Uso de polinômios fracionários nos modelos mistos
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Data
2019-02-21
Autores
Garcia, Edijane Paredes
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
A classe dos modelos de regressão incorporando polinômios fracionários - FPs (Fractional Polynomials), proposta por Royston & Altman (1994), tem sido amplamente estudada. O uso de FPs em modelos mistos constitui uma alternativa muito atrativa para explicar a dependência das medidas intra-unidades amostrais em modelos em que há não linearidade na relação entre a variável resposta e variáveis regressoras contínua. Tal característica ocorre devido aos FPs oferecerem, para a resposta média, uma variedade de formas funcionais não lineares para as variáveis regressoras contínuas, em que se destacam a família dos polinômios convencionais e algumas curvas assimétricas e com assíntotas. A incorporação dos FPs na estrutura dos modelos mistos tem sido investigada por diversos autores. Porém, não existem publicações sobre: a exploração da problemática da modelagem na parte fixa e na parte aleatória (principalmente na presença de várias variáveis regressoras contínuas e categóricas); o estudo da influência dos FPs na estrutura dos efeitos aleatórios; a investigação de uma adequada estrutura para a matriz de covariâncias do erro; ou, um ponto de fundamental importância para colaborar com a seleção do modelo, a realização da análise de diagnóstico dos modelos ajustados. Uma contribuição, do nosso ponto de vista, de grande relevância é a investigação e oferecimento de estratégias de ajuste dos modelos polinômios fracionários com efeitos mistos englobando os pontos citados acima com o objetivo de preencher essas lacunas e de despertar aos usuários o imenso potencial dos modelos mistos, agora ainda mais ampliado, para a modelagem de dados correlacionados. Nesta tese propomos uma estratégia para a implementação do uso de FPs para modelar tendências não lineares em dados agrupados, tais como medidas repetidas, dados longitudinais e multiníveis. Dentro dessa proposta possibilitamos a inclusão de componentes de interação entre as transformações FP e variáveis categóricas por meio do uso de uma adaptação do algoritmo MFPI (Multivariable Fractional Polynomial Interaction), proposto por Royston & Sauerbrei (2004), realizamos a modelagem da matriz de covariâncias e utilizamos ferramentas gráficas para análise de diagnóstico (Singer et al., 2017) com o objetivo de verificar possíveis violações das suposições do modelo, bem como avaliamos o efeito de observações influentes nos resultados do ajuste. Concluímos com análises de três exemplos práticos em que ajustes sob os modelos mistos são comparados. No desenvolvimento dessa tese utilizamos a linguagem de programação R como suporte computacional.
The class of regression models incorporating Fractional Polynomials (FPs), proposed by Royston & Altman (1994), has been extensively studied. The use of FPs in mixed models is a very attractive alternative to explain the within-subjects’ measurements dependence in models where there is non-linearity in the relationship between the response variable and continuous covariates. This characteristic occurs because the FPs offers a variety of non-linear functional forms for the continuous covariates in the average response, in which the family of the conventional polynomials and some asymmetric curves with asymptotes stand out. The incorporation of FPs into the structure of the mixed models has been investigated by several authors. However, there are no works about the following issues: the modeling of the fixed and random effects (mainly in the presence of several continuous and categorical covariates), the study of the influence of the FPs on the structure of the random effects, the investigation of an adequate structure for the covariance of the random errors, or, a point that has central importance to the selection of the model, to perform a diagnostic analysis of the fitted models. In our point of view, a contribution of great relevance is the investigation and the proposition of strategies for fitting FPs with mixed effects encompassing the points mentioned above, with the goals of filling these gaps and to awaken the users to the great potential of mixed models, now even more extended, for the modeling of correlated data. In this thesis we propose a strategy for the use of FPs to model nonlinear trends in grouped data, such as repeated measurements, longitudinal and multilevel data. In this proposal, we allow the inclusion of interaction components between FP transformations and categorical variables through the use of an adaptation of the MFPI (Multivariable Fractional Polynomial Interaction) algorithm proposed by Royston & Sauerbrei (2004). We also model the covariance structure and use graphical tools for diagnostic analysis (Singer et al., 2017) in order to verify possible violations of the model assumptions, as well as to evaluate the effect of influential observations on the fitting. We conclude with an analysis of three practical examples in which fittings under the mixed models are compared. We use the R programming language as computational support.
The class of regression models incorporating Fractional Polynomials (FPs), proposed by Royston & Altman (1994), has been extensively studied. The use of FPs in mixed models is a very attractive alternative to explain the within-subjects’ measurements dependence in models where there is non-linearity in the relationship between the response variable and continuous covariates. This characteristic occurs because the FPs offers a variety of non-linear functional forms for the continuous covariates in the average response, in which the family of the conventional polynomials and some asymmetric curves with asymptotes stand out. The incorporation of FPs into the structure of the mixed models has been investigated by several authors. However, there are no works about the following issues: the modeling of the fixed and random effects (mainly in the presence of several continuous and categorical covariates), the study of the influence of the FPs on the structure of the random effects, the investigation of an adequate structure for the covariance of the random errors, or, a point that has central importance to the selection of the model, to perform a diagnostic analysis of the fitted models. In our point of view, a contribution of great relevance is the investigation and the proposition of strategies for fitting FPs with mixed effects encompassing the points mentioned above, with the goals of filling these gaps and to awaken the users to the great potential of mixed models, now even more extended, for the modeling of correlated data. In this thesis we propose a strategy for the use of FPs to model nonlinear trends in grouped data, such as repeated measurements, longitudinal and multilevel data. In this proposal, we allow the inclusion of interaction components between FP transformations and categorical variables through the use of an adaptation of the MFPI (Multivariable Fractional Polynomial Interaction) algorithm proposed by Royston & Sauerbrei (2004). We also model the covariance structure and use graphical tools for diagnostic analysis (Singer et al., 2017) in order to verify possible violations of the model assumptions, as well as to evaluate the effect of influential observations on the fitting. We conclude with an analysis of three practical examples in which fittings under the mixed models are compared. We use the R programming language as computational support.
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Palavras-chave
Curvatura, Dados longitudinais, Forma funcional, Modelo polinomial fracionário, Seleção de variáveis, Curvature, Functional form, Fractional polynomial model, Longitudinal data, Variable selection