Dissertationen zum Thema „Proportion inference“

Purpose of this study was to propose a simple and effective method for cognitive diagnosis assessment (CDA) without heavy computational demand using Bayesian inference for binomial proportion (BIBP). In real data studies, BIBP was applied to a test data using two different item designs: four and ten attributes. Also, the BIBP method was compared with DINA and LCDM in the diagnosis result using the same four-attribute data set. There were slight differences in the attribute mastery probability estimate among the three model (DINA, LCDM, BIBP), which could result in different attribute mastery pattern. In Simulation studies, it was found that the general accuracy of the BIBP method in the true parameter estimation was relatively high. The DINA estimation showed slightly higher overall correct classification rate but the bigger overall biases and estimation errors than the BIBP estimation. The three simulation variables (Attribute Correlation, Attribute Difficulty, and Sample Size) showed impacts on the parameter estimations of both models. However, they affected differently the two models: Harder attributes showed the higher accuracy of attribute mastery classification in the BIBP estimation while easier attributes was associated with the higher accuracy of the DINA estimation. In conclusion, BIBP appears an effective method for CDA with the advantage of easy and fast computation and a relatively high accuracy of parameter estimation.

In longitudinal studies, data collected within a subject or cluster are somewhat correlated by their very nature and special cares are needed to account for such correlation in the analysis of data. Under the framework of longitudinal studies, three topics are being discussed in this thesis. In chapter 2, the joint modelling of multivariate longitudinal process consisting of different types of outcomes are discussed. In the large cohort study of UK north Stafforshire osteoarthritis project, longitudinal trivariate outcomes of continuous, binary and ordinary data are observed at baseline, year 3 and year 6. Instead of analysing each process separately, joint modelling is proposed for the trivariate outcomes to account for the inherent association by introducing random effects and the covariance matrix G. The influence of covariance matrix G on statistical inference of fixed-effects parameters has been investigated within the Bayesian framework. The study shows that by joint modelling the multivariate longitudinal process, it can reduce the bias and provide with more reliable results than it does by modelling each process separately. Together with the longitudinal measurements taken intermittently, a counting process of events in time is often being observed as well during a longitudinal study. It is of interest to investigate the relationship between time to event and longitudinal process, on the other hand, measurements taken for the longitudinal process may be potentially truncated by the terminated events, such as death. Thus, it may be crucial to jointly model the survival and longitudinal data. It is popular to propose linear mixed-effects models for the longitudinal process of continuous outcomes and Cox regression model for survival data to characterize the relationship between time to event and longitudinal process, and some standard assumptions have been made. In chapter 3, we try to investigate the influence on statistical inference for survival data when the assumption of mutual independence on random error of linear mixed-effects models of longitudinal process has been violated. And the study is conducted by utilising conditional score estimation approach, which provides with robust estimators and shares computational advantage. Generalised sufficient statistic of random effects is proposed to account for the correlation remaining among the random error, which is characterized by the data-driven method of modified Cholesky decomposition. The simulation study shows that, by doing so, it can provide with nearly unbiased estimation and efficient statistical inference as well. In chapter 4, it is trying to account for both the current and past information of longitudinal process into the survival models of joint modelling. In the last 15 to 20 years, it has been popular or even standard to assume that longitudinal process affects the counting process of events in time only through the current value, which, however, is not necessary to be true all the time, as recognised by the investigators in more recent studies. An integral over the trajectory of longitudinal process, along with a weighted curve, is proposed to account for both the current and past information to improve inference and reduce the under estimation of effects of longitudinal process on the risk hazards. A plausible approach of statistical inference for the proposed models has been proposed in the chapter, along with real data analysis and simulation study.

La compréhension du comportement chimique et mécanique des matériaux compactés (par exemple sol, sous-sol, matériaux ouvragés) nécessite de se baser sur une description quantitative de structuration du matériau, et en particulier de la nature des différentes phases minéralogiques et de leur relation spatiale. Or, les matériaux naturels sont composés de nombreux minéraux de petite taille, fréquemment mixés à petite échelle. Les avancées récentes en tomographie de diffraction des rayons X sur source synchrotron (à différencier de la tomographie en contraste de phase) permettent maintenant d'obtenir des volumes tomographiques avec des voxels de taille nanométrique, avec un diffractogramme pour chacun de ces voxels (là où le contraste de phase ne donne qu'un niveau de gris). En contrepartie, le volume de données (typiquement de l'ordre de 100~000 diffractogrammes par tranche d'échantillon), associé au grand nombre de phases présentes, rend le traitement quantitatif virtuellement impossible sans codes numériques appropriés. Cette thèse vise à combler ce manque, en utilisant des approches de type réseaux de neurones pour identifier et quantifier des minéraux dans un matériau. L'entrainement de tels modèles nécessite la construction de bases d'apprentissage de grande taille, qui ne peuvent pas être constituées uniquement de données expérimentales. Des algorithmes capables de synthétiser des diffractogrammes pour générer ces bases ont donc été développés. L'originalité de ce travail a également porté sur l'inférence de proportions avec des réseaux de neurones.Pour répondre à cette tâche, nouvelle et complexe, des fonctions de perte adaptées ont été conçues. Le potentiel des réseaux de neurones a été testé sur des données de complexités croissantes : (i) à partir de diffractogrammes calculés à partir des informations cristallographiques, (ii) en utilisant des diffractogrammes expérimentaux de poudre mesurés au laboratoire, (iii) sur les données obtenues par tomographie de rayons X. Différentes architectures de réseaux de neurones ont aussi été testées. Si un réseau de neurones convolutifs semble apporter des résultats intéressants, la structure particulière du signal de diffraction (qui n'est pas invariant par translation) a conduit à l'utilisation de modèles comme les Transformers. L'approche adoptée dans cette thèse a démontré sa capacité à quantifier les phases minérales dans un solide. Pour les données les plus complexes, tomographie notamment, des pistes d'amélioration ont été proposées
Understanding the chemical and mechanical behavior of compacted materials (e.g. soil, subsoil, engineered materials) requires a quantitative description of the material's structure, and in particular the nature of the various mineralogical phases and their spatial relationships. Natural materials, however, are composed of numerous small-sized minerals, frequently mixed on a small scale. Recent advances in synchrotron-based X-ray diffraction tomography (to be distinguished from phase contrast tomography) now make it possible to obtain tomographic volumes with nanometer-sized voxels, with a XRD pattern for each of these voxels (where phase contrast only gives a gray level). On the other hand, the sheer volume of data (typically on the order of 100~000 XRD patterns per sample slice), combined with the large number of phases present, makes quantitative processing virtually impossible without appropriate numerical codes. This thesis aims to fill this gap, using neural network approaches to identify and quantify minerals in a material. Training such models requires the construction of large-scale learning bases, which cannot be made up of experimental data alone.Algorithms capable of synthesizing XRD patterns to generate these bases have therefore been developed.The originality of this work also concerned the inference of proportions using neural networks. To meet this new and complex task, adapted loss functions were designed.The potential of neural networks was tested on data of increasing complexity: (i) from XRD patterns calculated from crystallographic information, (ii) using experimental powder XRD patterns measured in the laboratory, (iii) on data obtained by X-ray tomography. Different neural network architectures were also tested. While a convolutional neural network seemed to provide interesting results, the particular structure of the diffraction signal (which is not translation invariant) led to the use of models such as Transformers. The approach adopted in this thesis has demonstrated its ability to quantify mineral phases in a solid. For more complex data, such as tomography, improvements have been proposed

Often, Bayesian analyses for epidemiological applications use objective prior distributions. These prior distributions are chosen with the goal of allowing the posterior distribution to be determined by the observed data alone. While this is achieved in most situations, it is not the case for Bayesian estimation of a small proportion. Such a situation might arise, for example, when estimating the prevalence of a rare disease. Several candidate objective prior distributions have been proposed for a Binomial proportion, including the Uniform distribution and Jeffrey's distribution. Each of these prior distributions may lead to very different posterior inferences when the number of events in the Binomial experiment is small, but it is unclear which of these would lead to better estimates on average. We explore this question by examining the frequentist performance of the posterior credible interval in two problems: i) estimating a single proportion, ii) estimating the difference between two proportions. The credible intervals obtained when using standard objective prior distributions as well as informative prior distributions motivated by real-life examples are compared. To assess frequentist performance, numerous statistics, including average coverage and average length of the posterior credible intervals were considered.
Souvent des analyses bayésiennes de données épidémiologiques utilisent les distributions à priori objectives. Ces distributions à priori sont sélectionnées de sorte que les distributions à posteriori soient déterminées uniquement par les données observées. Bien que cette méthode soit efficace dans plusieurs situations, elle ne l'est pas dans le cas de l'estimation bayésienne de petites proportions. Cette situation peut survenir, par exemple lors de l'estimation de la prévalence d'une maladie rare. Plusieurs distributions à priori objectives ont été proposées pour l'estimation d'une proportion, telle que, par exemple la distribution uniforme de Jeffrey. Chacune de ces distributions à priori peut conduire à de différentes distributions à posteriori lorsque le nombre d'événements dans l'expérience binomiale est petit. Mais il n'est pas clair laquelle de ces distributions, en moyenne, donne de meilleurs estimés. Nous explorons cette question en examinant la performance fréquentiste des intervalles crédibles à posteriori obtenus, respectivement, avec chacune de ces distributions à priori. Pour évaluer cette performance, nous considèrons des statistiques comme la couverture moyenne et la longueur moyenne des intervalles crédibles à posteriori. Nous considérons aussi des distributions à priori plus informatives comme les distributions uniformes définies sur un sous-intervalle de l'intervalle [0, 1]. La performance des distributions à priori est évaluée en utilisant des données simulées de situations où l'intérêt de recherche est concentré sur l'estimation d'une seule proportion ou sur la différence entre deux proportions.

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Existem diferentes modelos de regressão que podem ser usados para modelar taxas, proporções e outras variáveis respostas que assumem valores no intervalo unitário padrão, (0,1). Quando só uma classe de modelos de regressão é considerada, a seleção do modelos pode ser baseada nos testes de hipóteses usuais. O objetivo da presente dissertação é apresentar e avaliar numericamente os desempenhos em amostras imitas de testes que podem ser usados quando há dois ou mais modelos que são plausíveis, são não-encaixados e pertencem a classes de modelos de regressão distintas. Os modelos competidores podem diferir nos regressores que utilizam, nas funções de ligação e/ou na distribuição assumida para a variável resposta. Através de simulações de Monte Cario nós estimamos as taxas de rejeição nulas e não-nulas dos testes sob diversos cenários. Avaliamos também o desempenho de um procedimento de seleção de modelos. Os resultados mostram que os testes podem ser bastante úteis na escolha do melhor modelo de regressão quando a variável resposta assume valores no intervalo unitário padrão.
There are several different regression models that can be used with rates, proportions and other continuous responses that assume values in the standard unit interval, (0,1). When only one class of models is considered, model selection can be based on standard hypothesis testing inference. In this dissertation, we develop tests that can be used when the practitioner has at his/her disposal more than one plausible model, the competing models are nonnested and possibly belong to different classes of models. The competing models can differ in the regressors they use, in the link functions and even in the response distribution. The finite sample performances of the proposed tests are numerically eval-uated. We evaluate both the null and nonnull behavior of the tests using Monte Cario simulations. The results show that the tests can be quite useful for selecting the best regression model when the response assumes values in the standard unit interval.

The PolyQ model is a large stochastic kinetic model that describes protein aggregation within human cells as they undergo ageing. The presence of protein aggregates in cells is a known feature in many age-related diseases, such as Huntington's. Experimental data are available consisting of the proportions of cell death over time. This thesis is motivated by the need to make inference for the rate parameters of the PolyQ model. Ideally observations would be obtained on all chemical species, observed continuously in time. More realistically, it would be hoped that partial observations were available on the chemical species observed discretely in time. However, current experimental techniques only allow noisy observations on the proportions of cell death at a few discrete time points. This presents an ambitious inference problem. The model has a large state space and it is not possible to evaluate the data likelihood analytically. However, realisations from the model can be obtained using a stochastic simulator such as the Gillespie algorithm. The time evolution of a cell can be repeatedly simulated, giving an estimate of the proportion of cell death. Various MCMC schemes can be constructed targeting the posterior distribution of the rate parameters. Although evaluating the marginal likelihood is challenging, a pseudo-marginal approach can be used to replace the marginal likelihood with an easy to construct unbiased estimate. Another alternative which allows for the sampling error in the simulated proportions is also considered. Unfortunately, in practice, simulation from the model is too slow to be used in an MCMC inference scheme. A fast Gaussian process emulator is used to approximate the simulator. This emulator produces fully probabilistic predictions of the simulator output and can be embedded into inference schemes for the rate parameters. The methods developed are illustrated in two smaller models; the birth-death model and a medium sized model of mitochondrial DNA. Finally, inference on the large PolyQ model is considered.

Background. Clustering of observations occurs frequently in epidemiological and clinical studies of time-to-event outcomes. However, only a few papers addressed the challenge of accounting for clustering while analyzing right-censored survival data. I propose two bootstrap-based approaches to correct standard errors of Cox's proportional hazards (PH) model estimates for clustering, and validate the approaches in simulations.
Methods. Both bootstrap-based approaches involve 2 stages of resampling the original data. The two methods share the same procedure at the first stage but employ different procedures at the second stage. At the first stage of both methods, the clusters (e.g. physicians) are resampled with replacement. At the second stage, one method resamples individual patients with replacement for each physician (i.e. units within-cluster) selected at the 1st stage, while another method picks up all the patients for each selected physician, without resampling. For both methods, each of the resulting bootstrap samples is then independently analyzed with standard Cox's PH model, and the standard errors (SE) of the regression parameters are estimated as the empirical standard deviation, of the corresponding estimates. Finally, 95% confidence intervals (CI) for the estimates are estimated using bootstrap-based SE and assuming normality.
Simulations design. I have simulated a hypothetical study with N patients clustered within practices of M physicians. Individual patients' times-to-events were generated from the exponential distribution with hazard conditional on (i) several patient-level variables, (ii) several cluster-level (physician's) variables, and (iii) physician's "random effects". Random right censoring was applied. Simulated data were analyzed using 4 approaches: the proposed two bootstrap methods, standard Cox's PH model and "classic" one-step bootstrap with direct resampling of the patients.
Results. Standard Cox's model and "Classic" 1-step bootstrap under-estimated variance of regression coefficients, leading to serious inflation of type I error rates and coverage rates of 95% CI as low as 60-70%. In contrast, the proposed approach that resamples both physicians and patients-within-physicians, with the 100 bootstrap resamples, resulted in slightly conservative estimates of standard errors, which yielded type I error rates between 2% and 6%, and coverage rates between 94% and 99%.
Conclusions. The proposed bootstrap approach offers an easy-to-implement method to account for interdependence of times-to-events in the inference about Cox model regression parameters in the context of analyses of right-censored clustered data.

Orientador: Caio Lucidius Naberezny Azevedo
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Neste trabalho desenvolvemos a distribuição beta retangular aumentada em zero e um, bem como um correspondente modelo de regressão beta retangular aumentado em zero e um para analisar dados limitados-aumentados (representados por variáveis aleatórias mistas com suporte limitado), que apresentam valores discrepantes. Desenvolvemos ferramentas de inferência sob as abordagens bayesiana e frequentista. No que diz respeito à inferência bayesiana, devido à impossibilidade de obtenção analítica das posteriores de interesse, utilizou-se algoritmos MCMC. Com relação à estimação frequentista, utilizamos o algoritmo EM. Desenvolvemos técnicas de análise de resíduos, utilizando o resíduo quantil aleatorizado, tanto sob o enfoque frequentista quanto bayesiano. Desenvolvemos, também, medidas de influência, somente sob o enfoque bayesiano, utilizando a medida de Kullback Leibler. Além disso, adaptamos métodos de checagem preditiva à posteriori existentes na literatura, ao nosso modelo, utilizando medidas de discrepância apropriadas. Para a comparação de modelos, utilizamos os critérios usuais na literatura, como AIC, BIC e DIC. Realizamos diversos estudos de simulação, considerando algumas situações de interesse prático, com o intuito de comparar as estimativas bayesianas com as frequentistas, bem como avaliar o comportamento das ferramentas de diagnóstico desenvolvidas. Um conjunto de dados da área psicométrica foi analisado para ilustrar o potencial do ferramental desenvolvido
Abstract: In this work we developed the zero-one augmented rectangular beta distribution, as well as a correspondent zero-one augmented rectangular beta regression model to analyze limited-augmented data (represented by mixed random variables with limited support), which present outliers. We develop inference tools under the Bayesian and frequentist approaches. Regarding to the Bayesian inference, due the impossibility of obtaining analytically the posterior distributions of interest, we used MCMC algorithms. Concerning the frequentist estimation, we use the EM algorithm. We develop techniques of residual analysis, by using the randomized quantile residuals, under both frequentist and Bayesian approaches. We also developed influence measures, only under the Bayesian approach, by using the measure of Kullback Leibler. In addition, we adapt methods of posterior predictive checking available in the literature, to our model, using appropriate discrepancy measures. For model selection, we use the criteria commonly employed in the literature, such as AIC, BIC and DIC. We performed several simulation studies, considering some situations of practical interest, in order to compare the Bayesian and frequentist estimates, as well as to evaluate the behavior of the developed diagnostic tools. A psychometric real data set was analyzed to illustrate the performance of the developed tools
Mestrado
Estatistica
Mestra em Estatística

In this dissertation, we develop several new inference procedures that are based on randomized nomination sampling (RNS). The first problem we consider is that of constructing distribution-free confidence intervals for quantiles for finite populations. The required algorithms for computing coverage probabilities of the proposed confidence intervals are presented. The second problem we address is that of constructing nonparametric confidence intervals for infinite populations. We describe the procedures for constructing confidence intervals and compare the constructed confidence intervals in the RNS setting, both in perfect and imperfect ranking scenario, with their simple random sampling (SRS) counterparts. Recommendations for choosing the design parameters are made to achieve shorter confidence intervals than their SRS counterparts. The third problem we investigate is the construction of tolerance intervals using the RNS technique. We describe the procedures of constructing one- and two-sided RNS tolerance intervals and investigate the sample sizes required to achieve tolerance intervals which contain the determined proportions of the underlying population. We also investigate the efficiency of RNS-based tolerance intervals compared with their corresponding intervals based on SRS. A new method for estimating ranking error probabilities is proposed. The final problem we consider is that of parametric inference based on RNS. We introduce different data types associated with different situation that one might encounter using the RNS design and provide the maximum likelihood (ML) and the method of moments (MM) estimators of the parameters in two classes of distributions; proportional hazard rate (PHR) and proportional reverse hazard rate (PRHR) models.

Este trabalho apresenta as principais causas para a mortalidade das empresas paulistas criadas entre 2003 e 2007 a partir de base de dados cedida pelo SEBRAE-SP para o desenvolvimento dessa pesquisa. A amostra final, construída a partir de dados disponibilizados pela primeira vez para estudos desta natureza, contou com 662 empresas e 33 variáveis coletadas por meio de questionário aplicado diretamente às próprias empresas. A análise consistiu no teste de modelos econométricos, baseados na literatura dos modelos de duração, de forma a traduzir quais fatores são mais críticos para a sobrevivência das empresas a ponto de distingui-las em dois grupos: o das empresas vencedoras, cuja longevidade está pautada em ações que promovem ganhos de produtividade e eficiência, e aquelas desprovidas dessas ações e que muito provavelmente deixarão o mercado. Os três tipos de modelos abordados neste trabalho - não paramétrico, semi-paramétrico (riscos proporcionais) e paramétrico - apresentaram resultados similares, sendo que na abordagem de riscos proporcionais os resultados foram segmentados por tamanho e setor de atuação das empresas. Para as micro empresas, a idade do empreendedor e a iniciativa em investir na qualificação da mão de obra dos funcionários mostraram-se importantes mitigadores do risco de falha desse grupo de empresa, enquanto que para as pequenas empresas, a inovação em processos e a elaboração de um plano de negócios se destacaram dentre o conjunto de variáveis. Entre empresas dos setores de comércio e serviços, as empresas do primeiro grupo que faziam o acompanhamento das finanças (fluxo de caixa) apresentaram menor risco de falhar. Para aquelas do setor de serviços, a idade do empreendedor, o investimento em qualificação dos funcionários e o tamanho da empresa ao nascer foram importantes para reduzir o risco de falha no tempo. Outro resultado encontrado, por meio do modelo paramétrico utilizando distribuição Weibull, foi que o risco de a empresa deixar o mercado mostrou-se crescente, pelo menos nos cinco primeiros anos de existência da empresa. Entretanto, esse resultado não deve ser generalizado para períodos de tempo maiores que cinco anos.
This thesis presents the main results that determined the bankruptcy of enterprises located in the São Paulo State from 2003 to 2007. The models used in this work were possible due to the partnership with SEBRAE, Small Business Service Supporting, located in the State of São Paulo. This institution provided the data basis for this research and its final version was compound by 662 enterprises and 33 variables, which were collected from a survey done by SEBRAE and the related enterprise. For first time available for research like this The research was supported by econometrics models, more precisely duration models, which identified the most important factors regarding enterprises survival. Two enterprise groups were distinguished: that one that will survive and grow and another will fail. In this work, three models were used: parametric, non-parametric and proportional risk with all of them presenting similar results. The proportional risk approach was applied for economic sectors and enterprises size. For the micro size business, the entrepreneurship\'s age and the resources applied on the employee\'s qualification were important to reduce the risk to fail in the time, whereas for small enterprises, variables like innovation and business plan building were the most important variables. For the commerce and service sectors, the enterprises related to the first one, the enterprises which kept attention on financial results (cash flow) presented lower risk to fail. For service sector, variables such as: entrepreneur\'s age, investment on the employee\'s qualification and enterprise\'s size were the most important variables to explain the difference the risk to fail between the enterprises. Another result presented was the risk to fail, which indicates the likelihood of an enterprise to leave its business activity. In this case, the parametric model using Weibull distribution concluded that the risk grows in the first five years. However, this result must be carefully evaluated since it would be necessary a longer term data to ensure this result.

O problema de Inferência sobre uma proporção, amplamente divulgado na literatura estatística, ocupa papel central no desenvolvimento das várias teorias de Inferência Estatística e, invariavelmente, é objeto de investigação e discussão em estudos comparativos entre as diferentes escolas de Inferência. Ademais, a estimação de proporções, bem como teste de hipóteses para proporções, é de grande importância para as diversas áreas do conhecimento, constituindo um método quantitativo simples e universal. Nesse trabalho, é feito um estudo comparativo entre as abordagens clássica e bayesiana do problema de testar as hipóteses de ocorrência ou não de 2º turno em um cenário típico de eleição majoritária (maioria absoluta) em dois turnos no Brasil.
The problem of inference about a proportion, widely explored in the statistical literature, plays a key role in the development of several theories of statistical inference and, invariably, is the object of investigation and discussion in comparative studies among different schools of inference. In addition, the estimation of proportions, as well as test of hypothesis for proportions, is very important in many areas of knowledge as it constitutes a simple and universal quantitative method. In this work a comparative study between the Classical and Bayesian approaches to the problem of testing the hypothesis of occurrence of second round (or not) in a typical scenario of a majoritarian election (absolute majority) in two rounds in Brazil is developed.

Orientador: Víctor Hugo Lachos Dávila
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Dados no intervalo (0,1) geralmente representam proporções, taxas ou índices. Porém, é possível observar situações práticas onde as proporções sejam zero e/ou um, representando ausência ou presença total da característica de interesse. Nesses casos, os modelos que analisam o efeito de covariáveis, tais como a regressão beta, beta retangular e simplex não são convenientes. Com o intuito de abordar este tipo de situações, considera-se como alternativa aumentar os valores zero e/ou um ao suporte das distribuições previamente mencionadas. Nesta tese, são propostos modelos de regressão de efeitos mistos para dados de proporções aumentados de zeros e uns, os quais permitem analisar o efeito de covariáveis sobre a probabilidade de observar ausência ou presença total da característica de interesse, assim como avaliar modelos com respostas correlacionadas. A estimação dos parâmetros de interesse pode ser via máxima verossimilhança ou métodos Monte Carlo via Cadeias de Markov (MCMC). Nesta tese, será adotado o enfoque Bayesiano, o qual apresenta algumas vantagens em relação à inferência clássica, pois não depende da teoria assintótica e os códigos são de fácil implementação, através de softwares como openBUGS e winBUGS. Baseados na distribuição marginal, é possível calcular critérios de seleção de modelos e medidas Bayesianas de divergência q, utilizadas para detectar observações discrepantes
Abstract: Continuous data in the unit interval (0,1) represent, generally, proportions, rates or indices. However, zeros and/or ones values can be observed, representing absence or total presence of a carachteristic of interest. In that case, regression models that analyze the effect of covariates such as beta, beta rectangular or simplex are not appropiate. In order to deal with this type of situations, an alternative is to add the zero and/or one values to the support of these models. In this thesis and based on these models, we propose the mixed regression models for proportional data augmented by zero and one, which allow analyze the effect of covariates into the probabilities of observing absence or total presence of the interest characteristic, besides of being possivel to deal with correlated responses. Estimation of parameters can follow via maximum likelihood or through MCMC algorithms. We follow the Bayesian approach, which presents some advantages when it is compared with classical inference because it allows to estimate the parameters even in small size sample. In addition, in this approach, the implementation is straightforward and can be done using software as openBUGS or winBUGS. Based on the marginal likelihood it is possible to calculate selection model criteria as well as q-divergence measures used to detect outlier observations
Doutorado
Estatistica
Doutora em Estatística

On s'intéresse à la construction et l'estimation - à partir d'observations incomplètes - de modèles de variables aléatoires à valeurs réelles sur un graphe. Ces modèles doivent être adaptés à un problème de régression non standard où l'identité des variables observées (et donc celle des variables à prédire) varie d'une instance à l'autre. La nature du problème et des données disponibles nous conduit à modéliser le réseau sous la forme d'un champ markovien aléatoire, choix justifié par le principe de maximisation d'entropie de Jaynes. L'outil de prédiction choisi dans ces travaux est l'algorithme Belief Propagation - dans sa version classique ou gaussienne - dont la simplicité et l'efficacité permettent son utilisation sur des réseaux de grande taille. Après avoir fourni un nouveau résultat sur la stabilité locale des points fixes de l'algorithme, on étudie une approche fondée sur un modèle d'Ising latent où les dépendances entre variables réelles sont encodées à travers un réseau de variables binaires. Pour cela, on propose une définition de ces variables basée sur les fonctions de répartition des variables réelles associées. Pour l'étape de prédiction, il est nécessaire de modifier l'algorithme Belief Propagation pour imposer des contraintes de type bayésiennes sur les distributions marginales des variables binaires. L'estimation des paramètres du modèle peut aisément se faire à partir d'observations de paires. Cette approche est en fait une manière de résoudre le problème de régression en travaillant sur les quantiles. D'autre part, on propose un algorithme glouton d'estimation de la structure et des paramètres d'un champ markovien gaussien, basé sur l'algorithme Iterative Proportional Scaling. Cet algorithme produit à chaque itération un nouveau modèle dont la vraisemblance, ou une approximation de celle-ci dans le cas d'observations incomplètes, est supérieure à celle du modèle précédent. Cet algorithme fonctionnant par perturbation locale, il est possible d'imposer des contraintes spectrales assurant une meilleure compatibilité des modèles obtenus avec la version gaussienne de Belief Propagation. Les performances des différentes approches sont illustrées par des expérimentations numériques sur des données synthétiques.

When categorical data are misplaced into the wrong category, we say the data is affected by misclassification. This is common for data collection. It is well-known that naive estimators of category probabilities and coefficients for regression that ignore misclassification can be biased. In this dissertation, we develop methods to provide improved estimators and confidence intervals for a proportion when only a misclassified proxy is observed, and provide improved estimators and confidence intervals for regression coefficients when only misclassified covariates are observed. Following the introduction and literature review, we develop two estimators for a proportion, one which reduces the bias, and one with smaller mean square error. Then we will give two methods to find a confidence interval for a proportion, one using optimization techniques, and the other one using Fieller’s method. After that, we will focus on developing methods to find corrected estimators for coefficients of regression with misclassified covariates, with or without perfectly measured covariates, and with a known estimated misclassification/reclassification model. These correction methods use the score function approach, regression calibration and a mixture model. We also use Fieller’s method to find a confidence interval for the slope of simple regression with misclassified binary covariates. Finally, we use simulation to demonstrate the performance of our proposed methods.

碩士
逢甲大學
統計學系統計與精算碩士班
106
In general, most people are reluctant to provide answers to sensitive questions. In this situation, the information collected by the investigator are likely to be untruthful and mislead the findings of the research. Therefore, we should use the randomized response technique (RRT) enables to protect the respondents’ privacies and increases the possibility for collecting honest answers from respondents. In this study, we employ the randomized response (RR) design of Greenberg et al. (1969) for unrelated question combined with the RR technique idea developed by Liu et al. (2016). In Liu et al. (2016), the conditional posterior distribution of proportion of the sensitive feature used only part of the potential variable information. Their idea can yield estimates of proportions which are inefficient. Thus, we need to improve the conditional posterior distribution, and then considers to different groups. The Gibbs sampling method is used to in this study. A simulation study is conducted to investigate the performance of the proposed methods where three different prior distributions are used. Furthermore, a real data about the social changes in Taiwan conducted by the Central Research Institute of Human and Social Research Center in 2012 is used to estimate of the proportions of affair between age groups and residential places.

碩士
逢甲大學
統計學系統計與精算碩士班
107
In general, when questionnaires are used to collect the data on sensitive issues, most respondents tend to become hostile to the interviewers. The respondents are more likely to provide untruthful responses or refuse to respond. Subsequently, this makes the collect of data difficile and the analysis results meaningless. Fortunately, the use of a randomized response question can protect the privacy of the respondent and further motivate the respondent to provide truthful responses. In this regards, Xiao (2018) used Greenberg, et al. (1969) uncorrelated randomized response technique (RRT) to provide conditional posterior distribution under all potential variable information. He used the Gibbs sampling technique to provide the estimator of the proportion of sensitive attributes under different exogenous groups. Hsieh and Perri (2019) gave the Bayesian inference of the proportion of sensitive attributes by combining Gibbs sampling technique with the randomized response techniques proposed by Christofides (2003). In this paper, we extends the Bayesian inference of Xiao (2018) to the RRT proposed by Christofides (2003) and derive the Bayesian inference of the proportion of sensitive attributes under different exogenous groups. Moreover, we use the Bayesian method combined with Gibbs sampling to provide different models for the sensitive attributes under different exogenous groups. Simulation studies is conducted in order to evaluate the performance of the proposed method. In addition, we use the RRT of Christofides (2003) to collect the data of the Sexual EQ questionnaire conducted by the Student Affairs Office of Feng Chia University during the second semester of the academic year 2016. The respondents basic information and the online dating experiences were used as external variables to obtain the network one-night ratio estimation under different group combinations.