The generalized linear model (GLM) is a popular model in many research areas. In the GLM, each outcome of the dependent variable is assumed to be generated from a particular distribution function in the exponential family. The mean of the distribution depends on the independent variables. The link function provides the relationship between the linear predictor and the mean of the distribution function. In this dissertation, two semiparametric extensions of the GLM will be developed. In the first part of this dissertation, we have proposed a new model, called a semiparametric generalized linear model with a log-concave random component (SGLM-L). In this model, the estimate of the distribution of the random component has a nonparametric form while the estimate of the systematic part has a parametric form. In the second part of this dissertation, we have proposed a model, called a generalized semiparametric single-index mixed model (GSSIMM). A nonparametric component with a single index is incorporated into the mean function in the generalized linear mixed model (GLMM) assuming that the random component is following a parametric distribution. In the first part of this dissertation, since most of the literature on the GLM deals with the parametric random component, we relax the parametric distribution assumption for the random component of the GLM and impose a log-concave constraint on the distribution. An iterative numerical algorithm for computing the estimators in the SGLM-L is developed. We construct a log-likelihood ratio test for inference. In the second part of this dissertation, we use a single index model to generalize the GLMM to have a linear combination of covariates enter the model via a nonparametric mean function, because the linear model in the GLMM is not complex enough to capture the underlying relationship between the response and its associated covariates. The marginal likelihood is approximated using the Laplace method. A penalized quasi-likelihood approach is proposed to estimate the nonparametric function and parameters including single-index coe±cients in the GSSIMM. We estimate variance components using marginal quasi-likelihood. Asymptotic properties of the estimators are developed using a similar idea by Yu (2008). A simulation example is carried out to compare the performance of the GSSIMM with that of the GLMM. We demonstrate the advantage of my approach using a study of the association between daily air pollutants and daily mortality adjusted for temperature and wind speed in various counties of North Carolina. / Ph. D.

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Longitudinal data occur frequently in medical studies and covariates measured by error are typical features of such data. Generalized linear mixed models (GLMMs) are commonly used to analyse longitudinal data. It is typically assumed that the random effects covariance matrix is constant across the subject (and among subjects) in these models. In many situations, however, this correlation structure may differ among subjects and ignoring this heterogeneity can cause the biased estimates of model parameters. In this thesis, following Lee et al. (2012), we propose an approach to properly model the random effects covariance matrix based on covariates in the class of GLMMs where we also have covariates measured by error. The resulting parameters from this decomposition have a sensible interpretation and can easily be modelled without the concern of positive definiteness of the resulting estimator. The performance of the proposed approach is evaluated through simulation studies which show that the proposed method performs very well in terms biases and mean square errors as well as coverage rates. The proposed method is also analysed using a data from Manitoba Follow-up Study. / February 2017

Motivados por um conjunto de dados de pacientes com insuficiência renal crônica (IRC), propomos uma nova modelagem bayesiana que envolve cópulas da família Arquimediana e um modelo misto para dados de sobrevivência com riscos semicompetitivos. A estrutura de riscos semicompetitivos é bastante comum em estudos clínicos em que dois eventos são de interesse, um intermediário e outro terminal, de forma tal que a ocorrência do evento terminal impede a ocorrência do intermediário mas não vice-versa. Nesta modelagem provamos que a distribuição a posteriori sob a cópula de Clayton é própria. Implementamos os algoritmos de dados aumentados e amostrador de Gibbs para a inferência bayesiana, assim como os criterios de comparação de modelos: LPML, DIC e BIC. Realizamos um estudo de simulação para avaliar o desempenho da modelagem e finalmente aplicamos a metodologia proposta para analisar os dados dos pacientes com IRC, além de outros de pacientes que receberam transplante de medula óssea. / Motivated by a dataset of patients with chronic kidney disease (CKD), we propose a new bayesian model including the Arquimedean copula and a mixed model for survival data with semicompeting risks. The structure of semicompeting risks appears frequently in clinical studies where two-types of events are involved: a nonterminal and a terminal event such that the occurrence of terminal event precludes the occurrence of the non-terminal event but not viceversa. In this work we prove that the posterior distribution is proper when the Clayton copula is used. We implement the data augmentation algorithm and Gibbs sampling for the bayesian inference, as well as some bayesian model selection criteria: LPML, BIC and DIC. We carry out a simulation study for assess the model performance and finally, our methodology is illustrated with the chronic kidney disease study.

Os P-splines de Eilers e Marx (1996) são métodos de suavização que é uma combinação de bases B-splines e uma penalização discreta sobre os coeficientes das bases utilizados para suavizar dados normais e não normais em uma ou mais dimensões, no caso de várias dimensões utiliza-se como suavização o produto tensor dos P-splines. Também os P-splines são utilizados como representação de modelos mistos Currie et al. (2006) pela presença de características tais como: efeitos fixos, efeitos aleatórios, correlação espacial ou temporal e utilizados em modelos mais generalizados tais como os modelos mistos lineares generalizados e modelos mistos aditivos generalizados. Neste trabalho apresentou-se toda a abordagem, metodologia e descrição dos P-splines como modelos mistos e como componentes das estruturas suavizadoras de variáveis unidimensionais e bidimensionais dos modelos mistos aditivos generalizados, mostrando essa abordagem e propondo seu uso em uma aplicação no comportamento dos níveis médios da produção de cana-de-açúcar sob a influência das alterações das variáveis climáticas como temperatura e precipitação, que foram medidos ao longo de 10 anos em cada mesorregião do Estado de São Paulo. O motivo de usar essa abordagem como método de suavização é que muitas vezes não é conhecido a tendência dessas covariáveis climáticas mas sabe-se que elas influenciam diretamente sobre a variável resposta. Além de permitir essa abordagem inclusão de efeitos fixos e aleatórios nos modelos a serem propostos, permitirá a inclusão do processo autoregressivo AR(1) como estrutura de correlação nos resíduos. / P-splines of Eilers e Marx (1996) are methods of smoothing that is a combination of B-splines bases and penalty the coefficients of the bases used to smooth normal and non-normal data in one or more dimensions; in the case of several dimensions it is used as smoothing the tensor product of the P-splines. Also the P-splines are used as representation of mixed models Currie et al. (2006) by the presence of characteristics such as: fixed effects, random effects, spatial or temporal correlation and used in more generalized models such as generalized linear mixed models and generalized additive mixed models. In this work the whole approach, methodology and description of the P-splines as mixed models and as components of the smoothing structures of one-dimensional and two-dimensional variables of generalized additive mixed models were presented, showing this approach and proposing its application in the behavior of the average levels of sugarcane production, which is influenced by changes in climatic variables such as temperature and precipitation , which were measured over 10 years in each mesoregion of the state of São Paulo. The reason for using this approach as a smoothing method is that the tendency of these climate covariables is not know for the most part, but is known that they influence directly the response variable, besides allowing this approach to include fixed and random effects in the models to be proposed, will allow the inclusion of the autoregressive process AR(1) as a correlation structure in the residuos.

Atualmente, um dos problemas mais importantes da Genética é a identificação de genes associados com doenças complexas. Um delineamento adequado para esta finalidade corresponde à coleta de dados de famílias e plataformas de marcadores moleculares do tipo SNP (do inglês, Single Nucleotide Polimorphism). Estas plataformas representam pontos de referência estrategicamente dispostos ao longo do genoma dos indivíduos e são de alta dimensão. A análise destes dados traz desafios analíticos como o problema de múltiplos testes e a seleção de variáveis preditoras. Nesta tese, propõe-se um critério para discriminar as variáveis preditoras genéticas em efeitos devidos ao componente aleatório poligênico e ao componente residual, sob a estrutura de um modelo linear misto. Também, considerando que o efeito individual das variáveis preditoras é esperado ser pequeno, é sugerido um método para encontrar subconjuntos ordenados destas variáveis e estudar o seu efeito simultâneo sobre a variável resposta em estudo. Neste contexto, utiliza-se a teoria associada ao Gráfico da Variável Adicionada em modelos mistos. As propostas são validadas por meio de um estudo de simulação, o qual é baseado em estruturas de famílias envolvidas no Projeto ``Corações de Baependi\" (InCor/USP), cujo objetivo é identificar genes associados a fatores de risco cardiovascular na população brasileira. Para a implementação dos procedimentos, usa-se o programa R e na geração das variáveis preditoras genéticas adota-se o aplicativo SimPed. / Recently, one of the most important problems in genetics is the identification of genes associated with complex diseases. A useful design for this proposal corresponds to collect data from extended families and molecular markers platforms SNPs (Single Nucleotide polymorphism). These platforms represent points of reference strategically placed along the genome of the individuals and are high dimensional. Analysis of these data brings analytical challenges as the problem of multiple testing and selection of predictive variables. In this thesis, we propose a criterion for discriminating predictors of genetic effects due to random polygenic component and the residual component, under the framework of a linear mixed model. Also, considering that the individual effects of predictor variables is expected to be small, it is suggested a method for finding ordered subsets of these variables and study their simultaneous effect on the response variable under study. In this context, is used the theory of the added variable plot under a mixed model framework. The proposals are validated through a simulation study, which is based on structures of families involved in the Project `` Baependi Heart Study (FAPESP Process 2007/58150-7), whose objective is to identify genes associated with cardiovascular risk factors in the Brazilian population. This proposal is implemented by using the R statistical environment and for the simulation of genetic predictors is adopted the SimPed application.

Motivados por um conjunto de dados de pacientes com insuficiência renal crônica (IRC), propomos uma nova modelagem bayesiana que envolve cópulas da família Arquimediana e um modelo misto para dados de sobrevivência com riscos semicompetitivos. A estrutura de riscos semicompetitivos é bastante comum em estudos clínicos em que dois eventos são de interesse, um intermediário e outro terminal, de forma tal que a ocorrência do evento terminal impede a ocorrência do intermediário mas não vice-versa. Nesta modelagem provamos que a distribuição a posteriori sob a cópula de Clayton é própria. Implementamos os algoritmos de dados aumentados e amostrador de Gibbs para a inferência bayesiana, assim como os criterios de comparação de modelos: LPML, DIC e BIC. Realizamos um estudo de simulação para avaliar o desempenho da modelagem e finalmente aplicamos a metodologia proposta para analisar os dados dos pacientes com IRC, além de outros de pacientes que receberam transplante de medula óssea. / Motivated by a dataset of patients with chronic kidney disease (CKD), we propose a new bayesian model including the Arquimedean copula and a mixed model for survival data with semicompeting risks. The structure of semicompeting risks appears frequently in clinical studies where two-types of events are involved: a nonterminal and a terminal event such that the occurrence of terminal event precludes the occurrence of the non-terminal event but not viceversa. In this work we prove that the posterior distribution is proper when the Clayton copula is used. We implement the data augmentation algorithm and Gibbs sampling for the bayesian inference, as well as some bayesian model selection criteria: LPML, BIC and DIC. We carry out a simulation study for assess the model performance and finally, our methodology is illustrated with the chronic kidney disease study.

Atualmente, um dos problemas mais importantes da Genética é a identificação de genes associados com doenças complexas. Um delineamento adequado para esta finalidade corresponde à coleta de dados de famílias e plataformas de marcadores moleculares do tipo SNP (do inglês, Single Nucleotide Polimorphism). Estas plataformas representam pontos de referência estrategicamente dispostos ao longo do genoma dos indivíduos e são de alta dimensão. A análise destes dados traz desafios analíticos como o problema de múltiplos testes e a seleção de variáveis preditoras. Nesta tese, propõe-se um critério para discriminar as variáveis preditoras genéticas em efeitos devidos ao componente aleatório poligênico e ao componente residual, sob a estrutura de um modelo linear misto. Também, considerando que o efeito individual das variáveis preditoras é esperado ser pequeno, é sugerido um método para encontrar subconjuntos ordenados destas variáveis e estudar o seu efeito simultâneo sobre a variável resposta em estudo. Neste contexto, utiliza-se a teoria associada ao Gráfico da Variável Adicionada em modelos mistos. As propostas são validadas por meio de um estudo de simulação, o qual é baseado em estruturas de famílias envolvidas no Projeto ``Corações de Baependi\" (InCor/USP), cujo objetivo é identificar genes associados a fatores de risco cardiovascular na população brasileira. Para a implementação dos procedimentos, usa-se o programa R e na geração das variáveis preditoras genéticas adota-se o aplicativo SimPed. / Recently, one of the most important problems in genetics is the identification of genes associated with complex diseases. A useful design for this proposal corresponds to collect data from extended families and molecular markers platforms SNPs (Single Nucleotide polymorphism). These platforms represent points of reference strategically placed along the genome of the individuals and are high dimensional. Analysis of these data brings analytical challenges as the problem of multiple testing and selection of predictive variables. In this thesis, we propose a criterion for discriminating predictors of genetic effects due to random polygenic component and the residual component, under the framework of a linear mixed model. Also, considering that the individual effects of predictor variables is expected to be small, it is suggested a method for finding ordered subsets of these variables and study their simultaneous effect on the response variable under study. In this context, is used the theory of the added variable plot under a mixed model framework. The proposals are validated through a simulation study, which is based on structures of families involved in the Project `` Baependi Heart Study (FAPESP Process 2007/58150-7), whose objective is to identify genes associated with cardiovascular risk factors in the Brazilian population. This proposal is implemented by using the R statistical environment and for the simulation of genetic predictors is adopted the SimPed application.

Thesis (M. Sc. (Statistics) / Most social and health science data are longitudinal and additionally multilevel in nature, which means that response data are grouped by attributes of some cluster. Ignoring the differences and similarities generated by these clusters results to misleading estimates, hence motivating for a need to assess variance components (VCs) using multilevel models (MLMs) or generalised linear mixed models (GLMMs). This study has explored and fitted teenage pregnancy census data that were gathered from 2011 to 2015 by the Africa Centre at Kwa-Zulu Natal, South Africa. The exploration of these data revealed a two level pure hierarchy data structure of teenage pregnancy status for some years nested within female teenagers. To fit these data, the effects that census year (year) and three female characteristics (namely age (age), number of household membership (idhhms), number of children before observation year (nch) have on teenage pregnancy were examined. Model building of this work, firstly, fitted a logit gen eralised linear model (GLM) under the assumption that teenage pregnancy measurements are independent between females and secondly, fitted a GLMM or MLM of female random effect. A better fit GLMM indicated, for an additional year on year, a 0.203 decrease on the log odds of teenage pregnancy while GLM suggested a 0.21 decrease and 0.557 increase for each additional year on age and year, respectively. A GLM with only year effect uncovered a fixed estimate which is higher, by 0.04, than that of a better fit GLMM. The inconsistency in the effect of year was caused by a significant female cluster variance of approximately 0.35 that was used to compute the VCs. Given the effect of year, the VCs suggested that 9.5% of the differences in teenage pregnancy lies between females while 0.095 similarities (scale from 0 to 1) are for the same female. It was also revealed that year does not vary within females. Apart from the small differences between observed estimates of the fitted GLM and GLMM, this work produced evidence that accounting for cluster effect improves accuracy of estimates. Keywords: Multilevel Model, Generalised Linear Mixed Model, Variance Components, Hier archical Data Structure, Social Science Data, Teenage Pregnancy

In this thesis, sensitivity and error analysis of a differential rectification method were performed by using digital images taken by a frame camera onboard BILSAT and pushbroom scanner on ASTER. Three methods were implemented for Sensitivity and Uncertainty analysis: Monte Carlo, covariance analysis and FAST (Fourier Amplitude Sensitivity Test). A parameter estimation procedure was carried out on the basis of so called Mixed Model extended by some suitable additional regularization parameters to stabilize the solution for improper geometrical conditions of the imaging system. The effectiveness and accuracy of the differential rectification method were compared with other rectification methods and the results were analyzed. Furthermore the differential method is adapted to the pushbroom scanners and software which provides rectified images from raw satellite images was developed.