The University of Michigan Department of Biostatistics Working Paper Series

VARYING INDEX COEFFICIENT MODELS

Shujie Ma et al. — Tue, 21 May 2013 14:59:50 PDT

It has been a long history of utilizing interactions in regression analysis to investigate interactive effects of covariates on response variables. In this paper we aim to address two kinds of new challenges resulted from the inclusion of such high-order effects in the regression model for complex data. The first kind arises from a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects. Generalizing the single index coefficient regression model (Xia and Li, 1999), we propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure utilizing both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by both simulation studies and an analysis of body fat dataset.

Surrogacy Assessment Using Principal Stratification When Surrogate and Outcome Measures are Multivariate Normal

Anna Conlon et al. — Tue, 21 May 2013 14:59:44 PDT

MISSING AT RANDOM AND IGNORABILITY FOR INFERENCES ABOUT SUBSETS OF PARAMETERS WITH MISSING DATA

Roderick J. Little et al. — Mon, 18 Feb 2013 08:31:31 PST

For likelihood-based inferences from data with missing values, Rubin (1976) showed that the missing data mechanism can be ignored when (a) the missing data are missing at random (MAR), in the sense that missingness does not depend on the missing values after conditioning on the observed data, and (b) the parameters of the data model and the missing-data mechanism are distinct; that is, there are no a priori ties, via parameter space restrictions or prior distributions, between the parameters of the data model and the parameters of the model for the mechanism. Rubin described (a) and (b) as the "weakest simple and general conditions under which it is always appropriate to ignore the process that causes missing data". However, these conditions are not always necessary. Also, they relate to the complete set of parameters in the model, but we argue that it would be useful to have definitions of MAR and ignorability for a subset of parameters of substantive interest. We propose such definitions, and apply them to a variety of examples where the missing data mechanism is missing not at random, but MAR or ignorable for the parameter subset.

In Praise of Simplicity not Mathematistry! Ten Simple Powerful Ideas for the Statistical Scientist

Roderick J. Little — Tue, 22 Jan 2013 06:06:10 PST

Ronald Fisher was by all accounts a first-rate mathematician, but he saw himself as a scientist, not a mathematician, and he railed against what George Box called (in his Fisher lecture) "mathematistry". Mathematics is the indispensable foundation for statistics, but our subject is constantly under assault by people who want to turn statistics into a branch of mathematics, making the subject as impenetrable to non-mathematicians as possible. Valuing simplicity, I describe ten simple and powerful ideas that have influenced my thinking about statistics, in my areas of research interest: missing data, causal inference, survey sampling, and statistical modeling in general. The overarching theme is that statistics is a missing data problem, and the goal is to predict unknowns with appropriate measures of uncertainty.

Subsample ignorable likelihood for accelerated failure time models with missing predictors

Nanhua Zhang et al. — Mon, 26 Nov 2012 13:31:33 PST

Proxy Pattern-Mixture Analysis for a Binary Variable Subject to Nonresponse.

Rebecca H. Andridge et al. — Mon, 26 Nov 2012 13:31:30 PST

We consider assessment of the impact of nonresponse for a binary survey

variable Y subject to nonresponse, when there is a set of covariates

observed for nonrespondents and respondents. To reduce dimensionality and

for simplicity we reduce the covariates to a continuous proxy variable X

that has the highest correlation with Y, estimated from a probit

regression analysis of respondent data. We extend our previously proposed

proxy-pattern mixture analysis (PPMA) for continuous outcomes to the binary

outcome using a latent variable approach. The method does not assume data

are missing at random, and creates a framework for sensitivity analyses.

Maximum likelihood, Bayesian, and multiple imputation versions of PPMA are

described, and robustness of these methods to model assumptions are

discussed. Properties are demonstrated through simulation and with data from

the Ohio Family Health Survey (OFHS).

A Phase I Bayesian Adaptive Design to Simultaneously Optimize Dose and Schedule Assignments Both Among and Within Patients

Thomas M. Braun et al. — Mon, 26 Nov 2012 13:31:28 PST

In traditional schedule or dose-schedule finding designs, patients are assumed to receive their assigned dose-schedule combination throughout the trial even though the combination may be found to have an undesirable toxicity profile, which contradicts actual clinical practice. Since no systematic approach exists to optimize intra-patient dose-schedule as- signment, we propose a Phase I clinical trial design that extends existing approaches that optimize dose and schedule solely among patients by incorporating adaptive variations to dose-schedule assignments within patients as the study proceeds. Our design is based on a Bayesian non-mixture cure rate model that incorporates multiple administrations each patient receives with the per-administration dose included as a covariate. Simulations demonstrate that our design identifies safe dose and schedule combinations as well as the traditional method that does not allow for intra-patient dose-schedule reassignments, but with a larger number of patients assigned to safe combinations.

The Bayesian Continual Reassessment Method Using a Mixture-of-Uniforms Prior

Thomas M. Braun — Mon, 26 Nov 2012 13:31:27 PST

Traditionally, the Bayesian formulation of the Continual Reassessment Method (CRM) is implemented with a one-parameter model describing the association of dose with the probability of dose-limiting toxicity (DLT). Determination of the appropriate value of the prior variance is often done via simulation over a grid search of possible values until suitable operating characteristics are found. However, it is under-appreciated that the operating characteristics for a given value of the prior variance vary by the “skeleton,” which is the vector of a priori probabilities of DLT for each dose. The skeleton implicitly leads to a set of indifference intervals, with one interval for each dose, that contain values of the model parameter that support each dose being the MTD. To remove the need of selecting a value for the prior variance, we propose placing a uniform distribution over each of the indifference intervals, making the prior distribution for the model parameter a mixture of uniform distributions. Via simulation, we compare the operating characteristics of the CRM using a traditional continuous prior to using the mixture-of-uniforms prior in a variety of settings and show that the mixture-of-uniforms prior leads to operating characteristics that are less sensitive to the chosen skeleton.

Analysis of Periodontal Data using Circular Statistics

Samopriyo Maitra et al. — Mon, 13 Aug 2012 05:20:32 PDT

Periodontal disease is a common cause of tooth loss in adults. The severity of periodontal disease is usually quantified based upon the magnitudes of several tooth-level clinical parameters, the most common of which is clinical attachment level (CAL). Re- cent clinical studies have presented data on the distribution of periodontal disease in hopes of providing information for localized treatments that can reduce the prevalence of periodontal disease. However, these findings have been descriptive without consid- eration of statistical modeling for estimation and inference. To this end, we visualize the mouth as a circle and the teeth as points located on the circumference of the circle to allow the use of circular statistical methods to determine the mean location of diseased teeth. We assume the directions of diseased teeth, as determined by their tooth averaged CAL values, to be observations from a von Mises distribution, the mean of which is a function of mouth-level covariates. Because multiple teeth from a subject are correlated, we use a bias-corrected generalized estimating equation approach (Mancl and DeRouen, 2001, Biometrics 57, 126–134) to obtain robust variance estimates for our parameter estimates. Via simulations of data motivated from an actual study of periodontal disease, we demonstrate that our methods have excellent performance in the moderately small sample sizes common to most periodontal studies.

Optimization and Simulation of an Evolving Kidney Paired Donation (KPD) Program

Yijiang Li et al. — Tue, 03 May 2011 12:37:44 PDT

The old concept of barter exchange has extended to the modern area of living-donor kidney transplantation, where one incompatible donor-candidate pair is matched to another pair with a complementary incompatibility, such that the donor from one pair gives an organ to a compatible candidate in the other pair and vice versa. Kidney paired donation (KPD) programs provide a unique and important platform for living incompatible donor-candidate pairs to exchange organs in order to achieve mutual benefit. We propose a novel approach to organizing kidney exchanges in an evolving KPD program with advantages, including (i) it allows for a more exible utility-based evaluation of potential kidney transplants; (ii) it takes into consideration stochastic features in managing a KPD program; and (iii) it exploits possible alternative exchanges when the originally planed allocation cannot be fully executed. Another primary contribution of this work is rooted in the development of a comprehensive microsimulation system for simulating and studying various aspects of an evolving KPD program. Various allocations can be obtained using integer programming (IP) techniques and microsimulation models can allow tracking of the evolving KPD over a series of match runs to evaluate different allocation strategies. Simulation studies are provided to illustrate the proposed method.

AN ANALYSIS OF NONIGNORABLE NONRESPONSE IN A SURVEY WITH A ROTATING PANEL DESIGN

Caterina Giusti et al. — Mon, 20 Dec 2010 13:44:44 PST

Missing values to income questions are common in survey data. When the probabilities of nonresponse are assumed to depend on the observed information and not on the underlining unobserved amounts, the missing income values are missing at random (MAR), and methods such as sequential multiple imputation can be applied. However, the MAR assumption is often considered questionable in this context, since missingness of income is thought to be related to the value of income itself, after conditioning on available covariates. In this article we describe a sensitivity analysis based on a pattern-mixture model for deviations from MAR, in the context of missing income values in a rotating panel survey. The sensitivity analysis avoids the well-known problems of underidentification of parameters of non-MAR models, is easy to carry out using existing sequential multiple imputation software and has a number of novel features.

Doubly Regularized REML for Estimation and Selection of Fixed and Random Effects in Linear Mixed-Effects Models

Sijian Wang et al. — Mon, 20 Dec 2010 13:44:44 PST

The linear mixed effects model (LMM) is widely used in the analysis of clustered or longitudinal data. In the practice of LMM, the inference on the structure of the random effects component is of great importance, not only to yield proper interpretation of subject-specific effects but also to draw valid statistical conclusions. This task of inference becomes significantly challenging when a large number of fixed effects and random effects are involved in the analysis. The difficulty of variable selection arises from the need of simultaneously regularizing both mean model and covariance structures, with possible parameter constraints between the two. In this paper, we propose a novel method of doubly regularized restricted maximum likelihood to select fixed and random effects simultaneously in the LMM. The Cholesky decomposition is invoked to ensure the positive-definiteness of the selected covariance matrix of random effects, and selected random effects are invariant with respect to the ordering of predictors appearing in the Cholesky decomposition. We then develop a new algorithm that solves the related optimization problem effectively, in which the computational cost is comparable with that of the Newton-Raphson algorithm for MLE or REML in the LMM. We also investigate large sample properties for the proposed method, including the oracle property. Both simulation studies and data analysis are included for illustration.

Modeling Menstrual Cycle Length and Variability at the Approach of Menopause Using Bayesian Changepoint Models

Xiaobi Huang et al. — Mon, 20 Dec 2010 13:44:43 PST

As women approach menopause, the patterns of their menstruation cycle lengths change. To study these changes, we need to jointly model both the mean and variability of the cycle length. The model incorporates separate mean and variance change points for each woman and a hierarchical model to link them together, along with regression components to include predictors of menopausal onset such as age at menarche and parity. Data are from TREMIN, an ongoing 70-year old longitudinal study that has obtained menstrual calendar data of women throughout their reproductive life course. An additional complexity arises from the fact that these calendars have substantial missingness due to hormone use, surgery, failure to report, and loss of contact. We integrate multiple imputation and time-to event modeling in our Bayesian estimation procedure to deal with different forms of the missingness. Posterior predictive model checks are applied to evaluate the model fit. Our method successfully modeled patterns of women’s menstrual cycle trajectories throughout their late reproductive life and identified the change points for mean and variability of segment length, which provides insight into the menopausal process. More generally, our model points the way toward increasing use of joint mean-variance models to predict health outcomes and better understand disease processes.

Predicting Treatment Efficacy via Quantitative MRI: A Bayesian Joint Model

Jincao Wu et al. — Mon, 20 Dec 2010 13:42:13 PST

The prognosis for patients with high-grade gliomas is poor, with a median survival of one year. Treatment efficacy assessment is typically unavailable until 5{6 months post diagnosis. Investigators hypothesize that quantitative MRI (qMRI) can assess treatment efficacy three weeks after therapy starts, thereby allowing salvage treatments to begin earlier. The purpose of this work is to build a predictive model of treatment efficacy using qMRI data and to assess its performance. The outcome is one-year survival status. We propose a joint, two-stage Bayesian model. In stage I, we smooth the image data with a multivariate spatio-temporal pairwise dierence prior. We propose four summary statistics that are functionals of posterior parameters from the first stage model. In stage II, these statistics enter a generalized non-linear model (GNLM)as predictors of survival status. We use the probit link and a multivariate adaptive regression spline basis. Gibbs sampling and reversible jump Markov chain monte carlo are applied iteratively between the two stages to estimate the posterior distribution. Through both simulation studies and model performance comparisons we find that we are able to attain higher overall correct classification rates by accounting for the spatio-temporal correlation in the images and by allowing for a more complex and flexible decision boundary provided by the GNLM.

WEIGHTING AND PREDICTION IN SAMPLE SURVEYS

Rod Little — Mon, 11 Jan 2010 09:30:56 PST

A fundamental technique in survey sampling is to weight included units by the inverse of their probability of inclusion, which may be known (as in the case of sampling weights) or estimated (as in the case of nonresponse weights). The technique is closely associated with the design-based approach to survey inference, with the idea that units in the sample are representing a certain number of units in the population. I discuss weighting from a modeling perspective. Some common misconceptions of weighting will be addressed, including the idea that modelers can ignore the sampling weights, or that weighting necessarily reduces bias at the expense of increased variance, or that units entering the calculation of nonresponse weights should be weighted by their sampling weights. A robust model-based perspective suggests that selection weights cannot be ignored, but there may be better ways of incorporating them in the inference than via the standard Horvitz-Thompson estimator and its variants.

Parametric Non-Mixture Cure Models for Schedule-Finding of Therapeutic Agents

Thomas M. Braun et al. — Mon, 11 Jan 2010 07:54:21 PST

We propose a Phase I clinical trial design that seeks to determine the cumulative safety of a series of administrations of a fixed dose of an investigational agent. In contrast to traditional Phase I trials that are designed to solely find the maximum tolerated dose (MTD) of the agent, our design instead identifies a maximum tolerated schedule (MTS) that includes an MTD as well as a vector of recommended administration times. Our model is based upon a non-mixture cure model that constrains the probability of toxicity for all subjects to monotonically increase with both dose and the number of administrations received. We assume a specific parametric hazard function for each administration and compute the total hazard of toxicity for a schedule as a sum of individual administration hazards. Throughout a variety of settings motivated by an actual study in allogeneic bone marrow transplant recipients, we demonstrate that our approach has excellent operating characteristics and performs as well as the only other currently published design for schedule-finding studies. We also present arguments for the preference of our non-mixture cure model over the existing model.

Composite likelihood Bayesian information criteria for model selection in high dimensional data

X Gao et al. — Tue, 05 May 2009 13:51:13 PDT

For high-dimensional data set with complicated dependency structures, the full likelihood approach often renders to intractable computational complexity. This imposes di±culty on model selection as most of the traditionally used information criteria require the evaluation of the full likelihood. We propose a composite likelihood version of the Bayesian information criterion (BIC) and establish its consistency property for the selection of the true underlying model. Under some mild regularity conditions, the proposed BIC is shown to be selection consistent, where the number of potential model parameters is allowed to increase to in¯nity at a certain rate of the sample size. Simulation studies demonstrate the empirical performance of this new BIC criterion, especially for the scenario that the number of parameters increases with the sample size.

Longitudinal Image Analysis of Tumor/Brain Change in Contrast Uptake Induced by Radiation

Xiaoxi Zhang et al. — Tue, 05 May 2009 13:51:11 PDT

This work is motivated by a quantitative Magnetic Resonance Imaging study of the differential tumor/healthy tissue change in contrast uptake induced by radiation. The goal is to determine the time in which there is maximal contrast uptake, a surrogate for permeability, in the tumor relative to healthy tissue. A notable feature of the data is its spatial heterogeneity. Zhang, Johnson, Little, and Cao (2008a and 2008b) discuss two parallel approaches to “denoise” a single image of change in contrast uptake from baseline to a single follow-up visit of interest. In this work we explore the longitudinal profile of the tumor/healthy tissue change in contrast uptake. In addition to the spatial correlation, we account for temporal correlation by jointly modeling multiple images on the individual subjects over time. We fit a two-stage model. First, we propose a longitudinal image model for each subject. This model simultaneously accounts for the spatial and temporal correlation and denoises the observed images by borrowing strength both across neighboring pixels and over time. We propose to use the area under the receiver operating characteristics (ROC) curve (AUC) to summarize the differential contrast uptake between tumor and healthy tissue. In the second stage, we fit a population model on the AUC values and estimate when it achieves the maximum.

A Bayesian Approach to Modeling Associations Between Pulsatile Hormones

Nichole E. Carlson et al. — Fri, 04 Apr 2008 06:30:37 PDT

Many hormones are secreted in pulses. The pulsatile relationship between hormones regulates many biological processes. To understand endocrine system regulation, time series of hormone concentrations are collected. The goal is to characterize pulsatile patterns and associations between hormones. Currently each hormone on each subject is fitted univariately. This leads to estimates of the number of pulses and estimates of the amount of hormone secreted; however, when the signal-to-noise ratio is small, pulse detection and parameter estimation remains di±cult with existing approaches. In this paper, we present a bivariate deconvolution model of pulsatile hormone data focusing on incorporating pulsatile associations. Through simulation, we exhibit that using the underlying pulsatile association between two hormones improves the estimation of the number of pulses and the other parameters de¯ning each hormone. We develop the one-to-one, driver-response case and show how birth-death MCMC can be used for estimation. We exhibit these features through a simulation study and on the relationship between luteinizing and follicle stimulating hormones.

Cluster Mass Inference Method via Random Field Theory

Hui Zhang et al. — Fri, 25 Jan 2008 07:07:11 PST

Cluster extent and voxel intensity are two widely used statistics in neuroimaging inference. Cluster extent is sensitive to spatially extended signals while voxel intensity is better for intense but focal signals. In order to leverage strength from both statistics, several nonparametric permutation methods have been proposed to combine the two methods. Simulation studies have shown that of the different cluster permutation methods, the cluster mass statistic is generally the best. However, to date, there is no parametric cluster mass inference available. In this paper, we propose a cluster mass inference method based on random field theory (RFT). We develop this method for Gaussian images, extend it to Student’s t-statistic images and investigate its statistical properties via simulation studies and real data. Simulation results show that the method is valid under the null hypothesis and demonstrate that it can be more powerful than the cluster extent inference method. Further, analyses with a single-subject and a group fMRI dataset demonstrate better power than traditional cluster size inference, and good accuracy relative to a gold-standard permutation test.