attempts to account for the changes in the reactions along the UFL boundary. Finally, the empirical method fits the boundaries with linear equations between the UFL or LFL and the intercept with the oxygen axis. xx Comparison of the models to experimental data of the flammability zone shows that the best model for estimating the flammability zone boundaries is the empirical method. It is shown that is fits the limiting oxygen concentration (LOC), upper oxygen limit (UOL), and the lower oxygen limit (LOL) quite well. The regression coefficient values for the fits to the LOC, UOL, and LOL are 0.672, 0.968, and 0.959, respectively. This is better than the fit of the "zyLFL" method for the LOC in which the regression coefficient’s value is 0.416.

Estimation of the degree of polarization through computational sensing

The degree of polarization of a refected field from active laser illumination can be used for object identifcation and classifcation. The goal of this study is to investigate methods for estimating the degree of polarization for refected fields with active laser illumination, which involves the measurement and processing of two orthogonal field components (complex amplitudes), two orthogonal intensity components, and the total field intensity. We propose to replace interferometric optical apparatuses with a computational approach for estimating the degree of polarization from two orthogonal intensity data and total intensity data. Cramer-Rao bounds for each of the three sensing modalities with various noise models are computed. Algebraic estimators and maximum-likelihood (ML) estimators are proposed. Active-set algorithm and expectation-maximization (EM) algorithm are used to compute ML estimates. The performances of the estimators are compared with each other and with their corresponding Cramer-Rao bounds. Estimators for four-channel polarimeter (intensity interferometer) sensing have a better performance than orthogonal intensities estimators and total intensity estimators. Processing the four intensities data from polarimeter, however, requires complicated optical devices, alignment, and four CCD detectors. It only requires one or two detectors and a computer to process orthogonal intensities data and total intensity data, and the bounds and estimator performances demonstrate that reasonable estimates may still be obtained from orthogonal intensities or total intensity data. Computational sensing is a promising way to estimate the degree of polarization.