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zE7XJ{&-)%T^MoppcEzq>!y@F9uAvIRdO@JTDhqDBI9i->Si;g^)eUjdXm1@Fk$F)x h?-BeqPD6!c?P!&bL61X>%*D&YgG5U!qb7^=e*g)SoYMdR diff --git a/docs/cap05_consideracoes-finais.Rnw b/docs/cap05_consideracoes-finais.Rnw index 16acc4c..fefaab7 100644 --- a/docs/cap05_consideracoes-finais.Rnw +++ b/docs/cap05_consideracoes-finais.Rnw @@ -2,4 +2,95 @@ % CAPÃTULO 5 - CONSIDERAÇÕES FINAIS % ------------------------------------------------------------------------ -\lipsum[1-3] \ No newline at end of file +O objetivo nesse trabalho foi a exploração, extensão e aplicação da +distribuição COM-Poisson na análise de dados de contagem cujo atendemos +com a apresentação de seis aplicações dos modelos COM-Poisson à +conjuntos de dados reais que percorreram caracterÃsticas de +equidispersão, subdispersão, superdispersão, contagens altas, excesso de +zeros e medidas repetidas mostrou-se a flexibilidade do modelo +COM-Poisson quando comparado à s demais abordagens comuns para tais +caracterÃsticas. + +Nos quatro primeiro conjuntos de dados, em que modelou-se as contagens +via modelos de regressão de efeitos fixos, tivemos os resultados dos +modelos COM-Poisson equivalentes a abordagem semi-paramétrica via +quasi-verossimilhança, quanto a significância dos efeitos e predição com +bandas de confiança. Porém ressalta-se que na abordagem por +quasi-verossimilhança, com a especificação de apenas dois momentos, i) +não recupera-se a distribuição de probabilidades da variável em estudo, +ii) a informação a respeito da média é igual ou inferior a uma abordagem +totalmente parâmetrica e iii) extensões como a modelagem de excesso de +zeros e modelagem do parâmetro de dispersão não são imediatas. Nos casos +de superdispersão exploramos também os resultados dos modelos baseados +na distribuição Binomial Negativa e nessa abordagem temos o +inconveniente de somente a caracterÃstica de superdispersão ser +contemplada. Nos estudos de caso os modelos Binomial Negativo +proporcionaram resultados, com relação a significância dos efeitos, +equivalentes ao COM-Poisson e Quasi-Poisson. Porém em um dos estudos de +caso com acentuada superdispersão, os valores preditos pontuais e +intervalares nessa abordagem diferiram dos modelos COM-Poisson e +Quasi-Poisson, isso devido a forma da relação média e variância dessa +distribuição, figura \ref{fig:mv-binomneg}. + +Nas extensões propostas tivemos resultados satisfatórios do desempenho a +distribuição COM-Poisson. No caso da inclusão de um componente de +barreira para modelagem de excesso de zeros, cujo denominamos como +modelo Hurdle COM-Poisson, os resultados dos testes de razão de +verossimilhanças para testar a significância dos efeitos foram +equivalentes ao modelo Hurdle Binomial Negativo assim como as +estimativas pontuais dos valores preditos. Na aplicação do modelo Hurdle +COM-Poisson não foi possÃvel a obtenção dos erros padrão das estimativas +dos efeitos devido a problemas numéricos na determinação da matriz +hessiano. Para o caso extendido do modelo COM-Poisson em que acomodamos +efeitos aleatórios, destacamos os procedimentos computacionalmente +intensivos que são empregados no algoritmo de estimação. A aplicação se +deu a um experimento que apresentou contagens com um grau não +significativo de subdispersão. Nessa aplicação os modelos em competição +foram o Poisson e o COM-Poisson de efeitos mistos. Dessa aplicação todos +os resultados em questões inferenciais foram equivalentes em ambos os +modelos, com poder de teste maior para o modelo COM-Poisson. + +Nas aplicações, em geral, tivemos caracterÃsticas que permearam a todos +os modelos baseados na distribuição COM-Poisson. A primeira delas, e +talvez a mais difÃcil de se contornar, é a determinação da constante de +normalização, pois essa depende do parâmetro em que associamos a um +preditor linear assim temos que calcular $n$ constantes a cada iteração +do algortimo de estimação. Em casos de contagens altas e superdispersão +dessa constante é extremamente demorado. Outra caracterÃstica que se +manisfestou em todas as aplicações foi a não ortogonalidade entre os +parâmetros de regressão e o parâmetro adicional $\phi$, observada pelas +correlações calculadas a partir da matriz hessiano. O que torna as +inferências dependentes. Em pesquisas não relatadas nesse trabalho +verificamos que a reparametrização do parâmetro $\lambda$, adotando a +aproximação para média contorna essa caracterÃstica com o preço de se +ter uma distribuição aproximada. Nas aplicações exploramos também os +prfis de verossimilhança para o parâmetro $\phi$ da COM-Poisson e o +comportamento aproximadamente simétrico em todos casos induz que +aproximações quadráticas da verossimilhança podem ter desempenhos +satisfatórios. + +Em geral, destaca-se o modelo Poisson, largamente utilizado na +estatÃstica aplicada, como uma alternativa reestritiva devido a sua +suposição de equidispersão (relação média igual a variância), que leva a +resultados incorretos quando essa suposição não é atendida. Como +alternativa sugere-se o modelo COM-Poisson que se apresenta como uma +alternativa paramétrica tão flexÃvel quanto a abordagem por +quasi-verossimilhança e que apresenta a distribuição Poisson como caso +particular facilitando a condução de testes de razão de verossimilhanças +para verificar a necessidade deste modelo mais flexÃvel. Em favor do +modelo COM-Poisson apresentamos extensões para modelagem de excesso de +zeros e inclusão de efeitos efeitos aleatórios mostrando que extensões +nesse modelo são feitas de maneira simples. + +Dado o escopo do trabalho foram vários os tópicos levantados para +pesquisas futuras. Estudo de reparametrizações que tornem os parâmetros +$\lambda$ e $\nu$ ortogonais no modelo COM-Poisson podem ser de grande +valia, pois tornaram as inferências entre eles independentes. Para +acelerar o algoritmo de estimação aproximações da constante normalização +podem resultar em ajustes satisfatórios. Estudos de simulação para +verificar a robustez do modelo a má especificação da distribuição da +variável resposta. Implementação da modelagem de excesso de zeros via +mistura de distribuições. Inclusão de efeitos aleatórios dependentes no +modelo misto COM-Poisson. São algumas das muitas possibilidades para +pesquisa envolvendo dados de contagem subdispersos ou superdispersos +modelados com a distribuição COM-Poisson. -- GitLab