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z@SYO)CETr0?MR@a+OEKhsv80uRhI=Cs?G^~7^1?Nq|z}_!L3Q+>K(x&9s-N4Y*;7$ zaD5~$YVRb@8M`F8$K*lb(WqOiqqnC%n1Av=V<giq*6Eg>T(Uf7CdLM)T&k+B{%%|V DC-1v5 diff --git a/docs/cap02_revisao-de-literatura.Rnw b/docs/cap02_revisao-de-literatura.Rnw index 5908a11..0c9025c 100644 --- a/docs/cap02_revisao-de-literatura.Rnw +++ b/docs/cap02_revisao-de-literatura.Rnw @@ -951,3 +951,97 @@ direita no ponto $y=1$ para estimação desta probabilidade, como explicam \section{Modelos de efeitos aleatórios} \label{cap02:aleatorio} + +Nas seções anteriores exploramos modelos que flexibilizam algumas +suposições do modelo Poisson. Basicamente pertimindo casos não +equidispersos e modelando conjuntamente um processo gerador de zeros +extra. Contudo uma suposição dos modelos de regressão para dados de +contagem vistos até aqui é que as variáveis aleatória $Y_1, Y_2, \cdots, +Y_n$ são condicionalmente indenpendentes, dado o vetor de +covariáveis. Porém não são raras as situações em que essa suposição não +se mostra adequada. \citeonline{Ribeiro2012} cita alguns exemplos: + +\begin{itemize} + \item as observações podem ser correlacionadas no espaço, + \item as observações podem ser correlacionadas no tempo, + \item interações complexas podem ser necessárias para modelar o efeito + conjunto de algumas covariáveis, + \item heterogeneidade entre indivíduos ou unidades podem não ser + suficientemente descrita por covariáveis. +\end{itemize} + +Nessas situações pode-se estender a classe de modelos de regressão +com a adição de efeitos aleatórios que incorporam variáveis não +observáveis (latentes) ao modelo, permitindo assim acomodar uma +variabilidade, que pode ser ou não estruturada, não prescrita pelo +modelo. De forma geral a especificação dos modelos de efeitos aleatórios +segue uma especificação hierárquica + +\begin{equation} + \label{eqn:reg-misto} + \begin{split} + Y_{ij} \mid b_{i},& X_{ij} \sim \textrm{D}(\mu_{ij}, \phi) \\ + g(&\mu_{ij}) = X_{ij}\beta + Z_ib_i \\ + & b \sim \textrm{K}(\Theta_b) + \end{split} +\end{equation} + +\noindent +para $i = 1, 2, \cdots, m$ (grupos com efeitos aleatórios comuns) e $j = +1, 2, \cdots, n$ (observações) com D$(\mu_{ij}, \phi)$, uma distribuição +considerada para as variáveis resposta condicionalmente independentes, +$g(\mu_{ij})$ uma função de ligação conforme definada na teoria dos +MLG's, $X_{ij}$ e $Z_{i}$ as vetores conhecidos representando os efeitos +das covariáveis de interesse, $b_i$ uma quantidade aleatória provida de +uma distribuição K$(\Theta_b)$. Note que nesses modelos uma quantidade +aleatória é somada ao preditor linear, diferentemente dos modelos de +efeitos fixos e a partir desta quantidade é possível induzir um +comportamento correlato entre as observações. + +Como temos duas quantidades aleatórias no modelo, $Y \mid X$ e $b$, a +verossimilhança para um modelo de efeito aleatório é dada integrando-se +os efeitos aleatórios + +\begin{equation} + \label{eqn:loglik-misto} + \Ell(\beta, \phi, \Theta_b \mid \underline{y}) = \prod_{i=1}^m \int_{\R^q} + \left ( \prod_{j=1}^{n_i} f_D(y_{ij}, \mu, b_i)\right ) \cdot f_K(b + \mid \Theta_b) db_i +\end{equation} + +Perceba que na avaliação da verossimilhança é necessário o cálculo de +$m$ integrais de dimensão $q$. Para muitos casos essa integral não tem +forma analítica sendo necessários métodos númericos de aproximação, que +são discutidos na seção \ref{cap03:metodos}. E as estimativas de máxima +verossimilhança são + +$$ +\hat{\Theta} = (\hat{\beta}, \hat{\Theta_b}) = +\underset{(\beta,\,\Theta_b)}{\textrm{arg max }} \log(\Ell(\beta, \phi, +\Theta_b \mid \underline{y})) +$$ + +\noindent +note que no processo de estimação dos modelos de efeitos aleatórios, +métodos numéricos são intensivamente utilizados, pois a cada iteração do +algoritmo de maximização da log-verossimilhança $m$ integrais de +dimensão $q$ são aproximadas, ou seja, métodos de aproximação de +integrais são utilizados concomitantemente ao método de maximização. + +Em modelos de contagem de efeitos mistos é comum adotar como +distribuição para os efeitos aleatórios uma Normal $q$-variada com média +0 e matriz de variância e covariâncias $\Sigma$, ou seja, na +especificação \ref{eqn:reg-misto} K$(\Theta_b) = NMV_q(0, \Sigma)$. Para +estes casos os principais métodos de aproximação da integral tem +desempenhos melhores \cite{Bates2015}. + +Como mencionado anteriormente modelos de efeitos aleatórios são +candidatos a modelagem de dados superdispersos. Quando não há uma +estrutura de delineamento experimental ou observacional pode-se incluir +efeitos aleatórios a nível de observação (e então $m=n$, ou seja, os +vetores $Y$ e $b$ tem mesma dimensão). Casos particulares de modelos de +efeitos aleatórios, onde o efeito aleatório é adiciona a nível de +observação são o modelo Binomial Negativo e o \textit{Inverse Gaussian + Model}, em ambos os casos a integral, definida em +\ref{eqn:loglik-misto} tem solução analítica e consequentemente a +marginal em $Y$ forma fechada. -- GitLab