Applied Mathematics Commons^™

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. …

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 …

A Hierarchical Multivariate Two-Part Model For Profiling Providers' Effects On Healthcare Charges, John W. Robinson, Scott L. Zeger, Christopher B. Forrest

Johns Hopkins University, Dept. of Biostatistics Working Papers

Procedures for analyzing and comparing healthcare providers' effects on health services delivery and outcomes have been referred to as provider profiling. In a typical profiling procedure, patient-level responses are measured for clusters of patients treated by providers that in turn, can be regarded as statistically exchangeable. Thus, a hierarchical model naturally represents the structure of the data. When provider effects on multiple responses are profiled, a multivariate model rather than a series of univariate models, can capture associations among responses at both the provider and patient levels. When responses are in the form of charges for healthcare services and sampled …

Non-Parametric Estimation Of Roc Curves In The Absence Of A Gold Standard, Xiao-Hua Zhou, Pete Castelluccio, Chuan Zhou

UW Biostatistics Working Paper Series

In evaluation of diagnostic accuracy of tests, a gold standard on the disease status is required. However, in many complex diseases, it is impossible or unethical to obtain such the gold standard. If an imperfect standard is used as if it were a gold standard, the estimated accuracy of the tests would be biased. This type of bias is called imperfect gold standard bias. In this paper we develop a maximum likelihood (ML) method for estimating ROC curves and their areas of ordinal-scale tests in the absence of a gold standard. Our simulation study shows the proposed estimates for the …

Differential Expression With The Bioconductor Project, Anja Von Heydebreck, Wolfgang Huber, Robert Gentleman

Bioconductor Project Working Papers

A basic, yet challenging task in the analysis of microarray gene expression data is the identification of changes in gene expression that are associated with particular biological conditions. We discuss different approaches to this task and illustrate how they can be applied using software from the Bioconductor Project. A central problem is the high dimensionality of gene expression space, which prohibits a comprehensive statistical analysis without focusing on particular aspects of the joint distribution of the genes expression levels. Possible strategies are to do univariate gene-by-gene analysis, and to perform data-driven nonspecific filtering of genes before the actual statistical analysis. …

A Graph Theoretic Approach To Testing Associations Between Disparate Sources Of Functional Genomic Data, Raji Balasubramanian, Thomas Laframboise, Denise Scholtens, Robert Gentleman

Bioconductor Project Working Papers

The last few years have seen the advent of high-throughput technologies to analyze various properties of the transcriptome and proteome of several organisms. The congruency of these different data sources, or lack thereof, can shed light on the mechanisms that govern cellular function. A central challenge for bioinformatics research is to develop a unified framework for combining the multiple sources of functional genomics information and testing associations between them, thus obtaining a robust and integrated view of the underlying biology.

We present a graph theoretic approach to test the significance of the association between multiple disparate sources of functional genomics …

Resampling Methods For Estimating Functions With U-Statistic Structure, Wenyu Jiang, Jack Kalbfleisch

The University of Michigan Department of Biostatistics Working Paper Series

Suppose that inference about parameters of interest is to be based on an unbiased estimating function that is U-statistic of degree 1 or 2. We define suitable studentized versions of such estimating functions and consider asymptotic approximations as well as an estimating function bootstrap (EFB) method based on resampling the estimated terms in the estimating functions. These methods are justified asymptotically and lead to confidence intervals produced directly from the studentized estimating functions. Particular examples in this class of estimating functions arise in La estimation as well as Wilcoxon rank regression and other related estimation problems. The proposed methods are …

Loss-Based Cross-Validated Deletion/Substitution/Addition Algorithms In Estimation, Sandra E. Sinisi, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In van der Laan and Dudoit (2003) we propose and theoretically study a unified loss function based statistical methodology, which provides a road map for estimation and performance assessment. Given a parameter of interest which can be described as the minimizer of the population mean of a loss function, the road map involves as important ingredients cross-validation for estimator selection and minimizing over subsets of basis functions the empirical risk of the subset-specific estimator of the parameter of interest, where the basis functions correspond to a parameterization of a specified subspace of the complete parameter space. In this article we …

Incorporating Death Into Health-Related Variables In Longitudinal Studies, Paula Diehr, Laura Lee Johnson, Donald L. Patrick, Bruce Psaty

UW Biostatistics Working Paper Series

Background: The aging process can be described as the change in health-related variables over time. Unfortunately, simple graphs of available data may be misleading if some people die, since they may confuse patterns of mortality with patterns of change in health. Methods have been proposed to incorporate death into self-rated health (excellent to poor) and the SF-36 profile scores, but not for other variables.

Objectives: (1) To incorporate death into the following variables: ADLs, IADLs, mini-mental state examination, depressive symptoms, body mass index (BMI), blocks walked per week, bed days, hospitalization, systolic blood pressure, and the timed walk. (2) To …

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Controlling Wound Healing Through Debridement, M. A. Jones, Baojun Song, D. M. Thomas

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

The formation of slough (dead tissue) on a wound is widely accepted as an inhibitor to natural wound healing. In this article, a system of differential equations that models slough/wound interaction is developed. We prove a threshold theorem that provides conditions on the amount of slough to guarantee wound healing. As a state-dependent time scale, debridement (the periodic removal of slough) is used as a control. We show that closure of the wound can be reached in infinite time by debriding.

Uniqueness Theorems In Bioluminescence Tomography, Ge Wang, Yi Li, Ming Jiang

Mathematics and Statistics Faculty Publications

Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. Our primary goal in this paper is to establish the solution uniqueness for BLT under practical constraints despite the ill-posedness of the inverse problem in the general case. After a review on the inverse source literature, we demonstrate that in the general case …

The Dual Spectral Set Conjecture, Steen Pedersen

Mathematics and Statistics Faculty Publications

Suppose that Λ = (aZ + b) ∪ (cZ + d) where a, b, c, d are real numbers such that a ≠ 0 and c ≠ 0. The union is not assumed to be disjoint. It is shown that the translates Ω + λ, λ is an element of Λ, tile the real line for some bounded measurable set Ω if and only if the exponentials e_λ(x) = e^i2πλx, λ is an element of Λ, form an orthogonal basis for some bounded measurable set Ω'.

Kernel Estimation Of Rate Function For Recurrent Event Data, Chin-Tsang Chiang, Mei-Cheng Wang, Chiung-Yu Huang

Johns Hopkins University, Dept. of Biostatistics Working Papers

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 …

Global Solutions To The Lake Equations With Isolated Vortex Regions, Chaocheng Huang

Mathematics and Statistics Faculty Publications

The vorticity formulation for the lake equations in R² is studied.

Unified Cross-Validation Methodology For Selection Among Estimators And A General Cross-Validated Adaptive Epsilon-Net Estimator: Finite Sample Oracle Inequalities And Examples, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

In Part I of this article we propose a general cross-validation criterian for selecting among a collection of estimators of a particular parameter of interest based on n i.i.d. observations. It is assumed that the parameter of interest minimizes the expectation (w.r.t. to the distribution of the observed data structure) of a particular loss function of a candidate parameter value and the observed data structure, possibly indexed by a nuisance parameter. The proposed cross-validation criterian is defined as the empirical mean over the validation sample of the loss function at the parameter estimate based on the training sample, averaged over …

Semi-Parametric Box-Cox Power Transformation Models For Censored Survival Observations, Tianxi Cai, Lu Tian, L. J. Wei

Harvard University Biostatistics Working Paper Series

No abstract provided.

Statistical Inferences Based On Non-Smooth Estimating Functions, Lu Tian, Jun S. Liu, Mary Zhao, L. J. Wei

Harvard University Biostatistics Working Paper Series

No abstract provided.

Maximum Likelihood Estimation Of Ordered Multinomial Parameters , Nicholas P. Jewell, Jack Kalbfleisch

The University of Michigan Department of Biostatistics Working Paper Series

The pool-adjacent violator-algorithm (Ayer et al., 1955) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see, Barlow et al., 1972). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of ‘ordered’ multinomial parameters pi = (p1i, p2i, . . . , pmi) for 1 < = I < = k, where ordered means that pj1 < = pj2 < = .. . < = pjk for each j with 1 < = j < = m-1. The data consist of k independent observations X1, . . . ,Xk where Xi has a multinomial distribution with probability parameter pi and known index ni > = 1. By making use of variants of the pool adjacent violator algorithm, …

A Population Pharmacokinetic Model With Time-Dependent Covariates Measured With Errors, Lang Lil, Xihong Lin, Mort B. Brown, Suneel Gupta, Kyung-Hoon Lee

The University of Michigan Department of Biostatistics Working Paper Series

We propose a population pharmacokinetic (PK) model with time-dependent covariates measured with errors. This model is used to model S-oxybutynin's kinetics following an oral administration of Ditropan, and allows the distribution rate to depend on time-dependent covariates blood pressure and heart rate, which are measured with errors. We propose two two-step estimation methods: the second order two-step method with numerical solutions of differential equations (2orderND), and the second order two-step method with closed form approximate solutions of differential equations (2orderAD). The proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear …

Exact Multiplicity For Periodic Solutions Of Duffing Type, Hongbin Chen, Yi Li, Xiaojie Hou

Mathematics and Statistics Faculty Publications

In this paper, we study the following Duffing-type equation:

x″+cx′+g(t,x)=h(t),

where g(t,x) is a 2π-periodic continuous function in t and concave–convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

Cross-Calibration Of Stroke Disability Measures: Bayesian Analysis Of Longitudinal Ordinal Categorical Data Using Negative Dependence, Giovanni Parmigiani, Heidi W. Ashih, Gregory P. Samsa, Pamela W. Duncan, Sue Min Lai, David B. Matchar

Johns Hopkins University, Dept. of Biostatistics Working Papers

It is common to assess disability of stroke patients using standardized scales, such as the Rankin Stroke Outcome Scale (RS) and the Barthel Index (BI). The Rankin Scale, which was designed for applications to stroke, is based on assessing directly the global conditions of a patient. The Barthel Index, which was designed for general applications, is based on a series of questions about the patient’s ability to carry out 10 basis activities of daily living. As both scales are commonly used, but few studies use both, translating between scales is important in gaining an overall understanding of the efficacy of …

An Extended General Location Model For Causal Inference From Data Subject To Noncompliance And Missing Values, Yahong Peng, Rod Little, Trivellore E. Raghuanthan

The University of Michigan Department of Biostatistics Working Paper Series

Noncompliance is a common problem in experiments involving randomized assignment of treatments, and standard analyses based on intention-to treat or treatment received have limitations. An attractive alternative is to estimate the Complier-Average Causal Effect (CACE), which is the average treatment effect for the subpopulation of subjects who would comply under either treatment (Angrist, Imbens and Rubin, 1996, henceforth AIR). We propose an Extended General Location Model to estimate the CACE from data with non-compliance and missing data in the outcome and in baseline covariates. Models for both continuous and categorical outcomes and ignorable and latent ignorable (Frangakis and Rubin, 1999) …

Locally Efficient Estimation Of Nonparametric Causal Effects On Mean Outcomes In Longitudinal Studies, Romain Neugebauer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Marginal Structural Models (MSM) have been introduced by Robins (1998a) as a powerful tool for causal inference as they directly model causal curves of interest, i.e. mean treatment-specific outcomes possibly adjusted for baseline covariates. Two estimators of the corresponding MSM parameters of interest have been proposed, see van der Laan and Robins (2002): the Inverse Probability of Treatment Weighted (IPTW) and the Double Robust (DR) estimators. A parametric MSM approach to causal inference has been favored since the introduction of MSM. It relies on correct specification of a parametric MSM to consistently estimate the parameter of interest using the IPTW …

On Adaptive Estimation In Orthogonal Saturated Designs, Weizhen Wang, Daniel T. Voss

Mathematics and Statistics Faculty Publications

A simple method is provided to construct a general class of individual and simultaneous confidence intervals for the effects in orthogonal saturated designs. These intervals use the data adaptively, maintain the confidence levels sharply at 1 - α at the least favorable parameter configuration, work effectively under effect sparsity, and include the intervals by Wang and Voss (2001) as a special case.