From e39fa27bb5fe99159801b99efdf1e54935684a23 Mon Sep 17 00:00:00 2001
From: Eduardo Junior <edujrrib@gmail.com>
Date: Wed, 15 Jun 2016 14:30:14 -0300
Subject: [PATCH] =?UTF-8?q?Adiciona=20interpreta=C3=A7=C3=A3o=20dos=20resu?=
 =?UTF-8?q?ltados=20modelos=20de=20regress=C3=A3o=20de=20efeito=20fixo=20n?=
 =?UTF-8?q?o=20cap04?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 docs/01-tcc.pdf                       | Bin 1232160 -> 1242779 bytes
 docs/cap03_materiais-e-metodos.Rnw    |   2 +-
 docs/cap04_resultados-e-discussao.Rnw | 722 +++++++++++++++++++++-----
 docs/compois.bib                      |  73 +--
 4 files changed, 642 insertions(+), 155 deletions(-)

diff --git a/docs/01-tcc.pdf b/docs/01-tcc.pdf
index d867ea20c1ad12d49056dd6ea654209b79844ca8..dc5f079139ea8545515850d502cfbb43624399bb 100644
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diff --git a/docs/cap03_materiais-e-metodos.Rnw b/docs/cap03_materiais-e-metodos.Rnw
index af642a4..2be7706 100644
--- a/docs/cap03_materiais-e-metodos.Rnw
+++ b/docs/cap03_materiais-e-metodos.Rnw
@@ -499,7 +499,7 @@ suppressWarnings({
 \subsubsection{Número de nematóides em raízes de feijoeiro}
 \label{sec:nematodes}
 
-<<data-nematodes, include=FALSE, echo=FALSE>>=
+<<data-nematodes, include=FALSE, cache=FALSE>>=
 
 data(nematodes, package = "tccPackage")
 
diff --git a/docs/cap04_resultados-e-discussao.Rnw b/docs/cap04_resultados-e-discussao.Rnw
index c2a4975..c3ea5af 100644
--- a/docs/cap04_resultados-e-discussao.Rnw
+++ b/docs/cap04_resultados-e-discussao.Rnw
@@ -7,6 +7,7 @@ ajustados aos dados apresentados na seção \ref{cap03:materiais-dados}.
 
 
 \section{Análise de dados de capulhos de algodão sob efeito de desfolha}
+\label{sec:analise-cottonBolls}
 
 <<ajuste-cottonBolls, include=FALSE, cache=TRUE>>=
 
@@ -57,23 +58,27 @@ Abaixo expressamos os cinco preditores considerados.
 \noindent
 Preditor 1: $g(\mu) = \beta_0$ \\
 Preditor 2: $g(\mu) = \beta_0 + \beta_1 \textrm{def}$ \\
-Preditor 3: $g(\mu) = \beta_0 + \beta_{1j} \textrm{def} + \beta_2
+Preditor 3: $g(\mu) = \beta_0 + \beta_{1} \textrm{def} + \beta_2
 \textrm{def}^2$ \\
-Preditor 3: $g(\mu) = \beta_0 + \beta_{1j} \textrm{def} + \beta_{2j}
+Preditor 4: $g(\mu) = \beta_0 + \beta_{1j} \textrm{def} + \beta_2
 \textrm{def}^2$ \\
+Preditor 5: $g(\mu) = \beta_0 + \beta_{1j} \textrm{def} + \beta_{2j}
+\textrm{def}^2$
 
 \noindent
-onde $j$ varia nos níveis de estágio fenológico da planta
-(\Sexpr{niveis.est}). A proposta desses preditores foi realizada de
-forma aninhada a fim de facilitar a condução de testes de hipóteses. O
-modelo 1 contêm apenas o intercepto, e é ajustado apenas como ponto de
-partida para verificar como modelos mais estruturados melhoram o
-ajuste. O modelo 2 apresenta apenas o efeito de desfolha de forma
-linear, o modelo 3 é o modelo 2 somado um efeito de segunda ordem. O
-modelo 4, apresenta o efeito de desfolha linear mudando de acordo com o
-estágio de crescimento, e por fim o modelo 5 diz que não somente o
-efeito de primeira ordem muda com o estágio de crescimento, mais também
-o efeito de segunda ordem.
+onde $j$ varia nos níveis de estágio fenológico da planta (1:
+vegetativo, 2: botão floral, 3: florecimento, 4: maça, 5: capulho) e
+$g(\mu)$ uma função de ligação entre o componente sistemático e o
+componente aleatório do modelo. A proposta desses preditores foi
+realizada de forma aninhada a fim de facilitar a condução de testes de
+hipóteses. O modelo 1 contêm apenas o intercepto, e é ajustado apenas
+como ponto de partida para verificar como modelos mais estruturados
+melhoram o ajuste. O modelo 2 apresenta apenas o efeito de desfolha de
+forma linear, o modelo 3 é o modelo 2 somado um efeito de segunda
+ordem. O modelo 4, apresenta o efeito de desfolha linear mudando de
+acordo com o estágio de crescimento, e por fim o modelo 5 diz que não
+somente o efeito de primeira ordem muda com o estágio de crescimento,
+mais também o efeito de segunda ordem.
 
 A seguir ajustamos os modelos Poisson e COM-Poisson como alternativas
 paramétricas à análise de dados e como alternativa semi-paramétrica a
@@ -92,27 +97,25 @@ anC <- myanova(m1C, m2C, m3C, m4C, m5C)
 anQ <- myanova(m1Q, m2Q, m3Q, m4Q, m5Q)
 
 ## Parâmetros de dispersão
-listQ <- list(m1Q, m2Q, m3Q, m4Q, m5Q)
-sigma <- sapply(listQ, function(x) summary(x)$dispersion)
+sigma <- quasitest(m1Q, m2Q, m3Q, m4Q, m5Q)
 phi <- cmptest(m1C, m2C, m3C, m4C, m5C)
 
 ## Tabelas com as medidas de ajuste
 tabP <- cbind(anP, matrix(nrow = nrow(anP), ncol = 2))
 tabC <- cbind(anC, phi)
-tabQ <- cbind(anQ, cbind(sigma, NA))
+tabQ <- cbind(anQ, sigma)
 
-rownames(tabP) <- paste("Preditor", 1:5)
-rownames(tabC) <- paste("Preditor", 1:5)
-rownames(tabQ) <- paste("Preditor", 1:5)
+etas <- rep(paste("Preditor", 1:5), 3)
 
 ## Empilhando as tabelas de ANODEV
 tab.ajuste <- rbind(tabP, tabC, tabQ)
+rownames(tab.ajuste) <- NULL
+tab.ajuste <- data.frame(etas, tab.ajuste)
 
 ## ##----------------------------------------------------------------------
 ## ## Copiar e colar o corpo do resultado na customização latex abaixo
-## digits <- c(1, 0, 3, 3, 3, 0, -3, 3, -3)
-## print(xtable(tabC, digits = digits), include.rownames = TRUE)
-
+## digits <- c(1, 0, 0, 2, 2, 2, 0, -2, 3, -2)
+## xtable(tab.ajuste, digits = digits)
 @
 
 \begin{table}[ht]
@@ -125,25 +128,25 @@ tab.ajuste <- rbind(tabP, tabC, tabQ)
   \toprule
  Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) &  &  \\
  \midrule
-  Preditor 1 & 1 & -279,933 & 561,866 &  &  &  &  &  \\
-  Preditor 2 & 2 & -272,001 & 548,001 & 15,864 & 1 & 6,805E-05 &  &  \\
-  Preditor 3 & 3 & -271,354 & 548,709 & 1,293 & 1 & 2,556E-01 &  &  \\
-  Preditor 4 & 7 & -258,674 & 531,348 & 25,360 & 4 & 4,258E-05 &  &  \\
-  Preditor 5 & 11 & -255,803 & 533,606 & 5,742 & 4 & 2,193E-01 &  &  \\[0.3cm]
+  Preditor 1 & 1 & -279,93 & 561,87 &  &  &  &  &  \\
+  Preditor 2 & 2 & -272,00 & 548,00 & 15,86 & 1 & 6,81E-05 &  &  \\
+  Preditor 3 & 3 & -271,35 & 548,71 & 1,29 & 1 & 2,56E-01 &  &  \\
+  Preditor 4 & 7 & -258,67 & 531,35 & 25,36 & 4 & 4,26E-05 &  &  \\
+  Preditor 5 & 11 & -255,80 & 533,61 & 5,74 & 4 & 2,19E-01 &  &  \\[0.3cm]
  COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ & P($>\rchi^2$) \\
   \midrule
-Preditor 1 & 2 & -272,479 & 548,959 &  &  &  & 0,551 & 1,130E-04 \\
-  Preditor 2 & 3 & -257,464 & 520,927 & 30,032 & 1 & 4,251E-08 & 0,794 & 6,966E-08 \\
-  Preditor 3 & 4 & -256,089 & 520,179 & 2,749 & 1 & 9,734E-02 & 0,816 & 3,287E-08 \\
-  Preditor 4 & 8 & -220,198 & 456,395 & 71,783 & 4 & 9,537E-15 & 1,392 & 1,751E-18 \\
-  Preditor 5 & 12 & -208,250 & 440,500 & 23,896 & 4 & 8,382E-05 & 1,585 & 1,804E-22 \\[0.3cm]
-  Quase-Poisson & np & deviance & AIC & F & diff np & P(>F) & $\hat{\sigma}^2$ & P($>F$) \\
+  Preditor 1 & 2 & -272,48 & 548,96 &  &  &  & 0,551 & 1,13E-04 \\
+  Preditor 2 & 3 & -257,46 & 520,93 & 30,03 & 1 & 4,25E-08 & 0,794 & 6,97E-08 \\
+  Preditor 3 & 4 & -256,09 & 520,18 & 2,75 & 1 & 9,73E-02 & 0,816 & 3,29E-08 \\
+  Preditor 4 & 8 & -220,20 & 456,40 & 71,78 & 4 & 9,54E-15 & 1,392 & 1,75E-18 \\
+  Preditor 5 & 12 & -208,25 & 440,50 & 23,90 & 4 & 8,38E-05 & 1,585 & 1,80E-22 \\[0.3cm]
+  Quase-Poisson & np & deviance & AIC & F & diff np & P($>F$) & $\hat{\sigma}^2$ & P($>\rchi^2$) \\
   \midrule
-  Preditor 1 & 1 & 75,514 &  &  &  &  & 0,567 &  \\
-  Preditor 2 & 2 & 59,650 &  & 65,820 & 1 & 6,343E-13 & 0,464 &  \\
-  Preditor 3 & 3 & 58,357 &  & 5,363 & 1 & 2,236E-02 & 0,460 &  \\
-  Preditor 4 & 7 & 32,997 &  & 26,305 & 4 & 1,846E-15 & 0,278 &  \\
-  Preditor 5 & 11 & 27,255 &  & 5,956 & 4 & 2,176E-04 & 0,241 &  \\
+  Preditor 1 & 1 & 75,51 &  &  &  &  & 0,567 & 3,66E-04 \\
+  Preditor 2 & 2 & 59,65 &  & 34,21 & 1 & 4,17E-08 & 0,464 & 5,13E-07 \\
+  Preditor 3 & 3 & 58,36 &  & 2,81 & 1 & 9,62E-02 & 0,460 & 3,66E-07 \\
+  Preditor 4 & 7 & 33,00 &  & 22,77 & 4 & 5,89E-14 & 0,278 & 9,15E-16 \\
+  Preditor 5 & 11 & 27,25 &  & 5,96 & 4 & 2,18E-04 & 0,241 & 3,57E-18 \\
  \bottomrule
 \end{tabular}
 \begin{tablenotes}
@@ -151,27 +154,110 @@ Preditor 1 & 2 & -272,479 & 548,959 &  &  &  & 0,551 & 1,130E-04 \\
 \item np, número de parâmetros, diff $\ell$, diferença entre
   log-verossimilhanças, F, estatística F baseada nas quasi-deviances,
   diff np, diferença entre o np. \\[0.1cm]
-\item Fonte: Elaborado pelo autor.
+\item Fonte: Elaborado pelo autor, adaptação de Zeviani et al. (2014,
+  Tabela 1).
 \end{tablenotes}
 \end{table}
 
-As estimativas dos parâmetros extras $phi$ e $sigma^2$ dos modelos
+As estimativas dos parâmetros extras $\phi$ e $\sigma^2$ dos modelos
 COM-Poisson e Quasi-Poisson respectivamente, também são apresentadas na
 \ref{tab:ajuste-cottonBolls} e indicam subdispersão ($\phi>0$ e
-$\sigma^2<1$). Note que mesmo quando não consideramos covariáveis,
-preditor 1, a hipótese de equidispersão foi rejeitada pelo modelo
-COM-Poisson.
-
-<<prof-cottonBolls, fig.height=4, fig.width=4, fig.show="hide", results="asis", fig.cap="Perfil de log-verossimilhança para o parâmetro extra da COM-Poisson">>=
+$\sigma^2<1$). Note que, mesmo quando não consideramos covariáveis,
+preditor 1, a hipótese de equidispersão foi rejeitada pelo modelos
+COM-Poisson e Quasi-Poisson. Isso se reflete nos níveis descritivos dos
+testes de razão de verossimilhaças realizados, em que o modelo Poisson,
+em discordância com os demais, não indicou significância do efeito
+quadrático por nível de desfolha, preditor 5, pois superestima a
+variabilidade do processo. Esses resultados estão de acordos com os
+apresentados por \citeonline{Zeviani2014}, onde um modelo
+\textit{Gamma-Count} foi ajustado, destaca-se a similaridade entre as
+medidas de ajuste dos modelos COM-Poisson e \textit{Gamma-Count}. Os
+valores das log-verossimilhanças maximizadas nos dois modelos difere
+somente nas casas decimais, para todos os preditores.
+
+<<prof-cottonBolls, fig.height=4, fig.width=4, fig.show="hide", results="asis", fig.cap="Perfil de log-verossimilhança para o parâmetro extra da COM-Poisson, estimado no modelo com o quinto preditor">>=
 
 myprof(prof.cottonBolls, par.settings = ps.sub)
 fonte.xy("Fonte: Elaborado pelo autor.")
 
 wrapfigure()
+@
+
+Na figura \ref{fig:prof-cottonBolls} apresentamos a avaliação do
+parâmetro $\phi$ do modelo COM-Poisson com efeito de desfolha artificial
+de primeira e segunda ordem para cada estágio fenológico, via
+verossimilhança perfilhada. O valor zero, que representa a não
+necessidade de um modelo COM-Poisson não é contemplado pelos intervalos
+de confiança de 90, 95 e até 99\%. A simetria do perfil de
+verossimilhança também é algo para se destacar, pois neste caso
+intervalos do tipo Wald (computacionalmente mais fáceis), via
+aproximação quadrática da verossimilhança, podem ser construídos, muito
+embora os construídos via perfil de log-verossimilhança sejam
+preferíveis. Em concordância com a figura, o teste de hipóteses via
+razão de verossimilhanças para $H_0: \phi = 0$, rejeitou a hipótese nula
+com um nível de significância muito próximo a zero, tabela
+\ref{tab:ajuste-cottonBolls}.
+
+<<coef-cottonBolls, include=FALSE>>=
+
+pnames <- c("phi", "beta0", paste0("beta1", 1:5),
+            paste0("beta2", 1:5))
+
+## Tabela com os coeficientes
+tab.coef <- coeftab(m5P, m5Q, m5C,
+                    rownames = paste0("$++", pnames, "$"))
+
+## ## Código latex para tabela com os coeficientes
+## print.xtable(xtable(tab.coef), include.rownames = TRUE)
 
 @
 
-\lipsum[1]
+\begin{table}[ht]
+\centering
+\caption{Estimativas dos parâmetros e razões entre as estimativa e erro
+  padrão para os três modelos em estudo}
+\label{tab:coef-cottonBolls}
+\begin{tabular}{lcccccc}
+  \toprule
+  & \multicolumn{2}{c}{Poisson} & \multicolumn{2}{c}{Quasi-Poisson} &  \multicolumn{2}{c}{COM-Poisson} \\
+  \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
+  Parâmetro  & Estimativa & Est/EP & Estimativa & Est/EP & Estimativa & Est/EP \\
+  \midrule
+  $\phi$ & -- & -- & 0,24 & -- & 1,58 & 12,42 \\
+  $\beta_{0}$ & 2,19 & 34,57 & 2,19 & 70,42 & 10,90 & 7,76 \\
+  $\beta_{11}$ & 0,44 & 0,85 & 0,44 & 1,73 & 2,02 & 1,77 \\
+  $\beta_{12}$ & 0,29 & 0,57 & 0,29 & 1,16 & 1,34 & 1,21 \\
+  $\beta_{13}$ & -1,24 & -2,06 & -1,24 & -4,19 & -5,75 & -3,89 \\
+  $\beta_{14}$ & 0,36 & 0,64 & 0,36 & 1,31 & 1,60 & 1,30 \\
+  $\beta_{15}$ & 0,01 & 0,02 & 0,01 & 0,04 & 0,04 & 0,03 \\
+  $\beta_{21}$ & -0,81 & -1,38 & -0,81 & -2,81 & -3,72 & -2,78 \\
+  $\beta_{22}$ & -0,49 & -0,86 & -0,49 & -1,75 & -2,26 & -1,80 \\
+  $\beta_{23}$ & 0,67 & 0,99 & 0,67 & 2,01 & 3,13 & 2,08 \\
+  $\beta_{24}$ & -1,31 & -1,95 & -1,31 & -3,97 & -5,89 & -3,66 \\
+  $\beta_{25}$ & -0,02 & -0,04 & -0,02 & -0,07 & -0,09 & -0,08 \\
+   \bottomrule
+\end{tabular}
+\begin{tablenotes}
+  \small
+\item Fonte: Elaborado pelo autor.
+\end{tablenotes}
+\end{table}
+
+As estimativas dos efeitos lineares e quadráticos de desfolha
+artificial, conforme notação do preditor 5, são apresentadas na tabela
+\ref{tab:coef-cottonBolls} para os modelos Poisson, Quasi-Poisson e
+COM-Poisson. As estimativas dos parâmetros para os modelos Poisson e
+Quasi-Poisson são idênticas, por construção \ref{cap02:poisson}, o que
+difere são as magnitudes dessas estimativas em comparação com seu erro
+padrão, que no caso Quasi-Poisson é corrigido pelo parâmetro
+$\sigma^2$. Considerando o modelo COM-Poisson as estimativas são
+notavelmente diferentes, pois o preditor linear é construído em
+$\lambda$, da expressão \ref{eqn:pmf-compoisson}, e este parâmetro não
+descreve, sozinho, a média da distribuição. Sendo assim as estimativas
+do COM-Poisson não podem ser comparadas com as demais
+estimativas. Contudo, a magnitude desses efeitos com relação ao efeito
+padrão sim e neste caso temos os modelos Quasi-Poisson e COM-Poisson
+levando as mesmas conclusões.
 
 <<corr-cottonBolls, fig.width=7, fig.height=7, fig.cap="Imagem da matriz de correlação entre os parâmetros do modelo COM-Poisson">>=
 
@@ -179,8 +265,6 @@ Vcov <- vcov(m5C)
 Corr <- cov2cor(Vcov)
 
 ## Usando expression não funciona! Estudar a função posteriormente
-pnames <- c("phi", "beta0", paste0("beta1", 1:5),
-            paste0("beta2", 1:5))
 dimnames(Corr) <- list(pnames, pnames)
 
 mycorrplot(Corr)
@@ -188,7 +272,16 @@ fonte("Fonte: Elaborado pelo autor.")
 
 @
 
-\lipsum[1]
+As covariâncias entre as estimativas dos parâmetros do modelo
+COM-Poisson são apresentadas, na escala da correlação, na figura
+\ref{fig:corr-cottonBolls}. Destaca-se nessa figura a forte correlação
+do parâmetro extra $\phi$ com os $\beta$'s da regressão. Embora seja uma
+representação empírica, observada a esse particular conjunto de dados,
+nota-se a não ortogonalidade na matriz de informação observada, o que
+implica que inferências sob o parâmetro $\phi$ influenciam as demais
+inferências a ser realizadas sobre os $\beta$'s. Esse comportamento dos
+modelos COM-Poisson é recorrente, como veremos também nos demais
+conjuntos de dados.
 
 <<pred-cottonBolls, fig.height=3.5, fig.width=7, fig.cap="Curva dos valores preditos com intervalo de confiança de (95\\%) como função do nível de desfolha e do estágio fenológico da planta.">>=
 
@@ -260,9 +353,28 @@ xyplot(ncap ~ des | est,
 
 @
 
-\lipsum[1]
+Essa característica de não ortogonalidade da matriz de informação
+observada teve de ser levada em consideração para cálculo dos valores
+preditos, uma vez que a informação sobre a incerteza das estimativas
+contida na matriz de variâncias e covariâncias não pôde ser
+marginalizada para os $\beta$'s, que efetivamente são utilizados para
+cálculo de $\hat{\lambda}_i$ e consequentemente $\hat{\mu}_i$. Portanto,
+para cálculo dos valores preditos utilizamos a matriz de variâncias e
+covariâncias condicionada a $\phi$ \cite[teorema 3.6,
+pág. 123]{Ferreira2011}. Essa é uma prática tomada também para cálculo
+dos valores preditos nos demais conjunto de dados.
+
+As médias com intervalos de confiança calculadas com os modelos
+COM-Poisson e Quasi-Poisson são praticamente idênticas, conforme pode
+ser visto na \ref{fig:pred-cottonBolls}. Contudo, ressaltamos que o
+modelo COM-Poisson é totalmente paramétrico e assim conseguimos
+recuperar a distribuição, calculando probabilidades o que não é possível
+com a formulação Quasi-Poisson. Ainda nota-se claramente que o modelo
+Poisson é inadequado a esse conjunto de dados e que inferências a partir
+deste seriam incorretas.
 
 \section{Análise de dados de capulhos de algodão sob efeito de Mosca-Branca}
+\label{sec:analise-cottonBolls2}
 
 <<ajuste-cottonBolls2, include=FALSE, cache=TRUE>>=
 
@@ -330,7 +442,16 @@ prof.nnos <- profile(m3C.nnos, which = "phi")
 
 @
 
-\lipsum[1]
+Nesse conjunto de dados também temos indícios de subdispersão para as
+três variáveis de interesse mensuradas no estudo, conforme apresentado
+na seção \ref{sec:cottonBolls2}. Para as três contagens procedeu-se com
+o ajuste dos modelos Poisson, Quasi-Poisson e COM-Poisson os preditores:
+
+\noindent
+Preditor 1: $g(\mu) = \beta_0$ \\
+Preditor 2: $g(\mu) = \beta_0 + \beta_1 \textrm{dexp}$ \\
+Preditor 3: $g(\mu) = \beta_0 + \beta_1 \textrm{dexp} + \beta_2
+\textrm{dexp}^2$
 
 <<loglik-cottonBolls2, include=FALSE>>=
 
@@ -369,6 +490,10 @@ tab.nnos <- data.frame(
 ## Empilhando
 tab.ajuste <- rbind(tab.ncapu, tab.nerep, tab.nnos)
 
+##-------------------------------------------
+## Estimativas e testes para o parametro phi
+phis <- cmptest(m3C.ncapu, m2C.nerep, m3C.nnos)
+
 ## ##----------------------------------------------------------------------
 ## ## Copiar e colar o corpo do resultado na customização latex abaixo
 ## digits <- c(1, 1, 0, 2, 2, -2, 2, 2, -2, 2, -2)
@@ -376,6 +501,12 @@ tab.ajuste <- rbind(tab.ncapu, tab.nerep, tab.nnos)
 
 @
 
+\noindent
+sendo \texttt{dexp} a variável aleatória dias de exposição à alta
+infestação de mosca-branca. Assim temos os preditores 1, 2, 3 que
+representam efeito nulo, linear e quadrático dos dias de exposição
+respectivamente.
+
 \begin{table}[ht]
 \centering
 \small
@@ -412,8 +543,41 @@ tab.ajuste <- rbind(tab.ncapu, tab.nerep, tab.nnos)
 \end{tablenotes}
 \end{table}
 
-
-\lipsum[1]
+Na tabela \ref{tab:ajuste-cottonBolls2} são exibidos as medidas de
+ajuste dos modelos para as três variáveis resposta. Em todos os casos o
+modelo COM-Poisson apresentou maiores log-verossimilhanças indicando um
+melhor ajuste, quando comparado ao Poisson, também indicado pelos os
+valores de AIC que ponderam a log-verossimilhança pelo número de
+parâmetros considerados no modelo. Para questões inferênciais novamente,
+temos um desacordo entre os modelos paramétricos. Pelos modelos Poisson
+não temos evidências para manutenção de nenhum efeito da variável número
+de dias sob infestação, em todos os casos, ao passo que no modelo
+COM-Poisson indicou efeito quadrático quando considerado o modelo para o
+número de nós da planta (nível descritivo de \Sexpr{1-anC.nnos[3,6]}) e
+o número de capulhos produzidos (nível descritivo de
+\Sexpr{1-anC.ncapu[3,6]}, na borda da região de significância, mas com
+uma diminuição do AIC em favor do efeito quadrático). Quando modelado o
+número de estruturas reprodutivas o modelo COM-Poisson também não
+indicou efeito quadrático, contudo o efeito linear de \texttt{dexp} pode
+ser discutido uma vez que a significância do TRV foi de
+\Sexpr{anC.ncapu[3,6]} e o AIC apresentou um pequeno aumento com relação
+ao modelo nulo. Consideramos nas demais inferências os preditores com
+efeitos linear, para o número de estruturas reprodutivas e quadrático,
+para o número de capulhos produzidos e número de nós da planta.
+
+A especificação do modelo via Quasi-Verossimilhança Poisson obteve
+níveis descritivos menos favoráveis a rejeição da hipótese nula que o
+modelo COM-Poisson. Contudo, para escolha de preditores as mesmas
+tendências apontadas pelo COM-Poisson foram seguidas.
+
+Para avaliação do parâmetro $\phi$ da COM-Poisson nos três modelos
+considerados, temos os intervalos de confiança construídos sob
+perfilhamento da verossimilhança na figura \ref{fig:prof-cottonBolls2}. Note
+que para nenhum dos modelos on intervalos de confiança de 90, 95 e 99\%
+de confiança contiveram o valor de $\phi = 0$. Os valores estimados dos
+parâmetros nos modelos para número de capulhos, número de estruturas
+reprodutivas e número de nós da planta foram de \Sexpr{phis[, 1]}
+respectivamente, indicando subdispersão.
 
 <<prof-cottonBolls2, fig.height=3.5, fig.width=7, fig.cap="Perfis de log-verossimilhança para o parâmetro extra da COM-Poisson nos modelos para número de capulhos produzidos (esquerda), número de estruturas reprodutivas (central) e número de nós (direira).">>=
 
@@ -459,7 +623,13 @@ xyplot(abs(z) ~ focal | param, data = da,
 
 @
 
-\lipsum[1]
+Na figura \ref{fig:corr-cottonBolls2a} temos a representação da matriz de
+covariância entre as estimativas dos modelos para número de capulhos, à
+esquerda e número de nós da plantas, à direita. A forte correlação entre
+o parâmetro extra $\phi$ e $\beta_0$ (principalmente) também foi
+observada no ajuste do modelo para esses conjuntos de dados. No modelo
+considerado para o número de estruturas reprodutivas, o mesmo
+comportamento é observado, \ref{fig:cor-cottonBolls2b}.
 
 <<corr-cottonBolls2a, fig.height=4, fig.width=8, fig.cap="Imagem da matriz de correlação entre os parâmetros do modelo COM-Poisson">>=
 
@@ -497,8 +667,13 @@ wrapfigure()
 
 @
 
-
-\lipsum[1]
+Finalmente calculando os valores preditos pelos modelos Poisson,
+COM-Poisson e Quasi-Poisson temos a representação gráfica na figura
+\ref{fig:pred-cottonBolls2} com intervalos de confiança para média com
+95\% de confiança. Assim como na análise realizada na seção
+\ref{sec:analise-cottonBolls} temos os valores preditos com bandas de
+confiança obtidos dos modelos COM-Poisson e Quasi-Poisson ajustados,
+praticamente idênticos levando as mesmas interpretações.
 
 <<pred-cottonBolls2, fig.height=3.5, fig.width=7, fig.cap="Curva dos valores preditos com intervalo de confiança de (95\\%) como função do nível de desfolha e do estágio fenológico da planta.">>=
 
@@ -634,10 +809,16 @@ fonte.xy("Fonte: Elaborado pelo autor.")
 
 @
 
-\lipsum[1]
+Com esse segundo exemplo de subdispersão, em que tivemos três
+contagens em um único experimento. Mostramos a flexibilidade do modelo
+COM-Poisson no que tange à característica de subdispersão, uma vez que
+seus resultados (predições pontuais e intervalares e testes de hipóteses
+para comparação de modelos) se equivalem a uma abordagem
+semi-paramétrica.
 
 \section{Análise de produção de soja sob efeito de umidade e adubação
   potássica}
+\label{sec:analise-soyaBeans}
 
 <<ajuste-soyaBeans, include=FALSE, cache=TRUE>>=
 
@@ -694,7 +875,47 @@ prof.ng <- profile(m2C.ng, which = "phi")
 
 @
 
-\lipsum[1]
+Nesse experimento apresentado em \ref{sec:soyaBeans}, também temos mais
+de uma variável de interesse em forma de contagem e pela descrição dos
+dados temos características relacionadas a dispersão da contagem
+distintas em ambas (equidispersão e superdispersão). Dos modelos
+apresentados no capítulo \ref{cap:modelos-para-dados-de-contagem}, o
+Poisson, COM-Poisson, Binomial-Negativo são as alternativas paramétricas
+avaliadas e o Quasi-Poisson é tomado como a alternativa
+semi-paramétrica. As variáveis de interesse números de grãos de soja e
+de vagens viáveis foram contabilizados por unidade experimental (vaso
+com duas plantas) e estão sob o efeito, controlado, de duas covariáveis,
+níveis de adubação potássica (\Sexpr{niveis.K}) e níveis de umidade do
+solo (\Sexpr{niveis.umid}), que consideramos na análise como fatores com
+5 e 3 níveis respectivamente. Ainda têm-se pela condução do experimento
+o efeito relacionado a blocagem realizada, foram cinco blocos utilizados
+para controle de variação local. Os preditores considerados são
+
+\noindent
+Preditor 1: $\eta_1 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k $ \\
+Preditor 1: $\eta_2 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k +
+\alpha_{jk}$
+
+\noindent
+em que $\tau_i$ é o efeito do i-ésimo bloco, $i=$1: bloco II, 2: bloco
+III, 3: bloco IV e 4: V; $\gamma_j$ o efeito do j-ésimo nível de umidade
+aplicado, $j=$1: 50\% e 2: 62,5\%; $\delta_k$ o efeito do k-ésimo nível
+de adubação potássica, $k=$ 30, 60, 120 e 180 mg dm$^{-3}$ e
+$\alpha_{jk}$ o efeito da interação entre o j-ésimo nível de umidade do
+solo e o k-ésimo nível de adubação potássica. Assim no modelo mais
+completo, com interação, teremos 19 parâmetros de locação a serem
+estimados.
+
+Para ajuste dos modelos COM-Poisson nesse exemplo tivemos um tempo
+computacional ligeiramente mais demorado (em torno de 10s para os quatro
+modelos considerando as duas contagens e os dois preditores). Isso se
+deve ao fato das contagens serem altas (variando entre
+\Sexpr{paste(range(soyaBeans[ , "ngra"]), collapse=" e ")} para o número
+de grãos e \Sexpr{paste(range(soyaBeans[ , "nvag"]), collapse=" e ")}
+para o número de vagens) e estarmos em um caso superdisperso
+($\phi<0$). Nesse cenário a constante normalizada $Z(\lambda_i, \nu =
+\exp(\phi))$, expressão \ref{eqn:constante-z}, converge para 0 mais
+lentamente.
 
 <<loglik-soyaBeans, include=FALSE>>=
 
@@ -751,6 +972,30 @@ tab.ng <- cbind(tab.ng, dispersions.ng)
 
 @
 
+Medidas de qualidade de ajuste calculadas sob os modelos Poisson,
+COM-Poisson, Binomial Negativo e Quasi-Poisson são apresentadas na
+tabela \ref{tab:ajuste-soyaBeans}. Considerando a variável resposta
+número de vagens viáveis, percebe-se que há indícios de equidispersão
+indicados i) pelos parâmetros extras dos modelos alternativos ao
+Poisson, em que estimativas $\hat{\phi}$, $\hat{\theta}$ e
+$\hat{\sigma^2}$ estão próximas dos valores 0, $\infty$ e 1 que
+compreendem o caso particular Poisson nos modelos COM-Poisson, Binomial
+Negativo e Quasi-Poisson, ii) pelas log-verossimilhanças dos modelos
+paramétricos que resultaram em valores muito próximos, iii) pelos
+valores de AIC que foram menores nos modelos Poisson, mostrando que não
+há ganho expressivo quando estimados os parâmetros extra dos modelos
+alternativos. Os \textit{p-valores} associados ao TRV entre os modelos
+COM-Poisson e Poisson com preditores 1 e 2 foram de
+\Sexpr{cmptest(m1C.nv, m2C.nv)[, 2]}, evidenciando a equidispersão dos
+dados. Na figura \ref{fig:prof-soyaBeans} à esquerda apresentamos os
+intervalos de confiança baseados no perfil de verossimilhança para
+$\phi$ no modelo COM-Poisson com efeito de interação, como esses
+intervalos exibidos contemplam o valor 0 o modelo COM-Poisson pode ser
+reduzido ao Poisson. Para avaliação dos preditores, temos novamente um
+caso de valores na borda de significância. Seguimos as análises
+permanecendo com o modelo mais completo que considera a interação entre
+adubação e umidade.
+
 \begin{table}[ht]
 \centering
 \small
@@ -761,19 +1006,19 @@ tab.ng <- cbind(tab.ng, dispersions.ng)
   \toprule
   & & \multicolumn{4}{c}{{\bfseries Número de vagens}} & \multicolumn{4}{c}{{\bfseries Número de grãos}} \\
   \cmidrule{3-6} \cmidrule{7-10}
-{\footnotesize Poisson} & np & $\ell$ & AIC & P($>\rchi^2$) &  & $\ell$ & AIC & P($>\rchi^2$) & \\
+{PO} & np & $\ell$ & AIC & P($>\rchi^2$) &  & $\ell$ & AIC & P($>\rchi^2$) & \\
   \midrule
   $\eta_1$ & 11 & -266,69 & 555,38 &  &  & -343,16 & 708,33 &  &  \\
   $\eta_2$ & 19 & -259,62 & 557,23 & 7,79E-02 &  & -321,67 & 681,34 & 8,83E-07 &  \\[0.3cm]
-{\footnotesize COM-Poisson} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ \\
+{CP} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ \\
   \midrule
   $\eta_1$ & 12 & -266,60 & 557,20 &  & -6,75E-02 & -326,61 & 677,21 &  & -8,17E-01 \\
   $\eta_2$ & 20 & -259,33 & 558,65 & 6,85E-02 & 1,29E-01 & -315,64 & 671,29 & 5,06E-03 & -5,18E-01 \\[0.3cm]
-{\footnotesize Binomial-Neg.} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ \\
+{BN} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\theta}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\theta}$ \\
   \midrule
   $\eta_1$ & 12 & -266,69 & 557,37 &  & 4,59E+03 & -326,54 & 677,07 &  & 1,42E+02 \\
   $\eta_2$ & 20 & -259,62 & 559,23 & 7,82E-02 & 1,03E+06 & -315,39 & 670,77 & 4,39E-03 & 2,61E+02 \\[0.3cm]
-{\footnotesize Quasi-Poisson} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\phi}$ \\
+{QP} & np & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\sigma^2}$ & $\ell$ & AIC & P($>\rchi^2$) & $\hat{\sigma^2}$ \\
   \midrule
   $\eta_1$ & 11 & 79,43 &  &  & 1,28E+00 & 167,71 &  &  & 2,71E+00 \\
   $\eta_2$ & 19 & 65,28 &  & 1,87E-01 & 1,20E+00 & 124,72 &  & 3,00E-02 & 2,29E+00 \\
@@ -781,12 +1026,23 @@ tab.ng <- cbind(tab.ng, dispersions.ng)
 \end{tabular}
 \begin{tablenotes}
   \footnotesize
-\item np, número de parâmetros.
+\item np, número de parâmetros, PO, Poisson, CP, COM-Poisson, BN,
+  Binomial Negativo, QP, Quasi-Poisson.
 \item Fonte: Elaborado pelo autor.
 \end{tablenotes}
 \end{table}
 
-\lipsum[1]
+No fragmento direito da tabela \ref{tab:ajuste-soyaBeans} são
+apresentados os resultados para os modelos que ajustam o número de grãos
+por parcela. Neste caso temos evidências de superdispersão, pois as
+estimativas dos parâmetros $\phi$ e $\sigma^2$ foram menores que zero e
+maiores que 1 respectivamente. Os valores de AIC se apresentam menores e
+as avaliações da log-verossimilhança no ponto máximo maiores para os
+modelos paramétricos alternativos ao Poisson. Ainda a evidência sobre o
+efeito de interação para essa variável resposta é mais contundente. Na
+\ref{fig:prof-soyaBeans} à direira temos a verossimilhança perfilhada
+com indicação dos intervalos de confiança para $\phi$ e estes não
+contemplam o zero.
 
 <<prof-soyaBeans, fig.height=3.5, fig.width=7, fig.cap="Perfis de log-verossimilhança para o parâmetro extra da COM-Poisson nos modelos para número de vagens viáveis por parcela (esquerda) e número grãos de soja por parcela (direira).">>=
 
@@ -800,6 +1056,12 @@ fonte.xy("Fonte: Elaborado pelo autor.")
 
 @
 
+A visualização da covariâncias entre as estimativas dos parâmetros no
+modelo COM-Poisson para o número de vagens por parcela é feita na figura
+\ref{corr-soyaBeansa} e para o número de grãos por parcela na figura
+\ref{fig:corr-soyaBeansa}. Em ambos os casos a correlação entre os
+parâmetros de locação ($\beta$'s) e dispersão ($\phi$) ganha destaque.
+
 <<corr-soyaBeansa, fig.height=7, fig.width=7, fig.cap="Imagem da matriz de correlação entre os parâmetros do modelo COM-Poisson ajustados ao número de vagens por parcela.">>=
 
 pnames <- c("phi", "beta0",
@@ -834,9 +1096,15 @@ fonte("Fonte: Elaborado pelo autor.")
 
 @
 
-\lipsum[1]
+Na figura \ref{fig:pred-soyaBeans} apresentamos as médias calculadas
+com com intervalos de confiança 95\% sob os modelos Poisson,
+COM-Poisson, Binomial-Negativo e Quasi-Poisson considerando efeito de
+interação entre os níveis de umidade do solo e adubação potássica de
+confiança. Tomou-se o efeito médio de bloco, uma vez que esse efeito não
+é de interesse prático.
 
 <<pred-soyaBeans, fig.height=5, fig.width=7.5, fig.cap="Valores preditos com intervalos de confiança (95\\%) como função do nível de adubação com potássio e do percentual de umidade do solo para cada variável de interesse mensurada (número de vagens e número de grãos por parcela).">>=
+
 library(multcomp)
 
 ## Predição pontual/intervalar
@@ -946,7 +1214,8 @@ da <- reshape2::melt(
     variable.name = "var", value.name = "count")
 
 key <- list(type = "o", divide = 1,
-            lines = list(pch = 1:nlevels(pred.all$model) + 4),
+            lines = list(pch = 1:nlevels(pred.all$model) + 4,
+                         cex = 0.8),
             text = list(c("Poisson", "COM-Poisson",
                           "Binomial Negativa", "Quasi-Poisson"))
             )
@@ -955,13 +1224,14 @@ useOuterStrips(
     segplot(
         K ~ lwr + upr | umid + var,
         key = key,
+        axis = axis.grid,
         ylab = "Contagens",
         xlab = "Nível de adubação Potássica",
         centers = fit, groups = modelo, data = pred.all,
         horizontal = FALSE, draw = FALSE,
-        lwd = 2, pch = 1:nlevels(pred.all$modelo) + 4,
+        lwd = 1.5, pch = 1:nlevels(pred.all$modelo) + 4,
         panel = panel.groups.segplot, as.table = TRUE,
-        gap = 0.3,
+        gap = 0.3, cex = 0.8,
         scales = list(
             y = list(relation = "free", rot = 0))
     ), strip = strip.custom(
@@ -983,10 +1253,23 @@ fonte.xy("Fonte: Elaborado pelo autor.")
 
 @
 
-
-\lipsum[1]
+Para a contagem do número de vagens, observa-se os intervalos
+com comprimento muito parecidos, ligeiramente menores para o caso
+COM-Poisson e Binomial Negativo. Para a contagem do número de grão por
+parcela, onde temos um caso superdisperso, percebe-se que o modelo
+Poisson nos leva a falsa confiança, uma vez que os intervalos são
+menores não por se ajustar melhor aos dados, mas sim por subestimar a
+variabilidade do processo. Para as formulações alternativas, temos os
+modelos paramétricos com intervalos menores que o semi-paramétrico
+Quasi-Poisson, isso é razoável, pois nos Quasi-Poisson temos somente a
+especificação de dois momentos, enquanto que nos parâmetricos
+especificamos a distribuição completa, ganhando informação
+\ref{eqn:quasi-informacao}. Os intervalos sob os modelos COM-Poisson e
+Binomial Negativa foram os mais parcimoniosos, sendo intervalos menores,
+porém fiéis a varibilidade inerente ao processo.
 
 \section{Análise de ninfas de mosca-branca em lavoura de soja}
+\label{sec:analise-whiteFly}
 
 <<ajuste-whiteFly, include=FALSE, cache=TRUE>>=
 
@@ -1021,7 +1304,47 @@ prof.ntot <- profile(m1C.ntot, which = "phi")
 
 @
 
-\lipsum[1]
+Neste experimento também temos fortes indícios de superdispersão,
+conforme visto na seção \ref{sec:whiteFly}. Assim os modelos Poisson,
+COM-Poisson, Binomial Negativo e Quasi-Poisson serão aplicados. A
+variável em estudo é a contagem de ninfas de Mosca-Branca nos folíolos
+de plantas de soja e têm-se interesse na avaliação dos fatores dias
+decorridos após a primeira avaliação da planta e cultivar. Como o
+experimento foi conduzido sob delineamento de blocos casualizados, os
+efeitos de bloco são considerados no modelo. As covariáveis serão
+tratadas como fator, assim como na aplicação anterior, com seis níveis
+para o número de dias decorridos a partir da primeira avaliação e quatro
+nível para o fator cultivar de soja. Os preditores em comparação são:
+
+\noindent
+Preditor 1: $\eta_1 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k $ \\
+Preditor 1: $\eta_2 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k +
+\alpha_{jk}$
+
+\noindent
+em que $\tau_i$ é o efeito do i-ésimo bloco, $i=$1: bloco II, 2: bloco
+III, 3: bloco IV e 4: V; $\gamma_j$ o efeito da j-ésima cultivar ,
+$j=$1: BRS 243 RR, 2: BRS 245 RR e 3: BRS 246 RR; $\delta_k$ o efeito do
+k-ésimo nível do número de dias após o início do experimento, $k=$ 8,
+13, 22, 31 e 38 dias e $\alpha_{jk}$ o efeito da interação entre a
+j-ésima cultivar e o k-ésimo nível do número de dias após o início do
+experimento. Como construído, a avaliação do efeito de interação é de
+interesse prático, pois informa se há desempenhos distintos das
+cultivares conforme o número de dias decorridos da primeira
+avaliação. No modelo com interação, temos 27 parâmetros de locação a
+serem estimados.
+
+Assim como na aplicação superdispersa apresentada na seção
+\ref{sec:analise-soyaBeans}, nesse exemplo temos um cenário com
+contagens altas (variando entre \Sexpr{paste(range(soyaBeans[ ,
+  "ngra"]), collapse=" e ")}) e ainda superdispersas (parâmetros $\phi$
+estimados próximos à -3). Isso torna a convergência da função $Z(\lambda_i, \nu =
+\exp(\phi))$ demorada e o valor dessa constante que normaliza a
+densidade é altíssimo. Em problemas com contagens altas e comportamento
+muito superdisperso a obtenção da constante Z pode se tornar probitiva
+computacionalmente, devido à \textit{overflow} (valores que ultrapassam o
+limite de capacidade de armazenamento da máquina) e consequentemente o
+modelo COM-Poisson não se ajusta.
 
 <<loglik-whiteFly, include=FALSE>>=
 
@@ -1041,13 +1364,6 @@ dispersions.ntot <- c(
     sapply(list(m1B.ntot, m2B.ntot), function(m) m$theta),
     sapply(list(m1Q.ntot, m2Q.ntot), function(m) summary(m)$dispersion))
 
-## Teste para o parâmetro de dispersão
-pvs <- rep(NA, 8)
-pvs[3:4] <- cmptest(m1C.ntot, m2C.ntot)[, 2]
-
-## Agrupando
-dispersions.ntot <- cbind(dispersions.ntot, pvs)
-
 ## Adicionando os parametros de dispersão à tabela
 tab.ntot <- cbind(tab.ntot, dispersions.ntot)
 rownames(tab.ntot) <- NULL
@@ -1059,33 +1375,47 @@ tab.ajuste <- data.frame(
 
 ## ##----------------------------------------------------------------------
 ## ## Copiar e colar o corpo do resultado na customização latex abaixo
-## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2, -2)
+## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2)
 ## print(xtable(tab.ajuste, digits = digits))
+
 @
 
+Nesse exemplo, os modelos COM-Poisson convergiram e seus resultados são
+exibidos na tabela \ref{tab:ajuste-whiteFly} em conjunto com os
+resultados proporcionados pelos modelos Poisson, Binomial Negativo e
+COM-Poisson. Todas as estimativas dos parâmetros extras nos modelos
+concorrentes ao Poisson, $\hat{\phi}$, $\hat{theta}$ e $\hat{sigma^2}$
+indicam expressivamente a superdispersão os dados. Em benefício dos
+modelos alternativos ao Poisson temos todas as medidas apresentadas
+indicando uma substancial melhora de ajuste quando flexibilizamos o
+modelo. Destaque para a magnitude dessas evidências, em que, por
+exemplo, o AIC obtido dos modelos alternativos é em torno de 0,47
+vezes o obtido do Poisson.
+
 \begin{table}[ht]
+\centering
 \small
 \caption{Medidas de ajuste para avaliação e comparação entre preditores
   e modelos ajustados}
 \label{tab:ajuste-whiteFly}
 \begin{tabular}{lcccccrcr}
   \toprule
- Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) &  &  \\
+ Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) &  \\
   \midrule
-  Preditor 1 & 12 & -922,98 & 1869,96 &  &  &  &  &  \\
-  Preditor 2 & 27 & -879,23 & 1812,46 & 87,50 & 15 & 2,90E-12 &  &  \\[0.3cm]
- COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ & P($>\rchi^2$) \\
+  Preditor 1 & 12 & -922,98 & 1869,96 &  &  &  &  \\
+  Preditor 2 & 27 & -879,23 & 1812,46 & 87,50 & 15 & 2,90E-12 &  \\[0.3cm]
+ COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$  \\
   \midrule
-  Preditor 1 & 13 & -410,44 & 846,89 &  &  &  & -3,08 & 6,39E-225 \\
-  Preditor 2 & 28 & -407,15 & 870,30 & 6,59 & 15 & 9,68E-01 & -2,95 & 2,47E-207 \\[0.3cm]
- Binomial Neg. & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\theta}$ & P($>F^2$) \\
+  Preditor 1 & 13 & -410,44 & 846,89 &  &  &  & -3,08 \\
+  Preditor 2 & 28 & -407,15 & 870,30 & 6,59 & 15 & 9,68E-01 & -2,95 \\[0.3cm]
+ Binomial Neg. & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\theta}$ \\
   \midrule
-  Preditor 1 & 13 & -406,16 & 838,31 &  &  &  & 3,44 &  \\
-  Preditor 2 & 28 & -400,55 & 857,10 & 11,21 & 15 & 7,38E-01 & 3,99 &  \\[0.3cm]
- Quase-Poisson & np & deviance & AIC & F & diff np & P(>F) & $\hat{\sigma}^2$ & P($>F$) \\
+  Preditor 1 & 13 & -406,16 & 838,31 &  &  &  & 3,44 \\
+  Preditor 2 & 28 & -400,55 & 857,10 & 11,21 & 15 & 7,38E-01 & 3,99 \\[0.3cm]
+ Quase-Poisson & np & deviance & AIC & F & diff np & P(>F) & $\hat{\sigma}^2$  \\
   \midrule
-  Preditor 1 & 12 & 1371,32 &  &  &  &  & 17,03 &  \\
-  Preditor 2 & 27 & 1283,82 &  & 0,31 & 15 & 9,93E-01 & 19,03 &  \\
+  Preditor 1 & 12 & 1371,32 &  &  &  &  & 17,03 \\
+  Preditor 2 & 27 & 1283,82 &  & 0,31 & 15 & 9,93E-01 & 19,03 \\
   \bottomrule
 \end{tabular}
 \begin{tablenotes}
@@ -1097,7 +1427,16 @@ tab.ajuste <- data.frame(
 \end{tablenotes}
 \end{table}
 
-\lipsum[1]
+Para tomada de decisão, observa-se que o modelo Poisson é claramente
+inadequado. Para avaliação dos preditores, na tabela
+\ref{tab:ajuste-whiteFly}, temos o modelo Poisson indicando (com uma
+significância inferior a 1E-10) que há efeito de interação entre os dias
+decorridos da primeira avaliação e as cultivares ao passo que nos
+modelos alternativos esse efeito é marcadamente não significativo. Essa
+discordância se deve, conforme já discutido, ao fato de o modelo Poisson
+subestimar a variabilidade por sua reestrição de equidispersão. Assim,
+com variâncias menores qualquer efeito acréscido no modelo passará por
+significativo.
 
 <<prof-whiteFly, fig.height=4, fig.width=4, fig.show="hide", results="asis", fig.cap="Perfil de log-verossimilhança para o parâmetro extra da COM-Poisson">>=
 
@@ -1108,8 +1447,15 @@ wrapfigure()
 
 @
 
-
-\lipsum[1]
+Enfatizando a superdispersão indicada pelo modelo COM-Poisson,
+considerando o preditor de efeitos aditivos, temos o perfil de
+verossimilhança para o parâmetro $\phi$ apresentado na figura
+\ref{fig:prof-whiteFly}. Podemos observar que os limites inferiores dos
+intervalos de confiança de 90, 95 e 99\% estão muito distantes do valor
+0, sob o qual temos equivalência entre os modelos Poisson e
+COM-Poisson. Outra característica desse gráfico é a leve assimétria à
+esquerda, o que atribuímos a forte característica de superdispersão dos
+dados.
 
 <<corr-whiteFly, fig.width=7, fig.height=7, fig.cap="Imagem da matriz de correlação entre os parâmetros do modelo COM-Poisson">>=
 
@@ -1127,11 +1473,15 @@ fonte("Fonte: Elaborado pelo autor.")
 
 @
 
+Também exibimos as covariâncias entre os efeitos estimados pelo modelo
+COM-Poisson conforme descrição do preditor 1 na figura
+\ref{fig:corr-whiteFly} na escala de correlação. Similarmente as
+análises anteriores observa-se a alta correlação entre $\hat{\phi}$ e os
+demais parâmetros de regressão. A soma dos valores absolutos das
+correlações observadas entre $\hat{\phi}$ e as demais estimativas é de
+\Sexpr{sum(abs(Corr[1, ]))} e a média \Sexpr{mean(abs(Corr[1, ]))}.
 
-\lipsum[1]
-
-
-<<pred-whiteFly, fig.height=4.5, fig.width=7.5, fig.cap="Valores preditos com intervalos de confiança (95\\%) em função das cultivares de soja e da data de avaliação da planta.">>=
+<<pred-whiteFly, fig.height=4, fig.width=7.5, fig.cap="Valores preditos com intervalos de confiança (95\\%) em função das cultivares de soja e da data de avaliação da planta.">>=
 
 library(multcomp)
 
@@ -1190,42 +1540,172 @@ pred.all <- pred.all[with(pred.all, order(cult, aval, modelo)), ]
 ## Gráfico
 
 key <- list(type = "o", divide = 1,
-            lines = list(pch = 1:nlevels(pred.all$model) + 4),
+            lines = list(pch = 1:nlevels(pred.all$model) + 4,
+                         cex = 0.8),
             text = list(c("Poisson", "COM-Poisson",
                           "Binomial Negativa", "Quasi-Poisson"))
             )
 
-xyplot(ntot ~ aval | cult,
-       data = whiteFly,
-       layout = c(NA, 1),
-       as.table = TRUE,
-       key = key,
-       alpha = 0.3,
-       type = c("p", "g"),
-       xlab = "Número de dias após o inicío do experimento",
-       ylab = "Número total de moscas-brancas",
-       par.settings = ps.sub) +
-    as.layer(
-        segplot(
-            aval ~ lwr + upr | cult,
-            layout = c(NA, 1),
-            key = key,
-            centers = fit, groups = modelo, data = pred.all,
-            horizontal = FALSE, draw = FALSE,
-            lwd = 2, pch = 1:nlevels(pred.all$modelo) + 4,
-            panel = panel.groups.segplot, gap = 0.3)
-    )
+## ## A escala do gráfico fica prejudicada, pois os limites do eixo y são
+## ## extendidos para exibir as observações mais distantes
+## xyplot(ntot ~ aval | cult,
+##        data = whiteFly,
+##        layout = c(NA, 1),
+##        as.table = TRUE,
+##        key = key,
+##        alpha = 0.3,
+##        type = c("p", "g"),
+##        xlab = "Número de dias após o inicío do experimento",
+##        ylab = "Número total de moscas-brancas",
+##        par.settings = ps.sub)
+
+segplot(
+    aval ~ lwr + upr | cult,
+    layout = c(NA, 1),
+    xlab = "Número de dias após o inicío do experimento",
+    ylab = "Número total de moscas-brancas",
+    key = key,
+    axis = axis.grid, cex = 0.8,
+    centers = fit, groups = modelo, data = pred.all,
+    horizontal = FALSE, draw = FALSE,
+    lwd = 1.5, pch = 1:nlevels(pred.all$modelo) + 4,
+    panel = panel.groups.segplot, gap = 0.3,
+    par.settings = ps.sub)
 
 fonte.xy("Fonte: Elaborado pelo autor.")
 
 @
 
-\lipsum[1]
+As médias com intervalos de confiança calculadas para cada combinação
+dos níveis de dias após a primeira avaliação e cultivar de soja
+considerando os modelos Poisson, COM-Poisson, Binomial-Negativo e
+Quasi-Poisson, são apresentadas na \ref{fig:pred-whiteFly}. Para o
+efeito de bloco consideramos o efeito médio para uma correta
+comparação. Podemos observar que o intervalo de confiança descrito pelo
+modelo Poisson é quase imperceptível quando comparados aos demais,
+mostrando novamente que seu uso pe inadequado a esses dados. Já para as
+outras alternativas não tivemos um comportamento padrão em todas as
+cultivares. Os intervalos pelo modelos Quasi-Poisson e COM-Poisson foram
+muito similares em todos os casos e os intervalos pelo modelo Binomial
+Negativo intervalos mais amplos. Um fato interessante é que não
+necessariamente as estimativas pontuais da média desses modelos
+alternativos serão iguais, isso ocorre, por construção, somente para nos
+modelos Poisson e Quasi-Poisson, esse exemplo ilustra na prática a
+constatação desse fato. Para o modelo Binomial Negativo tivemos
+médias visivelmente superiores que os demais para a cultivar BRS
+239. Para o modelo COM-Poisson as estimativas pontuais são visivelmente
+iguais as do modelo Poisson.
 
 \section{Análise de captura de peixes em um parque estadual}
-
+\label{sec:analise-fish}
 
 <<ajuste-fish, include=FALSE, cache=TRUE>>=
 
+## Dados
+data(fish, package = "tccPackage")
+
+## Preditores
+f1 <- npeixes ~  campista + npessoas |
+    campista + npessoas + ncriancas
+f2 <- npeixes ~ campista + npessoas * ncriancas |
+    campista + npessoas * ncriancas
+
+## Ajustando os modelos Hurdle Poisson
+library(pscl)
+m1HP <- hurdle(f1, data = fish, dist = "poisson")
+m2HP <- hurdle(f2, data = fish, dist = "poisson")
+
+## Hurdle Binomial Negativo
+m1HB <- hurdle(f1, data = fish, dist = "negbin")
+m2HB <- hurdle(f2, data = fish, dist = "negbin")
+
+## Ajustando os modelos Hurdle COM-Poisson
+m1HC <- hurdlecmp(f1, data = fish)
+m2HC <- hurdlecmp(f2, data = fish)
 
 @
+
+Nesse exemplo ilustramos a análise de um estudo observacional em que
+aparentemente temos uma quantidade excessiva de contagens nulas (veja a
+seção \ref{sec:fish}). O estudo tem por objetivo a modelagem do número
+de peixes capturados por grupos de visitantes em um Parque Estadual. As
+covariáveis mensuradas foram \texttt{np}, o número de pessoas no grupo,
+\texttt{nc}, o número de crianças e \texttt{ca} variável binária que
+indica a presença ou não de um campista no grupo. Com estrutura dos
+dados vamos modelar o número de peixes capturados em duas partes, as
+contagens nulas e as não nulas, seção \ref{cap02:zeros}. Abaixo
+definimos os preditores considerados para as duas partes
+
+\noindent
+Preditor 1: \quad $
+\begin{aligned}
+  g(\mu)     &= \beta_0 + \beta_1 \textrm{ca} +
+  \beta_2 \textrm{np} \\
+  logit(\pi) &= \gamma_0 + \gamma_1 \textrm{ca} +
+  \gamma_2 \textrm{np} + \gamma_3 \textrm{nc}
+\end{aligned}$ \\
+
+\noindent
+Preditor 2: \quad $
+\begin{aligned}
+  g(\mu)     &= \beta_0 + \beta_1 \textrm{ca} +
+  \beta_2 \textrm{np} + \beta_4 \textrm{nc} +
+  \beta_5 (\textrm{np} \cdot \textrm{nc}) \\
+  logit(\pi) &= \gamma_0 + \gamma_1 \textrm{ca} +
+  \gamma_2 \textrm{np} + \gamma_3 \textrm{nc} +
+  \gamma_5 (\textrm{np} \cdot \textrm{nc})
+\end{aligned}$ \\
+
+
+<<logLik-fish, include=FALSE>>=
+
+## Tabelas de ANOVA
+anHP <- myanova(m1HP, m2HP)
+anHC <- myanova(m1HC, m2HC)
+anHB <- myanova(m1HB, m2HB)
+
+## Obtem as estimativas dos parametros de dispersao
+dispHC <- sapply(list(m1HC, m2HC), function(m) m@coef[1])
+dispHB <- sapply(list(m1HB, m2HB), function(m) m$theta)
+dispersions <- c(NA, NA, dispHC, dispHB)
+
+## Empilhando
+tab <- data.frame(pred = rep(paste("Preditor", 1:2), 3))
+tab <- cbind(tab, rbind(anHP, anHC, anHB), dispersions)
+
+## ## Gerando o código latex
+## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2)
+xtable(tab, digits = digits)
+
+@
+
+\begin{table}[ht]
+\centering
+\small
+\caption{Medidas de ajuste para avaliação e comparação entre preditores
+  e modelos com componente de barreira ajustados}
+\label{tab:ajuste-fish}
+\begin{tabular}{lcccccrc}
+  \toprule
+Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & \\
+  \midrule
+  Preditor 1 & 7 & -857,48 & 1728,96 &  &  &  &  \\
+  Preditor 2 & 10 & -744,58 & 1509,17 & 225,79 & 3 & 1,12E-48 &  \\[0.3cm]
+COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ \\
+  \midrule
+  Preditor 1 & 8 & -409,85 & 835,71 &  &  &  & -8,77 \\
+  Preditor 2 & 11 & -402,30 & 826,59 & 15,12 & 3 & 1,72E-03 & -3,77 \\[0.3cm]
+Binomial Negativo & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\theta}$ \\
+  \midrule
+  Preditor 1 & 8 & -399,79 & 815,58 &  &  &  & 0,20 \\
+  Preditor 2 & 11 & -393,72 & 809,44 & 12,14 & 3 & 6,91E-03 & 0,37 \\
+  \bottomrule
+\end{tabular}
+\begin{tablenotes}
+  \footnotesize
+\item np, número de parâmetros, diff $\ell$, diferença entre
+  log-verossimilhanças, F, estatística F baseada nas quasi-deviances,
+  diff np, diferença entre o np. \\[0.1cm]
+\item Fonte: Elaborado pelo autor.
+\end{tablenotes}
+\end{table}
diff --git a/docs/compois.bib b/docs/compois.bib
index fb768f5..8b60dac 100644
--- a/docs/compois.bib
+++ b/docs/compois.bib
@@ -1,3 +1,10 @@
+@book{Ferreira2011,
+author = {Ferreira, Daniel Furtado},
+mendeley-groups = {TCC{\_}UFPR{\_}2015},
+publisher = {Editora UFLA},
+title = {{Estat{\'{i}}stica Multivariada}},
+year = {2011}
+}
 @book{Nocedal1995,
 abstract = {This is a book for people interested in solving optimization problems. Because of the wide (and growing) use of optimization in science, engineering, economics, and industry, it is essential for students and practitioners alike to develop an understanding of optimization algorithms. Knowledge of the capabilities and limitations of these algorithms leads to a better understanding of their impact on various applications, and points the way to future research on improving and extending optimization algorithms and software. Our goal in this book is to give a comprehensive description of the most powerful, state-of-the-art, techniques for solving continuous optimization problems. By presenting the motivating ideas for each algorithm, we try to stimulate the reader's intuition and make the technical details easier to follow. Formal mathematical requirements are kept to a minimum. Because of our focus on continuous problems, we have omitted discussion of important optimization topics such as discrete and stochastic optimization.},
 author = {Nocedal, Jorge and Wright, Stephen J.},
@@ -99,6 +106,16 @@ url = {http://www.jstor.org/stable/2111071},
 volume = {33},
 year = {1989}
 }
+@incollection{Kokonendji2014,
+author = {Kokonendji, Celestin C.},
+booktitle = {Methods and Applications of Statistics in Clinical Trials: Planning, Analysis, and Inferential Methods},
+file = {:home/eduardo/Documents/Mendeley Desktop/ch30{\_}kokonendji2014.pdf:pdf},
+mendeley-groups = {TCC{\_}UFPR{\_}2015},
+pages = {506--526},
+title = {{Over- and Underdisperson Models}},
+url = {https://lmb.univ-fcomte.fr/IMG/pdf/ch30{\_}kokonendji2014.pdf},
+year = {2014}
+}
 @article{Ridout1998,
 abstract = {We consider the problem of modelling count data with excess zeros and review some possible models. Aspects of model tting and inference are considered. An example from horticultural research is used for illustration.},
 author = {Ridout, Martin and Demetrio, Clarice G.B and Hinde, John},
@@ -111,16 +128,6 @@ pages = {1--13},
 title = {{Models for count data with many zeros}},
 year = {1998}
 }
-@incollection{Kokonendji2014,
-author = {Kokonendji, Celestin C.},
-booktitle = {Methods and Applications of Statistics in Clinical Trials: Planning, Analysis, and Inferential Methods},
-file = {:home/eduardo/Documents/Mendeley Desktop/ch30{\_}kokonendji2014.pdf:pdf},
-mendeley-groups = {TCC{\_}UFPR{\_}2015},
-pages = {506--526},
-title = {{Over- and Underdisperson Models}},
-url = {https://lmb.univ-fcomte.fr/IMG/pdf/ch30{\_}kokonendji2014.pdf},
-year = {2014}
-}
 @article{Sellers2016,
 abstract = {Excess zeroes are often thought of as a cause of data over-dispersion (i.e. when the variance exceeds the mean); this claim is not entirely accurate. In actuality, excess zeroes reduce the mean of a dataset, thus inflating the dispersion index (i.e. the variance divided by the mean). While this results in an increased chance for data over-dispersion, the implication is not guaranteed. Thus, one should consider a flexible distribution that not only can account for excess zeroes, but can also address potential over- or under-dispersion. A zero-inflated Conway-Maxwell-Poisson (ZICMP) regression allows for modeling the relationship between explanatory and response variables, while capturing the effects due to excess zeroes and dispersion. This work derives the ZICMP model and illustrates its flexibility, extrapolates the corresponding likelihood ratio test for the presence of significant data dispersion, and highlights various statistical properties and model fit through several examples.},
 author = {Sellers, Kimberly F. and Raim, Andrew},
@@ -138,22 +145,13 @@ url = {http://dx.doi.org/10.1016/j.csda.2016.01.007 http://linkinghub.elsevier.c
 volume = {99},
 year = {2016}
 }
-@misc{Winkelmann1994,
-abstract = {"This paper deals with the estimation of single equation models in which the counts are regressed on a set of observed individual characteristics such as age, gender, or nationality.... We propose a generalized event count model to simultaneously allow for a wide class of count data models and account for over- and underdispersion. This model is successfully applied to German data on fertility, divorces and mobility." (SUMMARY IN FRE)},
-author = {Winkelmann, R and Zimmermann, K F},
-booktitle = {Mathematical population studies},
-doi = {10.1080/08898489409525374},
-file = {:home/eduardo/Documents/Mendeley Desktop/41{\_}CountDataModel{\_}MathematicalPopulationStudies{\_}1993.pdf:pdf},
-isbn = {9780470510247},
-issn = {0889-8480},
-keywords = {Demographic Factors,Developed Countries,Divorce,Estimation Technics,Europe,Fertility,Germany,Mathematical Model,Migration,Models,Nuptiality,Population,Population Dynamics,Research Methodology,Theoretical,Western Europe},
+@phdthesis{Borges2012,
+author = {Borges, Patrick},
+file = {:home/eduardo/Documents/Mendeley Desktop/4552.pdf:pdf},
 mendeley-groups = {TCC{\_}UFPR{\_}2015},
-number = {3},
-pages = {205--221, 223},
-pmid = {12287090},
-title = {{Count data models for demographic data}},
-volume = {4},
-year = {1994}
+school = {Universidade Federal de S{\~{a}}o Carlos},
+title = {{Novos modelos de sobreviv{\^{e}}ncia com fra{\c{c}}{\~{a}}o de cura baseados no processo da carcinog{\^{e}}nese}},
+year = {2012}
 }
 @phdthesis{Ribeiro2012,
 author = {Ribeiro, Ang{\'{e}}lica Maria Tortola},
@@ -178,6 +176,23 @@ url = {http://www.jstor.org/stable/2334725?origin=crossref},
 volume = {61},
 year = {1974}
 }
+@misc{Winkelmann1994,
+abstract = {"This paper deals with the estimation of single equation models in which the counts are regressed on a set of observed individual characteristics such as age, gender, or nationality.... We propose a generalized event count model to simultaneously allow for a wide class of count data models and account for over- and underdispersion. This model is successfully applied to German data on fertility, divorces and mobility." (SUMMARY IN FRE)},
+author = {Winkelmann, R and Zimmermann, K F},
+booktitle = {Mathematical population studies},
+doi = {10.1080/08898489409525374},
+file = {:home/eduardo/Documents/Mendeley Desktop/41{\_}CountDataModel{\_}MathematicalPopulationStudies{\_}1993.pdf:pdf},
+isbn = {9780470510247},
+issn = {0889-8480},
+keywords = {Demographic Factors,Developed Countries,Divorce,Estimation Technics,Europe,Fertility,Germany,Mathematical Model,Migration,Models,Nuptiality,Population,Population Dynamics,Research Methodology,Theoretical,Western Europe},
+mendeley-groups = {TCC{\_}UFPR{\_}2015},
+number = {3},
+pages = {205--221, 223},
+pmid = {12287090},
+title = {{Count data models for demographic data}},
+volume = {4},
+year = {1994}
+}
 @article{Winkelmann1995,
 author = {Winkelmann, Rainer},
 doi = {10.1080/07350015.1995.10524620},
@@ -193,14 +208,6 @@ url = {http://www.tandfonline.com/doi/abs/10.1080/07350015.1995.10524620},
 volume = {13},
 year = {1995}
 }
-@phdthesis{Borges2012,
-author = {Borges, Patrick},
-file = {:home/eduardo/Documents/Mendeley Desktop/4552.pdf:pdf},
-mendeley-groups = {TCC{\_}UFPR{\_}2015},
-school = {Universidade Federal de S{\~{a}}o Carlos},
-title = {{Novos modelos de sobreviv{\^{e}}ncia com fra{\c{c}}{\~{a}}o de cura baseados no processo da carcinog{\^{e}}nese}},
-year = {2012}
-}
 @book{Hilbe2014,
 abstract = {This entry-level text offers clear and concise guidelines on how to select, construct, interpret and evaluate count data. Written for researchers with little or no background in advanced statistics, the book presents treatments of all major models using numerous tables, insets, and detailed modeling suggestions. It begins by demonstrating the fundamentals of linear regression and works up to an analysis of the Poisson and negative binomial models, and to the problem of overdispersion. Examples in Stata, R, and SAS code enable readers to adapt models for their own purposes, making the text an ideal resource for researchers working in public health, ecology, econometrics, transportation, and other related fields.},
 author = {Hilbe, Joseph M.},
-- 
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