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Academic Commons Search Resultsen-usStatistical Inference and Experimental Design for Q-matrix Based Cognitive Diagnosis Models
http://academiccommons.columbia.edu/catalog/ac:176169
Zhang, Stephaniehttp://dx.doi.org/10.7916/D8TQ5ZP5Mon, 07 Jul 2014 11:50:59 +0000There has been growing interest in recent years in using cognitive diagnosis models for diagnostic measurement, i.e., classification according to multiple discrete latent traits. The Q-matrix, an incidence matrix specifying the presence or absence of a relationship between each item in the assessment and each latent attribute, is central to many of these models. Important applications include educational and psychological testing; demand in education, for example, has been driven by recent focus on skills-based evaluation. However, compared to more traditional models coming from classical test theory and item response theory, cognitive diagnosis models are relatively undeveloped and suffer from several issues limiting their applicability. This thesis exams several issues related to statistical inference and experimental design for Q-matrix based cognitive diagnosis models.
We begin by considering one of the main statistical issues affecting the practical use of Q-matrix based cognitive diagnosis models, the identifiability issue. In statistical models, identifiability is prerequisite for most common statistical inferences, including parameter estimation and hypothesis testing. With Q-matrix based cognitive diagnosis models, identifiability also affects the classification of respondents according to their latent traits. We begin by examining the identifiability of model parameters, presenting necessary and sufficient conditions for identifiability in several settings.
Depending on the area of application and the researcher's degree of control over the experiment design, fulfilling these identifiability conditions may be difficult. The second part of this thesis proposes new methods for parameter estimation and respondent classification for use with non-identifiable models. In addition, our framework allows consistent estimation of the severity of the non-identifiability problem, in terms of the proportion of the population affected by it. The implications of this measure for the design of diagnostic assessments are also discussed.Statistics, Educational tests and measurements, Quantitative psychology and psychometricsStatisticsDissertationsUnbiased Penetrance Estimates with Unknown Ascertainment Strategies
http://academiccommons.columbia.edu/catalog/ac:175879
Gore, Kristenhttp://dx.doi.org/10.7916/D8KP8098Mon, 07 Jul 2014 11:39:52 +0000Allelic variation in the genome leads to variation in individuals' production of proteins. This, in turn, leads to variation in traits and development, and, in some cases, to diseases. Understanding the genetic basis for disease can aid in the search for therapies and in guiding genetic counseling. Thus, it is of interest to discover the genes with mutations responsible for diseases and to understand the impact of allelic variation at those genes.
A subject's genetic composition is commonly referred to as the subject's genotype. Subjects who carry the gene mutation of interests are referred to as carriers. Subjects who are afflicted with a disease under study (that is, subjects who exhibit the phenotype) are termed affected carriers. The age-specific probability that a given subject will exhibit a phenotype of interest, given mutation status at a gene is known as penetrance.
Understanding penetrance is an important facet of genetic epidemiology. Penetrance estimates are typically calculated via maximum likelihood from family data. However, penetrance estimates can be biased if the nature of the sampling strategy is not correctly reflected in the likelihood. Unfortunately, sampling of family data may be conducted in a haphazard fashion or, even if conducted systematically, might be reported in an incomplete fashion. Bias is possible in applying likelihood methods to reported data if (as is commonly the case) some unaffected family members are not represented in the reports.
The purpose here is to present an approach to find efficient and unbiased penetrance estimates in cases where there is incomplete knowledge of the sampling strategy and incomplete information on the full pedigree structure of families included in the data. The method may be applied with different conjectural assumptions about the ascertainment strategy to balance the possibly biasing effects of wishful assumptions about the sampling strategy with the efficiency gains that could be obtained through valid assumptions.StatisticsStatisticsDissertationsBook Reviews: Principles of Data Mining. By David Hand, Heikki Mannila, and Padhraic Smyth.
http://academiccommons.columbia.edu/catalog/ac:173915
Madigan, David B.http://dx.doi.org/10.7916/D8DZ06D8Thu, 15 May 2014 12:45:12 +0000"Principles of Data Mining. By David Hand, Heikki Mannila, and Padhraic Smyth. MIT Press, Cambridge, MA, 2001. $50.00. xxxii+546 pp., hardcover. ISBN 0-262-08290-X. Is data mining the same as statistics? The distinguished authors of Principles of Data Mining struggle to make a distinction between the two subjects. In the end, what they have written is a fine applied statistics text." -- page 501Statisticsdm2418StatisticsReviewsMedication-Wide Association Studies
http://academiccommons.columbia.edu/catalog/ac:173912
Ryan, P. B.; Stang, P. E.; Madigan, David B.; Schuemie, M. J.; Hripcsak, George M.http://dx.doi.org/10.7916/D8PG1PVXThu, 15 May 2014 12:30:39 +0000Undiscovered side effects of drugs can have a profound effect on the health of the nation, and electronic health-care databases offer opportunities to speed up the discovery of these side effects. We applied a “medication-wide association study” approach that combined multivariate analysis with exploratory visualization to study four health outcomes of interest in an administrative claims database of 46 million patients and a clinical database of 11 million patients. The technique had good predictive value, but there was no threshold high enough to eliminate false-positive findings. The visualization not only highlighted the class effects that strengthened the review of specific products but also underscored the challenges in confounding. These findings suggest that observational databases are useful for identifying potential associations that warrant further consideration but are unlikely to provide definitive evidence of causal effects.Pharmacology, Statistics, Bioinformaticsdm2418, gh13Statistics, Biomedical InformaticsArticlesAlgorithms for Sparse Linear Classifiers in the Massive Data Setting
http://academiccommons.columbia.edu/catalog/ac:173908
Balakrishnan, Suhrid; Bartlett, Peter; Madigan, David B.http://dx.doi.org/10.7916/D8Z0368XThu, 15 May 2014 12:25:33 +0000Classifiers favoring sparse solutions, such as support vector machines, relevance vector machines, LASSO-regression based classifiers, etc., provide competitive methods for classification problems in high dimensions. However, current algorithms for training sparse classifiers typically scale quite unfavorably with respect to the number of training examples. This paper proposes online and multi-pass algorithms for training sparse linear classifiers for high dimensional data. These algorithms have computational complexity and memory requirements that make learning on massive data sets feasible. The central idea that makes this possible is a straightforward quadratic approximation to the likelihood function.Statistics, Artificial intelligencedm2418StatisticsArticlesLearning Theory Analysis for Association Rules and Sequential Event Prediction
http://academiccommons.columbia.edu/catalog/ac:173905
Rudin, Cynthia; Letham, Benjamin; Madigan, David B.http://dx.doi.org/10.7916/D82N50C1Thu, 15 May 2014 12:19:33 +0000We present a theoretical analysis for prediction algorithms based on association rules. As part of this analysis, we introduce a problem for which rules are particularly natural, called “sequential event prediction." In sequential event prediction, events in a sequence are revealed one by one, and the goal is to determine which event will next be revealed. The training set is a collection of past sequences of events. An example application is to predict which item will next be placed into a customer's online shopping cart, given his/her past purchases. In the context of this problem, algorithms based on association rules have distinct advantages over classical statistical and machine learning methods: they look at correlations based on subsets of co-occurring past events (items a and b imply item c), they can be applied to the sequential event prediction problem in a natural way, they can potentially handle the “cold start" problem where the training set is small, and they yield interpretable predictions. In this work, we present two algorithms that incorporate association rules. These algorithms can be used both for sequential event prediction and for supervised classification, and they are simple enough that they can possibly be understood by users, customers, patients, managers, etc. We provide generalization guarantees on these algorithms based on algorithmic stability analysis from statistical learning theory. We include a discussion of the strict minimum support threshold often used in association rule mining, and introduce an “adjusted confidence" measure that provides a weaker minimum support condition that has advantages over the strict minimum support. The paper brings together ideas from statistical learning theory, association rule mining and Bayesian analysis.Statistics, Artificial intelligencedm2418StatisticsArticlesAnalysis of Variance of Cross-Validation Estimators of the Generalization Error
http://academiccommons.columbia.edu/catalog/ac:173902
Markatou, Marianthi; Tian, Hong; Biswas, Shameek; Hripcsak, George M.http://dx.doi.org/10.7916/D86D5R2XThu, 15 May 2014 11:58:33 +0000This paper brings together methods from two different disciplines: statistics and machine learning. We address the problem of estimating the variance of cross-validation (CV) estimators of the generalization error. In particular, we approach the problem of variance estimation of the CV estimators of generalization error as a problem in approximating the moments of a statistic. The approximation illustrates the role of training and test sets in the performance of the algorithm. It provides a unifying approach to evaluation of various methods used in obtaining training and test sets and it takes into account the variability due to different training and test sets. For the simple problem of predicting the sample mean and in the case of smooth loss functions, we show that the variance of the CV estimator of the generalization error is a function of the moments of the random variables Y=Card(Sj ∩ Sj') and Y*=Card(Sjc ∩ Sj'c), where Sj, Sj' are two training sets, and Sjc, Sj'c are the corresponding test sets. We prove that the distribution of Y and Y* is hypergeometric and we compare our estimator with the one proposed by Nadeau and Bengio (2003). We extend these results in the regression case and the case of absolute error loss, and indicate how the methods can be extended to the classification case. We illustrate the results through simulation.Statistics, Artificial intelligencemm168, ht2031, spb2003, gh13Biostatistics, Biomedical Informatics, StatisticsArticlesA One-Pass Sequential Monte Carlo Method for Bayesian Analysis of Massive Datasets
http://academiccommons.columbia.edu/catalog/ac:173899
Balakrishnan, Suhrid; Madigan, David B.http://dx.doi.org/10.7916/D8B56GTPThu, 15 May 2014 11:51:51 +0000For Bayesian analysis of massive data, Markov chain Monte Carlo (MCMC) techniques often prove infeasible due to computational resource constraints. Standard MCMC methods generally require a complete scan of the dataset for each iteration. Ridgeway and Madigan (2002) and Chopin (2002b) recently presented importance sampling algorithms that combined simulations from a posterior distribution conditioned on a small portion of the dataset with a reweighting of those simulations to condition on the remainder of the dataset. While these algorithms drastically reduce the number of data accesses as compared to traditional MCMC, they still require substantially more than a single pass over the dataset. In this paper, we present "1PFS," an efficient, one-pass algorithm. The algorithm employs a simple modification of the Ridgeway and Madigan (2002) particle filtering algorithm that replaces the MCMC based "rejuvenation" step with a more efficient "shrinkage" kernel smoothing based step. To show proof-of-concept and to enable a direct comparison, we demonstrate 1PFS on the same examples presented in Ridgeway and Madigan (2002), namely a mixture model for Markov chains and Bayesian logistic regression. Our results indicate the proposed scheme delivers accurate parameter estimates while employing only a single pass through the data.Mathematics, Statisticsdm2418StatisticsArticlesA Characterization of Markov Equivalence Classes for Acyclic Digraphs
http://academiccommons.columbia.edu/catalog/ac:173896
Andersson, Steen A.; Madigan, David B.; Perlman, Michael D.http://dx.doi.org/10.7916/D8FX77J3Thu, 15 May 2014 11:28:36 +0000Undirected graphs and acyclic digraphs (ADG's), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multiviarate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may be many ADG's that determine the same dependence (i.e., Markov) model. Thus, the family of all ADG's with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection of model averaging, that fail to take into account these equivalence classes may incur substantial computational or other inefficiences. Here it is show that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADG's in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsArticlesCorrection: Separation and completeness properties for AMP chain graph Markov models
http://academiccommons.columbia.edu/catalog/ac:173887
Madigan, David B.; Levitz, Michael; Perlman, Michael D.http://dx.doi.org/10.7916/D8QF8R05Wed, 14 May 2014 19:42:28 +0000Correction of table 2 on page 1757 of 'Separation and completeness properties for AMP chain graph Markov models', Annals of Statistics, volume 29 (2001).Mathematics, Statisticsdm2418StatisticsArticlesBayesian Hierarchical Rule Modeling for Predicting Medical Conditions
http://academiccommons.columbia.edu/catalog/ac:173882
McCormick, Tyler H.; Rudin, Cynthia; Madigan, David B.http://dx.doi.org/10.7916/D8V69GP1Wed, 14 May 2014 19:02:36 +0000We propose a statistical modeling technique, called the Hierarchical Association Rule Model (HARM), that predicts a patient’s possible future medical conditions given the patient’s current and past history of reported conditions. The core of our technique is a Bayesian hierarchical model for selecting predictive association rules (such as “condition 1 and condition 2 → condition 3”) from a large set of candidate rules. Because this method “borrows strength” using the conditions of many similar patients, it is able to provide predictions specialized to any given patient, even when little information about the patient’s history of conditions is available.Applied mathematics, Statistics, Medicinedm2418StatisticsArticles[Bayesian Analysis in Expert Systems]: Comment: What's Next?
http://academiccommons.columbia.edu/catalog/ac:173856
Madigan, David B.http://dx.doi.org/10.7916/D8W37TFJTue, 13 May 2014 17:59:40 +0000"These papers represent two of the many different graphical modeling camps that have emerged from a flurry of activity in the past decade. The paper by Cox and Wermuth falls within the statistical graphical modeling camp and provides a useful generalization of that body of work. There is, of course, a price to be paid for this generality, namely that the interpretation of the graphs is more complex...The paper by Spiegelhalter, Dawid, Lauritzen and Cowell falls within the probabilistic expert system camp. This is a tour de force by researchers responsible for much of the astonishing progress in this area. Ten years ago, probabilistic models were shunned by the artificial intelligence community. That they are now widely accepted and used is due in large measure to the insights and efforts of these authors, along with other pioneers such as Judea Pearl and Peter Cheeseman..." -- page 261Mathematics, Statisticsdm2418StatisticsArticlesBayesian Model Averaging: a Tutorial (with Comments by M. Clyde, David Draper and E. I. George, and a Rejoinder by the Authors)
http://academiccommons.columbia.edu/catalog/ac:173853
Hoeting, Jennifer A.; Madigan, David B.; Raftery, Adrian E.; Volinsky, Chris T.; Clyde, M.; Draper, David; George, E. I.http://dx.doi.org/10.7916/D84M92N7Tue, 13 May 2014 17:39:49 +0000Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This approach ignores the uncertainty in model selection, leading to over-confident inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA)provides a coherent mechanism for accounting for this model uncertainty. Several methods for implementing BMA have recently emerged. We discuss these methods and present a number of examples.In these examples, BMA provides improved out-of-sample predictive performance. We also provide a catalogue of currently available BMA software.Statisticsdm2418StatisticsArticlesSeparation and Completeness Properties for Amp Chain Graph Markov Models
http://academiccommons.columbia.edu/catalog/ac:173847
Levitz, Michael; Perlman, Michael D.; Madigan, David B.http://dx.doi.org/10.7916/D8X34VJGTue, 13 May 2014 16:30:46 +0000Pearl’s well-known d-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces p-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs (= adicyclic graphs), which include both ADGs and undirected graphs as special cases. The equivalence of p-separation to the augmentation criterion occurring in the AMP global Markov property is established, and p-separation is applied to prove completeness of the global Markov property for AMP chain graph models. Strong completeness of the AMP Markov property is established, that is, the existence of Markov perfect distributions that satisfy those and only those conditional independences implied by the AMP property(equivalently, by p-separation). A linear-time algorithm for determining p-separation is presented.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsArticles[Least Angle Regression]: Discussion
http://academiccommons.columbia.edu/catalog/ac:173841
Madigan, David B.; Ridgeway, Greghttp://dx.doi.org/10.7916/D81V5C29Tue, 13 May 2014 16:15:23 +0000Algorithms for simultaneous shrinkage and selection in regression and classification provide attractive solutions to knotty old statistical challenges. Nevertheless, as far as we can tell, Tibshirani's Lasso algorithm has had little impact on statistical practice. Two particular reasons for this may be the relative inefficiency of the original Lasso algorithm and the relative complexity of more recent Lasso algorithms [e.g., Osborne, Presnell and Turlach (2000)]. Efron, Hastie, Johnstone and Tibshirani have provided an efficient, simple algorithm for the Lasso as well as algorithms for stagewise regression and the new least angle regression. As such this paper is an important contribution to statistical computing.Mathematics, Statisticsdm2418StatisticsArticlesA Hierarchical Model for Association Rule Mining of Sequential Events: An Approach to Automated Medical Symptom Prediction
http://academiccommons.columbia.edu/catalog/ac:173838
McCormick, Tyler H.; Rudin, Cynthia; Madigan, David B.http://dx.doi.org/10.7916/D89C6VJDTue, 13 May 2014 15:27:01 +0000In many healthcare settings, patients visit healthcare professionals periodically and report multiple medical conditions, or symptoms, at each encounter. We propose a statistical modeling technique, called the Hierarchical Association Rule Model (HARM), that predicts a patient’s possible future symptoms given the patient’s current and past history of reported symptoms. The core of our technique is a Bayesian hierarchical model for selecting predictive association rules (such as “symptom 1 and symptom 2 → symptom 3 ”) from a large set of candidate rules. Because this method “borrows strength” using the symptoms of many similar patients, it is able to provide predictions specialized to any given patient, even when little information about the patient’s history of symptoms is available.Mathematics, Statistics, Medicinedm2418StatisticsArticlesGenerating Productive Dialogue between Consulting Statisticians and their Clients in the Pharmaceutical and Medical Research Settings
http://academiccommons.columbia.edu/catalog/ac:173832
Emir, Birol; Amaratunga, Dhammika; Beltangady, Mohan; Cabrera, Javier; Freeman, Roy; Madigan, David B.; Nguyen, Ha H.; Whalen, Edward Patrickhttp://dx.doi.org/10.7916/D8PK0D8NTue, 13 May 2014 15:09:40 +0000Due to the ever-increasing complexity of scientific technologies and resulting data, consulting statisticians are becoming more involved in the design, conduct, and analysis of biomedical research. This requires extensive collaboration between the consulting statistician and nonstatisticians, such as researchers, clinicians, and corporate executives. Consequently, a successful consulting career is becoming ever more dependent on the statistician's ability to effectively communicate with nonstatisticians. This is especially true when more complex, nontraditional analytical methods are required. In this paper, we examine the collaboration between statisticians and nonstatisticians from three different professional perspectives. Integrating these perspectives, we discuss ways to help the consulting statistician generate productive dialogue with clients. Finally, we examine how universities can better prepare students for careers in statistical consulting by incorporating more communication-based elements into their curriculum and by offering students ample opportunities to collaborate with nonstatisticians. Overall, we designed this exercise to help the consulting statistician generate dialogue with clients that results in more productive collaborations and a more satisfying work experience.Statistics, Bioinformatics, Medicinebe2166, dm2418, hhn2108, ew2320StatisticsArticlesA Note on Equivalence Classes of Directed Acyclic Independence Graphs
http://academiccommons.columbia.edu/catalog/ac:173826
Madigan, David B.http://dx.doi.org/10.7916/D8TB150CTue, 13 May 2014 14:46:04 +0000Directed acyclic independence graphs (DAIGs) play an important role in recent developments in probabilistic expert systems and influence diagrams (Chyu [1]). The purpose of this note is to show that DAIGs can usefully be grouped into equivalence classes where the members of a single class share identical Markov properties. These equivalence classes can be identified via a simple graphical criterion. This result is particularly relevant to model selection procedures for DAIGs (see, e.g., Cooper and Herskovits [2] and Madigan and Raftery [4]) because it reduces the problem of searching among possible orientations of a given graph to that of searching among the equivalence classes.Mathematics, Statisticsdm2418StatisticsArticlesLocation Estimation in Wireless Networks: A Bayesian Approach
http://academiccommons.columbia.edu/catalog/ac:173820
Madigan, David B.; Ju, Wen-Hua; Krishnan, P.; Krishnakumar, A. S. ; Zorych, Ivanhttp://dx.doi.org/10.7916/D82V2D74Tue, 13 May 2014 14:25:34 +0000We present a Bayesian hierarchical model for indoor location estimation in wireless networks. We demonstrate that out model achieves accuracy that is similar to other published models and algorithms. By harnessing prior knowledge, our model drastically reduces the requirement for training data as compared with existing approaches.Mathematics, Statistics, Applied mathematicsdm2418StatisticsArticlesA Flexible Bayesian Generalized Linear Model for Dichotomous Response Data with an Application to Text Categorization
http://academiccommons.columbia.edu/catalog/ac:173817
Eyheramendy, Susana; Madigan, David B.http://dx.doi.org/10.7916/D86M34ZFTue, 13 May 2014 14:04:25 +0000We present a class of sparse generalized linear models that include probit and logistic regression as special cases and offer some extra flexibility. We provide an EM algorithm for learning the parameters of these models from data. We apply our method in text classification and in simulated data and show that our method outperforms the logistic and probit models and also the elastic net, in general by a substantial margin.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsBook chaptersFit GFuseTLP penalized conditional logistic regression model for high-dimensional one-to- one matched case-control data
http://academiccommons.columbia.edu/catalog/ac:174087
Zhou, Hui; Wang, Shuang; Zheng, Tianhttp://dx.doi.org/10.7916/D8028PNJMon, 12 May 2014 16:26:54 +0000Fit GFuseTLP penalized conditional logistic regression model for high-dimensional one-to- one matched case-control dataStatisticshz2240, sw2206, tz33Biostatistics, StatisticsComputer softwareA Point Process Model for the Dynamics of Limit Order Books
http://academiccommons.columbia.edu/catalog/ac:171221
Vinkovskaya, Ekaterinahttp://dx.doi.org/10.7916/D88913WWFri, 28 Feb 2014 16:44:16 +0000This thesis focuses on the statistical modeling of the dynamics of limit order books in electronic equity markets. The statistical properties of events affecting a limit order book -market orders, limit orders and cancellations- reveal strong evidence of clustering in time, cross-correlation across event types and dependence of the order flow on the bid-ask spread. Further investigation reveals the presence of a self-exciting property - that a large number of events in a given time period tends to imply a higher probability of observing a large number of events in the following time period. We show that these properties may be adequately represented by a multivariate self-exciting point process with multiple regimes that reflect changes in the bid-ask spread.
We propose a tractable parametrization of the model and perform a Maximum Likelihood Estimation of the model using high-frequency data from the Trades and Quotes database for US stocks. We show that the model may be used to obtain predictions of order flow and that its predictive performance beats the Poisson model as well as Moving Average and Auto Regressive time series models.StatisticsStatisticsDissertationsMixed Methods for Mixed Models
http://academiccommons.columbia.edu/catalog/ac:169644
Dorie, Vincent J.http://dx.doi.org/10.7916/D8V40S5XWed, 22 Jan 2014 14:28:18 +0000This work bridges the frequentist and Bayesian approaches to mixed models by borrowing the best features from both camps: point estimation procedures are combined with priors to obtain accurate, fast inference while posterior simulation techniques are developed that approximate the likelihood with great precision for the purposes of assessing uncertainty. These allow flexible inferences without the need to rely on expensive Markov chain Monte Carlo simulation techniques. Default priors are developed and evaluated in a variety of simulation and real-world settings with the end result that we propose a new set of standard approaches that yield superior performance at little computational cost.StatisticsStatisticsDissertationsKernel-based association measures
http://academiccommons.columbia.edu/catalog/ac:167034
Liu, Yinghttp://hdl.handle.net/10022/AC:P:22154Thu, 07 Nov 2013 15:12:35 +0000Measures of associations have been widely used for describing the statistical relationships between two sets of variables. Traditional association measures tend to focus on specialized settings (specific types of variables or association patterns). Based on an in-depth summary of existing measures, we propose a general framework for association measures unifying existing methods and novel extensions based on kernels, including practical solutions to computational challenges. The proposed framework provides improved feature selection and extensions to a variety of current classifiers. Specifically, we introduce association screening and variable selection via maximizing kernel-based association measures. We also develop a backward dropping procedure for feature selection when there are a large number of candidate variables. We evaluate our framework using a wide variety of both simulated and real data. In particular, we conduct independence tests and feature selection using kernel association measures on diversified association patterns of different dimensions and variable types. The results show the superiority of our methods to existing ones. We also apply our framework to four real-word problems, three from statistical genetics and one of gender prediction from handwriting. We demonstrate through these applications both the de novo construction of new kernels and the adaptation of existing kernels tailored to the data at hand, and how kernel-based measures of associations can be naturally applied to different data structures including functional input and output spaces. This shows that our framework can be applied to a wide range of real world problems and work well in practice.Statistics, Computer scienceyl2802StatisticsDissertationsLow-rank graphical models and Bayesian inference in the statistical analysis of noisy neural data
http://academiccommons.columbia.edu/catalog/ac:166472
Smith, Carl Alexanderhttp://hdl.handle.net/10022/AC:P:21991Fri, 11 Oct 2013 16:56:29 +0000We develop new methods of Bayesian inference, largely in the context of analysis of neuroscience data. The work is broken into several parts. In the first part, we introduce a novel class of joint probability distributions in which exact inference is tractable. Previously it has been difficult to find general constructions for models in which efficient exact inference is possible, outside of certain classical cases. We identify a class of such models that are tractable owing to a certain "low-rank" structure in the potentials that couple neighboring variables. In the second part we develop methods to quantify and measure information loss in analysis of neuronal spike train data due to two types of noise, making use of the ideas developed in the first part. Information about neuronal identity or temporal resolution may be lost during spike detection and sorting, or precision of spike times may be corrupted by various effects. We quantify the information lost due to these effects for the relatively simple but sufficiently broad class of Markovian model neurons. We find that decoders that model the probability distribution of spike-neuron assignments significantly outperform decoders that use only the most likely spike assignments. We also apply the ideas of the low-rank models from the first section to defining a class of prior distributions over the space of stimuli (or other covariate) which, by conjugacy, preserve the tractability of inference. In the third part, we treat Bayesian methods for the estimation of sparse signals, with application to the locating of synapses in a dendritic tree. We develop a compartmentalized model of the dendritic tree. Building on previous work that applied and generalized ideas of least angle regression to obtain a fast Bayesian solution to the resulting estimation problem, we describe two other approaches to the same problem, one employing a horseshoe prior and the other using various spike-and-slab priors. In the last part, we revisit the low-rank models of the first section and apply them to the problem of inferring orientation selectivity maps from noisy observations of orientation preference. The relevant low-rank model exploits the self-conjugacy of the von Mises distribution on the circle. Because the orientation map model is loopy, we cannot do exact inference on the low-rank model by the forward backward algorithm, but block-wise Gibbs sampling by the forward backward algorithm speeds mixing. We explore another von Mises coupling potential Gibbs sampler that proves to effectively smooth noisily observed orientation maps.Statistics, Neurosciencescas2207Chemistry, StatisticsDissertationsGeneralized Volatility-Stabilized Processes
http://academiccommons.columbia.edu/catalog/ac:165162
Pickova, Radkahttp://hdl.handle.net/10022/AC:P:21616Fri, 13 Sep 2013 15:07:49 +0000In this thesis, we consider systems of interacting diffusion processes which we call Generalized Volatility-Stabilized processes, as they extend the Volatility-Stabilized Market models introduced in Fernholz and Karatzas (2005). First, we show how to construct a weak solution of the underlying system of stochastic differential equations. In particular, we express the solution in terms of time-changed squared-Bessel processes and argue that this solution is unique in distribution. In addition, we also discuss sufficient conditions under which this solution does not explode in finite time, and provide sufficient conditions for pathwise uniqueness and for existence of a strong solution.
Secondly, we discuss the significance of these processes in the context of Stochastic Portfolio Theory. We describe specific market models which assume that the dynamics of the stocks' capitalizations is the same as that of the Generalized Volatility-Stabilized processes, and we argue that strong relative arbitrage opportunities may exist in these markets, specifically, we provide multiple examples of portfolios that outperform the market portfolio. Moreover, we examine the properties of market weights as well as the diversity weighted portfolio in these models.
Thirdly, we provide some asymptotic results for these processes which allows us to describe different properties of the corresponding market models based on these processes.Statisticsrp2424Statistics, MathematicsDissertationsCredit Risk Modeling and Analysis Using Copula Method and Changepoint Approach to Survival Data
http://academiccommons.columbia.edu/catalog/ac:161682
Qian, Bohttp://hdl.handle.net/10022/AC:P:20510Thu, 30 May 2013 16:36:22 +0000This thesis consists of two parts. The first part uses Gaussian Copula and Student's t Copula as the main tools to model the credit risk in securitizations and re-securitizations. The second part proposes a statistical procedure to identify changepoints in Cox model of survival data. The recent 2007-2009 financial crisis has been regarded as the worst financial crisis since the Great Depression by leading economists. The securitization sector took a lot of blame for the crisis because of the connection of the securitized products created from mortgages to the collapse of the housing market. The first part of this thesis explores the relationship between securitized mortgage products and the 2007-2009 financial crisis using the Copula method as the main tool. We show in this part how loss distributions of securitizations and re-securitizations can be derived or calculated in a new model. Simulations are conducted to examine the effectiveness of the model. As an application, the model is also used to examine whether and where the ratings of securitized products could be flawed. On the other hand, the lag effect and saturation effect problems are common and important problems in survival data analysis. They belong to a general class of problems where the treatment effect takes occasional jumps instead of staying constant throughout time. Therefore, they are essentially the changepoint problems in statistics. The second part of this thesis focuses on extending Lai and Xing's recent work in changepoint modeling, which was developed under a time series and Bayesian setup, to the lag effect problems in survival data. A general changepoint approach for Cox model is developed. Simulations and real data analyses are conducted to illustrate the effectiveness of the procedure and how it should be implemented and interpreted.Statisticsbq2102StatisticsDissertationsStatistical Inference for Diagnostic Classification Models
http://academiccommons.columbia.edu/catalog/ac:160464
Xu, Gongjunhttp://hdl.handle.net/10022/AC:P:20058Tue, 30 Apr 2013 16:06:11 +0000Diagnostic classification models (DCM) are an important recent development in educational and psychological testing. Instead of an overall test score, a diagnostic test provides each subject with a profile detailing the concepts and skills (often called "attributes") that he/she has mastered. Central to many DCMs is the so-called Q-matrix, an incidence matrix specifying the item-attribute relationship. It is common practice for the Q-matrix to be specified by experts when items are written, rather than through data-driven calibration. Such a non-empirical approach may lead to misspecification of the Q-matrix and substantial lack of model fit, resulting in erroneous interpretation of testing results. This motivates our study and we consider the identifiability, estimation, and hypothesis testing of the Q-matrix. In addition, we study the identifiability of diagnostic model parameters under a known Q-matrix. The first part of this thesis is concerned with estimation of the Q-matrix. In particular, we present definitive answers to the learnability of the Q-matrix for one of the most commonly used models, the DINA model, by specifying a set of sufficient conditions under which the Q-matrix is identifiable up to an explicitly defined equivalence class. We also present the corresponding data-driven construction of the Q-matrix. The results and analysis strategies are general in the sense that they can be further extended to other diagnostic models. The second part of the thesis focuses on statistical validation of the Q-matrix. The purpose of this study is to provide a statistical procedure to help decide whether to accept the Q-matrix provided by the experts. Statistically, this problem can be formulated as a pure significance testing problem with null hypothesis H0 : Q = Q0, where Q0 is the candidate Q-matrix. We propose a test statistic that measures the consistency of observed data with the proposed Q-matrix. Theoretical properties of the test statistic are studied. In addition, we conduct simulation studies to show the performance of the proposed procedure. The third part of this thesis is concerned with the identifiability of the diagnostic model parameters when the Q-matrix is correctly specified. Identifiability is a prerequisite for statistical inference, such as parameter estimation and hypothesis testing. We present sufficient and necessary conditions under which the model parameters are identifiable from the response data.Statistics, Educational tests and measurementsgx2108StatisticsDissertationsBayesian Model Selection in terms of Kullback-Leibler discrepancy
http://academiccommons.columbia.edu/catalog/ac:158374
Zhou, Shouhaohttp://hdl.handle.net/10022/AC:P:19157Mon, 25 Feb 2013 13:36:40 +0000In this article we investigate and develop the practical model assessment and selection methods for Bayesian models, when we anticipate that a promising approach should be objective enough to accept, easy enough to understand, general enough to apply, simple enough to compute and coherent enough to interpret. We mainly restrict attention to the Kullback-Leibler divergence, a widely applied model evaluation measurement to quantify the similarity between the proposed candidate model and the underlying true model, where the true model is only referred to a probability distribution as the best projection onto the statistical modeling space once we try to understand the real but unknown dynamics/mechanism of interest. In addition to review and discussion on the advantages and disadvantages of the historically and currently prevailing practical model selection methods in literature, a series of convenient and useful tools, each designed and applied for different purposes, are proposed to asymptotically unbiasedly assess how the candidate Bayesian models are favored in terms of predicting a future independent observation. What's more, we also explore the connection of the Kullback-Leibler based information criterion to the Bayes factors, another most popular Bayesian model comparison approaches, after seeing the motivation through the developments of the Bayes factor variants. In general, we expect to provide a useful guidance for researchers who are interested in conducting Bayesian data analysis.Statisticssz2020StatisticsDissertationsMultiplicative Multiresolution Analysis for Lie-group Valued Data Indexed by a Euclidean Parameter
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Stodden, Victoria C.http://hdl.handle.net/10022/AC:P:15397Wed, 12 Dec 2012 15:17:09 +0000Lie-valued euclidean indexed data. These data might be: phase angles as functions of time or space, for example compass directions; 3D orientations of a rigid frame of reference as a function of time or space; or, quaternions as a function of time or space. This can also be extended to quotients of lie groups which gives us the ability to model points on S2, the unit sphere, as functions of time or space.Computer science, Statisticsvcs2115StatisticsPresentationsMultiple Imputation with Diagnostics (mi) in R: Opening Windows into the Black Box
http://academiccommons.columbia.edu/catalog/ac:154731
Su, Yu-Sung; Yajima, Masanao; Gelman, Andrew E.; Hill, Jenniferhttp://hdl.handle.net/10022/AC:P:15342Tue, 20 Nov 2012 16:49:06 +0000Our mi package in R has several features that allow the user to get inside the imputation process and evaluate the reasonableness of the resulting models and imputations. These features include: choice of predictors, models, and transformations for chained imputation models; standard and binned residual plots for checking the fit of the conditional distributions used for imputation; and plots for comparing the distributions of observed and imputed data. In addition, we use Bayesian models and weakly informative prior distributions to construct more stable estimates of imputation models. Our goal is to have a demonstration package that (a) avoids many of the practical problems that arise with existing multivariate imputation programs, and (b) demonstrates state-of-the-art diagnostics that can be applied more generally and can be incorporated into the software of others.Statisticsag389Statistics, Political ScienceArticlesR2WinBUGS: A Package for Running WinBUGS from R
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Sturtz, Sibylle; Ligges, Uwe; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:15341Tue, 20 Nov 2012 16:42:45 +0000The R2WinBUGS package provides convenient functions to call WinBUGS from R. It automatically writes the data and scripts in a format readable by WinBUGS for processing in batch mode, which is possible since version 1.4. After the WinBUGS process has finished, it is possible either to read the resulting data into R by the package itself—which gives a compact graphical summary of inference and convergence diagnostics—or to use the facilities of the coda package for further analyses of the output. Examples are given to demonstrate the usage of this package.Statisticsag389Statistics, Political ScienceArticlesBayesian Statistical Pragmatism
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Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:15340Tue, 20 Nov 2012 16:38:18 +0000I agree with Rob Kass’ point that we can and should make use of statistical methods developed under different philosophies, and I am happy to take the opportunity to elaborate on some of his arguments.Statisticsag389Statistics, Political ScienceArticlesSegregation in Social Networks Based on Acquaintanceship and Trust
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DiPrete, Thomas A.; Gelman, Andrew E.; McCormick, Tyler; Teitler, Julien O.; Zheng, Tianhttp://hdl.handle.net/10022/AC:P:15339Tue, 20 Nov 2012 16:17:57 +0000Using 2006 General Social Survey data, the authors compare levels of segregation by race and along other dimensions of potential social cleavage in the contemporary United States. Americans are not as isolated as the most extreme recent estimates suggest. However, hopes that “bridging” social capital is more common in broader acquaintanceship networks than in core networks are not supported. Instead, the entire acquaintanceship network is perceived by Americans to be about as segregated as the much smaller network of close ties. People do not always know the religiosity, political ideology, family behaviors, or socioeconomic status of their acquaintances, but perceived social divisions on these dimensions are high, sometimes rivaling racial segregation in acquaintanceship networks. The major challenge to social integration today comes from the tendency of many Americans to isolate themselves from others who differ on race, political ideology, level of religiosity, and other salient aspects of social identity.Statisticstad61, ag389, thm2105, jot8, tz33Sociology, Statistics, Political Science, Social WorkArticlesModeling Strategies for Large Dimensional Vector Autoregressions
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Zang, Pengfeihttp://hdl.handle.net/10022/AC:P:14666Tue, 11 Sep 2012 15:31:00 +0000The vector autoregressive (VAR) model has been widely used for describing the dynamic behavior of multivariate time series. However, fitting standard VAR models to large dimensional time series is challenging primarily due to the large number of parameters involved. In this thesis, we propose two strategies for fitting large dimensional VAR models. The first strategy involves reducing the number of non-zero entries in the autoregressive (AR) coefficient matrices and the second is a method to reduce the effective dimension of the white noise covariance matrix. We propose a 2-stage approach for fitting large dimensional VAR models where many of the AR coefficients are zero. The first stage provides initial selection of non-zero AR coefficients by taking advantage of the properties of partial spectral coherence (PSC) in conjunction with BIC. The second stage, based on $t$-ratios and BIC, further refines the spurious non-zero AR coefficients post first stage. Our simulation study suggests that the 2-stage approach outperforms Lasso-type methods in discovering sparsity patterns in AR coefficient matrices of VAR models. The performance of our 2-stage approach is also illustrated with three real data examples. Our second strategy for reducing the complexity of a large dimensional VAR model is based on a reduced-rank estimator for the white noise covariance matrix. We first derive the reduced-rank covariance estimator under the setting of independent observations and give the analytical form of its maximum likelihood estimate. Then we describe how to integrate the proposed reduced-rank estimator into the fitting of large dimensional VAR models, where we consider two scenarios that require different model fitting procedures. In the VAR modeling context, our reduced-rank covariance estimator not only provides interpretable descriptions of the dependence structure of VAR processes but also leads to improvement in model-fitting and forecasting over unrestricted covariance estimators. Two real data examples are presented to illustrate these fitting procedures.Statisticspz2146StatisticsDissertationsSome Models for Time Series of Counts
http://academiccommons.columbia.edu/catalog/ac:152149
Liu, Henghttp://hdl.handle.net/10022/AC:P:14561Wed, 29 Aug 2012 14:08:58 +0000This thesis focuses on developing nonlinear time series models and establishing relevant theory with a view towards applications in which the responses are integer valued. The discreteness of the observations, which is not appropriate with classical time series models, requires novel modeling strategies. The majority of the existing models for time series of counts assume that the observations follow a Poisson distribution conditional on an accompanying intensity process that drives the serial dynamics of the model. According to whether the evolution of the intensity process depends on the observations or solely on an external process, the models are classified into parameter-driven and observation-driven. Compared to the former one, an observation-driven model often allows for easier and more straightforward estimation of the model parameters. On the other hand, the stability properties of the process, such as the existence and uniqueness of a stationary and ergodic solution that are required for deriving asymptotic theory of the parameter estimates, can be quite complicated to establish, as compared to parameter-driven models. In this thesis, we first propose a broad class of observation-driven models that is based upon a one-parameter exponential family of distributions and incorporates nonlinear dynamics. The establishment of stability properties of these processes, which is at the heart of this thesis, is addressed by employing theory from iterated random functions and coupling techniques. Using this theory, we are also able to obtain the asymptotic behavior of maximum likelihood estimates of the parameters. Extensions of the base model in several directions are considered. Inspired by the idea of a self-excited threshold ARMA process, a threshold Poisson autoregression is proposed. It introduces a two-regime structure in the intensity process and essentially allows for modeling negatively correlated observations. E-chain, a non-standard Markov chain technique and Lyapunov's method are utilized to show the stationarity and a law of large numbers for this process. In addition, the model has been adapted to incorporate covariates, an important problem of practical and primary interest. The base model is also extended to consider the case of multivariate time series of counts. Given a suitable definition of a multivariate Poisson distribution, a multivariate Poisson autoregression process is described and its properties studied. Several simulation studies are presented to illustrate the inference theory. The proposed models are also applied to several real data sets, including the number of transactions of the Ericsson stock, the return times of Goldman Sachs Group stock prices, the number of road crashes in Schiphol, the frequencies of occurrences of gold particles, the incidences of polio in the US and the number of presentations of asthma in an Australian hospital. An array of graphical and quantitative diagnostic tools, which is specifically designed for the evaluation of goodness of fit for time series of counts models, is described and illustrated with these data sets.Statisticshl2494StatisticsDissertationsStatistical inference in two non-standard regression problems
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Seijo, Emilio Franciscohttp://hdl.handle.net/10022/AC:P:14317Wed, 08 Aug 2012 13:43:26 +0000This thesis analyzes two regression models in which their respective least squares estimators have nonstandard asymptotics. It is divided in an introduction and two parts. The introduction motivates the study of nonstandard problems and presents an outline of the contents of the remaining chapters. In part I, the least squares estimator of a multivariate convex regression function is studied in great detail. The main contribution here is a proof of the consistency of the aforementioned estimator in a completely nonparametric setting. Model misspecification, local rates of convergence and multidimensional regression models mixing convexity and componentwise monotonicity constraints will also be considered. Part II deals with change-point regression models and the issues that might arise when applying the bootstrap to these problems. The classical bootstrap is shown to be inconsistent on a simple change-point regression model, and an alternative (smoothed) bootstrap procedure is proposed and proved to be consistent. The superiority of the alternative method is also illustrated through a simulation study. In addition, a version of the continuous mapping theorem specially suited for change-point estimators is proved and used to derive the results concerning the bootstrap.Statistics, Applied mathematics, Mathematicsefs2113StatisticsDissertationsMultiscale Representations for Manifold-Valued Data
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Rahman, Inam Ur; Drori, Iddo; Stodden, Victoria C.; Donoho, David L.; Schroeder, Peterhttp://hdl.handle.net/10022/AC:P:11434Tue, 11 Oct 2011 15:45:58 +0000We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as: the sphere S2, the special orthogonal group SO(3), the positive definite matrices SPD(n), and the Grassmann manifolds G(n, k). The representations are based on the deployment of Deslauriers-Dubuc and Average Interpolating pyramids "in the tangent plane" of such manifolds, using the Exp and Log maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled much as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds, but is particularly suited to the manifolds we consider, i.e. Riemanian symmetric spaces, such as Sn−1, SO(n), G(n, k), where the Exp and Log maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.Statisticsvcs2115StatisticsArticlesWhen Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?
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Donoho, David L.; Stodden, Victoria C.http://hdl.handle.net/10022/AC:P:11433Tue, 11 Oct 2011 15:32:23 +0000We interpret non-negative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of "parts" and NMF correctly identifies the "parts". We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.Statisticsvcs2115StatisticsArticlesFast l1 Minimization for Genomewide Analysis of mRNA Lengths
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Drori, Iddo; Stodden, Victoria C.; Hurowitz, Evan H.Tue, 11 Oct 2011 15:19:48 +0000Application of the virtual northern method to human mRNA allows us to systematically measure transcript length on a genome-wide scale [1]. Characterization of RNA transcripts by length provides a measurement which complements cDNA sequencing. We have robustly extracted the lengths of the transcripts expressed by each gene for comparison with the Unigene, Refseq, and H-Invitational databases [2, 3]. Obtaining an accurate probability for each peak requires performing multiple bootstrap simulations, each involving a deconvolution operation which is equivalent to finding the sparsest non-negative solution of an underdetermined system of equations. This process is computationally intensive for a large number of simulations and genes. In this contribution we present an efficient approximation method which is faster than general purpose solvers by two orders of magnitude, and in practice reduces our processing time from a week to hours.Genetics, Statisticsvcs2115StatisticsArticlesBreakdown Point of Model Selection When the Number of Variables Exceeds the Number of Observations
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Donoho, David L.; Stodden, Victoria C.http://hdl.handle.net/10022/AC:P:11431Tue, 11 Oct 2011 15:07:17 +0000The classical multivariate linear regression problem assumes p variables X1, X2, ... , Xp and a response vector y, each with n observations, and a linear relationship between the two: y = X beta + z, where z ~ N(0, sigma2). We point out that when p > n, there is a breakdown point for standard model selection schemes, such that model selection only works well below a certain critical complexity level depending on n/p. We apply this notion to some standard model selection algorithms (Forward Stepwise, LASSO, LARS) in the case where pGtn. We find that 1) the breakdown point is well-de ned for random X-models and low noise, 2) increasing noise shifts the breakdown point to lower levels of sparsity, and reduces the model recovery ability of the algorithm in a systematic way, and 3) below breakdown, the size of coefficient errors follows the theoretical error distribution for the classical linear model.Statisticsvcs2115StatisticsArticlesSelf-controlled methods for postmarketing drug safety surveillance in large-scale longitudinal data
http://academiccommons.columbia.edu/catalog/ac:137551
Simpson, Shawn E.http://hdl.handle.net/10022/AC:P:10963Mon, 22 Aug 2011 13:01:55 +0000A primary objective in postmarketing drug safety surveillance is to ascertain the relationship between time-varying drug exposures and adverse events (AEs) related to health outcomes. Surveillance can be based on longitudinal observational databases (LODs), which contain time-stamped patient-level medical information including periods of drug exposure and dates of diagnoses. Due to its desirable properties, we focus on the self-controlled case series (SCCS) method for analysis in this context. SCCS implicitly controls for fixed multiplicative baseline covariates since each individual acts as their own control. In addition, only exposed cases are required for the analysis, which is computationally advantageous. In the first part of this work we present how the simple SCCS model can be applied to the surveillance problem, and compare the results of simple SCCS to those of existing methods. Many current surveillance methods are based on marginal associations between drug exposures and AEs. Such analyses ignore confounding drugs and interactions and have the potential to give misleading results. In order to avoid these difficulties, it is desirable for an analysis strategy to incorporate large numbers of time-varying potential confounders such as other drugs. In the second part of this work we propose the Bayesian multiple SCCS approach, which deals with high dimensionality and can provide a sparse solution via a Laplacian prior. We present details of the model and optimization procedure, as well as results of empirical investigations. SCCS is based on a conditional Poisson regression model, which assumes that events at different time points are conditionally independent given the covariate process. This requirement is problematic when the occurrence of an event can alter the future event risk. In a clinical setting, for example, patients who have a first myocardial infarction (MI) may be at higher subsequent risk for a second. In the third part of this work we propose the positive dependence self-controlled case series (PD-SCCS) method: a generalization of SCCS that allows the occurrence of an event to increase the future event risk, yet maintains the advantages of the original by controlling for fixed baseline covariates and relying solely on data from cases. We develop the model and compare the results of PD-SCCS and SCCS on example drug-AE pairs.Statisticsses2155StatisticsDissertationsSome Nonparametric Methods for Clinical Trials and High Dimensional Data
http://academiccommons.columbia.edu/catalog/ac:174242
Wu, Xiaoruhttp://hdl.handle.net/10022/AC:P:10335Wed, 11 May 2011 13:30:20 +0000This dissertation addresses two problems from novel perspectives. In chapter 2, I propose an empirical likelihood based method to nonparametrically adjust for baseline covariates in randomized clinical trials and in chapter 3, I develop a survival analysis framework for multivariate K-sample problems. (I): Covariate adjustment is an important tool in the analysis of randomized clinical trials and observational studies. It can be used to increase efficiency and thus power, and to reduce possible bias. While most statistical tests in randomized clinical trials are nonparametric in nature, approaches for covariate adjustment typically rely on specific regression models, such as the linear model for a continuous outcome, the logistic regression model for a dichotomous outcome, and the Cox model for survival time. Several recent efforts have focused on model-free covariate adjustment. This thesis makes use of the empirical likelihood method and proposes a nonparametric approach to covariate adjustment. A major advantage of the new approach is that it automatically utilizes covariate information in an optimal way without fitting a nonparametric regression. The usual asymptotic properties, including the Wilks-type result of convergence to a chi-square distribution for the empirical likelihood ratio based test, and asymptotic normality for the corresponding maximum empirical likelihood estimator, are established. It is also shown that the resulting test is asymptotically most powerful and that the estimator for the treatment effect achieves the semiparametric efficiency bound. The new method is applied to the Global Use of Strategies to Open Occluded Coronary Arteries (GUSTO)-I trial. Extensive simulations are conducted, validating the theoretical findings. This work is not only useful for nonparametric covariate adjustment but also has theoretical value. It broadens the scope of the traditional empirical likelihood inference by allowing the number of constraints to grow with the sample size. (II): Motivated by applications in high-dimensional settings, I propose a novel approach to testing equality of two or more populations by constructing a class of intensity centered score processes. The resulting tests are analogous in spirit to the well-known class of weighted log-rank statistics that is widely used in survival analysis. The test statistics are nonparametric, computationally simple and applicable to high-dimensional data. We establish the usual large sample properties by showing that the underlying log-rank score process converges weakly to a Gaussian random field with zero mean under the null hypothesis, and with a drift under the contiguous alternatives. For the Kolmogorov-Smirnov-type and the Cramer-von Mises-type statistics, we also establish the consistency result for any fixed alternative. As a practical means to obtain approximate cutoff points for the test statistics, a simulation based resampling method is proposed, with theoretical justification given by establishing weak convergence for the randomly weighted log-rank score process. The new approach is applied to a study of brain activation measured by functional magnetic resonance imaging when performing two linguistic tasks and also to a prostate cancer DNA microarray data set.Statisticsxw2144StatisticsDissertationsContagion and Systemic Risk in Financial Networks
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Moussa, Amalhttp://hdl.handle.net/10022/AC:P:10249Fri, 29 Apr 2011 18:12:27 +0000The 2007-2009 financial crisis has shed light on the importance of contagion and systemic risk, and revealed the lack of adequate indicators for measuring and monitoring them. This dissertation addresses these issues and leads to several recommendations for the design of an improved assessment of systemic importance, improved rating methods for structured finance securities, and their use by investors and risk managers. Using a complete data set of all mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, we explore in chapter 2 the structure and dynamics of the Brazilian financial system. We show that the Brazilian financial system exhibits a complex network structure characterized by a strong degree of heterogeneity in connectivity and exposure sizes across institutions, which is qualitatively and quantitatively similar to the statistical features observed in other financial systems. We find that the Brazilian financial network is well represented by a directed scale-free network, rather than a small world network. Based on these observations, we propose a stochastic model for the structure of banking networks, representing them as a directed weighted scale free network with power law distributions for in-degree and out-degree of nodes, Pareto distribution for exposures. This model may then be used for simulation studies of contagion and systemic risk in networks. We propose in chapter 3 a quantitative methodology for assessing contagion and systemic risk in a network of interlinked institutions. We introduce the Contagion Index as a metric of the systemic importance of a single institution or a set of institutions, that combines the effects of both common market shocks to portfolios and contagion through counterparty exposures. Using a directed scale-free graph simulation of the financial system, we study the sensitivity of contagion to a change in aggregate network parameters: connectivity, concentration of exposures, heterogeneity in degree distribution and network size. More concentrated and more heterogeneous networks are found to be more resilient to contagion. The impact of connectivity is more controversial: in well-capitalized networks, increasing connectivity improves the resilience to contagion when the initial level of connectivity is high, but increases contagion when the initial level of connectivity is low. In undercapitalized networks, increasing connectivity tends to increase the severity of contagion. We also study the sensitivity of contagion to local measures of connectivity and concentration across counterparties --the counterparty susceptibility and local network frailty-- that are found to have a monotonically increasing relationship with the systemic risk of an institution. Requiring a minimum (aggregate) capital ratio is shown to reduce the systemic impact of defaults of large institutions; we show that the same effect may be achieved with less capital by imposing such capital requirements only on systemically important institutions and those exposed to them. In chapter 4, we apply this methodology to the study of the Brazilian financial system. Using the Contagion Index, we study the potential for default contagion and systemic risk in the Brazilian system and analyze the contribution of balance sheet size and network structure to systemic risk. Our study reveals that, aside from balance sheet size, the network-based local measures of connectivity and concentration of exposures across counterparties introduced in chapter 3, the counterparty susceptibility and local network frailty, contribute significantly to the systemic importance of an institution in the Brazilian network. Thus, imposing an upper bound on these variables could help reducing contagion. We examine the impact of various capital requirements on the extent of contagion in the Brazilian financial system, and show that targeted capital requirements achieve the same reduction in systemic risk with lower requirements in capital for financial institutions. The methodology we proposed in chapter 3 for estimating contagion and systemic risk requires visibility on the entire network structure. Reconstructing bilateral exposures from balance sheets data is then a question of interest in a financial system where bilateral exposures are not disclosed. We propose in chapter 5 two methods to derive a distribution of bilateral exposures matrices. The first method attempts to recover the balance sheet assets and liabilities "sample by sample". Each sample of the bilateral exposures matrix is solution of a relative entropy minimization problem subject to the balance sheet constraints. However, a solution to this problem does not always exist when dealing with sparse sample matrices. Thus, we propose a second method that attempts to recover the assets and liabilities "in the mean". This approach is the analogue of the Weighted Monte Carlo method introduced by Avellaneda et al. (2001). We first simulate independent samples of the bilateral exposures matrix from a relevant prior distribution on the network structure, then we compute posterior probabilities by maximizing the entropy under the constraints that the balance sheet assets and liabilities are recovered in the mean. We discuss the pros and cons of each approach and explain how it could be used to detect systemically important institutions in the financial system. The recent crisis has also raised many questions regarding the meaning of structured finance credit ratings issued by rating agencies and the methodology behind them. Chapter 6 aims at clarifying some misconceptions related to structured finance ratings and how they are commonly interpreted: we discuss the comparability of structured finance ratings with bond ratings, the interaction between the rating procedure and the tranching procedure and its consequences for the stability of structured finance ratings in time. These insights are illustrated in a factor model by simulating rating transitions for CDO tranches using a nested Monte Carlo method. In particular, we show that the downgrade risk of a CDO tranche can be quite different from a bond with same initial rating. Structured finance ratings follow path-dependent dynamics that cannot be adequately described, as usually done, by a matrix of transition probabilities. Therefore, a simple labeling via default probability or expected loss does not discriminate sufficiently their downgrade risk. We propose to supplement ratings with indicators of downgrade risk. To overcome some of the drawbacks of existing rating methods, we suggest a risk-based rating procedure for structured products. Finally, we formulate a series of recommendations regarding the use of credit ratings for CDOs and other structured credit instruments.Finance, Statisticsam2810Statistics, Industrial Engineering and Operations ResearchDissertationsStatistical methods for indirectly observed network data
http://academiccommons.columbia.edu/catalog/ac:131447
McCormick, Tyler H.http://hdl.handle.net/10022/AC:P:10239Fri, 29 Apr 2011 16:32:12 +0000Social networks have become an increasingly common framework for understanding and explaining social phenomena. Yet, despite an abundance of sophisticated models, social network research has yet to realize its full potential, in part because of the difficulty of collecting social network data. In many cases, particularly in the social sciences, collecting complete network data is logistically and financially challenging. In contrast, Aggregated Relational Data (ARD) measure network structure indirectly by asking respondents how many connections they have with members of a certain subpopulation (e.g. How many individuals with HIV/AIDS do you know?). These data require no special sampling procedure and are easily incorporated into existing surveys. This research develops a latent space model for ARD. This dissertation proposes statistical methods for methods for estimating social network and population characteristics using one type of social network data collected using standard surveys. First, a method to estimate both individual social network size (i.e., degree) and the distribution of network sizes in a population is prosed. A second method estimates the demographic characteristics of hard-to-reach groups, or latent demographic profiles. These groups, such as those with HIV/AIDS, unlawful immigrants, or the homeless, are often excluded from the sampling frame of standard social science surveys. A third method develops a latent space model for ARD. This method is similar in spirit to previous latent space models for networks (see Hoff, Raftery and Handcock (2002), for example) in that the dependence structure of the network is represented parsimoniously in a multidimensional geometric space. The key distinction from the complete network case is that instead of conditioning on the (latent) distance between two members of the network, the latent space model for ARD conditions on the expected distance between a survey respondent and the center of a subpopulation in the latent space. A spherical latent space facilitates tractable computation of this expectation. This model estimates relative homogeneity between groups in the population and variation in the propensity for interaction between respondents and group members.Statisticsthm2105StatisticsDissertationsWhy we (usually) don't have to worry about multiple comparison
http://academiccommons.columbia.edu/catalog/ac:129500
Gelman, Andrew E.; Hill, Jennifer; Yajima, Masanaohttp://hdl.handle.net/10022/AC:P:9795Wed, 12 Jan 2011 16:09:19 +0000Applied researchers often find themselves making statistical inferences in settings that would seem to require multiple comparisons adjustments. We challenge the Type I error paradigm that underlies these corrections. Moreover we posit that the problem of multiple comparisons can disappear entirely when viewed from a hierarchical Bayesian perspective. We propose building multilevel models in the settings where multiple comparisons arise. Multilevel models perform partial pooling (shifting estimates toward each other), whereas classical procedures typically keep the centers of intervals stationary, adjusting for multiple comparisons by making the intervals wider (or, equivalently, adjusting the p-values corresponding to intervals of fixed width). Thus, multilevel models address the multiple comparisons problem and also yield more efficient estimates, especially in settings with low group-level variation, which is where multiple comparisons are a particular concern.Statisticsag389Statistics, Political Science, Columbia Population Research CenterWorking papersRejoinder: Struggles with survey weighting and regression modeling
http://academiccommons.columbia.edu/catalog/ac:125312
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8573Wed, 17 Mar 2010 11:54:10 +0000I was motivated to write this paper, with its controversial opening line, "Survey weighting is a mess," from various experiences as an applied statistician.Statisticsag389Statistics, Political ScienceArticlesStruggles with survey weighting and regression modeling
http://academiccommons.columbia.edu/catalog/ac:125309
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8572Wed, 17 Mar 2010 11:48:20 +0000The general principles of Bayesian data analysis imply that models for survey responses should be constructed conditional on all variables that affect the probability of inclusion and nonresponse, which are also the variables used in survey weighting and clustering. However, such models can quickly become very complicated, with potentially thousands of poststratification cells. It is then a challenge to develop general families of multilevel probability models that yield reasonable Bayesian inferences. We discuss in the context of several ongoing public health and social surveys. This work is currently open-ended, and we conclude with thoughts on how research could proceed to solve these problems.Statisticsag389Statistics, Political ScienceArticlesBayes: Radical, liberal, or conservative?
http://academiccommons.columbia.edu/catalog/ac:125306
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8571Wed, 17 Mar 2010 11:38:51 +0000Statisticsag389Statistics, Political ScienceArticlesComment: Bayesian Checking of the Second Levels of Hierarchical Models
http://academiccommons.columbia.edu/catalog/ac:125303
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8570Wed, 17 Mar 2010 11:17:18 +0000Bayarri and Castellanos (BC) have written an interesting paper discussing two forms of posterior model check, one based on cross-validation and one based on replication of new groups in a hierarchical model. We think both these checks are good ideas and can become even more effective when understood in the context of posterior predictive checking. For the purpose of discussion, however, it is most interesting to focus on the areas where we disagree with BC.Statisticsag389Statistics, Political ScienceArticlesBayesian hierarchical classes analysis
http://academiccommons.columbia.edu/catalog/ac:125300
Leenen, Iwin; Mechelen, Iven van; Gelman, Andrew E.; Knop, Stijn dehttp://hdl.handle.net/10022/AC:P:8569Wed, 17 Mar 2010 10:47:09 +0000Hierarchical classes models are models for N-way N-mode data that represent the association among the N modes and simultaneously yield, for each mode, a hierarchical classification of its elements. In this paper we present a stochastic extension of the hierarchical classes model for two-way two-mode binary data. In line with the original model, the new probabilistic extension still represents both the association among the two modes and the hierarchical classifications. A fully Bayesian method for fitting the new model is presented and evaluated in a simulation study. Furthermore, we propose tools for model selection and model checking based on Bayes factors and posterior predictive checks. We illustrate the advantages of the new approach with applications in the domain of the psychology of choice and psychiatric diagnosis.Statisticsag389Statistics, Political ScienceArticlesRich state, poor state, red state, blue state: What's the matter with Connecticut?
http://academiccommons.columbia.edu/catalog/ac:125297
Gelman, Andrew E.; Shor, Boris; Bafumi, Joseph; Park, David K.http://hdl.handle.net/10022/AC:P:8568Wed, 17 Mar 2010 10:03:16 +0000For decades, the Democrats have been viewed as the party of the poor, with the Republicans representing the rich. Recent presidential elections, however, have shown a reverse pattern, with Democrats performing well in the richer blue states in the northeast and coasts, and Republicans dominating in the red states in the middle of the country and the south. Through multilevel modeling of individual-level survey data and county- and state-level demographic and electoral data, we reconcile these patterns. Furthermore, we find that income matters more in red America than in blue America. In poor states, rich people are much more likely than poor people to vote for the Republican presidential candidate, but in rich states (such as Connecticut), income has a very low correlation with vote preference.Political science, Statisticsag389Statistics, Political ScienceArticlesPredicting and dissecting the seats-votes curve in the 2006 U.S. House election
http://academiccommons.columbia.edu/catalog/ac:125294
Kastellec, Jonathan P. ; Gelman, Andrew E.; Chandler, Jamie P.http://hdl.handle.net/10022/AC:P:8567Mon, 15 Mar 2010 15:24:38 +0000The 2008 U.S. House elections mark the first time since 1994 that the Democrats will seek to retain a majority. With the political climate favoring Democrats this year, it seems almost certain that the party will retain control, and will likely increase its share of seats. In five national polls taken in June of this year, Democrats enjoyed on average a 13-point advantage in the generic congressional ballot; as Bafumi, Erikson, and Wlezien (2007) point out, these early polls, suitably adjusted, are good predictors of the November vote. As of late July, bettors at intrade.com put the probability of the Democrats retaining a majority at about 95% (Intrade.com 2008). Elsewhere in this symposium, Klarner (2008) predicts an 11-seat gain for the Democrats, while Lockerbie (2008) forecasts a 25-seat pickup. In this paper we document how the electoral playing field has shifted from a Republican advantage between 1996 and 2004 to a Democratic tilt today. In an earlier article (Kastellec, Gelman, and Chandler 2008), we predicted the seats-votes curve in the 2006 election, showing how the Democrats faced an uphill battle in their effort to take control of the House and, their victory notwithstanding, ended up winning a lower percentage of seats than their average district vote nationwide. We follow up on this analysis by using the same method to predict the seats-votes curve in 2008. Due to the shift in incumbency advantage from the Republicans to the Democrats, compounded by a greater number of retirements among Republican members, we show that the Democrats now enjoy a partisan bias, and can expect to win more seats than votes for the first time since 1992. While this bias is not as large as the advantage the Republicans held in 2006, it will likely help the Democrats increase their share of seats.Statistics, Political sciencejpk2004, ag389Political Science, StatisticsArticlesPartisans without constraint: Political polarization and trends in American public opinion
http://academiccommons.columbia.edu/catalog/ac:125291
Baldassarri, Delia; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8566Mon, 15 Mar 2010 15:03:27 +0000Public opinion polarization is here conceived as a process of alignment along multiple lines of potential disagreement and measured as growing constraint in individuals' preferences. Using NES data from 1972 to 2004, the authors model trends in issue partisanship--the correlation of issue attitudes with party identification--and issue alignment--the correlation between pairs of issues--and find a substantive increase in issue partisanship, but little evidence of issue alignment. The findings suggest that opinion changes correspond more to a resorting of party labels among voters than to greater constraint on issue attitudes: since parties are more polarized, they are now better at sorting individuals along ideological lines. Levels of constraint vary across population subgroups: strong partisans and wealthier and politically sophisticated voters have grown more coherent in their beliefs. The authors discuss the consequences of partisan realignment and group sorting on the political process and potential deviations from the classic pluralistic account of American politics.Political science, Statisticsag389Statistics, Political ScienceArticlesDiscussion of the Article "Website Morphing"
http://academiccommons.columbia.edu/catalog/ac:125288
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8565Mon, 15 Mar 2010 14:57:13 +0000The article under discussion illustrates the trade-off between optimization and exploration that is fundamental to statistical experimental design. In this discussion, I suggest that the research under discussion could be made even more effective by checking the fit of the model by comparing observed data to replicated data sets simulated from the fitted model.Statisticsag389Statistics, Political ScienceArticlesThe playing field shifts: Predicting the seats-votes curve in the 2008 U.S. House election
http://academiccommons.columbia.edu/catalog/ac:125285
Kastellec, Jonathan P.; Gelman, Andrew E.; Chandler, Jamie P.http://hdl.handle.net/10022/AC:P:8564Mon, 15 Mar 2010 14:19:19 +0000The 2008 U.S. House elections mark the first time since 1994 that the Democrats will seek to retain a majority. With the political climate favoring Democrats this year, it seems almost certain that the party will retain control, and will likely increase its share of seats. In five national polls taken in June of this year, Democrats enjoyed on average a 13-point advantage in the generic congressional ballot; as Bafumi, Erikson, and Wlezien (2007) point out, these early polls, suitably adjusted, are good predictors of the November vote. As of late July, bettors at intrade.com put the probability of the Democrats retaining a majority at about 95% (Intrade.com 2008). Elsewhere in this symposium, Klarner (2008) predicts an 11-seat gain for the Democrats, while Lockerbie (2008) forecasts a 25-seat pickup. In this paper we document how the electoral playing field has shifted from a Republican advantage between 1996 and 2004 to a Democratic tilt today. In an earlier article (Kastellec, Gelman, and Chandler 2008), we predicted the seats-votes curve in the 2006 election, showing how the Democrats faced an uphill battle in their effort to take control of the House and, their victory notwithstanding, ended up winning a lower percentage of seats than their average district vote nationwide. We follow up on this analysis by using the same method to predict the seats-votes curve in 2008. Due to the shift in incumbency advantage from the Republicans to the Democrats, compounded by a greater number of retirements among Republican members, we show that the Democrats now enjoy a partisan bias, and can expect to win more seats than votes for the first time since 1992. While this bias is not as large as the advantage the Republicans held in 2006, it will likely help the Democrats increase their share of seats.Political science, Statisticsjpk2004, ag389Political Science, StatisticsArticlesBayes, Jeffreys, Prior Distributions and the Philosophy of Statistics
http://academiccommons.columbia.edu/catalog/ac:125279
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8563Mon, 15 Mar 2010 14:07:09 +0000I actually own a copy of Harold Jeffreys's Theory of Probability but have only read small bits of it, most recently over a decade ago to confirm that, indeed, Jeffreys was not too proud to use a classical chi-squared p-value when he wanted to check the misfit of a model to data (Gelman, Meng and Stern, 2006). I do, however, feel that it is important to understand where our probability models come from, and I welcome the opportunity to use the present article by Robert, Chopin and Rousseau as a platform for further discussion of foundational issues. In this brief discussion I will argue the following: (1) in thinking about prior distributions, we should go beyond Jeffreys's principles and move toward weakly informative priors; (2) it is natural for those of us who work in social and computational sciences to favor complex models, contra Jeffreys's preference for simplicity; and (3) a key generalization of Jeffreys's ideas is to explicitly include model checking in the process of data analysis.Statisticsag389Statistics, Political ScienceArticlesProtecting minorities in binary elections: A test of storable votes using field data
http://academiccommons.columbia.edu/catalog/ac:125276
Casella, Alessandra M.; Ehrenberg, Shuky; Gelman, Andrew E.; Shen, Jiehttp://hdl.handle.net/10022/AC:P:8562Fri, 12 Mar 2010 16:31:53 +0000Democratic systems are built, with good reason, on majoritarian principles, but their legitimacy requires the protection of strongly held minority preferences. The challenge is to do so while treating every voter equally and preserving aggregate welfare. One possible solution is storable votes: granting each voter a budget of votes to cast as desired over multiple decisions. During the 2006 student elections at Columbia University, we tested a simple version of this idea: voters were asked to rank the importance of the different contests and to choose where to cast a single extra "bonus vote," had one been available. We used these responses to construct distributions of intensities and electoral outcomes, both without and with the bonus vote. Bootstrapping techniques provided estimates of the probable impact of the bonus vote. The bonus vote performs well: when minority preferences are particularly intense, the minority wins at least one of the contests with 15-30 percent probability; and, when the minority wins, aggregate welfare increases with 85-95 percent probability. When majority and minority preferences are equally intense, the effect of the bonus vote is smaller and more variable but on balance still positive.Political science, Statisticsac186, ag389Economics, Statistics, Political ScienceWorking papersWhy we (usually) don't have to worry about multiple comparisons
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Gelman, Andrew E.; Hill, Jennifer; Yajima, Masanaohttp://hdl.handle.net/10022/AC:P:8561Fri, 12 Mar 2010 16:28:31 +0000Statisticsag389Statistics, Political SciencePresentationsWhy we (usually) don't have to worry about multiple comparisons
http://academiccommons.columbia.edu/catalog/ac:125255
Gelman, Andrew E.; Hill, Jennifer; Yajima, Masanaohttp://hdl.handle.net/10022/AC:P:8560Fri, 12 Mar 2010 16:26:16 +0000Statisticsag389Statistics, Political SciencePresentationsSampling for Bayesian computation with large datasets
http://academiccommons.columbia.edu/catalog/ac:125252
Huang, Zaiying; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8559Fri, 12 Mar 2010 16:15:17 +0000Multilevel models are extremely useful in handling large hierarchical datasets. However, computation can be a challenge, both in storage and CPU time per iteration of Gibbs sampler or other Markov chain Monte Carlo algorithms. We propose a computational strategy based on sampling the data, computing separate posterior distributions based on each sample, and then combining these to get a consensus posterior inference. With hierarchical data structures, we perform cluster sampling into subsets with the same structures as the original data. This reduces the number of parameters as well as sample size for each separate model fit. We illustrate with examples from climate modeling and newspaper marketing.Statisticsag389Statistics, Political ScienceArticlesWhat does "Do campaigns matter?" mean?
http://academiccommons.columbia.edu/catalog/ac:125249
Bafumi, Joseph; Gelman, Andrew E.; Park, David K.http://hdl.handle.net/10022/AC:P:8558Fri, 12 Mar 2010 16:10:11 +0000Scholars disagree over the extent to which presidential campaigns activate predispositions in voters or create vote preferences that could not be predicted. When campaign related information flows activate predispositions, election results are largely predetermined given balanced resources. They can be accurately forecast well before a campaign has run its course. Alternatively, campaigns may change vote outcomes beyond forcing predispositions to some equilibrium level. We find most evidence for the former: opinion poll data are consistent with Presidential campaigns activating predispositions, with fundamental variables increasing in importance as a presidential election draws near.Political science, Statisticsjb878, ag389Political Science, StatisticsArticlesFully Bayesian computing
http://academiccommons.columbia.edu/catalog/ac:125246
Kerman, Jouni; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8557Fri, 12 Mar 2010 16:03:04 +0000A fully Bayesian computing environment calls for the possibility of defining vector and array objects that may contain both random and deterministic quantities, and syntax rules that allow treating these objects much like any variables or numeric arrays. Working within the statistical package R, we introduce a new object-oriented framework based on a new random variable data type that is implicitly represented by simulations. We seek to be able to manipulate random variables and posterior simulation objects conveniently and transparently and provide a basis for further development of methods and functions that can access these objects directly. We illustrate the use of this new programming environment with several examples of Bayesian computing, including posterior predictive checking and the manipulation of posterior simulations. This new environment is fully Bayesian in that the posterior simulations can be handled directly as random variables.Computer science, Statisticsag389Statistics, Political ScienceArticlesFitting Multilevel Models When Predictors and Group Effects Correlate
http://academiccommons.columbia.edu/catalog/ac:125243
Bafumi, Joseph; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8556Fri, 12 Mar 2010 15:57:14 +0000Random effects models (that is, regressions with varying intercepts that are modeled with error) are avoided by some social scientists because of potential issues with bias and uncertainty estimates. Particularly, when one or more predictors correlate with the group or unit effects, a key Gauss-Markov assumption is violated and estimates are compromised. However, this problem can easily be solved by including the average of each individual-level predictors in the group-level regression. We explain the solution, demonstrate its effectiveness using simulations, show how it can be applied in some commonly-used statistical software, and discuss its potential for substantive modeling.Statisticsag389Statistics, Political ScienceArticlesGoing beyond the book: Toward critical reading in statistics teaching
http://academiccommons.columbia.edu/catalog/ac:125240
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8555Fri, 12 Mar 2010 15:50:02 +0000We can improve our teaching of statistical examples from books by collecting further data, reading cited articles, and performing further data analysis. This should not come as a surprise, but what might be new is the realization of how close to the surface these research opportunities are: even influential and celebrated books can have examples where more can be learned with a small amount of additional effort. We discuss three examples that have arisen in our own teaching: an introductory textbook that motivated us to think more carefully about categorical and continuous variables; a book for the lay reader that misreported a study of menstruation and accidents; and a monograph on the foundations of probability that overinterpreted statistically insignificant fluctuations in sex ratios.Political science, Statisticsag389Statistics, Political ScienceOne vote, many Mexicos: Income and vote choice in the 1994, 2000, and 2006 presidential elections
http://academiccommons.columbia.edu/catalog/ac:125237
Cortina, Jeronimo; Gelman, Andrew E.; Lasala Blanco, Maria Narayanihttp://hdl.handle.net/10022/AC:P:8554Fri, 12 Mar 2010 15:40:10 +0000Using multilevel modeling of state-level economic data and individual-level exit poll data from the 1994, 2000 and 2006 Mexican presidential elections, we find that income has a stronger effect in predicting the vote for the conservative party in poorer states than in richer states -- a pattern that has also been found in recent U.S. elections. In addition (and unlike in the U.S.), richer states on average tend to support the conservative party at higher rates than poorer states. Our findings raise questions regarding the role that income polarization and region play in vote choice. The electoral results since 1994 reveal that collapsing multiple states into large regions entails significant loss of information that otherwise may uncover sharper and quiet revealing differences in voting patterns between rich and poor states as well as rich and poor individuals within states.Political science, Statisticsag389, ml2362Statistics, Political ScienceArticlesThoughts on new statistical procedures for age-period-cohort analyses
http://academiccommons.columbia.edu/catalog/ac:125234
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8553Fri, 12 Mar 2010 15:35:41 +0000Statisticsag389Statistics, Political ScienceArticlesWhat will we know on Tuesday at 7pm?
http://academiccommons.columbia.edu/catalog/ac:125231
Gelman, Andrew E.; Silver, Natehttp://hdl.handle.net/10022/AC:P:8552Fri, 12 Mar 2010 15:27:57 +0000Political science, Statisticsag389Statistics, Political ScienceArticlesBayesian Combination of State Polls and Election Forecasts
http://academiccommons.columbia.edu/catalog/ac:125228
Lock, Kari; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8551Fri, 12 Mar 2010 15:17:50 +0000A wide range of potentially useful data are available for election forecasting: the results of previous elections, a multitude of pre-election polls, and predictors such as measures of national and statewide economic performance. How accurate are different forecasts? We estimate predictive uncertainty via analysis of data collected from past elections (actual outcomes, pre-election polls, and model estimates). With these estimated uncertainties, we use Bayesian inference to integrate the various sources of data to form posterior distributions for the state and national two-party Democratic vote shares for the 2008 election. Our key idea is to separately forecast the national popular vote shares and the relative positions of the states. More generally, such an approach could be applied to study changes in public opinion and other phenomena with wide national swings and fairly stable spatial distributions relative to the national average.Political science, Statisticsag389Statistics, Political ScienceArticlesWhy we (usually) don't have to worry about multiple comparisons
http://academiccommons.columbia.edu/catalog/ac:125225
Gelman, Andrew E.; Hill, Jennifer; Yajima, Masanaohttp://hdl.handle.net/10022/AC:P:8550Fri, 12 Mar 2010 15:04:57 +0000Applied researchers often find themselves making statistical inferences in settings that would seem to require multiple comparisons adjustments. We challenge the Type I error paradigm that underlies these corrections. Moreover we posit that the problem of multiple comparisons can disappear entirely when viewed from a hierarchical Bayesian perspective. We propose building multilevel models in the settings where multiple comparisons arise. Multilevel models perform partial pooling (shifting estimates toward each other), whereas classical procedures typically keep the centers of intervals stationary, adjusting for multiple comparisons by making the intervals wider (or, equivalently, adjusting the p-values corresponding to intervals of fixed width). Thus, multilevel models address the multiple comparisons problem and also yield more efficient estimates, especially in settings with low group-level variation, which is where multiple comparisons are a particular concern.Statisticsag389Statistics, Political ScienceArticlesSocial and political polarization, and some other topics in network analysis
http://academiccommons.columbia.edu/catalog/ac:125159
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8542Thu, 11 Mar 2010 16:05:49 +0000Statisticsag389Statistics, Political SciencePresentationsImproving the Presentation of Quantitative Results in Political Science
http://academiccommons.columbia.edu/catalog/ac:125095
Kastellec, John; Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8534Thu, 11 Mar 2010 15:56:47 +0000Political science, Statisticsag389Statistics, Political SciencePresentationsCulture wars, voting, and polarization: divisions and unities in modern American politics
http://academiccommons.columbia.edu/catalog/ac:125089
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8532Thu, 11 Mar 2010 15:51:49 +0000Political science, Statisticsag389Statistics, Political SciencePresentationsSome computational and modeling issues for hierarchical models
http://academiccommons.columbia.edu/catalog/ac:125092
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8533Thu, 11 Mar 2010 15:50:14 +0000Statisticsag389Statistics, Political SciencePresentationsPosterior predictive checking and generalized graphical models
http://academiccommons.columbia.edu/catalog/ac:125156
Gelman, Andrew E.http://hdl.handle.net/10022/AC:P:8541Thu, 11 Mar 2010 15:42:35 +0000Statisticsag389Statistics, Political SciencePresentations