Analyzing Panel Count Data with Informative Observation Times

In this paper, we study panel count data with informative observation times. We assume nonparametric and semiparametric proportional rate models for the underlying recurrent event process, where the form of the baseline rate function is left unspecified and a subject-specific frailty variable inflates or deflates the rate function multiplicatively. The proposed models allow the recurrent event processes and observation times to be correlated through their connections with the unobserved frailty; moreover, the distributions of both the frailty variable and observation times are considered as nuisance parameters. The baseline rate function and the regression parameters are estimated by maximizing a conditional likelihood function of observed event counts and solving estimation equations. Large sample properties of the proposed estimators are studied. Numerical studies demonstrate that the proposed estimation procedures perform well for moderate sample sizes. An application to a bladder tumor study is presented to illustrate the use of the proposed methods.

Computational Techniques for Spatial Logistic Regression with Large Datasets

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 features while limiting overfitting and being more efficient computationally than other Bayesian approaches. One of the contributions of this work is further development of this underused representation. The spectral basis model outperforms the penalized likelihood methods, which are prone to overfitting, but is slower to fit and not as easily implemented. Conclusions based on a real dataset of cancer cases in Taiwan are similar albeit less conclusive with respect to comparing the approaches. The success of the spectral basis with binary data and similar results with count data suggest that it may be generally useful in spatial models and more complicated hierarchical models.

MODELING FUNCTIONAL DATA WITH SPATIALLY HETEROGENEOUS SHAPE CHARACTERISTICS

We propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location. We use copulas so that the marginal distributions and the dependence structure can be modeled independently. Dependence is modeled with a Gaussian or t-copula, so that there is an underlying latent Gaussian process. We model the marginal distributions using the skew t family. The mean, variance, and shape parameters are modeled nonparametrically as functions of location. A computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy-tailed marginal distributions is introduced. This framework provides a new set of tools for increasingly complex data collected in medical and public health studies. Our methods were motivated by and are illustrated with a state-of-the-art study of neuronal tracts in multiple sclerosis patients and healthy controls. Using the tools we have developed, we were able to find those locations along the tract most affected by the disease. However, our methods are general and highly relevant to many functional data sets. In addition to the application to one-dimensional tract profiles illustrated here, higher-dimensional extensions of the methodology could have direct applications to other biological data including functional and structural MRI.

A unified approach to modeling multivariate binary data using copulas over partitions

Many seemingly disparate approaches for marginal modeling have been developed in recent years. We demonstrate that many current approaches for marginal modeling of correlated binary outcomes produce likelihoods that are equivalent to the proposed copula-based models herein. These general copula models of underlying latent threshold random variables yield likelihood based models for marginal fixed effects estimation and interpretation in the analysis of correlated binary data. Moreover, we propose a nomenclature and set of model relationships that substantially elucidates the complex area of marginalized models for binary data. A diverse collection of didactic mathematical and numerical examples are given to illustrate concepts.

Locally Efficient Estimation in Censored Data Models: Theory and Examples

In many applications the observed data can be viewed as a censored high dimensional full data random variable X. By the curve of dimensionality it is typically not possible to construct estimators that are asymptotically efficient at every probability distribution in a semiparametric censored data model of such a high dimensional censored data structure. We provide a general method for construction of one-step estimators that are efficient at a chosen submodel of the full-data model, are still well behaved off this submodel and can be chosen to always improve on a given initial estimator. These one-step estimators rely on good estimators of the censoring mechanism and thus will require a parametric or semiparametric model for the censoring mechanism. We present a general theorem that provides a template for proving the desired asymptotic results. We illustrate the general one-step estimation methods by constructing locally efficient one-step estimators of marginal distributions and regression parameters with right-censored data, current status data and bivariate right-censored data, in all models allowing the presence of time-dependent covariates. The conditions of the asymptotics theorem are rigorously verified in one of the examples and the key condition of the general theorem is verified for all examples.

Efficiency of the Sieved-NPMLE in CAR-Missing Data Models

We consider nonparametric missing data models for which the censoring mechanism satisfies coarsening at random and which allow complete observations on the variable X of interest. W show that beyond some empirical process conditions the only essential condition for efficiency of an NPMLE of the distribution of X is that the regions associated with incomplete observations on X contain enough complete observations. This is heuristically explained by describing the EM-algorithm. We provide identifiably of the self-consistency equation and efficiency of the NPMLE in order to make this statement rigorous. The usual kind of differentiability conditions in the proof are avoided by using an identity which holds for the NPMLE of linear parameters in convex models. We provide a bivariate censoring application in which the condition and hence the NPMLE fails, but where other estimators, not based on the NPMLE principle, are highly inefficient. It is shown how to slightly reduce the data so that the conditions hold for the reduced data. The conditions are verified for the univariate censoring, double censored, and Ibragimov-Has'minski models.

Fitting ACE Structural Equation Models to Case-Control Family Data

Investigators interested in whether a disease aggregates in families often collect case-control family data, which consist of disease status and covariate information for families selected via case or control probands. Here, we focus on the use of case-control family data to investigate the relative contributions to the disease of additive genetic effects (A), shared family environment (C), and unique environment (E). To this end, we describe a ACE model for binary family data and then introduce an approach to fitting the model to case-control family data. The structural equation model, which has been described previously, combines a general-family extension of the classic ACE twin model with a (possibly covariate-specific) liability-threshold model for binary outcomes. Our likelihood-based approach to fitting involves conditioning on the proband’s disease status, as well as setting prevalence equal to a pre-specified value that can be estimated from the data themselves if necessary. Simulation experiments suggest that our approach to fitting yields approximately unbiased estimates of the A, C, and E variance components, provided that certain commonly-made assumptions hold. These assumptions include: the usual assumptions for the classic ACE and liability-threshold models; assumptions about shared family environment for relative pairs; and assumptions about the case-control family sampling, including single ascertainment. When our approach is used to fit the ACE model to Austrian case-control family data on depression, the resulting estimate of heritability is very similar to those from previous analyses of twin data.