From 75fd60de8a44f7f2139cd9b58fd26d6aa5991eca Mon Sep 17 00:00:00 2001 From: Eduardo Junior <edujrrib@gmail.com> Date: Mon, 27 Jun 2016 10:42:01 -0300 Subject: [PATCH] =?UTF-8?q?Adiciona=20pr=C3=A9via=20da=20apresenta=C3=A7?= =?UTF-8?q?=C3=A3o?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- presentations/slides-tcc.Rmd | 1341 ++++++++++++++++++++++++++++++++++ presentations/slides-tcc.pdf | Bin 0 -> 518397 bytes 2 files changed, 1341 insertions(+) create mode 100644 presentations/slides-tcc.Rmd create mode 100644 presentations/slides-tcc.pdf diff --git a/presentations/slides-tcc.Rmd b/presentations/slides-tcc.Rmd new file mode 100644 index 0000000..b7240f5 --- /dev/null +++ b/presentations/slides-tcc.Rmd @@ -0,0 +1,1341 @@ +--- +date: "`r format(Sys.time(), '%d de %B de %Y')`" +fontsize: "10pt" +bibliography: ../docs/compois.bib +csl: abntcite.csl +output: + beamer_presentation: + fig_caption: yes + highlight: tango + includes: + in_header: preambulo-beamer.tex + keep_tex: yes + slide_level: 3 +--- + +<!-- Arquivos de estilo de bibliográfica e citação "Citation Style --> +<!-- Language (CSL)" disponÃveis em --> +<!-- <https://github.com/citation-style-language/styles> --> + +```{r, include = FALSE} + +source("../docs/_setup.R") +opts_chunk$set( + fig.width = 8, + out.width = "1\\textwidth" +) + +library(cmpreg) + +``` + +<!-- ### ### --> + +\titlepage + +### Sumário ### + +\tableofcontents[hideallsubsections] + +<!-- ------------------------------------------- --> +# Introdução # + +### Dados de contagem ### + +\begin{figure}[t] + \includegraphics[height=1cm]{./images/count} +\end{figure} + +São variáveis aleatórias aleatórias que representam o número de +ocorrências de um evento em um dominÃo discreto ou contÃnuo. +\vspace{0.5cm} + +Se $Y$ é uma v.a. de contagem, $y = 0, 1, 2, \ldots$ +\vspace{0.5cm} + +Exemplos: + + * Número de filhos por casal; + * Número de indivÃduos infectados por uma doença; + * Número de acidentes de trânsito em um mês; + * Número de _posts_ em uma rede social durante um dia; + * Número de frutos produzidos; + * ... + +### Análise de dados de contagem ### + + * Modelos de regressão Gaussianos com dados transformados + - Dificultam a interpretação dos resultados; + - Não contemplam a natureza discreta da variável; + - Não contemplam a relação média e variância; + - Transformação logarÃtmica é problemática para valores 0. + +\vspace{0.5cm} + + * Modelos de regressão Poisson [@Nelder1972] + - Fiel a natureza dos dados; + - Contempla a relação média e variância; + - Suposição de equidispersão. + +### ### + +```{r, fig.height = 4, fig.cap = "Ilustração de processos pontuais que levam a contagens com diferentes nÃveis de dispersão."} + +mygrid <- expand.grid(xc = 1:3, yc = 1:3) +mygrid <- data.frame(mygrid) +sp::coordinates(mygrid) <- ~xc + yc + +set.seed(201246) +equi <- sp::spsample(mygrid, n = 100, type = "random") +over <- sp::spsample(mygrid, n = 100, type = "clustered", nclusters = 20) +## unde <- sp::spsample(mygrid, n = 100, type = "nonaligned") +unde <- sp::spsample(mygrid, n = 100, type = "stratified") + +coords <- sapply(list("equi" = equi, "over" = over, "unde" = unde), + function(x) { + colnames(x@coords) <- c("x", "y") + x@coords + }) +da <- plyr::ldply(coords, .id="caso") + +col <- "gray50" +xyplot(y ~ x | caso, data = da, + layout = c(NA, 1), + as.table = TRUE, + pch = 19, + scales = list(draw = FALSE), + xlab = "", + ylab = "", + strip = strip.custom( + factor.levels = c("Equidispersão", + "Superdispersão", + "Subdispersão")), + panel = function(x, y, subscripts, ...) { + l <- seq(min(x), max(x), length.out = 10) + panel.abline(h = l, v = l, col = col, lty = 2) + panel.xyplot(x, y, ...) + }) + +``` + +### Distribuições de probabilidades para dados de contagem ### + +\begin{table} +\centering +\small +\caption{Distribuições de probabilidades para dados de contagem} +\only<1>{ +\begin{tabular}{lccc} + \toprule +\multirow{2}{*}{Distribuição} & \multicolumn{3}{c}{Contempla a caracterÃstica de} \\ + \cline{2-4} \\[-0.3cm] + & Equidispersão & Superdispersão & Subdispersão \\[0.1cm] + \hline +Poisson & \checkmark & & \\ +Binomial Negativa & \checkmark & \checkmark & \\ +\textit{Inverse Gaussian Poisson} & \checkmark & \checkmark & \\ +\textit{Compound Poisson} & \checkmark & \checkmark & \\ +Poisson Generalizada & \checkmark & \checkmark & \checkmark \\ +\textit{Gamma-Count} & \checkmark & \checkmark & \checkmark \\ +COM-Poisson & \checkmark & \checkmark & \checkmark \\ +Katz & \checkmark & \checkmark & \checkmark \\ +\textit{Poisson Polynomial} & \checkmark & \checkmark & \checkmark \\ +\textit{Double-Poisson} & \checkmark & \checkmark & \checkmark \\ +\textit{Lagrangian Poisson} & \checkmark & \checkmark & \checkmark \\ + \bottomrule +\end{tabular} +} +\only<2>{ +\begin{tabular}{lccc} + \toprule +\multirow{2}{*}{Distribuição} & \multicolumn{3}{c}{Contempla a caracterÃstica de} \\ + \cline{2-4} \\[-0.3cm] + & Equidispersão & Superdispersão & Subdispersão \\[0.1cm] + \hline + \rowcolor{teal!20} +\textbf{Poisson} & \checkmark & & \\ + \rowcolor{teal!20} +\textbf{Binomial Negativa} & \checkmark & \checkmark & \\ +\textit{Inverse Gaussian Poisson} & \checkmark & \checkmark & \\ +\textit{Compound Poisson} & \checkmark & \checkmark & \\ +Poisson Generalizada & \checkmark & \checkmark & \checkmark \\ +\textit{Gamma-Count} & \checkmark & \checkmark & \checkmark \\ + \rowcolor{teal!20} +\textbf{COM-Poisson} & \checkmark & \checkmark & \checkmark \\ +Katz & \checkmark & \checkmark & \checkmark \\ +\textit{Poisson Polynomial} & \checkmark & \checkmark & \checkmark \\ +\textit{Double-Poisson} & \checkmark & \checkmark & \checkmark \\ +\textit{Lagrangian Poisson} & \checkmark & \checkmark & \checkmark \\ + \bottomrule +\end{tabular} +} +\end{table} + +# Objetivos # + +### Objetivos gerais ### + +Colaborar com a literatura estatÃstica brasileira, no que diz respeito a +dados de contagem: + +* Apresentando e explorando o modelo de regressão COM-Poisson; +* Estendendo o modelo para modelagem de excesso de zeros e inclusão + de efeitos aleatórios; +* Discutindo o desempenho do modelo via análise de dados reais; +* Disponibilizando os recursos computacionais para ajuste dos modelos, + em formato de pacote R. + + +# Materiais e Métodos # + +## Materiais ## +### Conjuntos de dados ### + +Seis conjuntos de dados analisados: + +\begin{center} +\only<1>{ +\begin{itemize} + \tightlist + \item Capulhos de algodão sob desfolha artificial + \item Produtividade de algodão sob infestação de Mosca-branca + \item Produtividade de soja sob umidade e adubação potássica + \item Ocorrência de ninfas de Mosca-branca em lavoura de soja + \item Peixes capturados por visitantes de um parque Estadual + \item Número de nematoides em raizes de feijoeiro +\end{itemize} +} +\only<2>{ +\begin{itemize} + \tightlist + \setbeamercolor{itemize item}{fg=teal!30} + \item \textcolor{teal!30}{Capulhos de algodão sob desfolha artificial} + \setbeamercolor{itemize item}{fg=teal} + \item \textbf{Produtividade de algodão sob infestação de Mosca-branca} + \setbeamercolor{itemize item}{fg=teal!30} + \item \textcolor{teal!30}{Produtividade de soja sob umidade e adubação potássica} + \setbeamercolor{itemize item}{fg=teal} + \item \textbf{Ocorrência de ninfas de Mosca-branca em lavoura de soja} + \item \textbf{Peixes capturados por visitantes de um parque Estadual} + \item \textbf{Número de nematoides em raizes de feijoeiro} +\end{itemize} +} +\end{center} + +### Recursos Computacionais ### + +Software R versão `r with(R.version, paste0(major, ".", +minor))`. Principais pacotes: + +* `MASS` (modelo binomial negativo) +* `pscl` (modelagem de excesso de zeros) +* `lme4` (modelo Poisson com efeito aleatório Normal) +* `bbmle` (ajuste de modelos via máxima verossimilhança) + +## Métodos ## +### Estimação via máxima verossimilhança ### + +1. Escreva a função de verossimilhança - $\Ell(\Theta \mid \underline{y})$ +2. Tome seu logaritmo - $\ell(\Theta \mid \underline{y})$ +3. As estimativas dos parâmetros são +$$\hat{\Theta} = \underset{\Theta}{\textrm{arg max }} \ell(\Theta \mid +\underline{y})$$ + +* Algoritmo IWLS (_Interactive Weigthed Leasts Squares_) para os modelos + Poisson, Binomial Negativo e Quasi-Poisson. +* Método _BFGS_ para os modelos COM-Poisson. + +### Verossimilhança do modelo COM-Poisson ### + +* Reparametrizando $\phi = \log(\nu)$ + - $\phi < 0 \Rightarrow \textrm{Superdispersão}$ + - $\phi = 0 \Rightarrow \textrm{Equidispersão}$ + - $\phi > 0 \Rightarrow \textrm{Subdispersão}$ + +\begin{block}{log-verossimilhança} +\begin{equation} + \label{loglik-compoissonr} + \ell(\phi, \beta \mid \underline{y}) = \sum_{i=1}^n y_i \log(\lambda_i) - + e^\phi \sum_{i=1}^n \log(y!) - \sum_{i=1}^n \log(Z(\lambda_i, \phi)) +\end{equation} + +\noindent +em que $\lambda_i = e^{X_i\beta}$, com $X_i$ o vetor $(x_{i1}, x_{i2}, +\ldots x_{ip})$ de covariáveis da i-ésima observação, e $(\beta, \phi) +\in \R^{p+1}$. +\end{block} + +### Verossimilhança do modelo Hurdle COM-Poisson ### + +* $\underline{\pi}=\frac{\exp(G\gamma)}{1+\exp(G\gamma)}$ a + probabilidade de contagem nula. +* $\underline{\lambda}=\exp(X\beta)$ o parâmetro de locação da + distribuição COM-Poisson truncada. + +\begin{block}{verossimilhança} +\begin{equation} + \label{eqn:loglik-hurdlecmp} + \Ell(\phi, \beta, \gamma \mid \underline{y}) = + \ind [\underline{\pi}] \cdot (1-\ind) \left [ + (1-\underline{\pi})\left ( + \frac{\underline{\lambda}^y}{(y!)^{e^\phi} + Z(\underline{\lambda}, \phi)} + \right ) \left ( + 1-\frac{1}{Z(\underline{\lambda}, \phi)} + \right ) \right ] +\end{equation} + +\noindent +em que $\ind$ é uma função indicadora para $y = 0$ +\end{block} + +### Verossimilhança do modelo misto COM-Poisson ### + +$$\begin{aligned} + Y_{ij} \mid b_{i},& X_{ij} \sim \textrm{COM-Poisson}(\mu_{ij}, \phi) \\ + g(&\mu_{ij}) = X_{ij}\beta + Z_ib_i \\ + & b \sim \textrm{Normal}(0, \Sigma) +\end{aligned}$$ + +\begin{block}{Verossimilhança} +\begin{equation} + \label{eqn:loglik-mixedcmp} + \Ell(\phi, \Sigma, \beta \mid \underline{y}) = + \prod_{i=1}^m \int_{\R^q} \left ( + \prod_{j=1}^{n_i} \frac{\underline{\lambda}^y}{(y!)^{e^\phi} + Z(\underline{\lambda}, \phi)} + \right ) \cdot + (2\pi)^{q/2} |\Sigma| \exp \left ( + -\frac{1}{2}b^t \Sigma^{-1} b + \right ) db_i +\end{equation} + +\noindent +sendo $m$ o número de grupos que compartilham do mesmo efeito aleatório, +$q$ o número de efeitos aleatórios (intercepto aleatório, inclinação e +intercepto aleatórios, etc.) e $n_i$ o número de observações no i-ésimo +grupo. +\end{block} + +# Resultados # + +<!--------------------------------------------- --> +## Pacote R ## + +### `cmpreg`: Ajuste de Modelos de Regressões COM-Poisson ### + +Implementação em R de um _framework_ para ajuste dos modelos de +regressão COM-Poisson, pacote `cmpreg` + +```{r pacote, echo=TRUE, eval=FALSE} + +## Pode ser instalado do GitHub +devtools::install_git("https://github.com/JrEduardo/cmpreg.git") +library(cmpreg) + +## Regressão (efeitos fixos) +cmp(y ~ preditor, data = data) + +## Regressão com componente de barreira +hurdlecmp(y ~ count_pred | zero_pred, data = data) + +## Regressão (efeitos aleatórios) +mixedcmp(y ~ count_pred + (1 | ind.ranef), data = data) + +``` + +<!--------------------------------------------- --> +## Produtividade de algodão ## + +```{r ajuste-cottonBolls2, include=FALSE, cache=TRUE} + +## Dados +data(cottonBolls2) + +## Preditores considerados +f1.ncapu <- ncapu ~ 1 +f2.ncapu <- ncapu ~ dexp +f3.ncapu <- ncapu ~ dexp + I(dexp^2) + +f1.nerep <- nerep ~ 1 +f2.nerep <- nerep ~ dexp +f3.nerep <- nerep ~ dexp + I(dexp^2) + +f1.nnos <- nnos ~ 1 +f2.nnos <- nnos ~ dexp +f3.nnos <- nnos ~ dexp + I(dexp^2) + +## Ajustando os modelos Poisson +m1P.ncapu <- glm(f1.ncapu, data = cottonBolls2, family = poisson) +m2P.ncapu <- glm(f2.ncapu, data = cottonBolls2, family = poisson) +m3P.ncapu <- glm(f3.ncapu, data = cottonBolls2, family = poisson) + +m1P.nerep <- glm(f1.nerep, data = cottonBolls2, family = poisson) +m2P.nerep <- glm(f2.nerep, data = cottonBolls2, family = poisson) +m3P.nerep <- glm(f3.nerep, data = cottonBolls2, family = poisson) + +m1P.nnos <- glm(f1.nnos, data = cottonBolls2, family = poisson) +m2P.nnos <- glm(f2.nnos, data = cottonBolls2, family = poisson) +m3P.nnos <- glm(f3.nnos, data = cottonBolls2, family = poisson) + +## Ajustando os modelos quasiPoisson +m1Q.ncapu <- glm(f1.ncapu, data = cottonBolls2, family = quasipoisson) +m2Q.ncapu <- glm(f2.ncapu, data = cottonBolls2, family = quasipoisson) +m3Q.ncapu <- glm(f3.ncapu, data = cottonBolls2, family = quasipoisson) + +m1Q.nerep <- glm(f1.nerep, data = cottonBolls2, family = quasipoisson) +m2Q.nerep <- glm(f2.nerep, data = cottonBolls2, family = quasipoisson) +m3Q.nerep <- glm(f3.nerep, data = cottonBolls2, family = quasipoisson) + +m1Q.nnos <- glm(f1.nnos, data = cottonBolls2, family = quasipoisson) +m2Q.nnos <- glm(f2.nnos, data = cottonBolls2, family = quasipoisson) +m3Q.nnos <- glm(f3.nnos, data = cottonBolls2, family = quasipoisson) + +## Ajustando os modelos COM-Poisson +m1C.ncapu <- cmp(f1.ncapu, data = cottonBolls2, sumto = 30) +m2C.ncapu <- cmp(f2.ncapu, data = cottonBolls2, sumto = 30) +m3C.ncapu <- cmp(f3.ncapu, data = cottonBolls2, sumto = 30) + +m1C.nerep <- cmp(f1.nerep, data = cottonBolls2, sumto = 30) +m2C.nerep <- cmp(f2.nerep, data = cottonBolls2, sumto = 30) +m3C.nerep <- cmp(f3.nerep, data = cottonBolls2, sumto = 30) + +m1C.nnos <- cmp(f1.nnos, data = cottonBolls2, sumto = 30) +m2C.nnos <- cmp(f2.nnos, data = cottonBolls2, sumto = 30) +m3C.nnos <- cmp(f3.nnos, data = cottonBolls2, sumto = 30) + +##====================================================================== +## Perfil de log-verossimilhanca para o parametro phi +prof.ncapu <- profile(m3C.ncapu, which = "phi") +prof.nerep <- profile(m2C.nerep, which = "phi") +prof.nnos <- profile(m3C.nnos, which = "phi") + +``` + +### Experimento ### + +Conduzido na UFGD em casa de vegetação [@Martelli2008]. + +* Delineamento: inteiramente casualizado com cinco repetições +* Objetivo: avaliar o impacto da praga Mosca-branca na produção de + algodão. +* Unidade amostral: vaso com duas plantas. +* Covariável experimental: + - Tempo de exposição das plantas à praga, em dias. (`dexp`) +* Variáveis resposta: + - Número de capulhos produzidos + - Número de estruturas reprodutivas + - Número de nós + +### Modelagem ### + +Preditores considerados: + +* Preditor 1: $g(\mu_i) = \beta_0$ +* Preditor 2: $g(\mu_i) = \beta_0 + \beta_1 \textrm{dexp}_i$ +* Preditor 3: $g(\mu_i) = \beta_0 + \beta_1 \textrm{dexp}_i + \beta_2 +\textrm{dexp}_i^2$ +\vspace{0.5cm} + +Modelos concorrentes: + +* Poisson($\mu_i$) +* COM-Poisson($\lambda_i,\, \phi$) +* Quasi-Poisson($\mu_i,\, \sigma^2$) + +### Medidas de ajuste ### + +```{r loglik-cottonBolls2, include=FALSE} + +## Análise de desvios dos modelos +anP.ncapu <- myanova(m1P.ncapu, m2P.ncapu, m3P.ncapu) +anC.ncapu <- myanova(m1C.ncapu, m2C.ncapu, m3C.ncapu) +anQ.ncapu <- myanova(m1Q.ncapu, m2Q.ncapu, m3Q.ncapu) + +anP.nerep <- myanova(m1P.nerep, m2P.nerep, m3P.nerep) +anC.nerep <- myanova(m1C.nerep, m2C.nerep, m3C.nerep) +anQ.nerep <- myanova(m1Q.nerep, m2Q.nerep, m3Q.nerep) + +anP.nnos <- myanova(m1P.nnos, m2P.nnos, m3P.nnos) +anC.nnos <- myanova(m1C.nnos, m2C.nnos, m3C.nnos) +anQ.nnos <- myanova(m1Q.nnos, m2Q.nnos, m3Q.nnos) + +## Tabela com medidas de ajuste +tab.ncapu <- data.frame( + modelos = paste("Preditor", 1:3), + anP.ncapu[, c("np", "ll", "AIC", "P(>Chisq)")], + anC.ncapu[, c("ll", "AIC", "P(>Chisq)")], + anQ.ncapu[, c("dev", "P(>F)")]) + +tab.nerep <- data.frame( + modelos = paste("Preditor", 1:3), + anP.nerep[, c("np", "ll", "AIC", "P(>Chisq)")], + anC.nerep[, c("ll", "AIC", "P(>Chisq)")], + anQ.nerep[, c("dev", "P(>F)")]) + +tab.nnos <- data.frame( + modelos = paste("Preditor", 1:3), + anP.nnos[, c("np", "ll", "AIC", "P(>Chisq)")], + anC.nnos[, c("ll", "AIC", "P(>Chisq)")], + anQ.nnos[, c("dev", "P(>F)")]) + +## 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) +## print(xtable(tab.ajuste, digits = digits)) + +``` + +\begin{table}[ht] +\footnotesize +\centering +\caption{Medidas de ajuste para avaliação e comparação} +\begin{tabular}{lccccccccc} + \toprule + & & \multicolumn{3}{c}{Poisson} & \multicolumn{3}{c}{COM-Poisson} & \multicolumn{2}{c}{Quasi-Poisson} \\ + \cmidrule(lr){3-5} \cmidrule(lr){6-8} \cmidrule(lr){9-10} + & np & $\ell$ & AIC & P($>\rchi^2$) & $\ell$ & AIC & P($>\rchi^2$) & deviance & P($>F$) \\ + \midrule + \multicolumn{10}{l}{{\bfseries \footnotesize Número de capulhos produzidos}} \\[0.0cm] + \cline{1-5} \\[-0.2cm] + & 1 & -105,27 & 212,55 & --- & -92,05 & 188,09 & --- & 20,80 & --- \\ + & 2 & -105,03 & 214,05 & 4,83E-01 & -91,31 & 188,62 & 2,25E-01 & 20,31 & 2,23E-01 \\ + & 3 & -104,44 & 214,88 & 2,78E-01 & -89,47 & 186,95 & 5,52E-02 & 19,13 & 6,16E-02 \\[0.2cm] + \multicolumn{10}{l}{{\bfseries \footnotesize Número de estruturas reprodutivas}} \\[0.0cm] + \cline{1-5} \\[-0.2cm] + & 1 & -104,74 & 211,49 & --- & -86,41 & 176,82 & --- & 16,23 & --- \\ + & 2 & -104,27 & 212,54 & 3,32E-01 & -84,59 & 175,18 & 5,66E-02 & 15,29 & 6,19E-02 \\ + & 3 & -104,06 & 214,12 & 5,16E-01 & -83,73 & 175,47 & 1,90E-01 & 14,87 & 2,07E-01 \\[0.2cm] + \multicolumn{10}{l}{{\bfseries \footnotesize Número de nós da planta}} \\[0.0cm] + \cline{1-5} \\[-0.2cm] + & 1 & -143,79 & 289,59 & --- & -120,58 & 245,16 & --- & 12,69 & --- \\ + & 2 & -143,48 & 290,95 & 4,25E-01 & -119,03 & 244,06 & 7,87E-02 & 12,05 & 7,39E-02 \\ + & 3 & -142,95 & 291,89 & 3,04E-01 & -116,27 & 240,54 & 1,88E-02 & 11,00 & 2,23E-02 \\ + \bottomrule +\end{tabular} +\end{table} + +### Valores preditos ### + +```{r pred-cottonBolls2, fig.height=3.5, fig.width=7, fig.cap="Curva dos valores preditos com intervalo de confiança de (95\\%) como função dos dias de exposição a alta infestação de Mosca-branca."} + +data(cottonBolls2) +qn <- qnorm(0.975) * c(fit = 0, lwr = -1, upr = 1) + +## Predição pontual/intervalar +pred <- data.frame(dexp = with(cottonBolls2, + seq(min(dexp), max(dexp), l = 50))) + +##====================================================================== +## Considerando a Poisson +##------------------------------------------- +## Para ncapu +aux <- predict(m3P.ncapu, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Poisson", aux) +predP.ncapu <- cbind(var = "ncapu", pred, aux) +##------------------------------------------- +## Para nerep +aux <- predict(m2P.nerep, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Poisson", aux) +predP.nerep <- cbind(var = "nerep", pred, aux) +##------------------------------------------- +## Para nnos +aux <- predict(m3P.nnos, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Poisson", aux) +predP.nnos <- cbind(var = "nnos", pred, aux) +## +predP <- rbind(predP.ncapu, predP.nerep, predP.nnos) + +##====================================================================== +## Considerando a COM-Poisson +##------------------------------------------- +## Para ncapu +aux <- predict(m3C.ncapu, newdata = pred, type = "response", + interval = "confidence") +aux <- data.frame(modelo = "COM-Poisson", aux) +predC.ncapu <- cbind(var = "ncapu", pred, aux) +##------------------------------------------- +## Para nerep +aux <- predict(m2C.nerep, newdata = pred, type = "response", + interval = "confidence") +aux <- data.frame(modelo = "COM-Poisson", aux) +predC.nerep <- cbind(var = "nerep", pred, aux) +##------------------------------------------- +## Para nnos +aux <- predict(m3C.nnos, newdata = pred, type = "response", + interval = "confidence") +aux <- data.frame(modelo = "COM-Poisson", aux) +predC.nnos <- cbind(var = "nnos", pred, aux) +## +predC <- rbind(predC.ncapu, predC.nerep, predC.nnos) + +##====================================================================== +## Considerando a Quasi-Poisson +##------------------------------------------- +## Para ncapu +aux <- predict(m3Q.ncapu, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Quasi-Poisson", aux) +predQ.ncapu <- cbind(var = "ncapu", pred, aux) +##------------------------------------------- +## Para nerep +aux <- predict(m2Q.nerep, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Quasi-Poisson", aux) +predQ.nerep <- cbind(var = "nerep", pred, aux) +##------------------------------------------- +## Para nnos +aux <- predict(m3Q.nnos, newdata = pred, se.fit = TRUE) +aux <- with(aux, exp(fit + outer(se.fit, qn, FUN = "*"))) +aux <- data.frame(modelo = "Quasi-Poisson", aux) +predQ.nnos <- cbind(var = "nnos", pred, aux) +## +predQ <- rbind(predQ.ncapu, predQ.nerep, predQ.nnos) + +##====================================================================== +##------------------------------------------- +## Gráfico com os valores preditos +vars <- c("dexp", "vaso", "planta", "nerep", "ncapu", "nnos") +cottonBolls2 <- cottonBolls2[, vars] +da <- reshape2::melt(cottonBolls2, id = c("dexp", "vaso", "planta"), + variable.name = "va", value.name = "count") +cols <- trellis.par.get("superpose.line")$col[1:3] + +key <- list( + column = 3, + lines = list(lty = c(4, 1, 2), lwd = 1, col = cols), + text = list(c("Poisson", "COM-Poisson", "Quasi-Poisson"))) + +xyplot(count ~ dexp | va, data = da, + key = key, + type = c("p", "g"), + layout = c(NA, 1), + ylab = "Contagens", + xlab = "Dias de exposição a alta infestação de Mosca-branca", + scales = list( + y = list(relation = "free", rot = 0)), + strip = strip.custom( + factor.levels = c("Estruturas reprodutivas ", + "Capulhos produzidos", + "Nós da planta")), + alpha = 0.3, + spread = 0.15, + panel = panel.beeswarm, + par.settings = ps.sub)+ + as.layer( + xyplot(fit + lwr + upr ~ dexp | var, data = predP, + layout = c(NA, 1), + scales = list( + y = list(relation = "free", rot = 0)), + type = "l", col = cols[1], lty = c(1, 4, 4), lwd = 1) + ) + + as.layer( + xyplot(fit + lwr + upr ~ dexp | var, data = predC, + layout = c(NA, 1), + scales = list( + y = list(relation = "free", rot = 0)), + type = "l", col = cols[2], lty = c(1, 1, 1), lwd = 1) + ) + + as.layer( + xyplot(fit + lwr + upr ~ dexp | var, data = predQ, + layout = c(NA, 1), + scales = list( + y = list(relation = "free", rot = 0)), + type = "l", col = cols[3], lty = c(1, 2, 2), lwd = 1) + ) + +``` + +<!--------------------------------------------- --> +## Ocorrência de ninfas de Mosca-branca ## + +```{r ajuste-whiteFly, include=FALSE, cache=TRUE} + +data(whiteFly) +whiteFly <- droplevels(subset(whiteFly, grepl("BRS", x = cult))) +whiteFly$aval <- factor(whiteFly$dias) + +## Preditores considerados +f1 <- ntot ~ bloco + cult + aval +f2 <- ntot ~ bloco + cult * aval + +## Ajustando os modelos Poisson +m1P.ntot <- glm(f1, data = whiteFly, family = poisson) +m2P.ntot <- glm(f2, data = whiteFly, family = poisson) + +## Ajustando os modelos COM-Poisson +m1C.ntot <- cmp(f1, data = whiteFly, sumto = 800) +m2C.ntot <- cmp(f2, data = whiteFly, sumto = 800) + +## Ajustando os modelos Binomial Negativo +library(MASS) +m1B.ntot <- glm.nb(f1, data = whiteFly) +m2B.ntot <- glm.nb(f2, data = whiteFly) + +## Ajustando os modelos Quasi-Poisson +m1Q.ntot <- glm(f1, data = whiteFly, family = quasipoisson) +m2Q.ntot <- glm(f2, data = whiteFly, family = quasipoisson) + +##====================================================================== +## Perfil de log-verossimilhanca para o parametro phi +prof.ntot <- profile(m1C.ntot, which = "phi") + +``` + +### Experimento ### + +Conduzido na UFGD em casa de vegetação [@Suekane2011]. + +* Delineamento: blocos casualizados com quatro blocos. +* Objetivo: avaliar a propensão de cultivares de soja à praga + Mosca-branca. +* Unidade experimental: dois vasos com duas plantas. +* Covariáveis experimentais: + - Indicadora de bloco, I, II, III e IV, (`bloco`), + - Dias decorridos após a primeira avaliação, 0, 8, 13, 22, 31 e 38 + dias. (`dias`), + - Indicadora de cultivar de soja, BRS 239, BRS 243 RR, BRS 245 RR, + BRS246 RR, (`cult`). +* Variável resposta: + - Número de ninfas de Mosca-branca nos folÃolos dos terços superior, + médio e inferior. + +### Modelagem ### + +Preditores considerados: + +* $\eta_1 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k$ +* $\eta_2 = g(\mu_{ijk}) = \beta_0 + \tau_i + \gamma_j + \delta_k + +\alpha_{jk}$ + +\hspace{0.3cm} +\begin{minipage}[t]{0.9\textwidth} +\small +$\tau_i$ é o efeito do i-ésimo bloco, $i=1, 2, 3, 4$ + +$\gamma_j$ o efeito da j-ésima cultivar, $j=1,2,3,4$ + +$\delta_k$ o efeito do k-ésimo nÃvel de \texttt{dias}, $k= 1, 2, \ldots, +6$ e + +$\alpha_{jk}$ o efeito da interação entre a j-ésima cultivar e o k-ésimo +nÃvel de \texttt{dias} +\end{minipage} +\vspace{0.5cm} + +Modelos concorrentes: + +* Poisson($\mu_{ijk}$) +* COM-Poisson($\lambda_{ijk},\, \phi$) +* Binomial Negativo($\mu_{ijk},\, \theta$) +* Quasi-Poisson($\mu_{ijk},\, \sigma^2$) + +### Medidas de ajuste ### + +```{r loglik-whiteFly, include=FALSE} + +## Análise de desvios dos modelos +anP.ntot <- myanova(m1P.ntot, m2P.ntot) +anC.ntot <- myanova(m1C.ntot, m2C.ntot) +anB.ntot <- myanova(m1B.ntot, m2B.ntot) +anQ.ntot <- myanova(m1Q.ntot, m2Q.ntot) + +## Tabelas com medidas de ajuste +tab.ntot <- rbind(anP.ntot, anC.ntot, anB.ntot, anQ.ntot) + +## Obtem as estimativas dos parâmetros de dispersão/precisão +dispersions.ntot <- c( + c(rep(NA, 2)), + sapply(list(m1C.ntot, m2C.ntot), function(m) coef(m)[1]), + sapply(list(m1B.ntot, m2B.ntot), function(m) m$theta), + sapply(list(m1Q.ntot, m2Q.ntot), function(m) summary(m)$dispersion)) + +## Adicionando os parametros de dispersão à tabela +tab.ntot <- cbind(tab.ntot, dispersions.ntot) +rownames(tab.ntot) <- NULL + +## Juntando as tabelas +tab.ajuste <- data.frame( + pred = rep(paste("Preditor", 1:2), 4), + tab.ntot) + +## ##---------------------------------------------------------------------- +## ## Copiar e colar o corpo do resultado na customização latex abaixo +## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2) +## print(xtable(tab.ajuste, digits = digits)) + +``` + +\begin{table}[ht] +\centering +\footnotesize +\caption{Medidas de ajuste para avaliação e comparação} +\begin{tabular}{lcccccrcr} + \toprule + 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($>\rchi^2$) & $\hat{\phi}$ \\ + \midrule + 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) & $\hat{\sigma}^2$ \\ + \midrule + Preditor 1 & 12 & 1371,32 & & & & & 17,03 \\ + Preditor 2 & 27 & 1283,82 & & 0,31 & 15 & 9,93E-01 & 19,03 \\ + \bottomrule +\end{tabular} +\end{table} + +### Valores preditos ### + +```{r pred-whiteFly, fig.height=4.2, fig.width=7.5, fig.cap="Valores preditos com intervalos de confiança (95\\%)."} + +library(multcomp) + +## Predição pontual/intervalar +pred <- with(whiteFly, + expand.grid( + bloco = factor(levels(bloco)[1], + levels = levels(bloco)), + cult = levels(cult), + aval = levels(aval) + )) + +## Matrix de delineamento para predição, considera o efeito médio de +## bloco +X <- model.matrix(~bloco + cult + aval, data = pred) +bl <- attr(X, "assign") == 1 +X[, bl] <- X[, bl] * 0 + 1/(sum(bl) + 1) + +##------------------------------------------- +## Considerando a Poisson +aux <- exp(confint(glht(m1P.ntot, linfct = X), + calpha = univariate_calpha())$confint) +colnames(aux) <- c("fit", "lwr", "upr") +aux <- data.frame(modelo = "Poisson", aux) +predP <- cbind(pred, aux) + +##------------------------------------------- +## Considerando a COM-Poisson +aux <- predict(m1C.ntot, newdata = X, type = "response", + interval = "confidence") +aux <- data.frame(modelo = "COM-Poisson", aux[, c("fit", "lwr", "upr")]) +predC <- cbind(pred, aux) + +##------------------------------------------- +## Considerando a Binomial Negativa +aux <- family(m1B.ntot)$linkinv( + confint(glht(m1B.ntot, linfct = X), + calpha = univariate_calpha())$confint) +colnames(aux) <- c("fit", "lwr", "upr") +aux <- data.frame(modelo = "Binomial Negativa", aux) +predB <- cbind(pred, aux) + +##------------------------------------------- +## Considerando a Quasi-Poisson +aux <- exp(confint(glht(m1Q.ntot, linfct = X), + calpha = univariate_calpha())$confint) +colnames(aux) <- c("fit", "lwr", "upr") +aux <- data.frame(modelo = "Quasi-Poisson", aux) +predQ <- cbind(pred, aux) + +##------------------------------------------- +pred.all <- rbind(predP, predC, predB, predQ) +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$model) + 4, + cex = 0.7), + text = list(c("Poisson", "COM-Poisson", + "Binomial Negativa", "Quasi-Poisson")) +) + +## ## A escala do gráfico fica prejudicada, pois os limites do eixo y são +## ## extendidos para exibir as observações mais distantes +## xyplot(ntot ~ aval | cult, +## data = whiteFly, +## layout = c(NA, 1), +## as.table = TRUE, +## key = key, +## alpha = 0.3, +## type = c("p", "g"), +## xlab = "Número de dias após o inicÃo do experimento", +## ylab = "Número total de moscas-brancas", +## par.settings = ps.sub) + +segplot( + aval ~ lwr + upr | cult, + layout = c(NA, 1), + xlab = "Número de dias após o inicÃo do experimento", + ylab = "Número total de moscas-brancas", + key = key, + axis = axis.grid, cex = 0.7, + centers = fit, groups = modelo, data = pred.all, + horizontal = FALSE, draw = FALSE, + lwd = 1.5, pch = 1:nlevels(pred.all$modelo) + 4, + panel = panel.groups.segplot, gap = 0.3) + +``` + +<!--------------------------------------------- --> +## Peixes capturados ## + +```{r ajuste-fish, include=FALSE, cache=TRUE} + +## Dados +data(fish, package = "cmpreg") + +## 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) + +``` + +### Estudo ### + +Observacional conduzido por biólogos em um Parque Estadual [@UCLA]. + +* Delineamento: amostragem aleatória. +* Objetivo: modelar o número de peixes capturados pela atividade de + pesca esportiva. +* Unidade experimental: grupos de pescadores visitantes do parque. +* Covariáveis mensuradas: + - Número de pessoas, (`np`), + - Número de crianças. (`nc`), + - Indicador de campista no grupo, (`ca`). +* Variável resposta: + - Número de peixes capturados pelo grupo. + +### Modelagem ### + +Preditores considerados: + +* Preditor 1: \quad +$\begin{aligned} + g(\mu_i) &= \beta_0 + \beta_1 \textrm{ca}_i + + \beta_2 \textrm{np}_i \\ + \textrm{logit}(\pi_i) &= \gamma_0 + \gamma_1 \textrm{ca}_i + + \gamma_2 \textrm{np}_i + \gamma_3 \textrm{nc}_i +\end{aligned}$ +\vspace{0.2cm} + +* Preditor 2: \quad +$\begin{aligned} + g(\mu_i) &= \beta_0 + \beta_1 \textrm{ca}_i + + \beta_2 \textrm{np}_i + \beta_3 \textrm{nc}_i + + \beta_4 (\textrm{np}_i \cdot \textrm{nc}_i) \\ + \textrm{logit}(\pi_i) &= \gamma_0 + \gamma_1 \textrm{ca}_i + + \gamma_2 \textrm{np}_i + \gamma_3 \textrm{nc}_i + + \gamma_4 (\textrm{np}_i \cdot \textrm{nc}_i) +\end{aligned}$ +\vspace{0.5cm} + +Modelos concorrentes: + +* Hurdle Poisson($\pi_i,\, \mu_i$) +* Hurdle COM-Poisson($\pi_i,\, \lambda_i,\, \phi$) +* Hurdle Binomial Negativo($\pi_i,\, \mu_i,\, \theta$) + +### Medidas de ajuste ### + +```{r 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, dispHB, dispHC) + +## Empilhando +tab <- data.frame(pred = rep(paste("Preditor", 1:2), 3)) +tab <- cbind(tab, rbind(anHP, anHB, anHC), dispersions) + +## ## Gerando o código latex +## digits <- c(1, 1, 0, 2, 2, 2, 0, -2, 2) +## xtable(tab, digits = digits) + +``` + +\begin{table}[ht] +\centering +\footnotesize +\caption{Medidas de ajuste para avaliação e comparação} +\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] +Binomial Neg. & 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 \\[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 \\ + \bottomrule +\end{tabular} +\end{table} + +### Valores preditos ### + +```{r pred-fish, fig.height=4, fig.width=6.5, out.width="0.9\\textwidth", fig.cap="Valores preditos do número de peixes capturados."} + +##====================================================================== +## Preditos + +## Região para predição +pred <- expand.grid(campista = c("Não", "Sim"), + ncriancas = 0:3, npessoas = 1:4) + +##------------------------------------------- +## Estimando as médias + +## Pelo Hurdle Poisson +aux <- predict(m2HP, newdata = pred, type = "response") +predHP <- cbind(pred, fit = aux, model = "HP") + +## Pelo Hurdle Binomial Negativo +aux <- predict(m2HB, newdata = pred, type = "response") +predHB <- cbind(pred, fit = aux, model = "HB") + +## Pelo Hurdle COM-Poisson +aux <- predict(m2HC, newdata = pred, type = "response") +predHC <- cbind(pred, fit = aux, model = "HC") + +## +pred.all <- rbind(predHP, predHB, predHC) + +##------------------------------------------- +## Gráfico final + +key <- list( + lines = list(lty = 1:3), + text = list( + paste("Hurdle", c("Poisson", "COM-Poisson", + "Binomial Negativo"))) +) + +xyplot(npeixes ~ npessoas | campista, + groups = ncriancas, data = subset(fish, npeixes < 50), + jitter.x = TRUE, + jitter.y = TRUE, + type = c("p", "g"), + xlab = "Número de pessoas no grupo", + ylab = "Número de peixes capturados", + key = key, + alpha = 0.3, + strip = strip.custom( + strip.names = TRUE, var.name = "campista" + )) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHP, + type = "l", + lty = 1) + ) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHC, + type = "l", + lty = 2) + ) + + as.layer( + xyplot(fit ~ npessoas | campista, + groups = ncriancas, data = predHB, + type = "l", + lty = 3) + ) +##------------------------------------------- +## Legenda +cols <- trellis.par.get("superpose.line")$col[1:4] +draw.key( + key = list( + cex = 0.9, + columns = 1, + lines = list( + lty = 1, lwd = 2, col = cols), + text = list( + as.character(paste(unique(fish$ncriancas), "crianças"))) + ), draw = TRUE, + vp = grid::viewport( + x = grid::unit(0.22, "npc"), y = grid::unit(0.6, "npc"))) + +``` + +<!--------------------------------------------- --> +## Número de nematoides ## + +```{r ajuste-nematodes, include=FALSE, cache=FALSE} + +library(lme4) +data(nematodes) + +## Preditores +f1 <- nema ~ (1|cult) +f2 <- nema ~ log(off) + (1|cult) + +## Ajuste dos mixed Poisson +m1PM <- glmer(f1, data = nematodes, family = poisson) +m2PM <- glmer(f2, data = nematodes, family = poisson) + +## ## Ajuste dos mixed COM-Poisson +## m1CM <- mixedcmp(f1, data = nematodes, sumto = 50) +## m2CM <- mixedcmp(f2, data = nematodes, sumto = 50) +load("../docs/mixedcmp_models.rda") + +``` + +### Experimento ### + +Conduzido no IAPAR em casa de vegetação. + +* Delineamento: inteiramente casualizado com cinco repetições. +* Objetivo: avaliar a resistência à nematoides de linhagens de feijoeiro. +* Unidade amostral: alÃquota de 1ml da solução de raizes lavadas, + trituradas, peneiradas e diluÃdas em água provida por um vaso com duas + plantas. +* Covariáveis: + - Indicador de linhagem de feijoeiro, A, B, C, $\ldots$, S (`cult`) + - Concentração de raiz na solução. (`sol`) +* Variáveis resposta: + - Número de nematoides. + +### Modelagem ### + +Preditores considerados: + +* Preditor 1: $g(\mu_{ij}) = \beta_0 + b_j$ +* Preditor 2: $g(\mu_{ij}) = \beta_0 + \beta_1 \log(\textrm{sol})_i + + b_j$ + +\hspace{0.35cm} +\begin{minipage}[t]{0.9\textwidth} +$b_j \sim \textrm{Normal}(0,\, \sigma^2)$ +\end{minipage} +\vspace{0.5cm} + +Modelos concorrentes: + +* Poisson($\mu_{ij}$) +* COM-Poisson($\lambda_{ij},\, \phi$) + +### Medidas de ajuste ### + +\begin{table}[ht] +\centering +\footnotesize +\caption{Medidas de ajuste para avaliação e comparação} +\begin{tabular}{lcccccrcr} + \toprule + Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & \\ + \midrule + Preditor 1 & 2 & -237,20 & 478,40 & & & & & \\ + Preditor 2 & 3 & -234,66 & 475,32 & 5,07 & 1 & 2,43E-02 & & \\[0.3cm] + COM-Poisson & np & $\ell$ & AIC & 2(diff $\ell$) & diff np & P($>\rchi^2$) & $\hat{\phi}$ & P($>\rchi^2$) \\ + \midrule + Preditor 1 & 3 & -236,85 & 479,71 & & & & 0,15 & 4,06E-01 \\ + Preditor 2 & 4 & -233,86 & 475,72 & 5,99 & 1 & 1,44E-02 & 0,23 & 2,05E-01 \\ + \bottomrule +\end{tabular} +\end{table} + +### Valores preditos ### + +```{r pred-nematodes, fig.height=4.2, fig.width=7.4, fig.cap="Valores preditos nos modelos de efeitos mistos."} + +##------------------------------------------- +## Obtendo os efeitos aleatórios +ranefP <- ranef(m2PM)$cult[, 1] +ranefC <- mixedcmp.ranef(m2CM) +ranef.all <- rbind( + data.frame(model = "PM", ranef = ranefP), + data.frame(model = "CM", ranef = ranefC)) + +##------------------------------------------- +## Valores preditos +pred <- with( + nematodes, + expand.grid(off = seq(min(off), max(off), length.out = 20), + cult = levels(cult)) + ) +X <- model.matrix(~log(off), data = pred) + +## Pelo Poisson Mixed +aux <- predict(m2PM, newdata = pred, type = "link") +predPM <- data.frame(pred, y = exp(aux), model = "MP") +muPM <- data.frame(pred, mu = exp(X %*% fixef(m2PM)), model = "MP") + +## Pelo COM-Poisson Mixed +aux <- predict(m2CM, newdata = pred, type = "link") +predCM <- data.frame(pred, y = calc_mean_cmp(aux, phi = m2CM@coef[1]), + model = "MC") +muCM <- data.frame(pred, mu = calc_mean_cmp(X %*% m2CM@coef[-(1:2)], + phi = m2CM@coef[1]), + model = "MC") + +## Agrupa as predições +pred.all <- rbind(predPM, predCM) +mu.all <- rbind(muPM, muCM) + +xy1 <- densityplot( + ~ranef, groups = model, + auto.key = list( + column = 1, + text = c("Poisson", "COM-Poisson") + ), + xlab = "Predição dos efeitos\n aleatórios", + ylab = "Densidade", + data = ranef.all, + grid = TRUE, + par.settings = ps.sub) + +key <- list( + column = 1, + lines = list(lty = c(1, 2), lwd = c(3, 1)), + text = list(c("Perfil Médio", "Perfil por cultivar"))) + +## Faz o gráfico +nematodes2 <- rbind(nematodes, nematodes) +nematodes2$model <- rep(c("MP", "MC"), each = nrow(nematodes)) +xy2 <- xyplot(nema ~ off | model, + groups = cult, + data = nematodes2, + type = c("p", "g"), + alpha = 0.4, + key = key, + xlab = paste("Solução de massa fresca de raizes\n", + "pelo volume de água"), + ylab = "Contagem de nematoides", + strip = strip.custom( + factor.levels = c("Poisson", "COM-Poisson") + ), + par.settings = ps.sub) + + as.layer( + xyplot(y ~ off | model, groups = cult, data = pred.all, + type = "l", lty = 2) + ) + + as.layer( + xyplot(mu ~ off | model, groups = cult, data = mu.all, + type = "l", col = 1, lwd = 3) + ) + +## Organiza layout +print(xy2, position = c(0, 0, 0.6, 1), more = TRUE) +print(xy1, position = c(0.57, 0, 1, 1), more = FALSE) + +``` + +<!--------------------------------------------- --> +## Discussões ## + +### ### + +* Similaridade entre inferências via modelo Quasi-Poisson e COM-Poisson; +* Desempenho do modelo Binomial Negativo; +* Interpretação dos parâmetros nos modelos baseados na COM-Poisson +* Problemas numéricos para determinação da matriz hessiana no modelo + Hurdle COM-Poisson; +* Procedimentos computacionalmente intensivos na avaliação da + verossimilhança no caso COM-Poisson de efeitos aleatórios; +* Não ortogonalidade observada (empÃrica) entre os parâmetros de locação + e de precisão no modelo COM-Poisson; +* Comportamento simétrico dos perfis de log-verossimilhança para o + parâmetro $\phi$ da COM-Poisson. + +<!--------------------------------------------- --> +# Considerações finais # + +### Conclusões ### + +Aplicação do modelo COM-Poisson: + +* Resultados similares aos providos pela abordagem semi-paramétrica via + quasi-verossimilhança; +* A não ortogonalidade entre os parâmetros de locação e precisão no + modelo COM-Poisson se mostra como caracterÃstica da distrição; +* A simetria nos perfis de verossimilhança do parâmetro de precisão + também. +* A avaliação da constante de normalização é problemática no modelo. + + +### Conclusões ### + +Análise de dados de contagem: + +* Modelo Poisson inadequado na maioria das aplicações, mostrando que a + suposição de equidispersão é de fato restritiva; +* Modelos alternativos ao Poisson devem ser empregados na análise de + dados de contagem; +* Sugere-se o modelo COM-Poisson como alternativa totalmente + paramétrica e bastante flexÃvel. + +### Trabalhos futuros ### + +* Estudar reparametrizações do modelo COM-Poisson; +* Avaliar aproximações da constante de normalização; +* Realizar estudos de simulação para avaliar a robustez do modelo; +* Implementar o modelo COM-Poisson inflacionado de zeros; +* Implementar o modelo COM-Poisson com efeitos aleatórios dependentes. + +<!--------------------------------------------- --> +# # +### Publicização ### + +\begin{columns}[c] + \column{.3\textwidth} +\begin{flushright} + \includegraphics[scale=0.1]{./images/octocat} \\ +\end{flushright} + \hfill +\column{.7\textwidth} +\url{https://github.com/JrEduardo/cmpreg} +\url{https://github.com/JrEduardo/tccDocument} +\end{columns} + +\vfill +\includegraphics[scale=0.1]{./images/software} + +### Referências ### +\small diff --git a/presentations/slides-tcc.pdf b/presentations/slides-tcc.pdf new 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