Statistics seminars 2012-2013

In many scientific experiments we need to face analysis with functional data, where the observations are sampled from random process, together with a potentially large number of non-functional covariates. The complex nature of functional data makes it difficult to directly apply existing methods to model selection and estimation. We propose and study a new class of penalized semiparametric functional linear regression to characterize the regression relation between a scalar response and multiple covariates, including both functional covariates and scalar covariates. The resulting method provides a unified and flexible framework to jointly model functional and non-functional predictors, identify important covariates, and improve efficiency and interpretability of the estimates. Featured with two types of regularization: the shrinkage on the effects of scalar covariates and the truncation on principal components of the functional predictor, the new approach is flexible and effective in dimension reduction. One key contribution of this paper is to study theoretical properties of the regularized semiparametric functional linear model. We establish oracle and consistency properties under mild conditions by allowing possibly diverging number of scalar covariates and simultaneously taking the infinite-dimensional functional predictor into account. We illustrate the new estimator with extensive simulation studies, and then apply it to an image data analysis.

Fang Yao (http://www.utstat.utoronto.ca/fyao/) is Associate Professor, Department of Statistics, University of Toronto. His research interests include functional and longitudinal data analysis, nonparametric regression and smoothing methods, statistical modeling of high-dimensional and complex data, with applications involving functional objects (evolutional biology, human genetics,  finance and e-commerce, chemical engineering).

In this talk, I will provide an introduction to DTRs and an overview the state of the art (and science) of Q-learning, a popular tool in reinforcement learning. The use of Q-learning and its variance in randomized and non-randomized studies will be discussed, as well as issues concerning inference as the resulting estimators are not always regular. Current and future directions of interest will also be considered.

Using only bivariate copulas as building blocks, regular vines(R-vines) constitute a flexible class of high-dimensional dependence models. In this talk we introduce a Markov switching R-vine copula model, combining the flexibility of general R-vine copulas with the possibility for dependence structures to change over time. Frequentist as well as Bayesian parameter estimation is discussed. Further, we apply the newly proposed model to examine the dependence of exchange rates as well as stock and stock index returns. We show that changes in dependence are usually closely interrelated with periods of market stress. In such times the Value at Risk of an asset portfolio is significantly underestimated when changes in the dependence structure are ignored.

 Extreme Value Theory has been widely used for assessing risk for highly unusual events, either by using block maxima or peaks over the threshold (POT) methods. However, one of the main drawbacks of the POT method is the choice of a threshold, which plays an important role in the estimation since the parameter estimates strongly depend on this value. Bayesian inference is an alternative to handle these difficulties; the threshold can be treated as another parameter in the estimation, avoiding the classical empirical approach. In addition, it is possible to incorporate internal and external observations in combination with expert opinion, providing a natural, probabilistic framework in which to evaluate risk models. In this talk, we analyze operational risk data using a mixture model which combines a parametric form for the center and a GPD for the tail of the distribution, using all observations for inference about the unknown parameters from both distributions, the threshold included. A Bayesian analysis is performed and inference is carried out through Markov Chain Monte Carlo (MCMC) methods in order to determine the minimum capital requirement for operational risk.

Observational healthcare data, such as administrative claims and electronic health records, play an increasingly prominent role in healthcare.  Pharmacoepidemiologic studies in particular routinely estimate temporal associations between medical product exposure and subsequent health outcomes of interest, and such studies influence prescribing patterns and healthcare policy more generally.  Some authors have questioned the reliability and accuracy of such studies, but few previous efforts have attempted to measure their performance.

The Observational Medical Outcomes Partnership (OMOP, http://omop.fnih.org) has conducted a series of experiments to empirically measure the performance of various observational study designs with regard to predictive accuracy for discriminating between true drug effects and negative controls.  In this talk, I describe the past work of the Partnership, explore opportunities to expand the use of observational data to further our understanding of medical products, and highlight areas for future research and development.

(on behalf of the OMOP investigators)

Computer simulators are used widely to describe physical processes in lieu of physical observations. In some cases, more than one computer code can be used to explore the same physical system - each with different degrees of fidelity. In this work, we combine field observations and model runs from deterministic multi-fidelity computer simulators to build a predictive model for the real process. The resulting model can be used to perform sensitivity analysis for the system and make predictions with associated measures of uncertainty. Our approach is Bayesian and will be illustrated through a simple example, as well as a real application in predictive science at the Center for Radiative Shock Hydrodynamics at the University of Michigan.

Estimating extreme risks in a multivariate framework is highly connected with the estimation of the extremal dependence structure. This structure can be described via the stable tail dependence function L, for which several estimators have been introduced. Asymptotic normality is available for empirical estimates of L, with rate of convergence k^1/2, where k denotes the number of high order statistics used in the estimation. Choosing a higher k might be interesting for an improved accuracy of the estimation, but may lead to an increased asymptotic bias. We provide a bias correction procedure for the estimation of L. Combining estimators of L is done in such a way that the asymptotic bias term disappears. The new estimator of L is shown to allow more flexibility in the choice of k. Its asymptotic behavior is examined, and a simulation study is provided to assess its small sample behavior. This is a joint work with Cécile Mercadier (Université Lyon 1) and Laurens de Haan (Erasmus University Rotterdam).

Assessing tail risks using the asymptotic models provided by multivariate extreme value theory has the danger that when asymptotic independence is present (as with the Gaussian copula model), the
asymptotic model provides estimates of probabilities of joint tail regions that are zero. In diverse applications such as finance, telecommunications, insurance and environmental science, it may be difficult to believe in the absence of risk contagion. This problem can be partly ameliorated by using hidden regular variation which assumes a lower order asymptotic behavior on a subcone of the state space and this theory can be made more flexible by extensions in the following directions: (i) higher dimensions than two; (ii) where the lower order variation on a subcone is of extreme value type different from regular variation; and (iii) where the concept is extended to searching for lower order behavior on the complement of the support of the limit measure of regular variation. We discuss some challenges and potential applications to this ongoing effort.

In this paper we investigate a new approach of estimating a regression function based on copulas. The main idea behind this approach is to write the regression function in terms of a copula and marginal distributions. Once the copula and the marginal distributions are estimated we use the plug-in method to construct the new estimator. Because various methods are available in the literature for estimating both a copula and a distribution, this idea provides a rich and flexible alternative to many existing regression estimators. We provide some asymptotic results related to this copula-based regression modeling when the copula is estimated via profile likelihood and the marginals are estimated nonparametrically. We also study the finite sample performance of the estimator and illustrate its usefulness by analyzing data from air pollution studies.
 
Joint work with H.  Noh and A. El Ghouch from Université catholique de Louvain.

We propose a nonparametric Bayesian local clustering (NoB-LoC) approach for heterogeneous data.  Using genomics data as an example, the NoB-LoC clusters genes into gene sets and simultaneously creates multiple partitions of samples, one for each gene set. In other words, the sample partitions are nested within the gene sets.  Inference is guided by a joint probability model on all random elements. Biologically, the model formalizes the notion that biological samples cluster differently with respect to different genetic processes, and that each process is related to only a small subset of genes. These local features are importantly different from global clustering approaches such as hierarchical clustering, which create one partition of samples that applies for all genes in the data set. Furthermore, the NoB-LoC includes a special cluster of genes that do not give rise to any meaningful partition of samples. These genes could be irrelevant to the disease conditions under investigation. Similarly, for a given gene set, the NoB-LoC includes a subset of samples that do not co-cluster with other samples. The samples in this special cluster could, for example, be those whose disease subtype is not characterized by the particular gene set.

This is joint work with Juhee Lee and Yuan Ji.

While statistical agencies would like to share their data with researchers, they must also protect the confidentiality of the data provided by their respondents. To satisfy these two conflicting objectives, agencies use various techniques to restrict and modify the data before publication. Most of these techniques however share a common flaw: their confidentiality protection can not be rigorously measured. In this talk, I will present the criterion of differential privacy, a rigorous measure of the protection offered by such methods. Designed to guarantee confidentiality even in a worst-case scenario, differential privacy protects the information of any individual in the database against an adversary with complete knowledge of the rest of the dataset. I will first give a brief overview of recent and current research on the topic of differential privacy. I will then focus on the publication of differentially-private synthetic contingency tables and present some of my results on the methods for the generation
and proper analysis of such datasets.

Multiple testing problems arise in a variety of situations, notably in clinical trials with multiple endpoints. In such cases, it is often of interest to reject either all hypotheses or at least one of them. More generally, the question arises as to whether one can reject at least r out of m hypotheses. Statistical tools addressing this issue are rare in the literature. In this talk, I will recall well-known hypothesis testing concepts, both in a single- and in a multiple-hypothesis context. I will then present general power formulas for three important multiple comparison procedures: the Bonferroni and Hochberg procedures, as well as Holm’s sequential procedure. Next, I will describe an R package that we developed for sample size calculations in multiple endpoints trials where it is desired to reject at least r out of m hypotheses. This package covers the case where all the variables are continuous and four common variance-covariance patterns. I will show how to use this package to compute the sample size needed in a real-life application.

In the United States the preferred method of obtaining dietary intake data is the 24-hour dietary recall, yet the measure of most interest is usual or long-term average daily intake, which is impossible to measure. Thus, usual dietary intake is assessed with considerable measurement error. Also, diet represents numerous foods, nutrients and other components, each of which have distinctive attributes. Sometimes, it is useful to examine intake of these components separately, but increasingly nutritionists are interested in exploring them collectively to capture overall dietary patterns and their effect on various diseases. Consumption of these components varies widely: some are consumed daily by almost everyone on every day, while others are episodically consumed so that 24-hour recall data are zero-inflated. In addition, they are often correlated with each other. Finally, it is often preferable to analyze the amount of a dietary component relative to the amount of energy (calories) in a diet because dietary recommendations often vary with energy level.

We propose the first model appropriate for this type of data, and give the first workable solution to fit such a model. After describing the model, we use survey-weighted MCMC computations to fit the model, with uncertainty estimation coming from balanced repeated replication. The methodology is illustrated through an application to estimating the population distribution of the Healthy Eating Index-2005 (HEI-2005), a multi-component dietary quality index involving ratios of interrelated dietary components to energy, among children aged 2-8 in the United States. We pose a number of interesting questions about the HEI-2005, and show that it is a powerful predictor of the risk of developing colorectal cancer.

Statisticians are often faced with budget concerns when conducting studies. The collection of some covariates, such as genetic data, is very expensive. Other covariates, such as detailed histories, might be difficult or time-consuming to measure. This helped bring about the invention of the nested case-control study, and its more generalized version, risk-set sampled survival analysis. The literature has a good discussion of the properties of risk-set sampling in standard right-censored survival data. My interest is in extending the methods of risk-set sampling to left-truncated survival data, which arise in prevalent longitudinal studies. Since prevalent studies are easier and cheaper to conduct than incident studies, this extension is extremely practical and relevant. I will introduce the partial likelihood in this scenario, and briefly discuss the asymptotic properties of my estimator. I will also introduce Bayesian methods for standard survival analysis, and discuss methods for analyzing risk-set-sampled survival data using Bayesian methods.

We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances. Our analysis allows the number of controls to be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the "post-double-selection" method. Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties.

The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We illustrate the use of the developed methods with numerical simulations and an application to the effect of abortion on crime rates.
 
This is joint work with Alexandre Belloni, and Christian Hansen.

In an attempt to improve on the Metropolis algorithm, various MCMC methods with auxiliary variables, such as the multiple-try and delayed rejection Metropolis algorithms, have been proposed. These methods generate several candidates in a single iteration; accordingly they are computationally more intensive than the Metropolis algorithm. It is usually difficult to provide a general estimate for the computational cost of a method without being overly conservative; potentially efficient methods could thus be overlooked by relying on such estimates. In this talk, we describe three algorithms with auxiliary variables - the multiple-try Metropolis (MTM) algorithm, the multiple-try Metropolis hit-and-run (MTM-HR) algorithm, and the delayed rejection Metropolis algorithm with antithetic proposals (DR-A) - and investigate the net performance of these algorithms in various contexts. To allow for a fair comparison, the study is carried under optimal mixing conditions for each of these algorithms. The DR-A algorithm, whose proposal scheme introduces correlation in the pool of candidates, seems particularly promising. The algorithms are used in the contexts of Bayesian logistic regressions and classical inference for a linear regression model. This talk is based on work in collaboration with M. Mireuta, E. Moulines, and R. Douc.

I will consider the task of estimating high-dimensional Gaussian graphical models (or networks) corresponding to a single set of features under several distinct conditions. In other words, I wish to estimate several distinct but related networks. I assume that most aspects of the networks are shared, but that there are some structured differences between them. The goal is to exploit the similarity among the networks in order to obtain more accurate estimates of each individual network, as well as to identify the differences between the networks.

To begin, I will assume that network differences arise from edge perturbations. In this case, estimating the networks by maximizing the log likelihood subject to fused lasso or group lasso penalties on the differences between the precision matrices can lead to very good results. Next, I will discuss a more structured type of network difference that arises from node (rather than edge) perturbations. In order to estimate networks in this setting, I will present the "row-column overlap norm penalty", a type of overlapping group lasso penalty.

Finally, I will present an application of these network estimation techniques to a gene expression data set, in which the goal is to identify genes whose regulatory patterns are perturbed across various subtypes of brain cancer.

This is joint work with Pei Wang, Su-In Lee, Maryam Fazel, and others.

In the context of univariate association tests between a trait of interest and common genetic variants (SNPs) across the whole genome, corrections for multiple testing have been well-studied. Due to the patterns of correlation (i.e. linkage disequilibrium), the number of independent tests remains close to 1 million, even when many more common genetic markers are available. With the advent of the DNA sequencing era, however, newly-identified genetic variants tend to be rare or even unique, and consequently single-variant tests of association have little power. As a result, region-based tests of association are being developed that examine associations between the trait and all the genetic variability in a small pre-defined region of the genome. However, coping with multiple testing in this situation has had little attention. I will discuss two aspects of multiple testing for region-based tests. First, I will describe a method for estimating the effective number of independent tests, and second, I will discuss an approach for controlling type I error that is based stratified false discovery rates, where strata are defined by external information such as genomic annotation.

It is often crucial to know whether the dependence structure of a bivariate distribution belongs to the class of extreme-­‐value copulas. In this talk, I will describe a graphical tool that allows judgment regarding the existence of extreme-­‐value dependence. I will also present a data-­‐ driven nonparametric estimator of the Pickands dependence function. This estimator, which is constructed from constrained b-­‐splines, is intrinsic and differentiable, thereby enabling sampling from the fitted model. I will illustrate its properties via simulation. This will lead me to highlight some of the limitations associated with currently available tests of extremeness.

In this talk, we first provide an overview of issues arising from and methods dealing with complex survey data in the presence of missing observations, with a major focus on the estimating equation approach for analysis and imputation methods for missing data. We then propose a semiparametric fractional imputation method for handling item nonresponses, assuming certain baseline auxiliary variables can be observed for all units in the sample. The proposed strategy combines the strengths of conventional single imputation and multiple imputation methods, and is easy to implement even with a large number of auxiliary variables available, which is typically the case for large scale complex surveys. Simulation results and some general discussion on related issues will also be presented.

This talk is based partially on joint work with Jiahua Chen of University of British Columbia and Jaekwang Kim of Iowa State University.

Some problems of statistics can be reduced to extremal problems of minimizing functionals of smooth functions defined on the cube $[0,1]^m$, $m\geq 2$. In this talk, we consider a class of  extremal problems that is closely connected to the problem of testing multivariate independence. By solving the extremal problem, we provide a unified approach to establishing weak convergence for a wide class
of empirical processes which emerge in connection with testing multivariate independence. The use of our result will be also illustrated by describing the domain of local asymptotic optimality of some nonparametric tests of independence.

This is a joint work with Alexander Nazarov (St. Petersburg State University, Russia)

Join work with Yukun Liu (East China Normal University)

Population quantiles and their functions are important parameters in many applications. For example, the lower level quantiles often serve as crucial quality indices of forestry products and others. In the presence of several independent samples from populations satisfying density ratio model, we investigate the properties of the empirical likelihood (EL) based inferences of quantiles and their functions. In this paper, we first establish the consistency and asymptotic normality of the estimators of parameters and cumulative distributions. The induced EL quantile estimators are then shown to admit Bahadur representation. The results are used to construct asymptotically valid confidence intervals
for functions of quantiles. In addition, we rigorously prove that the EL quantiles based on all samples are more efficient than the empirical quantiles which can only utilize information from individual samples. Simulation study shows that the EL quantiles and their functions have superior performances both when the density ratio model assumption is satisfied and mildly violated. An application example is used to demonstrate the new methods and potential cost savings.

We give a characterization of the hyper Dirichlet distribution hyper Markov with respect to a decomposable graph $G$ (or equivalently a moral directed acyclic graph). For $X=(X_1,\ldots,X_d)$ following the hyper Dirichlet distribution, our characterization is through the so-called "local and global independence properties" for a carefully designed family of orders of the variables $X_1,\ldots,X_d$.

The hyper Dirichlet for general directed acyclic graphs was derived from a characterization of the Dirichlet distribution given by Geiger and Heckerman (1997). This characterization of the Dirichlet for $X=(X_1,\ldots,X_d)$ is obtained through a functional equation derived from the local and global independence properties for two different orders of the variables. These two orders are seemingly chosen haphazardly but, as our results show, this is not so. Our results generalize those of Geiger and Heckerman (1997) and are given without the assumption of existence of a positive density for $X$.

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