Numerical Analysis and Computation Commons^™

Gauss-Seidel Estimation Of Generalized Linear Mixed Models With Application To Poisson Modeling Of Spatially Varying Disease Rates, Subharup Guha, Louise Ryan

Harvard University Biostatistics Working Paper Series

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 …

Computational Techniques For Spatial Logistic Regression With Large Datasets, Christopher J. Paciorek, Louise Ryan

Harvard University Biostatistics Working Paper Series

In epidemiological work, outcomes are frequently non-normal, sample sizes may be large, and effects are often small. To relate health outcomes to geographic risk factors, fast and powerful methods for fitting spatial models, particularly for non-normal data, are required. We focus on binary outcomes, with the risk surface a smooth function of space. We compare penalized likelihood models, including the penalized quasi-likelihood (PQL) approach, and Bayesian models based on fit, speed, and ease of implementation.

A Bayesian model using a spectral basis representation of the spatial surface provides the best tradeoff of sensitivity and specificity in simulations, detecting real spatial …

Remarks On Risk-Sensitive Control Problems, José Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor α goes to zero. If u_α(θ, x) denotes the optimal cost function, being the risk factor, then it is shown that lim_α→0αu_α(θ, x) = ξ(θ) where ξ(θ) is the average on ]0, θ[ of the optimal cost of the (usual) in nite horizon risk-sensitive control problem.

Numerical Integration Strategy For Generating Root Migration Diagrams, Deepak H. Panduranga

Mechanical & Aerospace Engineering Theses & Dissertations

Evans root locus is a graphical method for analyzing or designing the stability and performance of a single channel closed-loop system. The method consists of plotting the locus of closed-loop characteristic equation roots in the complex plane as the loop gain parameter is varied over a wide range of values. Several methods for plotting the root locus graph exist and can be mainly classified into two types: analytical methods and numerical methods. The most common numerical method is based on polynomial factoring or root solving. A less common, alternative method for generating the root locus graph is based on numerical …

The Interquartile Range: Theory And Estimation., Dewey Lonzo Whaley

Electronic Theses and Dissertations

The interquartile range (IQR) is used to describe the spread of a distribution. In an introductory statistics course, the IQR might be introduced as simply the “range within which the middle half of the data points lie.” In other words, it is the distance between the two quartiles, IQR = Q₃ - Q₁. We will compute the population IQR, the expected value, and the variance of the sample IQR for various continuous distributions. In addition, a bootstrap confidence interval for the population IQR will be evaluated.

Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal

Mathematical Sciences Technical Reports (MSTR)

This paper considers the inverse problem of locating one or more circular inclusions in a two-dimensional domain using thermal boundary data, specifically, the input heat flux and measured boundary temperature. The forward problem is governed by the heat equation. We show how the position and size of such defects can be recovered using the boundary data and various approximations of the solution to the forward problem. We also consider the stability of the algorithm involved to recover the defects.

Investigation And Applications Of Latin Hypercube Sampling Methods, Prashant K. Patel

Mechanical & Aerospace Engineering Theses & Dissertations

The Latin hypercube sampling method, sometimes called the stratified sampling method, is a commonly used variance reduction method. Recently, the optimal Latin hypercube sampling method (OLH) and the modified optimal Latin hypercube sampling method (MOLH) are introduced in the literature, which use the genetic algorithm to adjust the distance between sampled points.

This paper will compare the basic Latin hypercube, the optimal Latin hypercube and the modified optimal Latin hypercube methods for generating the sample points for statistical application and curve fitting, in te1ms of accuracy and efficiency. Analytical and structural examples are used in this thesis to facilitate the …

Survival Model And Estimation For Lung Cancer Patients., Xingchen Yuan

Electronic Theses and Dissertations

Lung cancer is the most frequent fatal cancer in the United States. Following the notion in actuarial math analysis, we assume an exponential form for the baseline hazard function and combine Cox proportional hazard regression for the survival study of a group of lung cancer patients. The covariates in the hazard function are estimated by maximum likelihood estimation following the proportional hazards regression analysis. Although the proportional hazards model does not give an explicit baseline hazard function, the baseline hazard function can be estimated by fitting the data with a non-linear least square technique. The survival model is then examined …

Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su

All HMC Faculty Publications and Research

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their …

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N- Dimensions, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples.

Cluster Analysis Of Genomic Data With Applications In R, Katherine S. Pollard, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In this paper, we provide an overview of existing partitioning and hierarchical clustering algorithms in R. We discuss statistical issues and methods in choosing the number of clusters, the choice of clustering algorithm, and the choice of dissimilarity matrix. In particular, we illustrate how the bootstrap can be employed as a statistical method in cluster analysis to establish the reproducibility of the clusters and the overall variability of the followed procedure. We also show how to visualize a clustering result by plotting ordered dissimilarity matrices in R. We present a new R package, hopach, which implements the hybrid clustering method, …

Multiple Testing Procedures And Applications To Genomics, Merrill D. Birkner, Katherine S. Pollard, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

This chapter proposes widely applicable resampling-based single-step and stepwise multiple testing procedures (MTP) for controlling a broad class of Type I error rates, in testing problems involving general data generating distributions (with arbitrary dependence structures among variables), null hypotheses, and test statistics (Dudoit and van der Laan, 2005; Dudoit et al., 2004a,b; van der Laan et al., 2004a,b; Pollard and van der Laan, 2004; Pollard et al., 2005). Procedures are provided to control Type I error rates defined as tail probabilities for arbitrary functions of the numbers of Type I errors, V_n, and rejected hypotheses, R_n. These error rates include: …

Robust Inferences For Covariate Effects On Survival Time With Censored Linear Regression Models, Larry Leon, Tianxi Cai, L. J. Wei

Harvard University Biostatistics Working Paper Series

Various inference procedures for linear regression models with censored failure times have been studied extensively. Recent developments on efficient algorithms to implement these procedures enhance the practical usage of such models in survival analysis. In this article, we present robust inferences for certain covariate effects on the failure time in the presence of "nuisance" confounders under a semiparametric, partial linear regression setting. Specifically, the estimation procedures for the regression coefficients of interest are derived from a working linear model and are valid even when the function of the confounders in the model is not correctly specified. The new proposals are …

Penalty Approximation And Analytical Characterization Of The Problem Of Super-Replication Under Portfolio Constraints, Alain Bensoussan, Nizar Touzi, José Luis Menaldi

Mathematics Faculty Research Publications

In this paper, we consider the problem of super-replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super-replication value is the smallest function which lies above the Black-Scholes price function and which is stable for the so-called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.

Transient Non-Linear Heat Conduction Solution By A Dual Reciprocity Boundary Element Method With An Effective Posteriori Error Estimator, Eduardo Divo, Alain J. Kassab

Publications

A Dual Reciprocity Boundary Element Method is formulated to solve non-linear heat conduction problems. The approach is based on using the Kirchhoff transform along with lagging of the effective non-linear thermal diffusivity. A posteriori error estimate is used to provide effective estimates of the temporal and spatial error. A numerical example is used to demonstrate the approach.

Application Of Meshless Methods For Thermal Analysis, Darrell Pepper, Bozidar Sarler

Mechanical Engineering Faculty Research

Many numerical and analytical schemes exist for solving heat transfer problems. The meshless method is a particularly attractive method that is receiving attention in the engineering and scientific modeling communities. The meshless method is simple, accurate, and requires no polygonalisation. In this study, we focus on the application of meshless methods using radial basis functions (RBFs) — which are simple to implement — for thermal problems. Radial basis functions are the natural generalization of univariate polynomial splines to a multivariate setting that work for arbitrary geometry with high dimensions. RBF functions depend only on the distance from some center point. …

A Hybrid Newton-Type Method For The Linear Regression In Case-Cohort Studies, Menggang Yu, Bin Nan

The University of Michigan Department of Biostatistics Working Paper Series

Case-cohort designs are increasingly commonly used in large epidemiological cohort studies. Nan, Yu, and Kalbeisch (2004) provided the asymptotic results for censored linear regression models in case-cohort studies. In this article, we consider computational aspects of their proposed rank based estimating methods. We show that the rank based discontinuous estimating functions for case-cohort studies are monotone, a property established for cohort data in the literature, when generalized Gehan type of weights are used. Though the estimating problem can be formulated to a linear programming problem as that for cohort data, due to its easily uncontrollable large scale even for a …

Semiparametric Regression In Capture-Recapture Modelling, O. Gimenez, C. Barbraud, Ciprian M. Crainiceanu, S. Jenouvrier, B.T. Morgan

Johns Hopkins University, Dept. of Biostatistics Working Papers

Capture-recapture models were developed to estimate survival using data arising from marking and monitoring wild animals over time. Variation in the survival process may be explained by incorporating relevant covariates. We develop nonparametric and semiparametric regression models for estimating survival in capture-recapture models. A fully Bayesian approach using MCMC simulations was employed to estimate the model parameters. The work is illustrated by a study of Snow petrels, in which survival probabilities are expressed as nonlinear functions of a climate covariate, using data from a 40-year study on marked individuals, nesting at Petrels Island, Terre Adelie.

Survival Analysis Using Auxiliary Variables Via Nonparametric Multiple Imputation, Chiu-Hsieh Hsu, Jeremy Taylor, Susan Murray, Daniel Commenges

The University of Michigan Department of Biostatistics Working Paper Series

We develop an approach, based on multiple imputation, that estimates the marginal survival distribution in survival analysis using auxiliary variable to recover information for censored observations. To conduct the imputation, we use two working survival model to define the nearest neighbor imputing risk set. One model is for the event times and the other for the censoring times. Based on the imputing risk set, two nonparametric multiple imputation methods are considered: risk set imputation, and Kaplan-Meier estimator. For both methods a future event or censoring time is imputed for each censored observation. With a categorical auxiliary variable, we show that …

A Bayesian Mixture Model Relating Dose To Critical Organs And Functional Complication In 3d Conformal Radiation Therapy, Tim Johnson, Jeremy Taylor, Randall K. Ten Haken, Avraham Eisbruch

The University of Michigan Department of Biostatistics Working Paper Series

A goal of radiation therapy is to deliver maximum dose to the target tumor while minimizing complications due to irradiation of critical organs. Technological advances in 3D conformal radiation therapy has allowed great strides in realizing this goal, however complications may still arise. Critical organs may be adjacent to tumors or in the path of the radiation beam. Several mathematical models have been proposed that describe a relationship between dose and observed functional complication, however only a few published studies have successfully fit these models to data using modern statistical methods which make efficient use of the data. One complication …

A Bayesian Method For Finding Interactions In Genomic Studies, Wei Chen, Debashis Ghosh, Trivellore E. Raghuanthan, Sharon Kardia

The University of Michigan Department of Biostatistics Working Paper Series

An important step in building a multiple regression model is the selection of predictors. In genomic and epidemiologic studies, datasets with a small sample size and a large number of predictors are common. In such settings, most standard methods for identifying a good subset of predictors are unstable. Furthermore, there is an increasing emphasis towards identification of interactions, which has not been studied much in the statistical literature. We propose a method, called BSI (Bayesian Selection of Interactions), for selecting predictors in a regression setting when the number of predictors is considerably larger than the sample size with a focus …

Spatially Adaptive Bayesian P-Splines With Heteroscedastic Errors, Ciprian M. Crainiceanu, David Ruppert, Raymond J. Carroll

Johns Hopkins University, Dept. of Biostatistics Working Papers

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 …

Gllamm Manual, Sophia Rabe-Hesketh, Anders Skrondal, Andrew Pickles

U.C. Berkeley Division of Biostatistics Working Paper Series

This manual describes a Stata program gllamm that can estimate Generalized Linear Latent and Mixed Models (GLLAMMs). GLLAMMs are a class of multilevel latent variable models for (multivariate) responses of mixed type including continuous responses, counts, duration/survival data, dichotomous, ordered and unordered categorical responses and rankings. The latent variables (common factors or random effects) can be assumed to be discrete or to have a multivariate normal distribution. Examples of models in this class are multilevel generalized linear models or generalized linear mixed models, multilevel factor or latent trait models, item response models, latent class models and multilevel structural equation models. …

Data Adaptive Estimation Of The Treatment Specific Mean, Yue Wang, Oliver Bembom, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

An important problem in epidemiology and medical research is the estimation of the causal effect of a treatment action at a single point in time on the mean of an outcome, possibly within strata of the target population defined by a subset of the baseline covariates. Current approaches to this problem are based on marginal structural models, i.e., parametric models for the marginal distribution of counterfactural outcomes as a function of treatment and effect modifiers. The various estimators developed in this context furthermore each depend on a high-dimensional nuisance parameter whose estimation currently also relies on parametric models. Since misspecification …

Finding Cancer Subtypes In Microarray Data Using Random Projections, Debashis Ghosh

The University of Michigan Department of Biostatistics Working Paper Series

One of the benefits of profiling of cancer samples using microarrays is the generation of molecular fingerprints that will define subtypes of disease. Such subgroups have typically been found in microarray data using hierarchical clustering. A major problem in interpretation of the output is determining the number of clusters. We approach the problem of determining disease subtypes using mixture models. A novel estimation procedure of the parameters in the mixture model is developed based on a combination of random projections and the expectation-maximization algorithm. Because the approach is probabilistic, our approach provides a measure for the number of true clusters …

History-Adjusted Marginal Structural Models And Statically-Optimal Dynamic Treatment Regimes, Mark J. Van Der Laan, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a treatment. These models, introduced by Robins, model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at a final time point. However, the utility of these models for some applications has been limited by their inability to incorporate modification of the causal effect of treatment by time-varying covariates. …

Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh

Mathematical Sciences Technical Reports (MSTR)

This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.

Non-Destructive Testing Of Thermal Resistances For A Single Inclusion In A 2-Dimensional Domain, Nicholas Christian, Mathew A. Johnson

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the amount of corrosion/disbonding which has occurred on the boundary of a single circular (or nearly circular) inclusion D in a two dimensional domain W using Cauchy data for the steady-state heat equation. We develop an algorithm for reconsructing a function which qunatifies the level of corrosion/disbonding at each point in ¶W. We also address the issue of well-posedness and develop a simple regularization scheme. Then we provide several numerical examples. We shall show a simple procedure for recovering the center of D assuming that the boundary of W and …

On The Accelerated Failure Time Model For Current Status And Interval Censored Data, Lu Tian, Tianxi Cai

Harvard University Biostatistics Working Paper Series

This paper introduces a novel approach to making inference about the regression parameters in the accelerated failure time (AFT) model for current status and interval censored data. The estimator is constructed by inverting a Wald type test for testing a null proportional hazards model. A numerically efficient Markov chain Monte Carlo (MCMC) based resampling method is proposed to simultaneously obtain the point estimator and a consistent estimator of its variance-covariance matrix. We illustrate our approach with interval censored data sets from two clinical studies. Extensive numerical studies are conducted to evaluate the finite sample performance of the new estimators.

Studying Effects Of Primary Care Physicians And Patients On The Trade-Off Between Charges For Primary Care And Specialty Care Using A Hierarchical Multivariate Two-Part Model, John W. Robinson, Scott L. Zeger, Christopher B. Forrest

Johns Hopkins University, Dept. of Biostatistics Working Papers

Objective. To examine effects of primary care physicians (PCPs) and patients on the association between charges for primary care and specialty care in a point-of-service (POS) health plan.

Data Source. Claims from 1996 for 3,308 adult male POS plan members, each of whom was assigned to one of the 50 family practitioner-PCPs with the largest POS plan member-loads.

Study Design. A hierarchical multivariate two-part model was fitted using a Gibbs sampler to estimate PCPs' effects on patients' annual charges for two types of services, primary care and specialty care, the associations among PCPs' effects, and within-patient associations between charges for …